Affine Type Aq Functional Equation In Various Banach Spaces

1. Introduction

A inspirational and popular talk presented by S.M Ulam [37] in 1940, refreshed the reading of stability problems for various functional equations. He gave a wide range of talk before a Mathematical Colloquium at the University of Wisconsin in which he presented a list of unsolved problems.

The first assertive answer to Ulam’s question concerning the problem of stability of functional equations was given by D.H. Hyers [20] for the case of additive mappings in Banach spaces. In growth of time, the theorem delivered by Hyers was generalized by T. Aoki [3], Th.M Rassias [30], J.M. Rassias [28], P. Gavruta [19] for additive mappings and K. Ravi [32] for quadratic mappings.

The famous additive and quadratic functional equations are

$$\begin{array}{c}\hfill \mathcal{F}(w_1+w_2)=\mathcal{F}\left(w_1\right)+\mathcal{F}\left(w_2\right),\end{array}$$

(1)

and

$$\begin{array}{c}\hfill \mathcal{F}(w_1+w_2)+\mathcal{F}(w_1-w_2)=2\mathcal{F}\left(w_1\right)+2\mathcal{F}\left(w_2\right).\end{array}$$

(2)

The general solution and generalized Ulam - Hyers stability of several types of functional equations in various normed spaces were discussed by many authors one can see [2,16,18,21,22,31] and references there in.

Also, the general solution and Hyers-Ulam-Rassias stability of the several affine functional equations are discussed by L. Lucht, C. Methfessel [23], L. Cadariu, L. Gavruta, P. Gavruta [15], Md. Nasiruzzaman [26], M. Mursaleen, KJ. Ansari[25].

Infact, the general solution and generalized Hyers-Ulam stability of the several AQ functional equations are established in [4,5,6,7,8,9,10,11,12,14,29].

In this paper, the we analyze the generalized Ulam-Hyers stability of affine type AQ Functional Equation of the form

$$\begin{array}{cc}\hfill & \mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)\hfill \\ \hfill & =6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}-\sum _{\psi =1}^3\left\{\mathcal{F}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(w_{\psi }\right)+\mathcal{F}(-w_{\psi })\right]\right\}\hfill \end{array}$$

(3)

in various Banach Spaces using Direct and Fixed Methods .

Lemma 1.1. 

[26] Let A and B be real vector spaces. Suppose $\mathcal{F}:A\to B$ be an odd mapping satisfies (3) then $\mathcal{F}$ is additive.

Lemma 1.2. 

[17] Let A and B be real vector spaces. Suppose $\mathcal{F}:A\to B$ be an even mapping satisfies (3) then $\mathcal{F}$ is quadratic.

Now, we present the result due to Margolis, Diaz [24] and Radu [27] for fixed point theory.

Theorem 1.3. 

[24,27] Suppose that for a complete generalized metric space $(\Omega ,\delta )$ and a strictly contractive mapping $T:\Omega ⟶\Omega$ with Lipschitz constant L. Then, for each given $x\in \Omega$ , either

$$d(T^{n}x,T^{n+1}x)=\infty \forall n\ge 0,$$

or there exists a natural number $n_0$ such that

• (FPC1) $d(T^{n}x,T^{n+1}x)<\infty$ for all $n\ge n_0$ ;
• (FPC2) The sequence $\left(T^{n}x\right)$ is convergent to a fixed point $y^{*}$ of T
• (FPC3) $y^{*}$ is the unique fixed point of T in the set $\Delta =\{y\in \Omega :d(T^{n_0}x,y)<\infty \};$
• (FPC4) $d(y^{*},y)\le \frac{1}{1-L}d(y,Ty)$ for all $y\in \Delta .$

2. Stability In Banach Space of (3)

In this section, we explore the generalized Ulam - Hyers stability of the functional equation (3) in Banach space. To prove stability results, let us take ${\mathcal{W}}_1$ be a normed space and ${\mathcal{W}}_2$ be a Banach space. Suppose that $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ and $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ satisfying the following functional inequalities

$$\begin{array}{cc}\hfill & \parallel \mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)\hfill \\ \hfill & -6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)-\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}\hfill \\ \hfill & +\sum _{\psi =1}^3\left\{\mathcal{F}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(w_{\psi }\right)+\mathcal{F}(-w_{\psi })\right]\right\}\parallel \le \Psi \left(w_1,w_2,w_3\right),\hfill \end{array}$$

(1)

and

$$\begin{array}{cc}\hfill & \parallel \mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\hfill \\ \hfill & -\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{F}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(w_{\psi }\right)+\mathcal{F}(-w_{\psi })\right]\right\}\parallel \hfill \\ \hfill & \le \left\{\begin{array}{c}\delta ,\hfill \\ \delta {\displaystyle \sum _{\psi =1}^3}{\left|w_{\psi }\right|}^{\phi },\hfill \\ \delta {\displaystyle \sum _{\psi =1}^3}{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\hfill \\ \delta {\displaystyle \prod _{\psi =1}^3}{\left|w_{\psi }\right|}^{\phi },\hfill \\ \delta {\displaystyle \prod _{\psi =1}^3}{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\hfill \\ \delta \left\{{\displaystyle \sum _{\psi =1}^3}{\left|w_{\psi }\right|}^{3\phi }+{\displaystyle \prod _{\psi =1}^3}{\left|w_{\psi }\right|}^{\phi }\right\},\hfill \end{array}\right.\hfill \end{array}$$

(2)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and $\delta$ be a positive constant.

2.1. Oddness of $\mathcal{F}$: Additive Case Stability Results : Direct Method

Theorem 2.1. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the condition

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\frac{\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right)}{5^{\ell \mu }}=0;\mu =\pm 1\end{array}$$

(3)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥& \le \frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}{\Psi }_A\left(5^{\eta \mu }{w}_1\right)\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & =\frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+3\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \end{array}$$

(5)

and the mapping $\mathcal{A}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \mathcal{A}\left(w_1\right)=\underset{\ell \to \infty }{lim}\frac{1}{5^{\ell \mu }}\mathcal{F}\left(5^{\ell \mu }{w}_1\right)\end{array}$$

(6)

for all $w_1\in {\mathcal{W}}_1$.

Proof. 

Using oddness of $\mathcal{F}$ in (1), we get

$$\begin{array}{cc}\hfill \parallel \mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)& +\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\sum _{\psi =1}^3\mathcal{F}\left(w_{\psi }\right)\parallel \hfill \\ \hfill & \le \Psi \left(w_1,w_2,w_3\right),\forall w_1,w_2,w_3\in {\mathcal{W}}_1.\hfill \end{array}$$

(7)

Interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,w_1\right)$ in (7), we obtain

$$\begin{array}{c}\hfill \parallel 3\mathcal{F}\left(5w_1\right)-6\mathcal{F}\left(3w_1\right)+3\mathcal{F}\left(w_1\right)\parallel \le \Psi \left(w_1,w_1,w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(8)

Again interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,-w_1\right)$ in (7), we have

$$\begin{array}{cc}\hfill & \parallel 2\mathcal{F}\left(3w_1\right)-6\mathcal{F}\left(w_1\right)\parallel \le \Psi \left(w_1,w_1,-w_1\right)\hfill \\ \hfill & \Rightarrow \parallel 6\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right)\parallel \le 3\Psi \left(w_1,w_1,-w_1\right),\forall w_1\in {\mathcal{W}}_1.\hfill \end{array}$$

(9)

Combining (8) and (9), we arrive

$$\begin{array}{cc}\hfill \parallel 3\mathcal{F}\left(5w_1\right)-15\mathcal{F}\left(w_1\right)\parallel & \le \parallel 3\mathcal{F}\left(5w_1\right)-6\mathcal{F}\left(3w_1\right)+3\mathcal{F}\left(w_1\right)\parallel +\parallel 6\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right)\parallel \hfill \\ \hfill & \le \Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right),\forall w_1\in {\mathcal{W}}_1.\hfill \end{array}$$

(10)

One can see from (10) that

$$\begin{array}{c}\hfill \parallel \mathcal{F}\left(5w_1\right)-5\mathcal{F}\left(w_1\right)\parallel \le \frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}={\Psi }_A\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(11)

It follows from (11) that

$$\begin{array}{c}\hfill \parallel \frac{1}{5}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right)\parallel \le \frac{1}{5}{\Psi }_A\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(12)

Generalizing for a positive integer ℓ, we get

$$\begin{array}{c}\hfill \parallel \frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{F}\left(w_1\right)\parallel \le \frac{1}{5}\sum _{\eta =0}^{\ell }\frac{1}{5^{\eta }}{\Psi }_A\left(5^{\eta }{w}_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(13)

Now, changing $w_1$ by $5^{{\ell }_1}{w}_1$ in (13), we obtain

$$\begin{array}{cc}\hfill \parallel \frac{1}{5^{\ell +{\ell }_1}}\mathcal{F}\left(5^{\ell +{\ell }_1}{w}_1\right)-\frac{1}{5^{{\ell }_1}}\mathcal{F}\left(5^{{\ell }_1}{w}_1\right)\parallel & =\frac{1}{5^{{\ell }_1}}\parallel \frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell +{\ell }_1}{w}_1\right)-\mathcal{F}\left(5^{{\ell }_1}{w}_1\right)\parallel \hfill \\ \hfill & \le \frac{1}{5}\sum _{\eta =0}^{\ell }\frac{1}{5^{\eta +{\ell }_1}}{\Psi }_A\left(5^{\eta +{\ell }_1}{w}_1\right)\hfill \\ \hfill & \to 0as{\ell }_1\to \infty ,\forall w_1\in {\mathcal{W}}_1.\hfill \end{array}$$

(14)

Therefore, the sequence

$$\left\{\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)\right\},$$

is a Cauchy sequence and it converges to $\mathcal{A}\left(w_1\right)$ in ${\mathcal{W}}_2.$ So, we define

$$\begin{array}{c}\hfill \mathcal{A}\left(w_1\right)=\underset{\ell \to \infty }{lim}\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(15)

Taking limit $\ell \to \infty$ in (13), we have

$$\begin{array}{c}\hfill \parallel \mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right)\parallel \le \frac{1}{5}\sum _{\eta =0}^{\infty }\frac{1}{5^{\eta }}{\Psi }_A\left(5^{\eta }{w}_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(16)

Thus, (4) and (5) holds for $\mu =1$. Interchanging

$$\left(w_1,w_2,w_3\right)=\left(5^{\ell }{w}_1,5^{\ell }{w}_2,5^{\ell }{w}_3\right),$$

we arrive

$$\begin{array}{cc}\hfill & \frac{1}{5^{\ell }}\parallel \mathcal{F}\left(5^{\ell }(3w_1+w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+3w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+w_2+3w_3)\right)\hfill \\ \hfill & -6\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)-\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right\}\hfill \\ \hfill & +\sum _{\psi =1}^3\left\{\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)+\mathcal{F}(-5^{\ell }{w}_{\psi })\right]\right\}\parallel \hfill \\ \hfill & \le \frac{1}{5^{\ell }}\Psi \left(5^{\ell }{w}_1,5^{\ell }{w}_2,5^{\ell }{w}_3\right),\forall w_1,w_2,w_3\in {\mathcal{W}}_1.\hfill \end{array}$$

(17)

Taking limit $\ell \to \infty$ in (17), using (15) and (3), we get

$$\begin{array}{cc}\hfill & \mathcal{A}(3w_1+w_2+w_3)+\mathcal{A}(w_1+3w_2+w_3)+\mathcal{A}(w_1+w_2+3w_3)\hfill \\ \hfill & =6\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\frac{1}{2}\left\{\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{A}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}-\sum _{\psi =1}^3\left\{\mathcal{A}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{A}\left(w_{\psi }\right)+\mathcal{A}(-w_{\psi })\right]\right\}\hfill \end{array}$$

for all $w_1,w_2,w_3\in {\mathcal{W}}_1.$ So, $\mathcal{A}\left(w_1\right)$ satisfies (3). In order to confirm that $\mathcal{A}\left(w_1\right)$ is unique, suppose $\mathcal{B}\left(w_1\right)$ be another mapping (3), (15) and (16), we obtain

$$\begin{array}{cc}\hfill \parallel \mathcal{A}\left(w_1\right)-\mathcal{B}\left(w_1\right)\parallel & =\parallel \frac{1}{5^{\ell }}\mathcal{A}\left(5^{\ell }{w}_1\right)-\frac{1}{5^{\ell }}\mathcal{B}\left(5^{\ell }{w}_1\right)\parallel \hfill \\ \hfill & \le \frac{1}{5^{\ell }}\parallel \mathcal{A}\left(5^{\ell }{w}_1\right)-\mathcal{F}\left(5^{\ell }{w}_1\right)\parallel +\frac{1}{5^{\ell }}\parallel \mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{B}\left(5^{\ell }{w}_1\right)\parallel \hfill \\ \hfill & \le \frac{2}{5}\sum _{\eta =0}^{\infty }\frac{1}{5^{\eta +\ell }}{\Psi }_A\left(5^{\eta +\ell }{w}_1\right)\to 0as{\ell }_1\to \infty ,\hfill \end{array}$$

for all $w_1\in {\mathcal{W}}_1$. Therefore $\mathcal{A}\left(w_1\right)$ is unique. So, the Theorem holds for $\mu =1$.

Changing ${\displaystyle w_1=\frac{w_1}{5}}$ in (11), we have

$$\begin{array}{c}\hfill \parallel \mathcal{F}\left(w_1\right)-5\mathcal{F}\left(\frac{w_1}{5}\right)\parallel \le {\Psi }_A\left(\frac{w_1}{5}\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(18)

Generalizing for a positive integer ℓ, we get

$$\begin{array}{c}\hfill \parallel \mathcal{F}\left(w_1\right)-5^{\ell }\mathcal{F}\left(\frac{w_1}{5^{\eta }}\right)\parallel \le \frac{1}{5}\sum _{\eta =1}^{\ell }{5}^{\eta }{\Psi }_A\left(\frac{w_1}{5^{\eta }}\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(19)

The rest of the proof is similar to that of above case. So, the Theorem holds for $\mu =-1$. Hence the proof is complete □

Corollary 2.2. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥& \le \left\{\begin{array}{cc}\frac{\delta }{\left|3\right|},\hfill & \\ \frac{4\delta |w_1{|}^{\phi }}{|5-5^{\phi }|};\hfill & \phi \ne 1,\hfill \\ \frac{4\delta }{3}\sum _{\psi =1}^3\frac{|w_{\psi }{|}^{{\phi }_{{ }_{\psi }}}}{|5-5^{{\phi }_{{ }_{\psi }}}|};\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 1,\hfill \\ \frac{4\delta |w_1{|}^{3\phi }}{3|5-5^{3\phi }|};\hfill & 3\phi \ne 1,\hfill \\ \frac{4\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}}{3|5-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|};\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 1,\hfill \\ \frac{16\delta |w_1{|}^{3\phi }}{3|5-5^{3\phi }|};\hfill & 3\phi \ne 1,\hfill \end{array}\right.\hfill \end{array}$$

(20)

for all $w_1\in {\mathcal{W}}_1$.

