Existence and Stability for Fractional Differential Equations with a Regularized ψ–Hilfer Fractional Derivative

This article aims to explore the existence and stability of solutions to differential equations involving a regularized ψ−Hilfer fractional derivative, which, compared to standard ψ−Hilfer fractional derivatives, provide a clear physical interpretation when dealing with initial conditions. We discovered that the regularized ψ−Hilfer fractional derivative can be represented as the inverse operation of the ψ−Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a regularized ψ−Hilfer fractional derivative. Additionally, we applied the Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a regularized ψ−Hilfer fractional derivative, and further discussed Ulam-Hyers-Rassias stability and semi-Ulam-Hyers-Rassias stability of these solutions. Finally, we illustrated our results through case studies.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

1. Introduction

The history of fractional calculus can be traced back to 1965, and after many years of development, it has become an important mathematical tool [1]. As science and technology advance, the realm of fractional calculus is seeing an ever-widening scope of application. Researchers are increasingly turning to this sophisticated mathematical toolkit as a solution to the limitations inherent in traditional calculus methods when it comes to capturing the intricacies of certain complex phenomena and challenges. Currently, fractional calculus is mainly applied in viscoelastic materials [2], electrochemistry [3], control engineering [4], fluid mechanics [5], statistical mechanics [6], numerical schemes [7], etc.

Recently, research on the Hilfer fractional derivative has become a popular direction in fractional calculus. The Hilfer fractional derivative generalizes the Riemann-Liouville (R-L) and Caputo fractional derivatives [8]. Drawing inspiration from the

ψ

-Caputo fractional derivative [10], Sousa et al. introduced a novel fractional operator termed the

ψ

-Hilfer fractional derivative, which aims to unify the properties of the majority of fractional derivatives [11]. With the extensive usage of the

ψ

-Hilfer fractional derivative, there has been a problem regarding the selection of the order p. This arises from the lack of clear physical interpretation for the initial conditions associated with this derivative, unless

p = 1

. To address this confusion, Jajarmi et al. started from the definition of the

ψ

-Hilfer fractional derivative and extended it to introduce a new fractional derivative called the regularized

ψ

-Hilfer derivative [16].

Furthermore, the application of fixed point theorems to study the stability issues of

ψ

-Hilfer fractional nonlinear differential equations has also attracted significant attention from scholars. Salim et al. investigated the Ulam-Hyers-Rassias stability of k-generalized

ψ

-Hilfer fractional derivative by employing the Banach’s fixed-point theorem [12]. And researcher demonstrated that the

ψ

-Hilfer fractional boundary value problem, which models thermostat control, exhibits four distinct types of Ulam-Hyers stability [9].

Motivated by the above discussions, in this paper, we address the initial-value problem (IVP) for nonlinear differential equation with a regularized

ψ

–Hilfer fractional derivative

\{\begin{matrix} {}^{C}D_{a^{+}}^{α, β; ψ} x (t) = f (t, x (t)), t \in (a, b], \\ {x (t) |}_{t = a} = x_{0}, \end{matrix}

(1)

where

α \in (0, 1), β \in [0, 1], a < b

, and

{}^{C}D_{a^{+}}^{α, β; ψ} x (\cdot)

is the regularized version of

ψ

-Hilfer fractional derivative in the Caputo sense, see Definition 10. Let

I = [a, b]

, X be a Banach space which a norm

‖ \cdot ‖

, then

x \in C^{1} ((a, b], X)

. Let

f : I \times X \to X

be a given continuous function and

x_{0}

be the initial value of x.

In this article, we explore the relationship between the regularized

ψ -

Hilfer fractional derivative and the

ψ -

R-L fractional integral, identifying them as inverse operators. The flexibility in selecting

ψ

considerably extends the applications of this derivative. Building on this property, we have demonstrated the existence of solutions for differential equations that incorporate linear regularized

ψ -

Hilfer fractional derivatives. Differing from the strict Lipschitz condition employed by Jajarmi et al. to establish solution existence [16], our approach in Theorem 18 employs more relaxed conditions. We assume the nonlinear term

f (t, x (t))

is well-defined across the interval I and regulated by the continuous positive function

h (t) (1 + ∥ x (t) ∥)

, ensuring the existence of solutions. In addition, we use Banach’s fixed-point theorem to demonstrate the stability of the differential equation involving the regularized

ψ -

Hilfer fractional derivative. Unlike the equation in Zhou et al., which contains

ψ -

Hilfer fractional derivatives [24], the initial value conditions of the equation we studied have a clear physical meaning, which improves the accuracy and applicability of the model. Moreover, the solution operator is bounded at the initial conditions, which also makes the argument process significantly different from that of Zhou et al..

This article will discuss the following aspects. In Section 2, the properties of the measure of noncompactness, the definitions of several fractional operators, and the relationships among these fractional operators are provided. Besides that, the knowledge related to Ulam-Hyers-Rassias stability, semi-Ulam-Hyers-Rassias stability and fixed point theorems will also be elucidated in this section. In Section 3, we provide the proof process of the existence of the solution to (1). In Section 4, we discuss the Ulam-Hyers-Rassias and semi-Ulam-Hyers-Rassias stability results for the solution of (1). In the final section, we illustrate our results through two examples.

