1. Introduction
Fractional differential equations (Fra-Diff-Equs) [
1,
2,
3,
4,
5] are powerful tools for describing complex systems. In the early years, research was mainly focused on mathematical theoretical explorations, with less attention to time delays. They gained attention in the late 20th century and began to combine fractional calculus with time delays. Nowadays, fractional time delays have been applied in the fields of physics [
6,
7], solid mechanics [
8], electromechanics [
9], and finance [
10], and chemistry [
11], which help accurately describe the system dynamics, optimize the performance, improve the accuracy of the models, and assist in the decision-making process in these fields.
In order to ensure the solvability of the equations and to provide a prerequisite for numerical calculations, in recent years, scholars have devoted themselves to studying the properties of solutions of Fra-Diff-Equs [
12,
13,
14]. Benchohra M et al. [
15] explored the existence of solutions to Caputo Fra-Diff-Equs; Bao et al. [
16] explored the existence results of solutions to neutral stochastic Fra-Diff-Equs in Lp
.
However, due to the constraints of the time-delay conditions, the study of the stability of Fra-Diff-Equs [
17,
18,
19,
20,
21,
22] is extremely difficult. It requires scholars attempt to prove new inequalities according to the requirements of the conditions. Over the past years, numerous articles have been dedicated to studying the Hyers–Ulam stability (Hs-Um-St) of Fra-Diff-Equs.
In [
23], Dong et al. investigated a type of Riemann–Liouville Fra-Diff-Equs with time delay
where
,
,
represents the phase space. For Fra-Diff-Equs with non-zero initial values, the weighted-delay method was employed to investigate the properties of the solutions. Dong et al. proved new Gronwall inequalities, which laid the foundation for the proof of Hs-Um-St.
In [
24], Chen et al. investigated the existence, uniqueness, and Hs-Um-St of solutions for Fra-Diff-Equs with infinite delay
where
,
,
and
are a Caputo fractional derivative (Fr-De) with
,
is given that satisfies certain assumptions, the function
.
Inspired by the works in the discussions above, in this paper, we conduct an in-depth study on a class of nonlinear Fra-Diff-Equs with infinite delay and featuring the
-Caputo Fr-De
where
and
stand for
-Caputo Fr-Des with
,
is a constant, and the function
satisfies the specified hypotheses,
, and
is a phase space.
is a function defined on
as
Our approach mainly focuses on transforming delayed differential equations into integral equations. Subsequently, these integral equations are properly extended within the phase space. We use the Leray–Schauder alternative and Banach fixed-point theorems to examine the existence and uniqueness of solutions. Furthermore, we initiate an exploration of the Hs-Um-St of the solution. It is worth noting that this exploration is faced with substantial challenges. These challenges mainly originate from the specific constraints caused by the delay conditions, which increase the complexity of analyzing the stability of the solution. Specifically, the difficulty lies in the fact that the original Gronwall inequality is inapplicable. To overcome this difficulty, we need to extend new inequalities.
In
Section 2, a comprehensive review of some basic definitions and lemmas will be carried out.
Section 3 is devoted to investigating the existence and uniqueness of the solutions. In
Section 4, we discuss Hs-Um-St in detail. At the end of the article, two examples are provided to illustrate the conclusions.
2. Preliminaries and Lemmas
We first list several basic definitions and theorems. Subsequently, we proceed to prove a series of inequalities that are used to verify the Hs-Um-St. It should be emphasized that the function is increasing, positive, uniformly integrable, and . This makes the subsequent proofs more rigorous.
Definition 1 ([
25])
. Let and with . The Riemann–Liouville Fr-De with respect to Ψ with the order Θ of y is defined bywhere is defined by Definition 2 ([
26])
. Let and . The Caputo-type Fr-De with respect to Ψ with the order Θ of y is defined bywhere exists and Theorem 1 (i) If the function is continuous, then (ii) If and exists, then Definition 3 ([
28])
. Let be a Banach space. A linear topological space of functions from into , with the seminorm , is said to be an admissible phase space if has the following properties:(A1) There exists a constant and functions , such that is a continuous function and is locally bounded, such that for any constant with , if the function , and function is continuous on , then for every , the following conditions (i)-(iii) hold:
(i) ;
(ii) ;
(iii) .