2.2. Evenness of $\mathcal{F}$: Quadratic Case Stability Results : Direct Method

Theorem 2.3. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the condition

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\frac{\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right)}{{25}^{\ell \mu }}=0;\mu =\pm 1\end{array}$$

(21)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥& \le \frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}{\Psi }_Q\left(5^{\eta \mu }{w}_1\right)\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & =\frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+\frac{7}{2}\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \end{array}$$

(23)

and the mapping $\mathcal{Q}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \mathcal{Q}\left(w_1\right)=\underset{\ell \to \infty }{lim}\frac{1}{{25}^{\ell \mu }}\mathcal{F}\left(5^{\ell \mu }{w}_1\right)\end{array}$$

(24)

for all $w_1\in {\mathcal{W}}_1$.

Proof. 

Using evenness of $\mathcal{F}$ in (1), we get

$$\begin{array}{cc}\hfill \parallel \mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)& +\mathcal{F}(w_1+w_2+3w_3)-7\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)-4\sum _{\psi =1}^3\mathcal{F}\left(w_{\psi }\right)\parallel \hfill \\ \hfill & \le \Psi \left(w_1,w_2,w_3\right),\forall w_1,w_2,w_3\in {\mathcal{W}}_1.\hfill \end{array}$$

(25)

Interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,w_1\right)$ in (25), we obtain

$$\begin{array}{c}\hfill \parallel 3\mathcal{F}\left(5w_1\right)-7\mathcal{F}\left(3w_1\right)-12\mathcal{F}\left(w_1\right)\parallel \le \Psi \left(w_1,w_1,w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(26)

Again interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,-w_1\right)$ in (25), we have

$$\begin{array}{cc}\hfill & \parallel 2\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right)\parallel \le \Psi \left(w_1,w_1,-w_1\right)\hfill \\ \hfill & \Rightarrow \parallel 7\mathcal{F}\left(3w_1\right)-63\mathcal{F}\left(w_1\right)\parallel \le \frac{7}{2}\Psi \left(w_1,w_1,-w_1\right),\forall w_1\in {\mathcal{W}}_1.\hfill \end{array}$$

(27)

Combining 26 and (27), we arrive

$$\begin{array}{cc}\hfill \parallel 3\mathcal{F}\left(5w_1\right)-75\mathcal{F}\left(w_1\right)\parallel & \le \parallel 3\mathcal{F}\left(5w_1\right)-7\mathcal{F}\left(3w_1\right)-12\mathcal{F}\left(w_1\right)\parallel +\parallel 7\mathcal{F}\left(3w_1\right)-63\mathcal{F}\left(w_1\right)\parallel \hfill \\ \hfill & \le \Psi \left(w_1,w_1,w_1\right)+\frac{7}{2}\Psi \left(w_1,w_1,-w_1\right),\forall w_1\in {\mathcal{W}}_1.\hfill \end{array}$$

(28)

One can see from (28) that

$$\begin{array}{c}\hfill \parallel \mathcal{F}\left(5w_1\right)-25\mathcal{F}\left(w_1\right)\parallel \le \frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+\frac{7}{2}\Psi \left(w_1,w_1,-w_1\right)\right\}={\Psi }_Q\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(29)

It follows from (29) that

$$\begin{array}{c}\hfill \parallel \frac{1}{25}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right)\parallel \le \frac{1}{25}{\Psi }_Q\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(30)

The rest of the proof is similar to that of Theorem 2.1. Hence the proof is complete. □

Corollary 2.4. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥& \le \left\{\begin{array}{cc}\frac{3\delta }{2\left|24\right|},\hfill & \\ \frac{27\delta |w_1{|}^{\phi }}{6|25-5^{\phi }|};\hfill & \phi \ne 2,\hfill \\ \frac{9\delta }{6}\sum _{\psi =1}^3\frac{|w_{\psi }{|}^{{\phi }_{{ }_{\psi }}}}{|25-5^{{\phi }_{{ }_{\psi }}}|};\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 2,\hfill \\ \frac{9\delta |w_1{|}^{3\phi }}{6|25-5^{3\phi }|};\hfill & 3\phi \ne 2,\hfill \\ \frac{9\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}}{6|25-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|};\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 2,\hfill \\ {\displaystyle \frac{36\delta |w_1{|}^{3\phi }}{6|25-5^{3\phi }|};}\hfill & 3\phi \ne 2,\hfill \end{array}\right.\hfill \end{array}$$

(31)

for all $w_1\in {\mathcal{W}}_1$.

2.3. Oddness and Evenness of $\mathcal{F}$: Additive Quadratic Case Stability Results : Direct Method

Theorem 2.5. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the conditions (3) and (21) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill & ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥\hfill \\ \hfill & \le \frac{1}{2}\left\{\frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}\left\{{\Psi }_A\left(5^{\eta \mu }{w}_1\right)+{\Psi }_A\left(-5^{\eta \mu }{w}_1\right)\right\}+\frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}\left\{{\Psi }_Q\left(5^{\eta \mu }{w}_1\right)+{\Psi }_Q\left(-5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \\ \hfill & \le \frac{1}{2}\left\{\frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+3\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right.\right.\hfill \\ \hfill & \left.+\frac{1}{3}\left\{\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)+3\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \\ \hfill & +\frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+\frac{7}{2}\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right.\hfill \\ \hfill & \left.\left.+\frac{1}{3}\left\{\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)+\frac{7}{2}\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)\right\}\right\}\right\}\hfill \end{array}$$

(32)

and the mapping $\mathcal{A}\left(w_1\right)$ and $\mathcal{Q}\left(w_1\right)$ are given in (6) and (24) for all $w_1\in {\mathcal{W}}_1$.

Proof. 

Consider a function ${\mathcal{F}}_{odd}\left(w_1\right)$ by

$$\begin{array}{c}\hfill {\mathcal{F}}_{odd}\left(w_1\right)=\frac{1}{2}\left\{\mathcal{F}\left(w_1\right)-\mathcal{F}(-w_1)\right\},\forall w_1\in {\mathcal{W}}_1,\end{array}$$

(33)

which gives

$$\begin{array}{c}\hfill {\mathcal{F}}_{odd}\left(0\right)=0;{\mathcal{F}}_{odd}(-w_1)=-{\mathcal{F}}_{odd}\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(34)

By Theorem 2.1, it follows from (33), (1), (5) and (6), we arrive

$$\begin{array}{cc}\hfill & ∥{\mathcal{F}}_{odd}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{1}{2}·\frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}\left\{{\Psi }_A\left(5^{\eta \mu }{w}_1\right)+{\Psi }_A\left(-5^{\eta \mu }{w}_1\right)\right\}\hfill \\ \hfill & =\frac{1}{2}·\frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+3\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right.\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & \left.+\frac{1}{3}\left\{\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)+3\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \end{array}$$

(36)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Consider a function ${\mathcal{F}}_{even}\left(w_1\right)$ by

$$\begin{array}{c}\hfill {\mathcal{F}}_{even}\left(w_1\right)=\frac{1}{2}\left\{\mathcal{F}\left(w_1\right)+\mathcal{F}(-w_1)\right\},\forall w_1\in {\mathcal{W}}_1,\end{array}$$

(37)

which gives

$$\begin{array}{c}\hfill {\mathcal{F}}_{even}\left(0\right)=0;{\mathcal{F}}_{even}(-w_1)={\mathcal{F}}_{even}\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(38)

By Theorem 2.3, it follows from (37), (1), (22) and (23), we see

$$\begin{array}{cc}\hfill & ∥{\mathcal{F}}_{even}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{1}{2}·\frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}\left\{{\Psi }_Q\left(5^{\eta \mu }{w}_1\right)+{\Psi }_Q\left(-5^{\eta \mu }{w}_1\right)\right\}\hfill \\ \hfill & =\frac{1}{2}·\frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+\frac{7}{2}\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right.\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & \left.+\frac{1}{3}\left\{\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)+\frac{7}{2}\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \end{array}$$

(40)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Assume a function $\mathcal{F}\left(w_1\right)$ by

$$\begin{array}{c}\hfill \mathcal{F}\left(w_1\right)={\mathcal{F}}_{odd}\left(w_1\right)+{\mathcal{F}}_{even}\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(41)

Now, it follows from (35), (36), (39), (40) and (41), we have

$$\begin{array}{cc}\hfill & ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥\hfill \\ \hfill & \le ∥{\mathcal{F}}_{odd}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥+∥{\mathcal{F}}_{even}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥\hfill \\ \hfill & \le \frac{1}{2}\left\{\frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}\left\{{\Psi }_A\left(5^{\eta \mu }{w}_1\right)+{\Psi }_A\left(-5^{\eta \mu }{w}_1\right)\right\}+\frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}\left\{{\Psi }_Q\left(5^{\eta \mu }{w}_1\right)+{\Psi }_Q\left(-5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \\ \hfill & \le \frac{1}{2}\left\{\frac{1}{5}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{5^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+3\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right.\right.\hfill \\ \hfill & \left.+\frac{1}{3}\left\{\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)+3\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)\right\}\right\}\hfill \\ \hfill & +\frac{1}{25}\sum _{\eta =\frac{1-\mu }{2}}^{\infty }\frac{1}{{25}^{\eta \mu }}\left\{\frac{1}{3}\left\{\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)+\frac{7}{2}\Psi \left(5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)\right\}\right.\hfill \\ \hfill & \left.\left.+\frac{1}{3}\left\{\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1\right)+\frac{7}{2}\Psi \left(-5^{\eta \mu }{w}_1,-5^{\eta \mu }{w}_1,5^{\eta \mu }{w}_1\right)\right\}\right\}\right\}\hfill \end{array}$$

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. □

Corollary 2.6. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥& \le \left\{\begin{array}{cc}\frac{\delta }{\left|3\right|}+\frac{3\delta }{2\left|24\right|},\hfill & \\ \frac{4\delta |w_1{|}^{\phi }}{|5-5^{\phi }|}+\frac{27\delta |w_1{|}^{\phi }}{6|25-5^{\phi }|};\hfill & \phi \ne 1,2,\hfill \\ \frac{4\delta }{3}\sum _{\psi =1}^3\frac{|w_{\psi }{|}^{{\phi }_{{ }_{\psi }}}}{|5-5^{{\phi }_{{ }_{\psi }}}|}+\frac{9\delta }{6}\sum _{\psi =1}^3\frac{|w_{\psi }{|}^{{\phi }_{{ }_{\psi }}}}{|25-5^{{\phi }_{{ }_{\psi }}}|};\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 1,2,\hfill \\ \frac{4\delta |w_1{|}^{3\phi }}{3|5-5^{3\phi }|}+\frac{9\delta |w_1{|}^{3\phi }}{6|25-5^{3\phi }|};\hfill & 3\phi \ne 1,2,\hfill \\ \frac{4\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}}{3|5-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|}+\frac{9\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}}{6|25-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|};\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 1,2,\hfill \\ \frac{16\delta |w_1{|}^{3\phi }}{3|5-5^{3\phi }|}+\frac{36\delta |w_1{|}^{3\phi }}{6|25-5^{3\phi }|};\hfill & 3\phi \ne 1,2,\hfill \end{array}\right.\hfill \end{array}$$

(42)

for all $w_1\in {\mathcal{W}}_1$.

2.4. Oddness of $\mathcal{F}$: Additive Case Stability Results : Fixed Method

Theorem 2.7. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the condition

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\frac{\Psi \left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }^{\ell }{w}_2,{\tau }_{\nu }^{\ell }{w}_3\right)}{{\tau }_{\nu }^{\ell }}=0;{\tau }_{\nu }=\left\{\begin{array}{c}5;\nu =0\hfill \\ \frac{1}{5};\nu =1\hfill \end{array}\right.,\forall w_1,w_2,w_3\in {\mathcal{W}}_1.\end{array}$$

(43)

If there exists $L=L\left(\nu \right)$ be a function have the property

$$\begin{array}{c}\hfill {\Psi }_A\left(w_1\right)={\Psi }_A\left(\frac{w_1}{5}\right)and\frac{1}{{\tau }_{\nu }}{\Psi }_A\left({\tau }_{\nu }{w}_1\right)=L{\Psi }_A\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(44)

Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥& \le \frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right)\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \end{array}$$

(46)

and the mapping $\mathcal{A}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \mathcal{A}\left(w_1\right)=\underset{\ell \to \infty }{lim}\frac{1}{{\tau }_{\nu }^{\ell }}\mathcal{F}\left({\tau }_{\nu }^{\ell }{w}_1\right)\end{array}$$

(47)

for all $w_1\in {\mathcal{W}}_1$.

Proof. 

