[14]. Let an FMS is a set of $\mathbb{N}$ if we define Ð by, for all $\left(\mathsf{ȷ},\ell \right)\in \chi \times \chi$ with $f\left(v\right)=ln\left(v\right)$ and $\mu =ln\left(3\right),$ and not a FM. You should be aware that any measure on χ is an FM.

(ii) $k\in (0,1)$ occurs in such a way that Then g has a distinct fixed point ${\mathsf{ȷ}}^{*}\in \chi$. In addition, ${\mathsf{ȷ}}_0\in \chi$, the sequence $\left\{{\mathsf{ȷ}}_n\right\}\subset \chi$ defined by ${\mathsf{ȷ}}_{n+1}=h\left({\mathsf{ȷ}}_n\right)$, $n\in \mathbb{N}$, is $\mathcal{F}$-convergent for any to ${\mathsf{ȷ}}^{*}$ as well.

[12] Let $(\chi ,d)$ be a metric space. Let’s say that there are two functions $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$ and $S:\chi \to \chi$. A map S is considered as α-η-continuous map on $(\chi ,d)$ whenever given $\mathsf{ȷ}\in \chi$, and a sequence $\left\{{\mathsf{ȷ}}_n\right\}$ is as follows For more details see [20,21].

Let $(\chi ,$Ð) is be an FMS. Let there are two functions $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$ and $S:\chi \to \chi$. If there are constants $\lambda \in [0,1)$ and $\alpha ,\beta ,\gamma \in \left(0,1\right)$ such that whenever $\alpha (\mathsf{ȷ},\ell )\ge \eta (\mathsf{ȷ},\ell )$, we say that S is an interpolative convex Reich-type α-η-contraction. for all $\mathsf{ȷ},\ell \in \chi \setminus Fix\left(S\right)$, where þ$,q\in \left[1,\infty \right).$

Let $\chi =\left\{0,1,2,3\right\}$ be endowed with FMS given by with $f\left(v\right)=ln\left(v\right)$ and $\mu =ln\left(3\right).$ Define $S:\chi \to \chi$ by and $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$ by If $\mathsf{ȷ},\ell \in \chi .$ Clearly $\alpha (\mathsf{ȷ},\ell )\ge \eta (\mathsf{ȷ},\ell )$ then By taking any value of constants $\lambda \in [0,1),$ $\alpha ,\beta ,\gamma \in \left(0,1\right)$ and þ$,q\in \left[1,\infty \right).$ Clearly, (1) holds for all $\mathsf{ȷ},\ell \in \chi \setminus Fix\left(S\right).$ Keep in mind that S has two fixed points, 0 and 1. see for more information and examples [18].