(A2) For the function in (A1), is a -valued continuous function for .
(B1) The space is complete.
Definition 4 ([
29])
. The problem (1) is said to be Hs-Um-St if there exists a real number > 0, such that for each and for every solution of the inequalitiesthere exists a solution of the problem (1) with Lemma 1 ([
30])
. (Leray–Schauder alternative) Let be a Banach space, O is a subset of , and O is a closed and convex subset. Moreover, assume that is an open subset of O with . Suppose is a continuous and compact map. Then, either τ has a fixed point in , or there is a and with . Next, for conveniently proving Theorem 5, we will extend certain properties of the Caputo derivative to the -Caputo derivative and introduce some integral inequalities that can be regarded as an extended version of the Gronwall inequality.
Lemma 2. Suppose , a function , which is nonnegative and nondecreasing, and Ψ is an increasing function. Then, the functionis nondecreasing on . Proof. Assuming that the function
f satisfies the conditions mentioned above, we can obtain that
for all
. Let
. Then
. It follows that
Let
,
Then,
because Ψ is increasing, and
due to
is nondecreasing. So, we have
for all
and, hence,
is nondecreasing on
. □
To study the problem of infinite delay, we extend Lemma 5 in [
24] and prove the following inequality.
Lemma 3. For any function belonging to the space and any , the following integral inequalitycan be obtained. Proof. Fix
. Given the function
, which is nonnegative, and
, which is nondecreasing, this results in
also being nondecreasing according to Lemma 2. Then, for any
, we have
therefore,
Thus, the proof is completed. □
Next, we extend the Lemma 2.6 in [
1] to
-Caputo calculus.
Lemma 4. Let , . Consider as a non-negative continuous function that is defined on the domain , where . The function has the property of being non-decreasing in v and non-increasing in s. Let Ψ be an increasing and uniformly integrable function. Assume that the function is non-negative and integrable on the interval withThen, we have Proof. Let
, it follows that
, which implies
We now prove that
For
, the demonstration is straightforward. Assume that Equation (
11) is valid when (
). Then, when (
), we obtain
Through the method of mathematical induction, inequality (
11) holds for all
. Replacing
with
in (
11), we deduce that
,
. Analogous to the proof process of Lemma 3.4 in [
23], we are able to confirm that
as
n approaches infinity, uniformly in
. Finally, letting
n approach infinity in (
10), we obtain
The lemma is proved. □
Lemma 5. The solution of Equation (1) iswhere is a function. Proof. The proof is evident from Definition 1 and Theorem 1. □
3. Existence Results
In this part, we mainly use Lemma 1 and the Banach contraction principle to prove the existence results of solutions.
We list the following four assumptions.
(D1) The continuous function has a bounded subset , such that , on which it is uniformly continuous.
(D2) There exists a constant , such that for each and every .
(D3) There exist functions , such that for each and every .
(D4) There exists a nonnegative function with and a continuously non-decreasing function , such that for all and every .
Before the proof of the main result, let us initiate our analysis by transforming the problem of solutions into a fixed-point problem. According to Lemma 5, it is known that
y is a solution to (
1), precisely when
y satisfies
For any
, define the function
as
For
we define the function
as
We can decompose
into the sum of two functions. For
we have
. Moreover, for all
we can easily obtain that
, and note that
satisfies
Now, we define a set
It is not difficult to see that the set
is closed and, thus, completed. Based on this, we define an operator
by
Theorem 2. Suppose that (D1) and (D2) are satisfied. Ifthen Equation (1) has a unique solution on . Proof. Let
be given as (
13). For any
and each
, we have
As
where
, we get
Assumption (
14) shows that
T is a contraction. Applying the Banach contraction principle, we conclude that
T has a unique fixed point. Then Equation (
1) has a unique solution on
. □
Theorem 3. Suppose that (D1) and (D3) hold. Further, suppose thatholds. Then, Equation (1) has at least one solution on . Proof. To apply Lemma 1, we will demonstrate, through the subsequent steps, that the operator T is completely continuous.