Assume a set

$$\mathcal{G}=\{\mathcal{F}/\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2,\mathcal{F}\left(0\right)=0\}$$

(48)

and introduce the generalized metric on the above set $\mathcal{G}$ as

$$d(\mathcal{F},{\mathcal{F}}_1)=inf\{K\in (0,\infty ):∥\mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right)∥\le K\Psi (w_1,w_1,w_1),w_1\in {\mathcal{W}}_1\}.$$

(49)

It is easy to see that $(\mathcal{G},d)$ is complete. Define a function $\mathcal{H}:\mathcal{G}\to \mathcal{G}$ by

$$\mathcal{H}\mathcal{F}\left(w_1\right)=\frac{1}{{\tau }_{\nu }}\mathcal{F}\left({\tau }_{\nu }{w}_1\right),forall{w}_1\in {\mathcal{W}}_1.$$

(50)

Now $\mathcal{F},{\mathcal{F}}_1\in \mathcal{G}$ and $w_1\in {\mathcal{W}}_1$, we see

$$\begin{array}{cc}\hfill d(\mathcal{F},{\mathcal{F}}_1)\le K\Rightarrow & \Vert \mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right)\Vert \le K\Psi (w_1,w_1,w_1),\hfill \\ \hfill \Rightarrow & ∥\frac{1}{{\tau }_{\nu }}\mathcal{F}\left({\tau }_{\nu }{w}_1\right)-\frac{1}{{\tau }_{\nu }}{\mathcal{F}}_1\left({\tau }_{\nu }{w}_1\right)∥\le \frac{1}{{\tau }_{\nu }}K\Psi ({\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1),\hfill \\ \hfill \Rightarrow & \Vert \mathcal{H}\mathcal{F}\left(w_1\right)-\mathcal{H}{\mathcal{F}}_1\left(w_1\right)\Vert \le LK\Psi (w_1,w_1,w_1),\hfill \\ \hfill \Rightarrow & d(\mathcal{H}\mathcal{F},\mathcal{H}{\mathcal{F}}_1)\le LK,\hfill \end{array}$$

i.e., $\mathcal{H}$ is a strictly contractive mapping on $\mathcal{G}$ with Lipschitz constant L (see [24]).

For the case $\nu =0$, it follows from (12) and with the help of (44), (50), (49), we get

$$\begin{array}{c}\hfill \parallel \frac{1}{5}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right)\parallel \le \frac{1}{5}{\Psi }_A\left(w_1\right),\Rightarrow d(\mathcal{H}\mathcal{F},\mathcal{F})\le L=L^{1-\nu },\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(51)

For the case $\nu =1$, it follows from (18) and with the help of (44), (50), (49), we obtain

$$\begin{array}{c}\hfill \parallel \mathcal{F}\left(w_1\right)-5\mathcal{F}\left(\frac{w_1}{5}\right)\parallel \le {\Psi }_A\left(\frac{w_1}{5}\right),\Rightarrow d(\mathcal{F},\mathcal{H}\mathcal{F})\le 1=L^{1-\nu },\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(52)

Combining (51) and (52), we have

$$\begin{array}{c}\hfill d(\mathcal{F},\mathcal{H}\mathcal{F})\le 1=L^{1-\nu }.\end{array}$$

(53)

Therefore $\left(FPC1\right)$ of Theorem 1.3 holds. The rest of the proof follows by Theorem 1.3. Hence the proof is complete. □

Corollary 2.8. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality (20) for all $w_1\in {\mathcal{W}}_1$.

Proof. 

If we take

$$\begin{array}{c}\hfill \Psi \left(w_1,w_2,w_3\right)=\left\{\begin{array}{c}\delta ,\hfill \\ \delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi },\hfill \\ \delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\hfill \\ \delta \prod _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi },\hfill \\ \delta \prod _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\hfill \\ \delta \left\{\sum _{\psi =1}^3{\left|w_{\psi }\right|}^{3\phi }+\prod _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi }\right\},\hfill \end{array}\right.\end{array}$$

(54)

in Theorem 2.7 and changing $\left(w_1,w_2,w_3\right)$ by $\left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }^{\ell }{w}_2,{\tau }_{\nu }^{\ell }{w}_3\right)$ and dividing by ${\tau }_{\nu }^{\ell }$ in (54), one can see

$$\begin{array}{c}\hfill \frac{1}{{\tau }_{\nu }^{\ell }}\Psi \left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }{w}_2^{\ell },{\tau }_{\nu }^{\ell }{w}_3\right)=\left\{\begin{array}{cc}\frac{\delta }{{\tau }_{\nu }^{\ell }}\hfill & \to 0as\ell to\infty ,\hfill \\ \frac{\delta }{{\tau }_{\nu }^{\ell }}\sum _{\psi =1}^3{\left|{\tau }_{\nu }^{\ell }{w}_{\psi }\right|}^{\phi },\hfill & \to 0as\ell to\infty ,\hfill \\ \frac{\delta }{{\tau }_{\nu }^{\ell }}\sum _{\psi =1}^3{\left|{\tau }_{\nu }^{\ell }{w}_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\hfill & \to 0as\ell to\infty ,\hfill \\ \frac{\delta }{{\tau }_{\nu }^{\ell }}\prod _{\psi =1}^3{\left|{\tau }_{\nu }^{\ell }{w}_{\psi }\right|}^{\phi },\hfill & \to 0as\ell to\infty ,\hfill \\ \frac{\delta }{{\tau }_{\nu }^{\ell }}\prod _{\psi =1}^3{\left|{\tau }_{\nu }^{\ell }{w}_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\hfill & \to 0as\ell to\infty ,\hfill \\ \frac{\delta }{{\tau }_{\nu }^{\ell }}\left\{\sum _{\psi =1}^3{\left|{\tau }_{\nu }^{\ell }{w}_{\psi }\right|}^{3\phi }+\prod _{\psi =1}^3{\left|{\tau }_{\nu }^{\ell }{w}_{\psi }\right|}^{\phi }\right\},\hfill & \to 0as\ell to\infty .\hfill \end{array}\right.\end{array}$$

Therefore (43) holds for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Now, from (44), we have

$$\begin{array}{c}\hfill {\Psi }_A\left(w_1\right)={\Psi }_A\left(\frac{w_1}{5}\right)=\frac{1}{3}\left\{\Psi \left(\frac{w_1}{5},\frac{w_1}{5},\frac{w_1}{5}\right)+3\Psi \left(\frac{w_1}{5},\frac{w_1}{5},-\frac{w_1}{5}\right)\right\}=\left\{\begin{array}{c}\frac{4\delta }{3},\hfill \\ \frac{12|\frac{w_1}{5}{|}^{\phi }}{3},\hfill \\ \frac{4\delta }{3}\sum _{\psi =1}^3{\left|\frac{w_1}{5}\right|}^{{\phi }_{{ }_{\psi }}},\hfill \\ \frac{4\delta |\frac{w_1}{5}{|}^{3\phi }}{3},\hfill \\ \frac{4\delta |\frac{w_1}{5}{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}}{3},\hfill \\ \frac{16\delta |\frac{w_1}{5}{|}^{3\phi }}{3},\hfill \end{array}\right.\end{array}$$

(55)

$$\begin{array}{cc}\hfill \frac{1}{{\tau }_{\nu }}{\Psi }_A\left({\tau }_{\nu }{w}_1\right)& =\frac{1}{{\tau }_{\nu }}\frac{1}{3}\left\{\Psi \left({\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1\right)+3\Psi \left({\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1,-{\tau }_{\nu }{w}_1\right)\right\}\hfill \\ \hfill & =\left\{\begin{array}{c}\frac{4\delta }{{\tau }_{\nu }·3},\hfill \\ \frac{12\delta |{\tau }_{\nu }{w}_1{|}^{\phi }}{{\tau }_{\nu }·3},\hfill \\ \frac{4\delta }{{\tau }_{\nu }·3}\sum _{\psi =1}^3{\left|{\tau }_{\nu }{w}_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\hfill \\ \frac{4\delta |{\tau }_{\nu }{w}_1{|}^{3\phi }}{{\tau }_{\nu }·3},\hfill \\ \frac{4\delta |{\tau }_{\nu }{w}_1{|}^{{\sum }_{\psi =1}^3{\phi }_{{ }_{\psi }}}}{{\tau }_{\nu }·3},\hfill \\ \frac{16\delta |{\tau }_{\nu }{w}_1{|}^{3\phi }}{{\tau }_{\nu }·3},\hfill \end{array}\right.=\left\{\begin{array}{c}{\tau }_{\nu }^{-1}{\Psi }_A\left(w_1\right),\hfill \\ {\tau }_{\nu }^{\phi -1}{\Psi }_A\left(w_1\right),\hfill \\ \sum _{\psi =1}^3{\tau }_{\nu }^{{\phi }_{{ }_{\psi }}-1}{\Psi }_A\left(w_1\right),\hfill \\ {\tau }_{\nu }^{3\phi -1}{\Psi }_A\left(w_1\right),\hfill \\ {\tau }_{\nu }^{{\sum }_{\psi =1}^3{\phi }_{{ }_{\psi }}-1}{\Psi }_A\left(w_1\right),\hfill \\ {\tau }_{\nu }^{3\phi -1}{\Psi }_A\left(w_1\right),\hfill \end{array}\right.=\left\{\begin{array}{c}L{\Psi }_A\left(w_1\right)\hfill \\ L{\Psi }_A\left(w_1\right)\hfill \\ L{\Psi }_A\left(w_1\right)\hfill \\ L{\Psi }_A\left(w_1\right)\hfill \\ L{\Psi }_A\left(w_1\right)\hfill \\ L{\Psi }_A\left(w_1\right)\hfill \end{array}\right.\hfill \end{array}$$

(56)

for all $w_1\in {\mathcal{W}}_1$.

For the case $\nu =0$, we have $L={\tau }_0^{-1}=5^{-1}$ and from (46), we arrive

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥\le \frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right)& =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \\ \hfill & =\frac{{\left(5^{-1}\right)}^{1-0}}{1-5^{-1}}\left\{\frac{4\delta }{3}\right\}=\frac{\delta }{3}.\hfill \end{array}$$

For the case $\nu =1$, we have $L={\tau }_1^{-1}={\left(\frac{1}{5}\right)}^{-1}=5$ and from (46), we obtain

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥\le \frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right)& =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \\ \hfill & =\frac{{\left(5\right)}^{1-1}}{1-5}\left\{\frac{4\delta }{3}\right\}=\frac{\delta }{-3}.\hfill \end{array}$$

For the case $\nu =0$, we have $L={\tau }_0^{\phi -1}=5^{\phi -1}$ and from (46), we arrive

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥\le \frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right)& =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \\ \hfill & =\frac{{\left(5^{\phi -1}\right)}^{1-0}}{1-5^{\phi -1}}\left\{\frac{12\delta |\frac{w_1}{5}{|}^{\phi }}{3}\right\}=\frac{4\delta }{5-5^{\phi }}.\hfill \end{array}$$

For the case $\nu =1$, we have $L={\tau }_1^{\phi -1}={\left(\frac{1}{5}\right)}^{\phi -1}=5^{1-\phi }$ and from (46), we get

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥\le \frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right)& =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \\ \hfill & =\frac{{\left(5^{1-\phi }\right)}^{1-1}}{1-5^{1-\phi }}\left\{\frac{12\delta |\frac{w_1}{5}{|}^{\phi }}{3}\right\}=\frac{4\delta }{5^{\phi }-5}.\hfill \end{array}$$

For the case $\nu =0$, we have $L={\tau }_0^{3\phi -1}=5^{3\phi -1}$ and from (46), we arrive

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥\le \frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right)& =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \\ \hfill & =\frac{{\left(5^{3\phi -1}\right)}^{1-0}}{1-5^{3\phi -1}}\left\{\frac{4\delta |\frac{w_1}{5}{|}^{3\phi }}{3}\right\}=\frac{4\delta |w_1{|}^{3\phi }}{3(5-5^{3\phi })}.\hfill \end{array}$$

For the case $\nu =1$, we have $L={\tau }_1^{3\phi -1}={\left(\frac{1}{5}\right)}^{3\phi -1}=5^{1-3\phi }$ and from (46), we obtain

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)∥\le \frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right)& =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \\ \hfill & =\frac{{\left(5^{1-3\phi }\right)}^{1-1}}{1-5^{1-3\phi }}\left\{\frac{4\delta |\frac{w_1}{5}{|}^{3\phi }}{3}\right\}=\frac{4\delta |w_1{|}^{3\phi }}{3(5^{3\phi }-5)}.\hfill \end{array}$$

Similarly, we can prove for rest of the cases. □

2.5. Evenness of $\mathcal{F}$: Quadratic Case Stability Results : Fixed Method

Theorem 2.9. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the condition

$$\begin{array}{c}\hfill \underset{\ell \to \infty }{lim}\frac{\Psi \left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }^{\ell }{w}_2,{\tau }_{\nu }^{\ell }{w}_3\right)}{{\tau }_{\nu }^{2\ell }}=0;{\tau }_{\nu }=\left\{\begin{array}{c}5;\nu =0\hfill \\ \frac{1}{5};\nu =1\hfill \end{array}\right.,\forall w_1,w_2,w_3\in {\mathcal{W}}_1.\end{array}$$

(57)

If there exists $L=L\left(\nu \right)$ be a function have the property

$$\begin{array}{c}\hfill {\Psi }_Q\left(w_1\right)={\Psi }_Q\left(\frac{w_1}{5}\right)and\frac{1}{{\tau }_{\nu }^2}{\Psi }_Q\left({\tau }_{\nu }{w}_1\right)=L{\Psi }_Q\left(w_1\right),\forall w_1\in {\mathcal{W}}_1.\end{array}$$

(58)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill ∥\mathcal{F}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥& \le \frac{L^{1-\nu }}{1-L}{\Psi }_Q\left(w_1\right)\hfill \end{array}$$

(59)

$$\begin{array}{cc}\hfill & =\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\left\{\Psi \left(w_1,w_1,w_1\right)+\frac{7}{2}\Psi \left(w_1,w_1,-w_1\right)\right\}\right\}\hfill \end{array}$$

(60)

and the mapping $\mathcal{Q}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \mathcal{Q}\left(w_1\right)=\underset{\ell \to \infty }{lim}\frac{1}{{\tau }_{\nu }^{2\ell }}\mathcal{F}\left({\tau }_{\nu }^{\ell }{w}_1\right)\end{array}$$

(61)

for all $w_1\in {\mathcal{W}}_1$.

Proof. 