Let ${\mathsf{ȷ}}_0$ in $\chi$ such that $\alpha ({\mathsf{ȷ}}_0,S{\mathsf{ȷ}}_0)\ge \eta ({\mathsf{ȷ}}_0,S{\mathsf{ȷ}}_0).$ For ${\mathsf{ȷ}}_0\in \chi ,$ we construct a sequence ${\left\{{\mathsf{ȷ}}_n\right\}}_{n=1}^{\infty }$such that ${\mathsf{ȷ}}_1=S{\mathsf{ȷ}}_0$, ${\mathsf{ȷ}}_2=S{\mathsf{ȷ}}_1=S^2{\mathsf{ȷ}}_0$. Continue this approach until for every $n\in \mathbb{N}, {\mathsf{ȷ}}_{n+1}=S{\mathsf{ȷ}}_n=S^{n+1}{\mathsf{ȷ}}_0$. . Because of (i), S is an $\alpha$-admissible in term of $\eta$ after that $\alpha ({\mathsf{ȷ}}_0, {\mathsf{ȷ}}_1)=\alpha ({\mathsf{ȷ}}_0,S{\mathsf{ȷ}}_0)\ge \eta ({\mathsf{ȷ}}_0,S{\mathsf{ȷ}}_0)=\eta ({\mathsf{ȷ}}_0, {\mathsf{ȷ}}_1)$. By carrying out this procedure further, we have On condition that ${\mathsf{ȷ}}_{n+1}={\mathsf{ȷ}}_n$ a few $n\in \mathbb{N},$ afterwards ${\mathsf{ȷ}}_n={\mathsf{ȷ}}^{*},$ is a fixed point of S. Thus, we presume ${\mathsf{ȷ}}_n\ne {\mathsf{ȷ}}_{n+1}$ with Since S is an interpolative convex Reich-type $\alpha$-$\eta$-contraction, for any $n\in \mathbb{N},$ give us we acquire Afterward decide that $\left\{\text{Ð}({\mathsf{ȷ}}_{n-1},{\mathsf{ȷ}}_n)\right\}$ is decreasing terms. As a result there is a positive term $\varrho$ s.t. ${lim}_{n\to \infty }$Ð$({\mathsf{ȷ}}_{n-1},{\mathsf{ȷ}}_n)=\varrho$. Take note $\varrho \ge 0,$ if we extrapolate use $(2.3)$, gives Which provide Subsequently There are some $N\in \mathbb{N}$ therefore Let $ϵ>0$ be fixed and $\left(g,\mu \right)\in \mathcal{F}\times {\mathbb{R}}^{+}$ be satisfied the (Ð${}_3)$. Next to $\left({\mathcal{F}}_2\right),$ there exist $\delta >0$ such that Due to (4) and $\left({\mathcal{F}}_1\right)$, we obtained where Ð$({\mathsf{ȷ}}_n,{\mathsf{ȷ}}_m)>0$ and $m,n\in \mathbb{N}$ in like a way that $m>n\ge N.$ Consequently, combining (5) and (Ð${}_3)$ gives that which suggest that $\left({\mathcal{F}}_1\right),$ we obtain As a result $\left\{{\mathsf{ȷ}}_n\right\}$ is an $\mathcal{F}$-Cauchy sequence. There exists ${\mathsf{ȷ}}^{*}\in \chi$ such that ${\mathsf{ȷ}}_n$ is $\mathcal{F}$-convergent to ${\mathsf{ȷ}}^{*},$ because $\left(\chi ,\text{Ð}\right)$ is a F-Complete MS that is, S is $\alpha$-$\eta$-continuous and has the properties $\alpha ({\mathsf{ȷ}}_{n-1}, {\mathsf{ȷ}}_n)\ge \eta ({\mathsf{ȷ}}_{n-1}, {\mathsf{ȷ}}_n),$ as every $n\in \mathbb{N}$ afterwards apply limit approaches to infinity ${\mathsf{ȷ}}_{n+1}=S{\mathsf{ȷ}}_n\to S{\mathsf{ȷ}}^{*}$ a certian ${\mathsf{ȷ}}^{*}=S{\mathsf{ȷ}}^{*}$. We will now demonstrate that ${\mathsf{ȷ}}^{*}$ is a fixed point of S. We use contradiction to argue by assuming that Ð$(S{\mathsf{ȷ}}^{*},{\mathsf{ȷ}}^{*})>0$. (Ð$3)$, gives us By using (${\mathcal{F}}_1$) and the contractive condition gives for every $n\in \mathbb{N}.$ In otherway, by using (${\mathcal{F}}_2$) together with (6), possess that result in a contradiction. In light of the fact that Ð$(S{\mathsf{ȷ}}^{*},{\mathsf{ȷ}}^{*})=0.$ Finally, ${\mathsf{ȷ}}^{*}$ is a fixed point of S. □

In a manner similar to the proof of Theorem 2.3. We obtain $\alpha ({\mathsf{ȷ}}_n, {\mathsf{ȷ}}^{*})\ge \eta ({\mathsf{ȷ}}_n, {\mathsf{ȷ}}^{*})$ for every $n\in \mathbb{N}$. (D3) gives us (1) and $\left({\mathcal{F}}_1\right)$, give us Using (6) information we obtain Using $\left({\mathcal{F}}_2\right)$,gives that that result in a contradiction. In light of the fact that Ð$({\mathsf{ȷ}}^{*},S{\mathsf{ȷ}}^{*})=0$ it is an established point ${\mathsf{ȷ}}^{*}$ possess fixed point of S. □

Assume $\chi =\mathbb{R}$ to FMS Ð$:\chi \times \chi \to {\mathbb{R}}^{+}$ by with$\mu =ln\left(100\right)$ and $f\left(v\right)=ln\left(v\right).$ Define $S:\chi \to \chi$ by and $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$ by

Assume that $(\chi ,$Ð) is an FMS and $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$ be two functions. The FMS in χ is considered to be α-η-complete iff each $\mathcal{F}$-Cauchy sequence $\left\{{\mathsf{ȷ}}_n\right\}$ must contain in $\chi ,$$\mathcal{F}$ -converges.

Let $(\chi ,$Ð) is an FMS. Let there are two functions $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$ and $S:\chi \to \chi$. If there are constants $\lambda \in [0,1)$ and $\alpha ,\beta \in \left(0,1\right)$ such that whenever $\alpha (\mathsf{ȷ},\ell )\ge \eta (\mathsf{ȷ},\ell )$, we say that S is an convex interpolative convex Kannan-type α-η-contraction. where þ$,q\in \left[1,\infty \right)$ for all $\mathsf{ȷ},\ell \in \chi$ with $\mathsf{ȷ}\ne S\mathsf{ȷ}.$

The same procedures are used in the proof of Theorem 3.1.