2. Preliminaries

Definition 1

([13]). Let B be a metric space, and let J denote a bounded subset of B. The Kuratowski measure of noncompactness is defined in the following manner:

μ (J) = i n f \{ξ > 0 | J = ⋃_{i = 1}^{t} J_{i}, d i a m J_{i} \leq ξ\} .

Lemma 2

([14]). In the Banach space, there exist subsets J,

J_{1}

and

J_{2}

. We provide the following properties:

$0 \leq μ (J) < + \infty;$
$μ (J) = μ (\bar{J});$
$μ (J) = 0 \Leftrightarrow J$ is relatively compact;
$J_{1} \subset J_{2} \Rightarrow μ (J_{1}) \leq μ (J_{2});$
$μ (J_{1} + J_{2}) \leq μ (J_{1}) + μ (J_{2}),$ where $J_{1} + J_{2} = {t | t = m + n, m \in J_{1} n \in J_{2}};$
$μ (c J) = ∥ c ∥ μ (J), c \in R,$ where $c J = {t | t = c q, q \in J} .$

In the sections that follow, we will explore various properties and outcomes associated with fractional calculus that incorporates the kernel function

ψ

Definition 3

([1]). Let

α \in (0, + \infty)

t \in I

x (t)

be an integrable function and

ψ (t) \in C^{1} (I, X)

be a strictly monotonically increasing function, the αth order ψ-R-L integral of

x (t)

is defined as follows

I_{a +}^{α; ψ} x (t) = \frac{1}{Γ (α)} \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} x (τ) d τ .

The definition of the ψ-R-L fractional derivate of order

α > 0

with respect to t for an integrable function x is given as follows

\begin{matrix} D_{a +}^{α; ψ} x (t) = δ_{ψ}^{n} I_{a +}^{n - α, ψ} x (t) = \frac{1}{Γ (n - α)} δ_{ψ}^{n} \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{n - α - 1} x (τ) d τ, \end{matrix}

where

n = [α] + 1, δ_{ψ}^{n} = {(\frac{1}{ψ^{'} (t)} \frac{d}{d t})}^{n}

and this expression is defined in the same manner throughout the subsequent text.

Lemma 4

([1]). Let

α_{1}, α_{2} \in (0, + \infty)

. Then the

ψ -

R-L integral has the following semigroup property

I_{a +}^{α_{1}; ψ} I_{a +}^{α_{2}; ψ} x (t) = I_{a +}^{α_{1} + α_{2}; ψ} x (t) .

Lemma 5

([1]). For any

α, β > 0

and

x (t) = {(ψ (t) - ψ (a))}^{β - 1}

, the following equality holds

I_{a +}^{α; ψ} x (t) = \frac{Γ (β)}{Γ (α + β)} {(ψ (t) - ψ (a))}^{α + β - 1} .

Definition 6

([15]). Let

α > 0, n \in N, t \in I, x, ψ \in C^{n} (I, X)

, and

ψ (t) \in C^{1} (I, X)

be a strictly monotonically increasing function. Then the αth order ψ-Caputo fractional derivative with respect to x can be expressed as

{}^{C}D_{a^{+}}^{α; ψ} x (t) = I_{a +}^{n - α, ψ} δ_{ψ}^{n} [x (t)],

where

n = [α] + 1, δ_{ψ}^{n} [x (t)] = {(\frac{1}{ψ^{'} (t)} \frac{d}{d t})}^{n} x (t) .

In particular, if

α \in (0, 1]

, then

\begin{matrix} {}^{C}D_{a^{+}}^{α; ψ} x (t) = \frac{1}{Γ (1 - α)} \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} δ_{ψ} [x (τ)] d τ . \end{matrix}

Lemma 7

([15]). If

r \in (0, 1)

and

x \in C^{n} (I, X)

, then

\begin{matrix} I_{a^{+}}^{r; ψ} {}^{C}D_{a^{+}}^{r; ψ} x (t) = x (t) - x (a) . \end{matrix}

(2)

Theorem 8

([16]). Let

α \in (0, 1)

and

x \in C^{1} (I, X)

, then

{}^{C}D_{a^{+}}^{α; ψ} x (t) = D_{a +}^{α; ψ} x (t) - \frac{1}{Γ (1 - q)} δ_{ψ} (x_{0}) {(ψ (t) - ψ (a))}^{- α} .

Next, we will introduce the

ψ

-Hilfer fractional derivative and the regularized

ψ

-Hilfer fractional derivative, and compare these two types of fractional derivatives.