By Theorem 2.7, define a function $\mathcal{H}:\mathcal{G}\to \mathcal{G}$ by

$$\mathcal{H}\mathcal{F}\left(w_1\right)=\frac{1}{{\tau }_{\nu }^2}\mathcal{F}\left({\tau }_{\nu }{w}_1\right),forall{w}_1\in {\mathcal{W}}_1.$$

(62)

Now $\mathcal{F},{\mathcal{F}}_1\in \mathcal{G}$ and $w_1\in {\mathcal{W}}_1$, we see

$$\begin{array}{cc}\hfill d(\mathcal{F},{\mathcal{F}}_1)\le K\Rightarrow & \Vert \mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right)\Vert \le K\Psi (w_1,w_1,w_1),\hfill \\ \hfill \Rightarrow & ∥\frac{1}{{\tau }_{\nu }^2}\mathcal{F}\left({\tau }_{\nu }{w}_1\right)-\frac{1}{{\tau }_{\nu }^2}{\mathcal{F}}_1\left({\tau }_{\nu }{w}_1\right)∥\le \frac{1}{{\tau }_{\nu }^2}K\Psi ({\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1),\hfill \\ \hfill \Rightarrow & \Vert \mathcal{H}\mathcal{F}\left(w_1\right)-\mathcal{H}{\mathcal{F}}_1\left(w_1\right)\Vert \le LK\Psi (w_1,w_1,w_1),\hfill \\ \hfill \Rightarrow & d(\mathcal{H}\mathcal{F},\mathcal{H}{\mathcal{F}}_1)\le LK,\hfill \end{array}$$

i.e., $\mathcal{H}$ is a strictly contractive mapping on $\mathcal{G}$ with Lipschitz constant L (see [24]). The rest of the proof is similar to that of Theorem 2.7. Hence the proof is complete. □

Corollary 2.10. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality (31) for all $w_1\in {\mathcal{W}}_1$.

2.6. Oddness and Evenness of $\mathcal{F}$: Additive Quadratic Case Stability Results : Fixed Method

Theorem 2.11. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the conditions (43) and (57) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. If there exists $L=L\left(\nu \right)$ be function have the properties (44) and (58) Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{cc}\hfill & ∥\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right)∥\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le \frac{1}{2}·\frac{L^{1-\nu }}{1-L}\left\{{\Psi }_A\left(w_1\right)+{\Psi }_A\left(-w_1\right)+{\Psi }_Q\left(w_1\right)+{\Psi }_Q\left(-w_1\right)\right\}\hfill \\ \hfill & =\frac{1}{2}·\frac{L^{1-\nu }}{1-L}\left\{\frac{1}{3}\{\Psi \left(w_1,w_1,w_1\right)+3\Psi \left(w_1,w_1,-w_1\right)+\Psi \left(-w_1,-w_1,-w_1\right)+3\Psi \left(-w_1,-w_1,w_1\right)\right.\hfill \end{array}$$

(63)

$$\begin{array}{cc}\hfill & \left.+\Psi \left(w_1,w_1,w_1\right)+\frac{7}{2}\Psi \left(w_1,w_1,-w_1\right)+\Psi \left(-w_1,-w_1,-w_1\right)+\frac{7}{2}\Psi \left(-w_1,-w_1,w_1\right)\}\right\}\hfill \end{array}$$

(64)

and the mapping $\mathcal{A}\left(w_1\right)$ and $\mathcal{Q}\left(w_1\right)$ are given in (47) and (61) for all $w_1\in {\mathcal{W}}_1$.

Proof. 

The proof is similar ideas to that of Theorem 2.5. □

Corollary 2.12. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality (42) for all $w_1\in {\mathcal{W}}_1$.

3. Stability In Intuitionistic Fuzzy Banach Space of (3)

In this section, we explore the generalized Ulam - Hyers stability of the functional equations (3) in Intuitionistic Fuzzy Banach Space.

In order to prove stability results, assume $\left({\mathcal{W}}_1,\mu ,\nu \right)$ and $\left({\mathcal{W}}_2,{\mu }^{\prime },{\nu }^{\prime }\right)$ are Intuitionistic Fuzzy normed space and Intuitionistic Fuzzy Banach space respectively. Suppose that $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ and $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ satisfying the following functional inequalities

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ \left.{\displaystyle -\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{F}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(w_{\psi }\right)+\mathcal{F}(-w_{\psi })\right]\right\},\Lambda }\right)\hfill \\ \hfill \ge {\mu }^{\prime }\left(\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\\ \nu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ \left.{\displaystyle -\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{F}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(w_{\psi }\right)+\mathcal{F}(-w_{\psi })\right]\right\},\Lambda }\right)\hfill \\ \hfill \le {\nu }^{\prime }\left(\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\end{array}\right\}\end{array}$$

(1)

and

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ \left.{\displaystyle -\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{F}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(w_{\psi }\right)+\mathcal{F}(-w_{\psi })\right]\right\},\Lambda }\right)\hfill \\ \hfill \ge \left\{\begin{array}{c}{\mu }^{\prime }\left(\delta ,\Lambda \right),\hfill \\ {\mu }^{\prime }\left(\delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi },\Lambda \right),\hfill \\ {\mu }^{\prime }\left(\delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\Lambda \right),\hfill \\ {\mu }^{\prime }\left(\delta \prod _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi },\Lambda \right),\hfill \\ {\mu }^{\prime }\left(\delta \prod _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\Lambda \right),\hfill \\ {\mu }^{\prime }\left(\delta \left\{\sum _{\psi =1}^3{\left|w_{\psi }\right|}^{3\phi }+\prod _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi }\right\},\Lambda \right),\hfill \end{array}\right.\\ \nu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ \left.{\displaystyle -\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{F}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(w_{\psi }\right)+\mathcal{F}(-w_{\psi })\right]\right\},\Lambda }\right)\hfill \\ \hfill \le \left\{\begin{array}{c}{\nu }^{\prime }\left(\delta ,\Lambda \right),\hfill \\ {\nu }^{\prime }\left(\delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi },\Lambda \right),\hfill \\ {\nu }^{\prime }\left(\delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\Lambda \right),\hfill \\ {\nu }^{\prime }\left(\delta \prod _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi },\Lambda \right),\hfill \\ {\nu }^{\prime }\left(\delta \prod _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\Lambda \right),\hfill \\ {\nu }^{\prime }\left(\delta \left\{\sum _{\psi =1}^3{\left|w_{\psi }\right|}^{3\phi }+\prod _{\psi =1}^3{\left|w_{\psi }\right|}^{\phi }\right\},\Lambda \right),\hfill \end{array}\right.\end{array}\right\}\end{array}$$

(2)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$ with $\delta$ be a positive constant.

3.1. Definitions and Notations of Intuitionistic Fuzzy Banach Space

Now, we recall the basic definitions and notations in the setting of intuitionistic fuzzy normed space given in [33].

Definition 3.1. 

[33] A binary operation $*:\left[0,1\right]\times \left[0,1\right]⟶\left[0,1\right]$ is said to be continuous t-norm if * satisfies the following conditions:

(*1) 

* is commutative and associative;

(*2) 

* is continuous;

(*3) 

$a*1=a$ for all $a\in \left[0,1\right]$;

(*4) 

$a*b\le c*d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in \left[0,1\right].$

Definition 3.2. 

[33] A binary operation $\diamond :\left[0,1\right]\times \left[0,1\right]⟶\left[0,1\right]$ is said to be continuous t-conorm if ⋄ satisfies the following conditions:

(⋄1) 

⋄ is commutative and associative;

(⋄2) 

⋄ is continuous;

(⋄3) 

$a\diamond 0=a$ for all $a\in \left[0,1\right]$;

(⋄4) 

$a\diamond b\le c\diamond d$ whenever $a\le c$ and $b\le d$ for all $a,b,c,d\in \left[0,1\right].$

Definition 3.3. 

[33] The five-tuple $(X,\mu ,\nu ,*,\diamond )$ is said to be an intuitionistic fuzzy normed space (for short, IFNS) if X is a vector space, * is a continuous t-norm, ⋄ is a continuous $t-$ conorm, and $\mu ,\nu$ are fuzzy sets on $X\times (0,\infty )$ satisfying the following conditions. For every $x,y\in X$ and $s,t>0$

(IFN1) 

$\mu (x,t)+\nu (x,t)\le 1$;

(IFN2) 

$\mu (x,t)>0$;

(IFN3) 

$\mu (x,t)=1$, if and only if $x=0$;

(IFN4) 

$\mu (dx,t)=\mu \left(x,\frac{t}{d}\right)$ for each $d\ne 0$;

(IFN5) 

$\mu (x,t)*\mu (y,s)\le \mu (x+y,t+s)$;

(IFN6) 

$\mu (x,·):(0,\infty )\to [0,1]$ is continuous;

(IFN7) 

$\underset{t\to \infty }{lim}\mu (x,t)=1$ and $\underset{t\to 0}{lim}\mu (x,t)=0$;

(IFN8) 

$\nu (x,t)<1$;

(IFN9) 

$\nu (x,t)=0$, if and only if $x=0$;

(IFN10) 

$\nu (dx,t)=\nu \left(x,\frac{t}{d}\right)$ for each $d\ne 0$;

(IFN11) 

$\nu (x,t)\diamond \nu (y,s)\ge \nu (x+y,t+s)$;

(IFN12) 

$\nu (x,·):(0,\infty )\to [0,1]$ is continuous;

(IFN13) 

$\underset{t\to \infty }{lim}\nu (x,t)=0$ and $\underset{t\to 0}{lim}\nu (x,t)=1$.

In this case, $(\mu ,\nu )$ is called an intuitionistic fuzzy norm.

Example 3.4. 

[33] Let $\left(X,∥·∥\right)$ be a normed space. Let $a*b=ab$ and $a\diamond d=min\left\{a+b,1\right\}$ for all $a,b\in [0,1]$. For all $x\in X$ and every $t>0$, consider

$$\begin{array}{c}\hfill \mu (x,t)=\left\{\begin{array}{ccc}\frac{t}{t+∥x∥}\hfill & if\hfill & t>0;\hfill \\ 0\hfill & if\hfill & t\le 0;\hfill \end{array}\right.and\nu (x,t)=\left\{\begin{array}{ccc}\frac{∥x∥}{t+∥x∥}\hfill & if\hfill & t>0;\hfill \\ 0\hfill & if\hfill & t\le 0.\hfill \end{array}\right.\end{array}$$

Then $\left(X,\mu ,\nu ,*,\diamond \right)$ is an IFN-space.

Definition 3.5. 

[33] Let $\left(X,\mu ,\nu ,*,\diamond \right)$ be an IFNS. Then, a sequence $x=\left\{x_k\right\}$ is said to be intuitionistic fuzzy convergent to a point $L\in X$ if

$$lim\mu (x_k-L,t)=1andlim\nu (x_k-L,t)=0$$

for all $\rho >0$. In this case, we write

$$x_k\stackrel{IF}{⟶}Lask\to \infty$$

Definition 3.6. 

[33] Let $\left(X,\mu ,\nu ,*,\diamond \right)$ be an IFN-space. Then, $x=\left\{x_k\right\}$ is said to be intuitionistic fuzzy Cauchy sequence if

$$\mu \left(x_{k+p}-x_k,t\right)=1and\nu \left(x_{k+p}-x_k,t\right)=0$$

for all $\rho >0$, and $p=1,2\cdots$.

Definition 3.7. 

[33] Let $\left(X,\mu ,\nu ,*,\diamond \right)$ be an IFN-space. Then $\left(X,\mu ,\nu ,*,\diamond \right)$ is said to be complete if every intuitionistic fuzzy Cauchy sequence in $\left(X,\mu ,\nu ,*,\diamond \right)$ is intuitionistic fuzzy convergent $\left(X,\mu ,\nu ,*,\diamond \right)$.

3.2. Oddness of $\mathcal{F}$: Additive Case Stability Results : Direct Method

Theorem 3.8. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the conditions

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\mu }^{\prime }\left(\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right),\Lambda \right)\ge {\mu }^{\prime }\left(I^{\ell \mu }\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\hfill \\ {\nu }^{\prime }\left(\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right),\Lambda \right)\le {\nu }^{\prime }\left(I^{\ell \mu }\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\hfill \end{array}\right\}\end{array}$$

(3)

and

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}{\mu }^{\prime }\left(\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right),5^{\ell \mu }\Lambda \right)=1\hfill \\ \underset{\ell \to \infty }{lim}{\nu }^{\prime }\left(\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right),5^{\ell \mu }\Lambda \right)=0\hfill \end{array}\right\}\end{array}$$

(4)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$ with $\mu =\pm 1$ and $0<{\left(\frac{I}{5}\right)}^{\mu }<1$ . Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}|5-I|\right)}\hfill \\ {\displaystyle ={\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}|5-I|\right)*{\mu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}|5-I|\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}(5-I)\right)}\hfill \\ {\displaystyle ={\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}(5-I)\right)\diamond {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}(5-I)\right)}\hfill \end{array}\right\}\end{array}$$

(5)

and the mapping $\mathcal{A}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}\mu \left({\displaystyle \frac{1}{5^{\ell \mu }}\mathcal{F}\left(5^{\ell \mu }{w}_1\right)-\mathcal{A}\left(w_1\right),\Lambda }\right)=1\hfill \\ \underset{\ell \to \infty }{lim}\nu \left({\displaystyle \frac{1}{5^{\ell \mu }}\mathcal{F}\left(5^{\ell \mu }{w}_1\right)-\mathcal{A}\left(w_1\right),\Lambda }\right)=0\hfill \end{array}\right\}\end{array}$$

(6)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$.

Proof. 