Since S is convex interpolative Kannan-type $\alpha$-$\eta$-contraction, give us It is implied from equation (8) that The rest of the proof follows the structure of Theorem 2.1 and proceeds in a similar manner. □

Carried out in a manner similar to that of Theorem 2.2. (7) and (F1) give us Utilizing (6) and the information We achieve This is incongruous. Ð$({\mathsf{ȷ}}^{*},S{\mathsf{ȷ}}^{*})=0$ as a result, meaning that is a fixed point of S. □

Let S be a continuous self-map on χ and $(\chi ,$Ð) be a F-Complete FMS. Assume $r\in \left[0,1\right)$ and $\alpha ,\beta ,\gamma \in \left(0,1\right)$ are present and in such a way that where þ$,q\in \left[1,\infty \right),$ for every $\mathsf{ȷ},\ell \in \chi .$

Set two functions $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$ by and $\beta ,\gamma \in \left(0,1\right),$ and $r\in \left[0,1\right).$ It is clear that that is, our Theorem 2.1’s criteria (i) through (iii) are satisfied. Let it suggests a contractive condition As a result, Theorem 3’s criteria are all satisfied. Hence S attain a fixed point in $\mathsf{ȷ}.$ □

Suppose a continuous map S and $(\chi ,$Ð) be a F-Complete FMS. Assume $r\in \left[0,1\right)$ and $\alpha ,\beta \in \left(0,1\right)$ are present and in such a way that where þ$,q\in \left[1,\infty \right),$ for all $\mathsf{ȷ},\ell \in \chi .$

Suppose a continuous map S and $(\chi ,$Ð) be a F-Complete FMS. Assume $r\in \left[0,1\right)$ in such a way that for all $\mathsf{ȷ},\ell \in \chi .$ Then S possess a fixed point.

Set $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$, by $\alpha (\mathsf{ȷ},\ell )=3$ on $O\left(\omega \right)\times O\left(\omega \right)$ and $\alpha (\mathsf{ȷ},\ell )=0$ otherwise and $\eta (\mathsf{ȷ},\ell )=1$ for all $\mathsf{ȷ},\ell \in \mathsf{ȷ}$( see Remark 6 [10]). Then $(\chi ,$Ð) is an $\alpha$-$\eta$-complete $\mathcal{F}$-metric and S is an $\alpha$-admissible regard to $\eta$. Let $\alpha (\mathsf{ȷ},\ell )\ge \eta (\mathsf{ȷ},\ell ),$ later $\mathsf{ȷ},\ell \in O\left(\omega \right),$ afterwards from (ii) give us That is S is Interpolative convex $\alpha$-$\eta$-contraction of the Reich-type. Let a sequence $\left\{{\mathsf{ȷ}}_n\right\}$ applies it read $\alpha ({\mathsf{ȷ}}_n, {\mathsf{ȷ}}_{n+1})\ge \eta ({\mathsf{ȷ}}_n, {\mathsf{ȷ}}_{n+1})$ and ${lim}_{n\to \infty }$${\mathsf{ȷ}}_n={\mathsf{ȷ}}^{*}.$ Therefore $\left\{{\mathsf{ȷ}}_n\right\}\subseteq O\left(\omega \right).$ The expression it taken from (iii) ${\mathsf{ȷ}}^{*}\in O\left(\omega \right),$$\alpha ({\mathsf{ȷ}}_n,{\mathsf{ȷ}}^{*})\ge \eta ({\mathsf{ȷ}}_n,{\mathsf{ȷ}}^{*}).$ As a result, Theorem 2.2’s criteria are all fulfilled. S therefore has a fixed point. □

Similar to Theorem 4.3’s hypotheses, satisfies Therefore S attain a fixed point.

Define $\alpha ,\eta :\chi \times \chi \to [0,+\infty )$, by $\alpha (\mathsf{ȷ},\ell )=3$ on $O\left(\omega \right)\times O\left(\omega \right)$ and $\alpha (\mathsf{ȷ},\ell )=0$ otherwise and $\eta (\mathsf{ȷ},\ell )=1$( see Remark 1.1 [11]), we know S is $\alpha$-$\eta$-continuous map. Assume $\alpha (\mathsf{ȷ},\ell )\ge \eta (\mathsf{ȷ},\ell ),$ afterwards $\mathsf{ȷ},\ell \in O\left(\omega \right).$ Therefore $S\mathsf{ȷ},S\ell \in O\left(\omega \right)$ that is $\alpha (S\mathsf{ȷ},S\ell )\ge \eta (S\mathsf{ȷ},S\ell ).$ In light of, S is therefore a mapping that is $\alpha$-admissible. We have from (i) That is to say, S is a Reich-type interpolative convex $\alpha$-$\eta$-contraction. As a result, Theorem 2.1’s entire premise is true. S therefore attains a fixed point. □

Theorem 4.5 (i) through (ii)hypotheses’s are true then Consequently S attains a fixed point.

[13] Let $\alpha >0$, $n=-[-\alpha ],g\in L[c,d]$ and ${}_a{I}_h^{\alpha }g\in A{C}_h^n[c,d]$. Then We are thinking about the ensuing boundary value problem