Definition 9

([11]). Let

α \in (n - 1, n)

with

n \in N, ψ \in C^{n} (I, X)

, and

ψ (t) \in C^{1} (I, X)

be a strictly monotonically increasing function, for all

t \in I

. The

ψ -

Hilfer fractional derivative of order α and type

β \in [0, 1]

can be written as

\begin{matrix} {}^{H}D_{a^{+}}^{α, β; ψ} x (t) = I_{a +}^{β (n - α); ψ} δ_{ψ}^{n} I_{a +}^{(1 - β) (n - α); ψ} x (t) . \end{matrix}

(3)

Building upon the definition of the ψ-Hilfer fractional derivative provided earlier, it can also be articulated in the following manner

\begin{matrix} {}^{H}D_{a^{+}}^{α, β; ψ} x (t) & = I_{a +}^{r - α; ψ} D_{a +}^{r; ψ} x (t) \\ = \frac{1}{Γ (r - α)} \int_{a}^{t} {(ψ (t) - ψ (τ))}^{r - α - 1} ψ^{'} (τ) D_{a^{+}}^{r; ψ} x (τ) d τ, \end{matrix}

(4)

where

r = α + β (n - α) .

Definition 10

([16]). Let

α \in (0, + \infty)

t \in I

x (t)

be an integrable function and

ψ (t) \in C^{1} (I, X)

be a strictly monotonically increasing function. Then the regularized ψ-Hilfer fractional derivative is defined by

\begin{matrix} {}^{C}D_{a^{+}}^{α, β; ψ} x (t) & = I_{a^{+}}^{r - α; ψ} {}^{C}D_{a^{+}}^{α; ψ} x (t) \\ = \frac{1}{Γ (r - α)} \int_{a}^{t} {(ψ (t) - ψ (τ))}^{r - α - 1} ψ^{'} (τ) {}^{C}D_{a^{+}}^{r; ψ} x (τ) d τ, \end{matrix}

(5)

where

r = α + β (n - α) .

Theorem 11.

α \in (0, 1)

β \in [0, 1]

and

x \in C^{1} (I, X)

then

\begin{matrix} {}^{C}D_{a^{+}}^{α, β; ψ} I_{a^{+}}^{α; ψ} x (t) & = I_{a^{+}}^{r - α; ψ} {}^{C}D_{a^{+}}^{r; ψ} I_{a^{+}}^{α; ψ} x (t) \\ = I_{a^{+}}^{r - α; ψ} I_{a^{+}}^{n - r; ψ} δ_{ψ}^{n} I_{a^{+}}^{α; ψ} x (t) \\ = I_{a^{+}}^{r - α; ψ} I_{a^{+}}^{n - r; ψ} I_{a^{+}}^{α - n; ψ} x (t) \\ = x (t) . \end{matrix}

(6)

Remark 12.

This property is the same as that of the

ψ -

Caputo fractional derivative, as detailed in Theorem 5 of [15]. Utilizing this property, we can derive other important conclusions related to the regularized

ψ -

Hilfer derivative.

By Theorem 8, Lemma 5, and equation 4, we have established the relationship between the order

α \in (0, 1)

ψ -

Hilfer fractional derivative and the regularized

ψ

-Hilfer fractional derivative

{}^{C}D_{a^{+}}^{α, β; ψ} x (t) = {}^{H}D_{a^{+}}^{α, β; ψ} x (t) - \frac{1}{Γ (1 - α)} δ_{ψ} [x_{0}] {(ψ (t) - ψ (a))}^{- α} .

Definition 13

([18]). If for each

x (t) \in C^{1} ((a, b], X)

satisfying

∥ x (t) - I_{a +}^{α; ψ} f (t, x (t)) ∥ \leq I_{a +}^{α; ψ} Φ (t),

where

Φ (t) > 0

consistently increases as a continuous function across all t, and assuming the presence of a solution

x_{1} (t)

to (1), alongside a chosen positive constant C, this ensures the fulfillment of the subsequent inequality for all t

∥ x (t) - x_{1} (t) ∥ \leq C Φ (t),

then it implies that (1) has the Ulam-Hyers-Rassias stability. In this case, if

Φ (t)

takes on a constant value, then (1) exhibits Ulam-Hyers stability.

Definition 14

([19]). If for each

x (t) \in C^{1} ((a, b], X)

satisfying

∥ x (t) - I_{a +}^{α; ψ} f (t, x (t)) ∥ \leq σ,

where

σ > 0

, it follows that (1) possesses a solution

x_{1} (t)

, and a positive constant C can be identified such that

∥ x (t) - x_{1} (t) ∥ \leq C Φ (t),

where

Φ (t) > 0

for all t, then it implies that (1) has the semi-Ulam-Hyers-Rassias stability.

Lemma 15

([20]). Suppose

(G, d)

constitutes a generalized complete metric space, and consider Q as a self-mapping on G with a Lipschitz constant

N < 1

. Given any

g \in G

, should there be an

i \in N

for which

d (Q^{i + 1} g, Q^{i} g) < \infty

, then

The sequence denoted by ${Q^{i} g_{0}}$ tends toward a fixed point, $g^{*}$ , of the mapping Q, starting from an initial point $g_{0}$ within G;
In the set $G^{*} = {k \in G ∣ d (Q^{i} k, g) < \infty}$ , $k^{*}$ represents the sole fixed point of the function Q;
If $g \in G^{*}$ ,then $d (g, k^{*}) \leq \frac{1}{1 - N} d (T g, g)$ .