Using oddness of $\mathcal{F}$ in (1), we get

$$\begin{array}{c}\hfill \left.\begin{array}{c}\hfill {\displaystyle \mu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\sum _{\psi =1}^3\mathcal{F}\left(w_{\psi }\right),\Lambda \right)}\hfill \\ \hfill \ge {\mu }^{\prime }\left(\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\\ \hfill {\displaystyle \nu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-6\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\sum _{\psi =1}^3\mathcal{F}\left(w_{\psi }\right),\Lambda \right)}\hfill \\ \hfill \le {\nu }^{\prime }\left(\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\end{array}\right\}\end{array}$$

(7)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$ . Interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,w_1\right)$ in (7), we obtain

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(3\mathcal{F}\left(5w_1\right)-6\mathcal{F}\left(3w_1\right)+3\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \\ \nu \left(3\mathcal{F}\left(5w_1\right)-6\mathcal{F}\left(3w_1\right)+3\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \end{array}\right\}\end{array}$$

(8)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$ . Again interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,-w_1\right)$ in (7) and using (IFN4), (IFN10), we have

$$\begin{array}{cc}\hfill & \left.\begin{array}{c}\mu \left(2\mathcal{F}\left(3w_1\right)-6\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \\ \nu \left(2\mathcal{F}\left(3w_1\right)-6\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill & \Rightarrow \left.\begin{array}{c}\mu \left(6\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right),3\Lambda \right)\ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \\ \nu \left(6\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right),3\Lambda \right)\le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)\hfill \end{array}\right\}\hfill \end{array}$$

(9)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Combining (8) and (9) using (IFN5), (IFN11), we arrive

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(3\mathcal{F}\left(5w_1\right)-15\mathcal{F}\left(w_1\right),4\Lambda \right)\ge \mu \left(3\mathcal{F}\left(5w_1\right)-6\mathcal{F}\left(3w_1\right)+3\mathcal{F}\left(w_1\right),\Lambda \right)*\mu \left(6\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right),3\Lambda \right)\hfill \\ \ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)*{\mu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)={\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)\hfill \\ \nu \left(3\mathcal{F}\left(5w_1\right)-15\mathcal{F}\left(w_1\right),4\Lambda \right)\le \nu \left(3\mathcal{F}\left(5w_1\right)-6\mathcal{F}\left(3w_1\right)+3\mathcal{F}\left(w_1\right),\Lambda \right)\diamond \nu \left(6\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right),3\Lambda \right)\hfill \\ \le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\diamond {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)={\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)\hfill \end{array}\right\}\end{array}$$

(10)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Using (IFN4), (IFN10), one can see from (10) that

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{5}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right),\frac{4}{3}·\frac{1}{5}\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\frac{1}{5}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right),\frac{4}{3}·\frac{1}{5}\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \end{array}\right\}\end{array}$$

(11)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Changing $w_1$ by $5^{\ell }{w}_1$ in (11), and using (IFN4), (IFN10), (3), we get

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{5^{\ell +1}}\mathcal{F}\left(5^{\ell +1}{w}_1\right)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right),\frac{4}{3·5}·\frac{1}{5^{\ell }}\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(5^{\ell }{w}_1\right),\Lambda \right)\ge {\mu }^{\prime }\left(I^{\ell }{\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle ={\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{1}{I^{\ell }}\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\frac{1}{5^{\ell +1}}\mathcal{F}\left(5^{\ell +1}{w}_1\right)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right),\frac{4}{3·5}·\frac{1}{5^{\ell }}\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(5^{\ell }{w}_1\right),\Lambda \right)\le {\nu }^{\prime }\left(I^{\ell }{\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle ={\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{1}{I^{\ell }}\Lambda \right)}\hfill \end{array}\right\}\end{array}$$

(12)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$ also $\ell >0$. Changing $\Lambda$ by $I^{\ell }\Lambda$ in (112), we see

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{5^{\ell +1}}\mathcal{F}\left(5^{\ell +1}{w}_1\right)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right),\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\ell }\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\frac{1}{5^{\ell +1}}\mathcal{F}\left(5^{\ell +1}{w}_1\right)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right),\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\ell }\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \end{array}\right\}\end{array}$$

(13)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. It is easy to check that

$$\begin{array}{c}\hfill \frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{F}\left(w_1\right)=\sum _{\eta =0}^{\ell -1}\frac{1}{5^{\eta +1}}\mathcal{F}\left(5^{\eta +1}{w}_1\right)-\frac{1}{5^{\eta }}\mathcal{F}\left(5^{\eta }{w}_1\right)\end{array}$$

(14)

for all $w_1\in {\mathcal{W}}_1$. Using (IFN5), (IFN11), it follows from (13) and (14), we obtain

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{F}\left(w_1\right),\sum _{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta }\Lambda \right)}\hfill \\ {\displaystyle =\mu \left(\sum _{\eta =0}^{\ell -1}\frac{1}{5^{\eta +1}}\mathcal{F}\left(5^{\eta +1}{w}_1\right)-\frac{1}{5^{\eta }}\mathcal{F}\left(5^{\eta }{w}_1\right),\sum _{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta }\Lambda \right)}\hfill \\ {\displaystyle \ge \prod _{\eta =0}^{\ell -1}\mu \left(\frac{1}{5^{\eta +1}}\mathcal{F}\left(5^{\eta +1}{w}_1\right)-\frac{1}{5^{\eta }}\mathcal{F}\left(5^{\eta }{w}_1\right),\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta }\Lambda \right)}\hfill \\ \ge \prod _{\eta =0}^{\ell -1}{\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)={\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)\hfill \\ {\displaystyle \nu \left(\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{F}\left(w_1\right),\sum _{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta }\Lambda \right)}\hfill \\ {\displaystyle =\nu \left(\sum _{\eta =0}^{\ell -1}\frac{1}{5^{\eta +1}}\mathcal{F}\left(5^{\eta +1}{w}_1\right)-\frac{1}{5^{\eta }}\mathcal{F}\left(5^{\eta }{w}_1\right),\sum _{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta }\Lambda \right)}\hfill \\ {\displaystyle \le \coprod _{\eta =0}^{\ell -1}\nu \left(\frac{1}{5^{\eta +1}}\mathcal{F}\left(5^{\eta +1}{w}_1\right)-\frac{1}{5^{\eta }}\mathcal{F}\left(5^{\eta }{w}_1\right),\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta }\Lambda \right)}\hfill \\ \le \coprod _{\eta =0}^{\ell -1}{\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)={\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)\hfill \end{array}\right\}\end{array}$$

(15)

where

$$\prod _{\eta =0}^{\ell -1}\mu =\mu *\mu *\mu *...and\coprod _{\eta =0}^{\ell -1}\nu =\nu \diamond \nu \diamond \nu \diamond ...$$

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Again changing $w_1$ by $5^{{\ell }_1}{w}_1$ in (15), and using (IFN4), (IFN10), (3) in that changing $\Lambda$ by $I^{{\ell }_1}\Lambda$, we have

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{5^{\ell +{\ell }_1}}\mathcal{F}\left(5^{\ell +{\ell }_1}{w}_1\right)-\frac{1}{5^{{\ell }_1}}\mathcal{F}\left({\ell }_1{w}_1\right),\sum _{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta +{\ell }_1}\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\frac{1}{5^{\ell +{\ell }_1}}\mathcal{F}\left(5^{\ell +{\ell }_1}{w}_1\right)-\frac{1}{5^{{\ell }_1}}\mathcal{F}\left({\ell }_1{w}_1\right),\sum _{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta +{\ell }_1}\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \end{array}\right\}\end{array}$$

(16)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$ also $\ell ,{\ell }_1>0$. It follows from (16) that

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{5^{\ell +{\ell }_1}}\mathcal{F}\left(5^{\ell +{\ell }_1}{w}_1\right)-\frac{1}{5^{{\ell }_1}}\mathcal{F}\left({\ell }_1{w}_1\right),\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{\Lambda }{{\sum }_{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta +{\ell }_1}}\right)}\hfill \\ {\displaystyle \nu \left(\frac{1}{5^{\ell +{\ell }_1}}\mathcal{F}\left(5^{\ell +{\ell }_1}{w}_1\right)-\frac{1}{5^{{\ell }_1}}\mathcal{F}\left({\ell }_1{w}_1\right),\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{\Lambda }{{\sum }_{\eta =0}^{\ell -1}\frac{4}{3·5}·{\left(\frac{I}{5}\right)}^{\eta +{\ell }_1}}\right)}\hfill \end{array}\right\}\end{array}$$

(17)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. By data, the Cauchy criterion for convergence in Intuitionistic Fuzzy normed space gives that the sequence $\left\{\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)\right\},$ is Cauchy in $\left({\mathcal{W}}_2,{\mu }^{\prime },{\nu }^{\prime }\right)$ and it is a complete Intuitionistic Fuzzy normed space, this sequence converges to some point $\mathcal{A}\left(w_1\right)$ in $\left({\mathcal{W}}_2,{\mu }^{\prime },{\nu }^{\prime }\right)$ for all $w_1\in {\mathcal{W}}_1$. So, by notation, we write

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}\mu \left({\displaystyle \frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{A}\left(w_1\right),\Lambda }\right)=1\hfill \\ \underset{\ell \to \infty }{lim}\nu \left({\displaystyle \frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{A}\left(w_1\right),\Lambda }\right)=0\hfill \end{array}\right\}\end{array}$$

(18)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Letting ${\ell }_1=0$ and $\ell \to \infty$ in (17) and using (18), we arrive

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}(5-I)\right)}\hfill \\ {\displaystyle ={\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}(5-I)\right)*{\mu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}(5-I)\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}(5-I)\right)}\hfill \\ {\displaystyle ={\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}(5-I)\right)\diamond {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}(5-I)\right)}\hfill \end{array}\right\}\end{array}$$

(19)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Thus, (5) and (6) holds for $\mu =1$. Interchanging

$$\left(w_1,w_2,w_3\right)=\left(5^{\ell }{w}_1,5^{\ell }{w}_2,5^{\ell }{w}_3\right),$$

in (1) and using (IFN4), (IFN10), we have

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\frac{1}{5^{\ell }}\{\mathcal{F}\left(5^{\ell }(3w_1+w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+3w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+w_2+3w_3)\right)-6\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right.\hfill \\ \left.-\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)+\mathcal{F}(-5^{\ell }{w}_{\psi })\right]\}\right\},\Lambda \right)\hfill \\ \hfill \ge {\mu }^{\prime }\left(\Psi \left(5^{\ell }{w}_1,5^{\ell }{w}_2,5^{\ell }{w}_3\right),5^{\ell }\Lambda \right)\\ \nu \left(\frac{1}{5^{\ell }}\{\mathcal{F}\left(5^{\ell }(3w_1+w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+3w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+w_2+3w_3)\right)-6\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right.\hfill \\ \left.-\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)+\mathcal{F}(-5^{\ell }{w}_{\psi })\right]\}\right\},\Lambda \right)\hfill \\ \hfill \le {\nu }^{\prime }\left(\Psi \left(5^{\ell }{w}_1,5^{\ell }{w}_2,5^{\ell }{w}_3\right),5^{\ell }\Lambda \right)\end{array}\right\}\end{array}$$

(20)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$. Now,

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{A}(3w_1+w_2+w_3)+\mathcal{A}(w_1+3w_2+w_3)+\mathcal{A}(w_1+w_2+3w_3)-6\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ -\left.\frac{1}{2}\left\{\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)-\mathcal{A}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{A}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{A}\left(w_{\psi }\right)+\mathcal{A}(-w_{\psi })\right]\right\},\Lambda \right)\hfill \\ \ge \mu \left(\mathcal{A}(3w_1+w_2+w_3)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }(3w_1+w_2+w_3)\right),\frac{\Lambda }{7}\right)*\hfill \\ \mu \left(\mathcal{A}(w_1+3w_2+w_3)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }(w_1+3w_2+w_3)\right),\frac{\Lambda }{7}\right)*\hfill \\ \mu \left(\mathcal{A}(w_1+w_2+3w_3)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }(w_1+w_2+3w_3)\right),\frac{\Lambda }{7}\right)*\hfill \\ \mu \left(-6\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\frac{1}{5^{\ell }}6\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi },\frac{\Lambda }{7}\right)\right)*\hfill \\ \mu \left(\frac{1}{2}\left\{\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{A}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}\right.\hfill \\ \left.-\frac{1}{5^{\ell }}\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right\},\frac{\Lambda }{7}\right)*\hfill \\ \mu \left(\sum _{\psi =1}^3\left\{\mathcal{A}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{A}\left(w_{\psi }\right)+\mathcal{A}(-w_{\psi })\right]\right\}\right.\hfill \\ \left.-\frac{1}{5^{\ell }}\sum _{\psi =1}^3\left\{\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)+\mathcal{F}(-5^{\ell }{w}_{\psi })\right]\right\},\frac{\Lambda }{7}\right)*\hfill \\ \mu \left(\frac{1}{5^{\ell }}\{\mathcal{F}\left(5^{\ell }(3w_1+w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+3w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+w_2+3w_3)\right)\right.\hfill \\ -6\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)-\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right\}\hfill \\ \left.+\sum _{\psi =1}^3\left\{\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)+\mathcal{F}(-5^{\ell }{w}_{\psi })\right]\}\right\},\frac{\Lambda }{7}\right)\hfill \\ \nu \left(\mathcal{A}(3w_1+w_2+w_3)+\mathcal{A}(w_1+3w_2+w_3)+\mathcal{A}(w_1+w_2+3w_3)-6\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ -\left.\frac{1}{2}\left\{\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)-\mathcal{A}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{A}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{A}\left(w_{\psi }\right)+\mathcal{A}(-w_{\psi })\right]\right\},\Lambda \right)\hfill \\ \le \nu \left(\mathcal{A}(3w_1+w_2+w_3)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }(3w_1+w_2+w_3)\right),\frac{\Lambda }{7}\right)\diamond \hfill \\ \nu \left(\mathcal{A}(w_1+3w_2+w_3)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }(w_1+3w_2+w_3)\right),\frac{\Lambda }{7}\right)\diamond \hfill \\ \nu \left(\mathcal{A}(w_1+w_2+3w_3)-\frac{1}{5^{\ell }}\mathcal{F}\left(5^{\ell }(w_1+w_2+3w_3)\right),\frac{\Lambda }{7}\right)\diamond \hfill \\ \nu \left(-6\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\frac{1}{5^{\ell }}6\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi },\frac{\Lambda }{7}\right)\right)\diamond \hfill \\ \nu \left(\frac{1}{2}\left\{\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{A}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}\right.\hfill \\ \left.-\frac{1}{5^{\ell }}\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right\},\frac{\Lambda }{7}\right)\diamond \hfill \\ \nu \left(\sum _{\psi =1}^3\left\{\mathcal{A}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{A}\left(w_{\psi }\right)+\mathcal{A}(-w_{\psi })\right]\right\}\right.\hfill \\ \left.-\frac{1}{5^{\ell }}\sum _{\psi =1}^3\left\{\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)+\mathcal{F}(-5^{\ell }{w}_{\psi })\right]\right\},\frac{\Lambda }{7}\right)\diamond \hfill \\ \nu \left(\frac{1}{5^{\ell }}\{\mathcal{F}\left(5^{\ell }(3w_1+w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+3w_2+w_3)\right)+\mathcal{F}\left(5^{\ell }(w_1+w_2+3w_3)\right)\right.\hfill \\ -6\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)-\frac{1}{2}\left\{\mathcal{F}\left(\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)+\mathcal{F}\left(-\sum _{\psi =1}^3{5}^{\ell }{w}_{\psi }\right)\right\}\hfill \\ \left.+\sum _{\psi =1}^3\left\{\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)-\frac{5}{2}\left[\mathcal{F}\left(5^{\ell }{w}_{\psi }\right)+\mathcal{F}(-5^{\ell }{w}_{\psi })\right]\}\right\},\frac{\Lambda }{7}\right)\hfill \end{array}\right\}\end{array}$$