To obtain various stability results for Ulam-Hyers on the interval I, we establish the following Banach spaces

d_{1} (\cdot)

and

d_{2} (\cdot)

[21].

\begin{matrix} d_{1} (x, y) = i n f \{C | ∥ x (t) - y (t) ∥ \leq C Φ_{1} (t), t \in I, C > 0\}, \end{matrix}

where

Φ_{1} (t) > 0

is a nondecreasing continuous function on I.

\begin{matrix} d_{2} (x, y) = s u p \{C | \frac{∥ x (t) - y (t) ∥}{Φ_{2} (t)} \leq C, t \in I, C > 0\}, \end{matrix}

where

Φ_{2} (t) > 0

is a nonincreasing continuous function on I.

For ease of computation in subsequent calculations, we assume

D = sup_{t \in I} \frac{{(ψ (t) - ψ (a))}^{α}}{Γ (α + 1)} .

Next, let’s consider the linear problem corresponding to (1).

Lemma 16.

Let

α \in (0, 1)

and

p (t)

is a continuous function on

(a, b]

. Then the linear initial value problem for (1)

\{\begin{matrix} {}^{C}D_{a^{+}}^{α, β; ψ} x (t) = p (t), t \in (a, b], \\ {x (t) |}_{t = a} = x_{0}, \end{matrix}

(7)

has a unique solution as follows

x (t) = I_{a^{+}}^{α; ψ} p (t) + x_{0}, t \in I .

(8)

Proof.

According to the definition of the regularized

ψ

-Hilfer fractional derivative and using semigroup property, the following equation holds

I_{a^{+}}^{α; ψ} {}^{C}D_{a^{+}}^{α, β; ψ} x (t)) = I_{a^{+}}^{α; ψ} (I_{a^{+}}^{r - α; ψ} {}^{C}D_{a^{+}}^{r; ψ} x (t)) = I_{a^{+}}^{r; ψ} {}^{C}D_{a^{+}}^{r; ψ} x (t) .

where

r = α + β (1 - α)

, and based on Lemma 7, it can be seen that

I_{a^{+}}^{r; ψ} {}^{C}D_{a^{+}}^{r; ψ} x (t) = x (t) - x_{0} .

Then, we get

x (t) = I_{a^{+}}^{α; ψ} p (t) + x_{0} .

Therefore, if

x (t)

satisfies (8), it is easy to obtain

{}^{C}D_{a^{+}}^{α, β; ψ} x (t) = p (t), t \in I .

□

Lemma 17.([17]) Suppose U is a subset that is bounded, closed, and convex within a Banach space, with 0 included in U, and

Q : U \to U

is continuous. Then for any subset V of U, Q possesses a fixed point if and only if

V = \bar{c o n v} Q (V)

, or

V = Q (V) \cup {0} \Rightarrow μ (V) = 0

3. Main Result

In this section, we will prove the existence of (1) by applying the knowledge related to Mönch’s fixed-point theorem. Before proving the existence, we need the following assumptions.

Theorem 18.

Let’s consider that the following assumptions are valid:

(H_{1})

The function

f (t, x)

maintains continuity in relation to X for a.e.

t \in I

, and with respect to

x \in X

, it is measurable on I.

(H_{2})

For a.e.

t \in I

and

x \in X

, the subsequent inequality holds

∥ f (t, x) ∥ \leq h (t) (1 + ∥ x ∥),

where h is a continuous function and

h : I \to [0, \infty)

(H_{3})

Let E be arbitrary bounded subset of X, then f satisfies

μ (f (t, E)) \leq h (t) μ (E),

where

t \in I

and

h (t)

corresponds to the aforementioned assumption.

M : = h_{m} D < 1,

(9)

where

h_{m} = {sup}_{t \in I} h (t)

and D has been defined above, then (1) has a solution.

Proof.

Let operator

Q : C^{1} (I, X) \to C^{1} (I, X)

defined by

(Q x) (t) = \frac{1}{Γ (α)} \int_{a}^{t} f (τ, x (τ)) {(ψ (t) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ + x_{0} .

(10)

From the above assumptions, it can be seen that the fixed points of the operator Q correspond to solution of (1).

For each

ω \in C^{1} (I, X)

, let

K \geq \frac{∥ x_{0} ∥ + M}{1 - M} > 0,

which satisfies the ball

B_{K} : = {{∥ ω ∥}_{\infty} \leq K}

For any

x \in C^{1} (I, X)

and each

t \in I

, we have

\begin{matrix} ∥ (Q x) (t) ∥ & \leq \frac{1}{Γ (α)} \int_{a}^{t} {∥ f (τ, x (τ)) ∥ (ψ (t) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ + ∥ x_{0} ∥ \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} {h (τ) ∥ 1 + x (τ) ∥ (ψ (t) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ + ∥ x_{0} ∥ \\ \leq \frac{h_{m} (1 + K)}{Γ (α)} \int_{a}^{t} {(ψ (t) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ + ∥ x_{0} ∥ \\ \leq h_{m} (1 + K) D + ∥ x_{0} ∥ \\ = M (1 + K) + ∥ x_{0} ∥ \leq K . \end{matrix}

Thus

{∥ Q (x) ∥}_{\infty} \leq K .