(21)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$. Taking limit $\ell \to \infty$ in (21), using (18) and (20), we get

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{A}(3w_1+w_2+w_3)+\mathcal{A}(w_1+3w_2+w_3)+\mathcal{A}(w_1+w_2+3w_3)-6\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ \left.-\frac{1}{2}\left\{\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{A}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{A}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{A}\left(w_{\psi }\right)+\mathcal{A}(-w_{\psi })\right]\right\},\Lambda \right)=1\hfill \\ \nu \left(\mathcal{A}(3w_1+w_2+w_3)+\mathcal{A}(w_1+3w_2+w_3)+\mathcal{A}(w_1+w_2+3w_3)-6\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)\right.\hfill \\ \left.-\frac{1}{2}\left\{\mathcal{A}\left(\sum _{\psi =1}^3{w}_{\psi }\right)+\mathcal{A}\left(-\sum _{\psi =1}^3{w}_{\psi }\right)\right\}+\sum _{\psi =1}^3\left\{\mathcal{A}\left(w_{\psi }\right)-\frac{5}{2}\left[\mathcal{A}\left(w_{\psi }\right)+\mathcal{A}(-w_{\psi })\right]\right\},\Lambda \right)=0\hfill \end{array}\right\}\end{array}$$

(22)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$. Using (IFN3), (IFN9) in (22), we see, $\mathcal{A}\left(w_1\right)$ satisfies (3). In order to confirm that $\mathcal{A}\left(w_1\right)$ is unique, suppose $\mathcal{B}\left(w_1\right)$ be another mapping (3), (18) and (19), we obtain

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{A}\left(w_1\right)-\mathcal{B}\left(w_1\right),2\Lambda \right)=\mu \left(\mathcal{A}\left(5^{\ell }{w}_1\right)-\mathcal{B}\left(5^{\ell }{w}_1\right),5^{\ell }2\Lambda \right)\hfill \\ \ge \mu \left(\mathcal{A}\left(5^{\ell }{w}_1\right)-\mathcal{F}\left(5^{\ell }{w}_1\right),5^{\ell }\Lambda \right)*\mu \left(\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{B}\left(5^{\ell }{w}_1\right),5^{\ell }\Lambda \right)\hfill \\ \ge {\mu }^{\prime }\left({\Psi }_A\left(5^{\ell }{w}_1\right),\frac{3\Lambda }{4}{5}^{\ell }(5-I)\right)*{\mu }^{\prime }\left({\Psi }_A\left(5^{\ell }{w}_1\right),\frac{3\Lambda }{4}{5}^{\ell }(5-I)\right)\hfill \\ \ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}\frac{5^{\ell }}{I^{\ell }}(5-I)\right)\hfill \\ \nu \left(\mathcal{A}\left(w_1\right)-\mathcal{B}\left(w_1\right),2\Lambda \right)=\nu \left(\mathcal{A}\left(5^{\ell }{w}_1\right)-\mathcal{B}\left(5^{\ell }{w}_1\right),5^{\ell }2\Lambda \right)\hfill \\ \le \nu \left(\mathcal{A}\left(5^{\ell }{w}_1\right)-\mathcal{F}\left(5^{\ell }{w}_1\right),5^{\ell }\Lambda \right)\diamond \nu \left(\mathcal{F}\left(5^{\ell }{w}_1\right)-\mathcal{B}\left(5^{\ell }{w}_1\right),5^{\ell }\Lambda \right)\hfill \\ \le {\nu }^{\prime }\left({\Psi }_A\left(5^{\ell }{w}_1\right),\frac{3\Lambda }{4}{5}^{\ell }(5-I)\right)\diamond {\nu }^{\prime }\left({\Psi }_A\left(5^{\ell }{w}_1\right),\frac{3\Lambda }{4}{5}^{\ell }(5-I)\right)\hfill \\ \le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}\frac{5^{\ell }}{I^{\ell }}(5-I)\right)\hfill \end{array}\right\}\end{array}$$

(23)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Taking limit $\ell \to \infty$ in (23), and using (IFN7), (IFN13), we arrive

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{A}\left(w_1\right)-\mathcal{B}\left(w_1\right),2\Lambda \right)=1\hfill \\ \nu \left(\mathcal{A}\left(w_1\right)-\mathcal{B}\left(w_1\right),2\Lambda \right)=0\hfill \end{array}\right\}\end{array}$$

(24)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. By (IFN4) and (IFN10), we get $\mathcal{A}\left(w_1\right)$ is unique. So, the Theorem holds for $\mu =1$.

Changing ${\displaystyle w_1=\frac{w_1}{5}}$ in (10) and using (IFN4), (IFN10), (3), in that changing $\Lambda$ by $\frac{\Lambda }{I}$, we have

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{F}\left(w_1\right)-5\mathcal{F}\left(\frac{w_1}{5}\right),\frac{4}{3·I}\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\mathcal{F}\left(w_1\right)-5\mathcal{F}\left(\frac{w_1}{5}\right),\frac{4}{3·I}\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \end{array}\right\}\end{array}$$

(25)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Changing $w_1$ by ${\displaystyle \frac{w_1}{5^{\ell }}}$ in (25), and using (IFN4), (IFN10), (3) in that changing $\Lambda$ by ${\displaystyle \frac{\Lambda }{I^{\ell }}}$, we get

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(5^{\ell }\mathcal{F}\left(\frac{w_1}{5^{\ell }}\right)-5^{\ell +1}\mathcal{F}\left(\frac{w_1}{5^{\ell +1}}\right),\frac{4}{3·I}{\left(\frac{5}{I}\right)}^{\ell }\Lambda \right)\le {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(5^{\ell }\mathcal{F}\left(\frac{w_1}{5^{\ell }}\right)-5^{\ell +1}\mathcal{F}\left(\frac{w_1}{5^{\ell +1}}\right),\frac{4}{3·I}{\left(\frac{5}{I}\right)}^{\ell }\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \end{array}\right\}\end{array}$$

(26)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$ also $\ell >0$. It is easy to check that

$$\begin{array}{c}\hfill \mathcal{F}\left(w_1\right)-5^{\ell }\mathcal{F}\left(\frac{w_1}{5^{\ell }}\right)=\sum _{\eta =0}^{\ell -1}{5}^{\eta }\mathcal{F}\left(\frac{w_1}{5^{\eta }}\right)-5^{\eta +1}\mathcal{F}\left(\frac{w_1}{5^{\eta +1}}\right)\end{array}$$

(27)

for all $w_1\in {\mathcal{W}}_1$. The rest of the proof is similar to that of above case. So, the Theorem holds for $\mu =-1$. Hence the proof is complete □

Corollary 3.9. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\ge \left\{{\displaystyle \begin{array}{cc}{\mu }^{\prime }\left(\delta ,\left|3\right|\Lambda \right),\hfill & \\ {\mu }^{\prime }\left(\delta |w_1{|}^{\phi },\frac{\Lambda }{4}|5-5^{\phi }|\right),\hfill & \phi \ne 1,\hfill \\ {\mu }^{\prime }\left(\delta \sum _{\psi =1}^3|w_1{|}^{{\phi }_{{ }_{\psi }}},\frac{3\Lambda }{4}\sum _{\psi =1}^3|5-5^{{\phi }_{{ }_{\psi }}}|\right),\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 1,\hfill \\ {\mu }^{\prime }\left(\delta |w_1{|}^{3\phi },\frac{3\Lambda }{4}|5-5^{3\phi }|\right),\hfill & 3\phi \ne 1,\hfill \\ {\mu }^{\prime }\left(\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}},\frac{3\Lambda }{4}|5-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|\right),\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 1,\hfill \\ {\mu }^{\prime }\left(2\delta |w_1{|}^{3\phi },\frac{3\Lambda }{4}|5-5^{3\phi }|\right),\hfill & 3\phi \ne 1,\hfill \end{array}}\right.}\hfill \\ \\ {\displaystyle \nu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\le \left\{{\displaystyle \begin{array}{cc}{\nu }^{\prime }\left(\delta ,\left|3\right|\Lambda \right),\hfill & \\ {\nu }^{\prime }\left(\delta |w_1{|}^{\phi },\frac{\Lambda }{4}|5-5^{\phi }|\right),\hfill & \phi \ne 1,\hfill \\ {\nu }^{\prime }\left(\delta \sum _{\psi =1}^3|w_1{|}^{{\phi }_{{ }_{\psi }}},\frac{3\Lambda }{4}\sum _{\psi =1}^3|5-5^{{\phi }_{{ }_{\psi }}}|\right),\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 1,\hfill \\ {\nu }^{\prime }\left(\delta |w_1{|}^{3\phi },\frac{3\Lambda }{4}|5-5^{3\phi }|\right),\hfill & 3\phi \ne 1,\hfill \\ {\nu }^{\prime }\left(\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}},\frac{3\Lambda }{4}|5-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|\right),\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 1,\hfill \\ {\nu }^{\prime }\left(2\delta |w_1{|}^{3\phi },\frac{3\Lambda }{4}|5-5^{3\phi }|\right),\hfill & 3\phi \ne 1,\hfill \end{array}}\right.}\hfill \end{array}\right\}\end{array}$$

(28)

for all $w_1\in {\mathcal{W}}_1$.

3.3. Evenness of $\mathcal{F}$: Quadratic Case Stability Results : Direct Method

Theorem 3.10. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the conditions (3) and

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}{\mu }^{\prime }\left(\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right),{25}^{\ell \mu }\Lambda \right)=1\hfill \\ \underset{\ell \to \infty }{lim}{\nu }^{\prime }\left(\Psi \left(5^{\ell \mu }{w}_1,5^{\ell \mu }{w}_2,5^{\ell \mu }{w}_3\right),{25}^{\ell \mu }\Lambda \right)=0\hfill \end{array}\right\}\end{array}$$

(29)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$ with $\mu =\pm 1$ and $0<{\left(\frac{I}{25}\right)}^{\mu }<1$ . Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{Q}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}|25-I|\right)}\hfill \\ {\displaystyle ={\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}|25-I|\right)*{\mu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}|25-I|\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{Q}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}(25-I)\right)}\hfill \\ {\displaystyle ={\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}(25-I)\right)\diamond {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}(25-I)\right)}\hfill \end{array}\right\}\end{array}$$

(30)

and the mapping $\mathcal{Q}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}\mu \left({\displaystyle \frac{1}{{25}^{\ell \mu }}\mathcal{F}\left(5^{\ell \mu }{w}_1\right)-\mathcal{Q}\left(w_1\right),\Lambda }\right)=1\hfill \\ \underset{\ell \to \infty }{lim}\nu \left({\displaystyle \frac{1}{{25}^{\ell \mu }}\mathcal{F}\left(5^{\ell \mu }{w}_1\right)-\mathcal{Q}\left(w_1\right),\Lambda }\right)=0\hfill \end{array}\right\}\end{array}$$

(31)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$.

Proof. 

Using evenness of $\mathcal{F}$ in (1), we get

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-7\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)-4\sum _{\psi =1}^3\mathcal{F}\left(w_{\psi }\right),\Lambda \right)}\hfill \\ \hfill \ge {\mu }^{\prime }\left(\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\\ {\displaystyle \nu \left(\mathcal{F}(3w_1+w_2+w_3)+\mathcal{F}(w_1+3w_2+w_3)+\mathcal{F}(w_1+w_2+3w_3)-7\mathcal{F}\left(\sum _{\psi =1}^3{w}_{\psi }\right)-4\sum _{\psi =1}^3\mathcal{F}\left(w_{\psi }\right),\Lambda \right)}\hfill \\ \hfill \le {\nu }^{\prime }\left(\Psi \left(w_1,w_2,w_3\right),\Lambda \right)\end{array}\right\}\end{array}$$

(32)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$ . Interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,w_1\right)$ in (32), we obtain

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(3\mathcal{F}\left(5w_1\right)-7\mathcal{F}\left(3w_1\right)-12\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \\ \nu \left(3\mathcal{F}\left(5w_1\right)-7\mathcal{F}\left(3w_1\right)-12\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \end{array}\right\}\end{array}$$

(33)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$ . Again interchanging $\left(w_1,w_2,w_3\right)$ by $\left(w_1,w_1,-w_1\right)$ in (32) and using (IFN4), (IFN10), we have

$$\begin{array}{cc}\hfill & \left.\begin{array}{c}\mu \left(2\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \\ \nu \left(2\mathcal{F}\left(3w_1\right)-18\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill & \Rightarrow \left.\begin{array}{c}\mu \left(7\mathcal{F}\left(3w_1\right)-63\mathcal{F}\left(w_1\right),\frac{2}{7}\Lambda \right)\ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\hfill \\ \nu \left(7\mathcal{F}\left(3w_1\right)-63\mathcal{F}\left(w_1\right),\frac{2}{7}\Lambda \right)\le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)\hfill \end{array}\right\}\hfill \end{array}$$

(34)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Combining (33) and (34) using (IFN5), (IFN11), we arrive

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(3\mathcal{F}\left(5w_1\right)-75\mathcal{F}\left(w_1\right),\frac{9}{7}\Lambda \right)\ge \mu \left(3\mathcal{F}\left(5w_1\right)-7\mathcal{F}\left(3w_1\right)-12\mathcal{F}\left(w_1\right),\Lambda \right)*\mu \left(7\mathcal{F}\left(3w_1\right)-63\mathcal{F}\left(w_1\right),\frac{2}{7}\Lambda \right)\hfill \\ \ge {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)*{\mu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)={\mu }^{\prime }\left({\Psi }_Q\left(w_1\right),\Lambda \right)\hfill \\ \nu \left(3\mathcal{F}\left(5w_1\right)-15\mathcal{F}\left(w_1\right),\frac{9}{7}\Lambda \right)\le \nu \left(3\mathcal{F}\left(5w_1\right)-7\mathcal{F}\left(3w_1\right)-12\mathcal{F}\left(w_1\right),\Lambda \right)\diamond \nu \left(7\mathcal{F}\left(3w_1\right)-63\mathcal{F}\left(w_1\right),\frac{2}{7}\Lambda \right)\hfill \\ \le {\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\Lambda \right)\diamond {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\Lambda \right)={\nu }^{\prime }\left({\Psi }_Q\left(w_1\right),\Lambda \right)\hfill \end{array}\right\}\end{array}$$