(11)

The above process proves that Q is a transformation of

B_{K}

onto itself.

Subsequently, we aim to demonstrate that the operator

Q : B_{K} \to B_{K}

fulfills every criterion stipulated by Mönch’s fixed point theorem, as outlined in Lemma 17.

Consider a sequence

{x_{n}}_{n \in N}

converging to x within

B_{K}

. For every

t \in I

, it follows that

\begin{matrix} ∥ (Q x_{n}) (t) - (Q x) (t) ∥ \\ \leq \frac{1}{Γ (α)} \int_{a}^{t} ∥ f (τ, x_{n} (τ)) - f (τ, x (τ)) ∥ {(ψ (t) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ . \end{matrix}

(12)

(H_{1})

and the Lebesgue dominated convergence theorem, (12) implies

∥ Q x_{n} {- Q x ∥}_{\infty} \to 0 as n \to \infty .

The above proves that

Q : B_{K} \to B_{K}

is continuous.

Next, we aim to demonstrate that

Q (B_{K})

is both bounded and equicontinuous. Given that

Q (B_{K})

is a subset of the bounded set

B_{K}

, it follows that

Q (B_{K})

is also bounded. Now, let

a_{1}, a_{2} \in I, a_{1} < a_{2}

and let

f \in B_{K}

. Thus, we have

\begin{matrix} ∥ (Q x) (a_{2}) & - (Q x) (a_{1}) ∥ \\ \leq \frac{1}{Γ (α)} ∥ \int_{a}^{a_{2}} f (τ, x (τ)) {(ψ (a_{2}) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ \\ - \int_{a}^{a_{1}} f (τ, x (τ)) {(ψ (a_{1}) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ ∥ \\ \leq \frac{1}{Γ (α)} \int_{a_{1}}^{a_{2}} ∥ f (τ, x (τ)) ∥ {(ψ (a_{2}) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ \\ + \frac{1}{Γ (α)} \int_{a}^{a_{1}} ∥ f (τ, x (τ)) ∥ \cdot | {(ψ (a_{2}) - ψ (τ))}^{α - 1} - {(ψ (a_{1}) - ψ (τ))}^{α - 1} | ψ^{'} (τ) d τ \\ \leq \frac{1}{Γ (α)} \int_{a_{1}}^{a_{2}} h (τ) (1 + ∥ x (τ) ∥) {(ψ (a_{2}) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ \\ + \frac{1}{Γ (α)} \int_{a}^{a_{1}} h (τ) (1 + ∥ x (τ) ∥) \cdot | {(ψ (a_{2}) - ψ (τ))}^{α - 1} - {(ψ (a_{1}) - ψ (τ))}^{α - 1} | ψ^{'} (τ) d τ, \end{matrix}

Hence,

\begin{matrix} ∥ (Q x) (a_{2}) - (Q x) (a_{1}) ∥ & \leq \frac{h_{m} (1 + K)}{Γ (α + 1)} {(ψ (a_{2}) - ψ (a_{1}))}^{α} \\ + \frac{h_{m} (1 + K)}{Γ (α + 1)} \int_{a}^{a_{1}} | {(ψ (a_{2}) - ψ (τ))}^{α - 1} - {(ψ (a_{1}) - ψ (τ))}^{α - 1} | ψ^{'} (τ) d τ . \end{matrix}

a_{1} \to a_{2}

, the entire expression tends to zero.

Now, consider W as a subset of

B_{K}

such that

W \subset \bar{Q (W)} \cup {0}

. The function

t \to ω (t) = μ (W (t))

remains on Q as W is bounded and equicontinuous. By the properties of measures and the content of

H_{3}

, we can obtain

\begin{matrix} ω (t) & \leq μ ((Q W) (t) \cup {0}) \\ \leq μ ((Q W) (t)) \\ \leq \frac{1}{Γ (α)} \int_{0}^{t} {(ψ (t) - ψ (τ))}^{α - 1} h (τ) μ (W (τ)) d τ \\ \leq \frac{1}{Γ (α)} \int_{0}^{t} {(ψ (t) - ψ (τ))}^{α - 1} h (τ) ω (τ) d τ \\ \leq h_{m} D {∥ ω ∥}_{\infty} . \end{matrix}

where

t \in I

. Thus

{∥ ω ∥}_{\infty} \leq M {∥ ω ∥}_{\infty} .

According to the value of M in (9), we obtain

{∥ ω ∥}_{\infty} = 0

, further leading to

ω (t) = μ (W (t)) = 0

for any

t \in I

. Further, by the properties of noncompactness measure, we can directly deduce that

W (t)

is relatively compact in X. By virtue of the Ascoli-Arzeà theorem [23], it can be concluded that W is relatively compact in

B_{K}

. With Theorem 17 applied, we can conclude that Q possesses a fixed point, representing a solution to (1).