(35)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Using (IFN4), (IFN10), one can see from (35) that

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{25}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right),\frac{9}{7·3}·\frac{1}{25}\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_Q\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\frac{1}{25}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right),\frac{9}{7·3}·\frac{1}{25}\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_Q\left(w_1\right),\Lambda \right)}\hfill \end{array}\right\}\end{array}$$

(36)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. The rest of the proof is similar to that of Theorem 3.8. Hence the proof is complete. □

Corollary 3.11. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$. Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{Q}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\ge \left\{{\displaystyle \begin{array}{cc}{\mu }^{\prime }\left(\delta ,\left|8\right|7\Lambda \right),\hfill & \\ {\mu }^{\prime }\left(\delta |w_1{|}^{\phi },\frac{7\Lambda }{9}|25-5^{\phi }|\right),\hfill & \phi \ne 2,\hfill \\ {\mu }^{\prime }\left(\delta \sum _{\psi =1}^3|w_1{|}^{{\phi }_{{ }_{\psi }}},\frac{7\Lambda }{3}\sum _{\psi =1}^3|25-5^{{\phi }_{{ }_{\psi }}}|\right),\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 2,\hfill \\ {\mu }^{\prime }\left(\delta |w_1{|}^{3\phi },\frac{7\Lambda }{3}|25-5^{3\phi }|\right),\hfill & 3\phi \ne 2,\hfill \\ {\displaystyle {\mu }^{\prime }\left(\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}},\frac{7\Lambda }{3}|25-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|\right),}\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 2,\hfill \\ {\mu }^{\prime }\left(2\delta |w_1{|}^{3\phi },\frac{7\Lambda }{3}|25-5^{3\phi }|\right),\hfill & 3\phi \ne 2,\hfill \end{array}}\right.}\hfill \\ \\ {\displaystyle \nu \left(\mathcal{Q}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\le \left\{{\displaystyle \begin{array}{cc}{\nu }^{\prime }\left(\delta ,\left|8\right|7\Lambda \right),\hfill & \\ {\nu }^{\prime }\left(\delta |w_1{|}^{\phi },\frac{7\Lambda }{9}|25-5^{\phi }|\right),\hfill & \phi \ne 2,\hfill \\ {\nu }^{\prime }\left(\delta \sum _{\psi =1}^3|w_1{|}^{{\phi }_{{ }_{\psi }}},\frac{7\Lambda }{3}\sum _{\psi =1}^3|25-5^{{\phi }_{{ }_{\psi }}}|\right),\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 2,\hfill \\ {\nu }^{\prime }\left(\delta |w_1{|}^{3\phi },\frac{7\Lambda }{3}|25-5^{3\phi }|\right),\hfill & 3\phi \ne 2,\hfill \\ {\nu }^{\prime }\left(\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}},\frac{7\Lambda }{3}|25-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|\right),\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 2,\hfill \\ {\nu }^{\prime }\left(2\delta |w_1{|}^{3\phi },\frac{7\Lambda }{3}|25-5^{3\phi }|\right),\hfill & 3\phi \ne 2,\hfill \end{array}}\right.}\hfill \end{array}\right\}\end{array}$$

(37)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$.

3.4. Oddness and Evenness of $\mathcal{F}$: Additive Quadratic Case Stability Results : Direct Method

Theorem 3.12. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the conditions (3), (4), and (29) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$ with $\mu =\pm 1$ and $0<{\left(\frac{I}{5}\right)}^{\mu }<1$, $0<{\left(\frac{I}{25}\right)}^{\mu }<1$ . Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)\hfill \\ \ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}|5-I|\right)*{\mu }^{\prime }\left({\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}|5-I|\right)*\hfill \\ {\mu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}|25-I|\right)*{\mu }^{\prime }\left({\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}|25-I|\right)\hfill \\ ={\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}|5-I|\right)*{\mu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}|5-I|\right)*\hfill \\ {\mu }^{\prime }\left(\Psi \left(-w_1,-w_1,-w_1\right),\frac{3\Lambda }{4}|5-I|\right)*{\mu }^{\prime }\left(\Psi \left(-w_1,-w_1,w_1\right),\frac{3\Lambda }{4}|5-I|\right)*\hfill \\ {\mu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}|5-I|\right)*{\mu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}|5-I|\right)*\hfill \\ {\mu }^{\prime }\left(\Psi \left(-w_1,-w_1,-w_1\right),\frac{7\Lambda }{3}|5-I|\right)*{\mu }^{\prime }\left(\Psi \left(-w_1,-w_1,w_1\right),\frac{7\Lambda }{3}|5-I|\right)\hfill \\ \nu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)\hfill \\ \le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond {\nu }^{\prime }\left({\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond \hfill \\ {\nu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}|25-I|\right)\diamond {\nu }^{\prime }\left({\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}|25-I|\right)\hfill \\ ={\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond \hfill \\ {\nu }^{\prime }\left(\Psi \left(-w_1,-w_1,-w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond {\nu }^{\prime }\left(\Psi \left(-w_1,-w_1,w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond \hfill \\ {\nu }^{\prime }\left(\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}|5-I|\right)\diamond {\nu }^{\prime }\left(\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}|5-I|\right)\diamond \hfill \\ {\nu }^{\prime }\left(\Psi \left(-w_1,-w_1,-w_1\right),\frac{7\Lambda }{3}|5-I|\right)\diamond {\nu }^{\prime }\left(\Psi \left(-w_1,-w_1,w_1\right),\frac{7\Lambda }{3}|5-I|\right)\hfill \end{array}\right\}\end{array}$$

(38)

and the mapping $\mathcal{A}\left(w_1\right)$ and $\mathcal{Q}\left(w_1\right)$ are given in (6) and (31) for all $w_1\in {\mathcal{W}}_1$.

Proof. 

By Theorem 3.8, it follows from (33), (1) and (5), we arrive

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{A}\left(w_1\right)-{\mathcal{F}}_{odd}\left(w_1\right),2\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}|5-I|\right)*{\mu }^{\prime }\left({\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}|5-I|\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{A}\left(w_1\right)-{\mathcal{F}}_{odd}\left(w_1\right),2\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond {\nu }^{\prime }\left({\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}|5-I|\right)}\hfill \end{array}\right\}\end{array}$$

(39)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. By Theorem 3.10, it follows from (37), (1), and (30), we see

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{Q}\left(w_1\right)-{\mathcal{F}}_{even}\left(w_1\right),2\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}|25-I|\right)*{\mu }^{\prime }\left({\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}|25-I|\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{Q}\left(w_1\right)-{\mathcal{F}}_{even}\left(w_1\right),2\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}|25-I|\right)\diamond {\nu }^{\prime }\left({\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}|25-I|\right)}\hfill \end{array}\right\}\end{array}$$

(40)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Now, it follows from (39), (40) and (40), we have

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)}\hfill \\ {\displaystyle \ge \mu \left(\mathcal{A}\left(w_1\right)-{\mathcal{F}}_{odd}\left(w_1\right),2\Lambda \right)*\mu \left(\mathcal{Q}\left(w_1\right)-{\mathcal{F}}_{even}\left(w_1\right),2\Lambda \right)}\hfill \\ {\displaystyle \ge {\mu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}|5-I|\right)*{\mu }^{\prime }\left({\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}|5-I|\right)*}\hfill \\ {\displaystyle {\mu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}|25-I|\right)*{\mu }^{\prime }\left({\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}|25-I|\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)}\hfill \\ {\displaystyle \le \nu \left(\mathcal{A}\left(w_1\right)-{\mathcal{F}}_{odd}\left(w_1\right),2\Lambda \right)\diamond \nu \left(\mathcal{Q}\left(w_1\right)-{\mathcal{F}}_{even}\left(w_1\right),2\Lambda \right)}\hfill \\ {\displaystyle \le {\nu }^{\prime }\left({\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond {\nu }^{\prime }\left({\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}|5-I|\right)\diamond }\hfill \\ {\displaystyle {\nu }^{\prime }\left({\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}|25-I|\right)\diamond {\nu }^{\prime }\left({\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}|25-I|\right)}\hfill \end{array}\right\}\end{array}$$

(41)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. □

Corollary 3.13. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)}\hfill \\ \ge \left\{{\displaystyle \begin{array}{cc}{\mu }^{\prime }\left(2\delta ,\left(\left|3\right|+7\left|8\right|\right)\Lambda \right)\hfill & \\ {\mu }^{\prime }\left(2\delta |w_1{|}^{\phi },\Lambda \left\{\frac{1}{4}|5-5^{\phi }|+\frac{7}{9}|25-5^{\phi }|\right\}\right),\hfill & \phi \ne 1,2,\hfill \\ {\mu }^{\prime }\left(2\delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\left\{\frac{3}{4}\sum _{\psi =1}^3|5-5^{{\phi }_{{ }_{\psi }}}|+\frac{7}{3}\sum _{\psi =1}^3|25-5^{{\phi }_{{ }_{\psi }}}|\right\}\right),\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 1,2,\hfill \\ {\mu }^{\prime }\left(2\delta |w_1{|}^{3\phi },\Lambda \left\{\frac{3}{4}|5-5^{3\phi }|+\frac{7}{3}|25-5^{3\phi }|\right\}\right),\hfill & 3\phi \ne 1,2,\hfill \\ {\mu }^{\prime }\left(4\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}},2\Lambda \left\{\frac{3}{4}|5-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|+\frac{7}{3}|25-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|\right\}\right)\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 1,2,\hfill \\ {\mu }^{\prime }\left(4\delta |w_1{|}^{3\phi },\Lambda \left\{\frac{3}{4}|5-5^{3\phi }|+\frac{7}{3}|25-5^{3\phi }|\right\}\right)\hfill & 3\phi \ne 1,2,\hfill \end{array}}\right.\hfill \\ {\displaystyle \nu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)}\hfill \\ \le \left\{{\displaystyle \begin{array}{cc}{\nu }^{\prime }\left(2\delta ,\left(\left|3\right|+7\left|8\right|\right)\Lambda \right)\hfill & \\ {\nu }^{\prime }\left(2\delta |w_1{|}^{\phi },\Lambda \left\{\frac{1}{4}|5-5^{\phi }|+\frac{7}{9}|25-5^{\phi }|\right\}\right),\hfill & \phi \ne 1,2,\hfill \\ {\nu }^{\prime }\left(2\delta \sum _{\psi =1}^3{\left|w_{\psi }\right|}^{{\phi }_{{ }_{\psi }}},\left\{\frac{3}{4}\sum _{\psi =1}^3|5-5^{{\phi }_{{ }_{\psi }}}|+\frac{7}{3}\sum _{\psi =1}^3|25-5^{{\phi }_{{ }_{\psi }}}|\right\}\right),\hfill & {\phi }_1,{\phi }_2,{\phi }_3\ne 1,2,\hfill \\ {\nu }^{\prime }\left(2\delta |w_1{|}^{3\phi },\Lambda \left\{\frac{3}{4}|5-5^{3\phi }|+\frac{7}{3}|25-5^{3\phi }|\right\}\right),\hfill & 3\phi \ne 1,2,\hfill \\ {\nu }^{\prime }\left(4\delta |w_{\psi }{|}^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}},2\Lambda \left\{\frac{3}{4}|5-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|+\frac{7}{3}|25-5^{\sum _{\psi =1}^3{\phi }_{{ }_{\psi }}}|\right\}\right)\hfill & \sum _{\psi =1}^3{\phi }_{{ }_{\psi }}\ne 1,2,\hfill \\ {\nu }^{\prime }\left(4\delta |w_1{|}^{3\phi },\Lambda \left\{\frac{3}{4}|5-5^{3\phi }|+\frac{7}{3}|25-5^{3\phi }|\right\}\right)\hfill & 3\phi \ne 1,2,\hfill \end{array}}\right.\hfill \end{array}\right\}\end{array}$$

(42)

for all $w_1\in {\mathcal{W}}_1$.

3.5. Oddness of $\mathcal{F}$: Additive Case Stability Results : Fixed Method

Theorem 3.14. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the condition

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}{\mu }^{\prime }\left(\Psi \left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }^{\ell }{w}_2,{\tau }_{\nu }^{\ell }{w}_3\right),{\tau }_{\nu }^{\ell }\Lambda \right)=1\hfill \\ \underset{\ell \to \infty }{lim}{\nu }^{\prime }\left(\Psi \left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }^{\ell }{w}_2,{\tau }_{\nu }^{\ell }{w}_3\right),{\tau }_{\nu }^{\ell }\Lambda \right)=0\hfill \end{array}\right\};{\tau }_{\nu }=\left\{\begin{array}{c}5;\nu =0\hfill \\ \frac{1}{5};\nu =1\hfill \end{array}\right.\end{array}$$

(43)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$ If there exists $L=L\left(\nu \right)$ be a function have the property

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left({\Psi }_A\left(w_1\right),\Lambda \right)=\mu \left({\Psi }_A\left(\frac{w_1}{5}\right),\Lambda \right)\hfill \\ \nu \left({\Psi }_A\left(w_1\right),\Lambda \right)=\nu \left({\Psi }_A\left(\frac{w_1}{5}\right),\Lambda \right)\hfill \end{array}\right\}and\left.\begin{array}{c}\mu \left(\frac{1}{{\tau }_{\nu }}{\Psi }_A\left({\tau }_{\nu }{w}_1\right),\Lambda \right)=\mu \left(L{\Psi }_A\left(w_1\right),\Lambda \right)\hfill \\ \nu \left(\frac{1}{{\tau }_{\nu }}{\Psi }_A\left({\tau }_{\nu }{w}_1\right),\Lambda \right)=\nu \left(L{\Psi }_A\left(w_1\right),\Lambda \right)\hfill \end{array}\right\},\end{array}$$

(44)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}\right)}\hfill \\ {\displaystyle ={\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{A}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}\right)}\hfill \\ {\displaystyle ={\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}\right)}\hfill \end{array}\right\}\end{array}$$

(45)

and the mapping $\mathcal{A}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}\mu \left({\displaystyle \frac{1}{{\tau }_{\nu }^{\ell }}\mathcal{F}\left({\tau }_{\nu }^{\ell }{w}_1\right)-\mathcal{A}\left(w_1\right),\Lambda }\right)=1\hfill \\ \underset{\ell \to \infty }{lim}\nu \left({\displaystyle \frac{1}{{\tau }_{\nu }^{\ell }}\mathcal{F}\left({\tau }_{\nu }^{\ell }{w}_1\right)-\mathcal{A}\left(w_1\right),\Lambda }\right)=0\hfill \end{array}\right\}\end{array}$$

(46)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$.