□

4. Stability Analysis

In this section, we will present the Ulam-Hyers-Rassias stability, Ulam-Hyers stability, and semi-Ulam-Hyers-Rassias stability of the solution to (1). Before proving the stability, we need to impose the following special growth conditions on the function f to ensure a more effective and reliable proof process.

(H_{4})

Let the continuous function

N (\cdot) \geq 0

, the continuous function

f : I \times X \to X

and for arbitrary

x, g \in X

satisfying

∥ f (t, x (t)) - f (t, g (t)) ∥ \leq N (t) ∥ x (t) - g (t) ∥, \forall t \in I .

(H_{5})

Consider a constant P in the interval

(0, 1)

which satisfies

sup_{t \in I} \frac{1}{Γ (α)} \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} N (τ) d τ \leq P .

Theorem 19.

Let

(H_{4})

and

(H_{5})

hold,

Φ_{1}

is a positive increasing continuous function and

t \in I, x \in C^{1} (I, X)

satisfying

\begin{matrix} ∥ x (t) - (T x) (t) ∥ \leq I_{a +}^{α; ψ} Φ_{1} (t), \end{matrix}

(13)

where

t \in I

and

(Q x) (t)

defined by (10), then (1) has a unique solution

x_{1} (t)

satisfying the following inequality

∥ x (t) - x_{1} (t) ∥ \leq \frac{D}{1 - N} Φ_{1} (t),

where

t \in I, N \in (0, 1)

. That means (1) exhibits Ulam-Hyers-Rassias stability.

Proof.

(Q x) (t) = \frac{1}{Γ (α)} \int_{a}^{t} f (τ, x (τ)) {(ψ (t) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ + x_{0} .

For

x, y \in I

, from conditions

(H_{4})

(H_{5})

and metric

d_{1} (\cdot)

we can obtain

\begin{matrix} ∥ (Q x) (t) - (Q y) (t) ∥ & \leq \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} ∥ f (τ, x (τ)) - f (τ, y (τ)) ∥ d τ \\ \leq \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} L (τ) ∥ x (τ) - y (τ) ∥ d τ \\ \leq P C Φ_{1} (t) . \end{matrix}

Therefore,

d_{1} (Q x, Q y) \leq P C = P d_{1} (x, y) .

Otherwise, (13) imply that

\begin{matrix} ∥ x (t) - (Q x) (t) ∥ & \leq \frac{1}{Γ (α)} \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} Φ_{1} (τ) d τ \\ \leq \frac{Φ_{1} (t)}{Γ (α)} \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} d τ \\ \leq sup \frac{{(ψ (t) - ψ (a))}^{α}}{Γ (α + 1)} Φ_{1} (t) \\ = D Φ_{1} (t) . \end{matrix}

From the above equation, we can obtain

d_{1} (x, Q x) \leq D < \infty .

By combining

(i)

and

(i i)

of Theorem 6, we can conclude the existence of a unique fixed point

x_{1}

, which satisfies

Q x_{1} = x_{1}

. According to

(i i i)

of Theorem 6, we can obtain

d_{1} (x, x_{1}) \leq \frac{1}{1 - N} d_{1} (Q x, x) \leq \frac{D}{1 - N} .

In summary, we conclude that (1) possesses Ulam-Hyers-Rassias stability. □

Remark 20.

Φ_{1} (t)

represents an arbitrary constant function in Theorem 19, then the (1) has the Ulam-Hyers stability.

Theorem 21.

Let

(H_{4})

and

(H_{5})

hold,

Φ_{2}

is a positive nonincreasing continuous function and the function

x \in C^{1} (I, X)

satisfying

\begin{matrix} ∥ x (t) - (Q x) (t) ∥ \leq I_{a +}^{α; ψ} ξ, \end{matrix}

(14)

where

t \in I

ξ > 0

and

(Q x) (t)

defined by (10), then (1) has a unique solution

x_{1} (t)

satisfying the following inequality

∥ x (t) - x_{1} (t) ∥ \leq \frac{ξ D C_{1}}{1 - N} Φ_{2} (t),

where

t \in I, N \in (0, 1)

. This indicates that (1) has the semi-Ulam-Hyers-Rassias stability.

Proof.

(Q x) (t) = \frac{1}{Γ (α)} \int_{a}^{t} f (τ, x (τ)) {(ψ (t) - ψ (τ))}^{α - 1} ψ^{'} (τ) d τ + x_{0} .

From the contents of

(H_{4}), (H_{5})

and metric

d_{2} (\cdot)

, it is straightforward to establish that the following inequality holds for any

x, y \in X

\begin{matrix} \frac{∥ (Q x) (t) - (Q y) (t) ∥}{Φ_{2} (t)} & \leq \frac{\int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} ∥ f (τ, x (τ)) - f (τ, y (τ)) ∥ d τ}{Φ_{2} (t)} \\ \leq \frac{\int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} L (τ) ∥ x (τ) - y (τ) ∥ d τ}{Φ_{2} (t)} \\ \leq P C . \end{matrix}

It is obvious that

d_{2} (Q x, Q y) \leq P C = P d_{2} (x, y) .

Based on the functional properties of

Φ_{2} (t)

, we define

C_{1} > 0

such that it satisfies the following:

\frac{1}{Φ_{2} (t)} \leq C_{1} .

From (14), we have

\begin{matrix} \frac{∥ x (t) - (Q x) (t) ∥}{Φ_{2} (t)} & \leq \frac{I^{α; ψ} ξ}{Φ_{2} (t)} \\ \leq \frac{ξ \int_{a}^{t} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} d τ}{Γ (α) Φ_{2} (t)} \\ = ξ D C_{1} . \end{matrix}

From the above equation, we can obtain

d_{2} (x, Q x) \leq ξ D C_{1} < \infty .

By combining

(i)

and

(i i)

of Theorem 6, we can conclude the existence of a unique fixed point

x_{1}

, which satisfies

Q x_{1} = x_{1}

. According to

(i i i)

of Theorem 6, we can obtain

d_{2} (x, x_{1}) \leq \frac{1}{1 - N} d_{2} (Q x, x) \leq \frac{ξ D C_{1}}{1 - N} .

In summary, we conclude that (1) possesses semi-Ulam-Hyers-Rassias stability. □

5. Examples

Consider the fractional initial value problem(IVP) involving the regularized

ψ -

Hilfer fractional differential equations of the form:

\{\begin{matrix} {}^{C}D_{a^{+}}^{α, β; ψ} x (t) = f (t, x (t)), t \in I, \\ {x (t) |}_{t = a} = 1, \end{matrix}

(15)

where

α \in (0, 1), β \in [0, 1], I = [a, b]

{}^{C}D_{a^{+}}^{α, β; ψ} x (\cdot)

is the regularized version of

ψ

-Hilfer fractional derivative and

{x (t) |}_{t = a}

is initial value. The subsequent two instances represent specific scenarios of (15).

Example 1. Consider the (15). Taking

X = I, ψ (t) = t, a = 1, b = \frac{3}{2}, α = β = \frac{1}{2}, f (t, x (t)) = s i n x (t)

and substituting it into

(H_{2})

, it is clear that

h (t)

taking 1.

When

h_{m} = 1

{max}_{t \in I} ψ (t) = \frac{3}{2}

and we have

M = h_{m} D \leq \frac{{(\frac{1}{2})}^{\frac{1}{2}}}{Γ (\frac{3}{2})} \approx 0.7979 .

Obviously,

M \leq 1

. From Theorem 18, we can deduce that (15) has a solution on I.

Example 2. We set

X = I

, and

f (t, x (t)) = \frac{t^{\frac{1}{2}}}{(1 + ∣ x (t) ∣)}, t \in I

Now

f (t, s) = \frac{1}{5 e^{t} (1 + ∣ s ∣)},

where,

t \in I, s \in [0, \infty)

. In this case,

α

and

β

both equal

\frac{1}{2}

Since the function f is continuous, and for

s, v \in [0, \infty)

and

t \in I

, then

∣ f (t, s) - f (t, v) ∣ \leq t^{\frac{1}{2}} (∣ s - v ∣) .

From the above expression, it is evident that

N (t) = t^{\frac{1}{2}}

(H_{4})

.Thus,

\begin{matrix} sup_{t \in I} \frac{1}{Γ (α)} \int_{a}^{t} ψ^{'} (s) {(ψ (t) - ψ (τ))}^{α - 1} N (τ) d τ \\ = sup_{t \in I} \frac{1}{Γ (\frac{1}{2})} \int_{1}^{\frac{3}{2}} ψ^{'} (τ) {(ψ (t) - ψ (τ))}^{α - 1} t^{\frac{1}{2}} d τ \\ = \frac{{(\frac{3}{4})}^{\frac{1}{2}}}{Γ (\frac{3}{2})} \approx 0.9772 . \end{matrix}

Hence, the condition

(H_{5})

is satisfied with

Q = 0.98 < 1

6. Conclusions

Compared to other fractional derivatives, the

ψ

-Hilfer fractional derivative provides a robust tool for addressing complex system problems and holds a significant position in the development of fractional calculus. This study focuses on the

ψ

-Hilfer fractional derivative in the Caputo sense, which addresses the issue of ambiguous physical meaning of initial conditions in initial value problems. We investigate the existence of solutions in Banach spaces using the Mönch’s fixed-point theorem and establish the Ulam-Hyers-Rassias stability and semi-Ulam-Hyers-Rassias stability of solutions through growth conditions and stability theorems. This further enhances the properties of

ψ

-Hilfer derivatives, expands their application scope, and enriches the theory of fractional calculus.