Proof. 

Assume a set $\mathcal{G}$ as in Theorem 2.7 of (48) and introduce the generalized metric on the above set $\mathcal{G}$ as

$$\begin{array}{c}\hfill d(\mathcal{F},{\mathcal{F}}_1)=inf\left\{K\in (0,\infty ):\left\{\begin{array}{c}\mu \left(\mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\ge \mu \left(K\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \\ \nu \left(\mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\le \nu \left(K\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \end{array}\right\}\right\}.\end{array}$$

(47)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. It is easy to see that $(\mathcal{G},d)$ is complete. Define a function $\mathcal{H}:\mathcal{G}\to \mathcal{G}$ as by Theorem 2.7 of (50) and for $\mathcal{F},{\mathcal{F}}_1\in \mathcal{G}$ and $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$, we see

$$\begin{array}{cc}\hfill d(\mathcal{F},{\mathcal{F}}_1)\le K\Rightarrow & \left\{\begin{array}{c}\mu \left(\mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\ge \mu \left(K\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \\ \nu \left(\mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\le \nu \left(K\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill \Rightarrow & \left\{\begin{array}{c}\mu \left(∥\frac{1}{{\tau }_{\nu }}\mathcal{F}\left({\tau }_{\nu }{w}_1\right)-\frac{1}{{\tau }_{\nu }}{\mathcal{F}}_1\left({\tau }_{\nu }{w}_1\right)∥,\Lambda \right)\ge \mu \left({\tau }_{\nu }K\Psi (\frac{1}{{\tau }_{\nu }}{w}_1,{\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1),\Lambda \right)\hfill \\ \nu \left(∥\frac{1}{{\tau }_{\nu }}\mathcal{F}\left({\tau }_{\nu }{w}_1\right)-\frac{1}{{\tau }_{\nu }}{\mathcal{F}}_1\left({\tau }_{\nu }{w}_1\right)∥,\Lambda \right)\le \nu \left({\tau }_{\nu }K\Psi (\frac{1}{{\tau }_{\nu }}{w}_1,{\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill \Rightarrow & \left\{\begin{array}{c}\mu \left(\mathcal{H}\mathcal{F}\left(w_1\right)-\mathcal{H}{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\ge \mu \left(LK\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \\ \nu \left(\mathcal{H}\mathcal{F}\left(w_1\right)-\mathcal{H}{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\le \nu \left(LK\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill \Rightarrow & d(\mathcal{H}\mathcal{F},\mathcal{H}{\mathcal{F}}_1)\le LK,\hfill \end{array}$$

i.e., $\mathcal{H}$ is a strictly contractive mapping on $\mathcal{G}$ with Lipschitz constant L (see [24]).

For the case $\nu =0$, it follows from (11) and with the help of (44), (50), (47), we get

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\frac{1}{5}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right),\frac{4}{3}\Lambda \right)\ge {\mu }^{\prime }\left(\frac{1}{5}{\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\frac{1}{5}\mathcal{F}\left(5w_1\right)-\mathcal{F}\left(w_1\right),\frac{4}{3}\Lambda \right)\le {\nu }^{\prime }\left(\frac{1}{5}{\Psi }_A\left(w_1\right),\Lambda \right)}\hfill \end{array}\right\}\Rightarrow d(\mathcal{H}\mathcal{F},\mathcal{F})\le L=L^{1-\nu },\end{array}$$

(48)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$.

For the case $\nu =1$, it follows from (17) and with the help of (44), (50), (47), we obtain

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{F}\left(w_1\right)-5\mathcal{F}\left(\frac{w_1}{5}\right),\frac{4}{3·I}\Lambda \right)\ge {\mu }^{\prime }\left({\Psi }_A\left(\frac{w_1}{5}\right),\Lambda \right)}\hfill \\ {\displaystyle \nu \left(\mathcal{F}\left(w_1\right)-5\mathcal{F}\left(\frac{w_1}{5}\right),\frac{4}{3·I}\Lambda \right)\le {\nu }^{\prime }\left({\Psi }_A\left(\frac{w_1}{5}\right),\Lambda \right)}\hfill \end{array}\right\}\Rightarrow d(\mathcal{F},\mathcal{H}\mathcal{F})\le 1=L^{1-\nu },\end{array}$$

(49)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Combining (48) and (49), we have

$$\begin{array}{c}\hfill d(\mathcal{F},\mathcal{H}\mathcal{F})\le 1=L^{1-\nu }.\end{array}$$

(50)

Therefore $\left(FPC1\right)$ of Theorem 1.3 holds. The rest of the proof follows by Theorem 1.3. Hence the proof is complete. □

Corollary 3.15. 

Suppose that an odd function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality (20) for all $w_1\in {\mathcal{W}}_1$.

3.6. Evenness of $\mathcal{F}$: Quadratic Case Stability Results : Fixed Method

Theorem 3.16. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the condition

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}{\mu }^{\prime }\left(\Psi \left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }^{\ell }{w}_2,{\tau }_{\nu }^{\ell }{w}_3\right),{\tau }_{\nu }^{2\ell }\Lambda \right)=1\hfill \\ \underset{\ell \to \infty }{lim}{\nu }^{\prime }\left(\Psi \left({\tau }_{\nu }^{\ell }{w}_1,{\tau }_{\nu }^{\ell }{w}_2,{\tau }_{\nu }^{\ell }{w}_3\right),{\tau }_{\nu }^{2\ell }\Lambda \right)=0\hfill \end{array}\right\};{\tau }_{\nu }=\left\{\begin{array}{c}5;\nu =0\hfill \\ \frac{1}{5};\nu =1\hfill \end{array}\right.\end{array}$$

(51)

for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$. If there exists $L=L\left(\nu \right)$ be function have the property

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left({\Psi }_Q\left(w_1\right),\Lambda \right)=\mu \left({\Psi }_Q\left(\frac{w_1}{5}\right),\Lambda \right)\hfill \\ \nu \left({\Psi }_Q\left(w_1\right),\Lambda \right)=\nu \left({\Psi }_Q\left(\frac{w_1}{5}\right),\Lambda \right)\hfill \end{array}\right\}and\left.\begin{array}{c}\mu \left(\frac{1}{{\tau }_{\nu }^2}{\Psi }_Q\left({\tau }_{\nu }{w}_1\right),\Lambda \right)=\mu \left(L{\Psi }_Q\left(w_1\right),\Lambda \right)\hfill \\ \nu \left(\frac{1}{{\tau }_{\nu }^2}{\Psi }_Q\left({\tau }_{\nu }{w}_1\right),\Lambda \right)=\nu \left(L{\Psi }_Q\left(w_1\right),\Lambda \right)\hfill \end{array}\right\},\end{array}$$

(52)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}{\displaystyle \mu \left(\mathcal{Q}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\ge {\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}\right)}\hfill \\ {\displaystyle ={\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}\right)}\hfill \\ {\displaystyle \nu \left(\mathcal{Q}\left(w_1\right)-\mathcal{F}\left(w_1\right),\Lambda \right)\le {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}\right)}\hfill \\ {\displaystyle ={\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}\right)}\hfill \end{array}\right\}\end{array}$$

(53)

and the mapping $\mathcal{Q}\left(w_1\right)$ is obtained by

$$\begin{array}{c}\hfill \left.\begin{array}{c}\underset{\ell \to \infty }{lim}\mu \left({\displaystyle \frac{1}{{\tau }_{\nu }^{2\ell }}\mathcal{F}\left({\tau }_{\nu }^{\ell }{w}_1\right)-\mathcal{Q}\left(w_1\right),\Lambda }\right)=1\hfill \\ \underset{\ell \to \infty }{lim}\nu \left({\displaystyle \frac{1}{{\tau }_{\nu }^{2\ell }}\mathcal{F}\left({\tau }_{\nu }^{\ell }{w}_1\right)-\mathcal{Q}\left(w_1\right),\Lambda }\right)=0\hfill \end{array}\right\}\end{array}$$

(54)

for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$.

Proof. 

Define a function $\mathcal{H}:\mathcal{G}\to \mathcal{G}$ as by Theorem 2.9 of (62) and for $\mathcal{F},{\mathcal{F}}_1\in \mathcal{G}$ and $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$, we see

$$\begin{array}{cc}\hfill d(\mathcal{F},{\mathcal{F}}_1)\le K\Rightarrow & \left\{\begin{array}{c}\mu \left(\mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\ge \mu \left(K\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \\ \nu \left(\mathcal{F}\left(w_1\right)-{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\le \nu \left(K\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill \Rightarrow & \left\{\begin{array}{c}\mu \left(∥\frac{1}{{\tau }_{\nu }^2}\mathcal{F}\left({\tau }_{\nu }{w}_1\right)-\frac{1}{{\tau }_{\nu }^2}{\mathcal{F}}_1\left({\tau }_{\nu }{w}_1\right)∥,\Lambda \right)\ge \mu \left({\tau }_{\nu }K\Psi ({\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1),\Lambda \right)\hfill \\ \nu \left(∥\frac{1}{{\tau }_{\nu }^2}\mathcal{F}\left({\tau }_{\nu }{w}_1\right)-\frac{1}{{\tau }_{\nu }^2}{\mathcal{F}}_1\left({\tau }_{\nu }{w}_1\right)∥,\Lambda \right)\le \nu \left({\tau }_{\nu }^{2}K\Psi ({\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1,{\tau }_{\nu }{w}_1),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill \Rightarrow & \left\{\begin{array}{c}\mu \left(\mathcal{H}\mathcal{F}\left(w_1\right)-\mathcal{H}{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\ge \mu \left(LK\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \\ \nu \left(\mathcal{H}\mathcal{F}\left(w_1\right)-\mathcal{H}{\mathcal{F}}_1\left(w_1\right),\Lambda \right)\le \nu \left(LK\Psi (w_1,w_1,w_1),\Lambda \right)\hfill \end{array}\right\}\hfill \\ \hfill \Rightarrow & d(\mathcal{H}\mathcal{F},\mathcal{H}{\mathcal{F}}_1)\le LK,\hfill \end{array}$$

i.e., $\mathcal{H}$ is a strictly contractive mapping on $\mathcal{G}$ with Lipschitz constant L (see [24]). The rest of the proof is similar to that of Theorem 3.14. Hence the proof is complete. □

Corollary 3.17. 

Suppose that an even function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality (37) for all $w_1\in {\mathcal{W}}_1$.

3.7. Oddness and Evenness of $\mathcal{F}$: Additive Quadratic Case Stability Results : Fixed Method

Theorem 3.18. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (1) where $\Psi :{\mathcal{W}}_1^3\to [0,\infty )$ with the conditions (43) and (51) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$ and all $\Lambda >0$. If there exists $L=L\left(\nu \right)$ be function have the properties (44) and (52) for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality

$$\begin{array}{c}\hfill \left.\begin{array}{c}\mu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)\hfill \\ \ge {\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}\right)*\hfill \\ {\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}\right)\hfill \\ ={\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}\right)*\hfill \\ {\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,-w_1\right),\frac{3\Lambda }{4}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,w_1\right),\frac{3\Lambda }{4}\right)*\hfill \\ {\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}\right)*\hfill \\ {\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,-w_1\right),\frac{7\Lambda }{3}\right)*{\mu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,w_1\right),\frac{7\Lambda }{3}\right)\hfill \\ \nu \left(\mathcal{F}\left(w_1\right)-\mathcal{A}\left(w_1\right)-\mathcal{Q}\left(w_1\right),4\Lambda \right)\hfill \\ \le {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_A\left(w_1\right),\frac{3\Lambda }{4}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_A\left(-w_1\right),\frac{3\Lambda }{4}\right)\diamond \hfill \\ {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_Q\left(w_1\right),\frac{7\Lambda }{3}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}{\Psi }_Q\left(-w_1\right),\frac{7\Lambda }{3}\right)\hfill \\ ={\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{3\Lambda }{4}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{3\Lambda }{4}\right)\diamond \hfill \\ {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,-w_1\right),\frac{3\Lambda }{4}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,w_1\right),\frac{3\Lambda }{4}\right)\diamond \hfill \\ {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,w_1\right),\frac{7\Lambda }{3}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(w_1,w_1,-w_1\right),\frac{7\Lambda }{3}\right)\diamond \hfill \\ {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,-w_1\right),\frac{7\Lambda }{3}\right)\diamond {\nu }^{\prime }\left(\frac{L^{1-\nu }}{1-L}\Psi \left(-w_1,-w_1,w_1\right),\frac{7\Lambda }{3}\right)\hfill \end{array}\right\}\end{array}$$

(55)

and the mapping $\mathcal{A}\left(w_1\right)$ and $\mathcal{Q}\left(w_1\right)$ are given in (45) and (54) for all $w_1\in {\mathcal{W}}_1$ and all $\Lambda >0$.

Proof. 

The proof is similar ideas to that of Theorem 3.12. □

Corollary 3.19. 

Suppose that a function $\mathcal{F}:{\mathcal{W}}_1\to {\mathcal{W}}_2$ satisfying the functional inequality (2) for all $w_1,w_2,w_3\in {\mathcal{W}}_1$. Then there exists a unique additive mapping $\mathcal{A}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ and a unique quadratic mapping $\mathcal{Q}\left(w_1\right):{\mathcal{W}}_1\to {\mathcal{W}}_2$ which satisfying (3) and the functional inequality (42) for all $w_1\in {\mathcal{W}}_1$.