Author Contributions

Conceptualization, W.H. and J.M.; Methodology, W.H.; Validation, W.H.; Writing-original draft, W.H.; Writing-review & editing, W.H. and J.M.; Investigation, Y.J. and L.W.; Project administration, Y.J. and L.W.; Supervision, J.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by Fundamental Research Funds for the Central Universities (31920230052) and Innovation Team of Intelligent Computing and Dynamical System Analysis and Application of Northwest Minzu University.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflicts of interest.

References

Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and applications of fractional differential equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
Shitikova, M.V. Fractional operator viscoelastic models in dynamic problems of mechanics of solids: A review. Mech. Solids 2022, 57, 1–33. [Google Scholar] [CrossRef]
Wang, Y.J.; Zhao, G.H. A comparative study of fractional-order models for lithium-ion batteries using Runge Kutta optimizer and electrochemical impedance spectroscopy. Control Eng. Pract. 2023, 133, 105451. [Google Scholar] [CrossRef]
Jamil, A.A.; Tu, W.F.; Ali, S.W.; Terriche, Y.; Guerrero, J. M. Fractional-order PID controllers for temperature control: A review. Energies. 2022, 15, 3800. [Google Scholar] [CrossRef]
Yavuz, M.; Sene, N.; Yıldız, M. Analysis of the influences of parameters in the fractional second-grade fluid dynamics. Mathematics 2022, 10, 1125. [Google Scholar] [CrossRef]
Tarasov, V.E. Nonlocal statistical mechanics: General fractional Liouville equations and their solutions. Phys. A Stat. Mech. Appl. 2023, 609, 128366. [Google Scholar] [CrossRef]
Kumar, K.; Pandey, R.K.; Sharma, S.; Xu, Y.F. Numerical scheme with convergence for a generalized time-fractional Telegraph-type equation. Numer. Methods Partial Differ. Equ. 2019, 35, 1164–1183. [Google Scholar] [CrossRef]
Hilfer, R. Applications of fractional calculus in physics; World Scientific, Singapore, 2000.
Thaiprayoon, C.; Sudsutad, W.; Alzabut, J.; Etemad, S.; Rezapour, S. On the qualitative analysis of the fractional boundary value problem describing thermostat control model via ψ-Hilfer fractional operator. Adv. Differ. Equ. 2021, 2021, 1–28. [Google Scholar] [CrossRef]
Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 2017, 44, 406–481. [Google Scholar] [CrossRef]
Sousa, J.V.d.C.; De Oliveira, E.C.d. On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 2018, 60, 72–91. [Google Scholar] [CrossRef]
Salim, A.; Lazreg, J.E.; Ahmad, B.; Benchohra, M.; Nieto, J.J. A study on k-generalized ψ-Hilfer derivative operator. Vietnam J. Math. 2024, 52, 25–43. [Google Scholar] [CrossRef]
Kuratowski, K. Sur les espaces complets. Fund. Math. 1930, 1, 31–39. [Google Scholar] [CrossRef]
Banaś, J. On measures of noncompactness in Banach spaces. Comment. Math. Univ. Carolin.. 1980, 21, 131–143. [Google Scholar]
Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonl. Sci. Numer. Simult. 2017, 44, 460–481. [Google Scholar] [CrossRef]
Jajarmi, A.; Baleanu, D.; Sajjadi, S. S.; Nieto, J. J. Analysis and some applications of a regularized ψ–Hilfer fractional derivative. J. Comput. Appl. Math. 2022, 415, 114476. [Google Scholar] [CrossRef]
Mönch, H. Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces. Nonlinear Anal. 1980, 4, 985–999. [Google Scholar] [CrossRef]
Sousa, J.V.C.; Oliveira, E.C. Ulam–Hyers stability of a nonlinear fractional Volterra integro-differential equation. Appl. Math. Lett. 2018, 81, 50–56. [Google Scholar] [CrossRef]
Oliveira, E.C.; Sousa, J.V.C. Ulam–Hyers–Rassias stability for a class of fractional integro-differential equations. Results Math. 2018, 73, 111. [Google Scholar] [CrossRef]
Diaz, J.B.; Margolis, B. A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 1968, 74, 305–309. [Google Scholar] [CrossRef]
Cădariu, L.; Găvruţa, L.; Găvruţa, P. Weighted space method for the stability of some nonlinear equations. Appl. Anal. Discrete Math. 2012, 6, 126–139. [Google Scholar] [CrossRef]
Zhou, J.L.; Zhang, S.Q.; He, Y.B. Existence and stability of solution for a nonlinear fractional differential equation. J. Math. Anal. Appl. 2021, 498, 124921. [Google Scholar] [CrossRef]
Yosida, K. Functional analysis; Springer Science & Business Media, 2012.
Zhou, J.L.; Zhang, S.Q.; He, Y.B. Existence and stability of solution for nonlinear differential equations with ψ-Hilfer fractional derivative. Appl. Math. Lett. 2021, 121, 107457. [Google Scholar] [CrossRef]

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Existence and Stability for Fractional Differential Equations with a Regularized ψ–Hilfer Fractional Derivative

Abstract

1. Introduction

2. Preliminaries

3. Main Result

4. Stability Analysis

5. Examples

6. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Conflicts of Interest

References

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