Let
be a sequence, such that
in
. Then, for each
, we have
Set
. From (D1), it can be observed that the function
f is uniformly continuous with respect to
. This means that for each
, there exists
, such that for each
with
, we have
. As
,
, such that for each
, we have
. Hence,
for all
. Based on the previously introduced definition
, it follows that
, so we get
Hence,
as
. Due to the arbitrariness of
, we obtain that
, which implies that
T is continuous.
We now show that
T maps bounded subsets into bounded subsets in
. In fact, it is enough to prove that for any
, there exists a positive constant
, such that for each
, we have
. Let
. As
f is a continuous function, for each
, we have
As
where
,
,
, we have
This can be proven.
Next, we prove that
T maps bounded subsets into equicontinuous subsets of
. For all
and
,
, we have
As , , it is easy to see that as . As a consequence, is equicontinuous. According to the Arzela-Ascolli theorem, we can conclude that is a completely continuous mapping.
Now, we have to verify that there exists at least one fixed point
of
T. We assume that there exists an open set
, such that for every
and for each
, the inequality
holds. Define the set
. So, the operator
satisfies the complete continuity. Assume
holds for some
and
. Then, we obtain
Hence, the following inequalities can be obtained based on the conditions (
15).
holds, which contradicts with the fact that
. Thus, we obtain
for any
and
.
Utilizing Lemma 1, we infer that there exists at least one fixed point of T. This completes the proof. □
Theorem 4. Suppose that (D1) and (D4) hold. Additionally, suppose thatholds, where , . Then, the Equation (1) has at least one solution on . Proof. With the aid of the Lebesgue dominated convergence theorem, it is straightforward to confirm that T is continuous.
We now show that
T maps bounded subsets into bounded subsets in
. Let
. Then, for any
and
, we have
As
where
,
and
is a constant. By H
ölder inequality, we have
where
,
,
,
and
.
Next, we prove that
T maps bounded subsets into equicontinuous subsets of
. Consider the set
and take an arbitrary
. Then, for
with
, we have
As
,
,
, we can obtain
where
,
,
. As
,
, it is easy to see that
as
. By the Arzela–Ascolli theorem, we can conclude that
is a completely continuous mapping.
Now, we have to verify that there exists at least one fixed point
of
T. We assume that there exists an open set
, such that for every
and for each
, the inequality
holds. Define the set
. So, the operator
satisfies the complete continuity. Assume
holds for some
and
. Then, we obtain
Hence, the following inequalities can be obtained based on the conditions (
16).
holds, which contradicts
. Thus,
for any
and
.
As a consequence of Lemma 1, we deduce that there exists at least one fixed point of T. The proof is complete. □
4. Stability Analysis
In this section, we conduct an analysis of the Hs-Um-St of the Fra-Diff-Equs (
1) with infinite delay.
Theorem 5. Assume that all the conditions stated in Theorem 2 are met, and inequality (6) admits at least one solution. Under such circumstances, the problem (1) exhibits Hs-Um-St. Proof. For each
, and each function ℷ that satisfies the following inequalities
let
. Then we have
According to Theorem 2, there is a unique solution
of problem (
1), and the function ℸ can be expressed as
So, we have
As
together with Definition 3, we obtain
which indicates that
According to Lemma 3,
Let
,
and
We can obtain
It is not difficult to show that
Then, the inequality
holds for
which implies that the Hs-Um-St of problem (
1) is proved. □
5. Examples
For any real constant
, we set
and set
By [
31],
satisfies the conditions
, and, hence,
is a phase space.
Example 1. Consider the following nonlinear Ψ-Caputo-type Fra-Diff-Equs with infinite delay of the formwhere . In this case, it is readily observable that (D1) and (D3) are satisfied with the function
. Due to
Thus, all the conditions of Theorem 3 hold. Therefore, the problem (
17) has at least one solution on
Example 2. Consider the following problemwhere , . Set Then, for any
x,
we have
Hence,
, condition (D2) holds. Thus, problem (
18) has a unique solution to
by applying Theorem 2.
Next, we discuss the Hs-Um-St of problem (
18). For any
, and each function
y that satisfies the following inequalities
let
represent the left side of the inequality above and let
be the unique solution of problem (
18); then, we have
let
,
and
By Lemma 4,
let
, it follows that
, which implies that the problem (
18) is Hs-Um-St.
6. Conclusions
In this paper, we mainly discuss a class of nonlinear Fra-Diff-Equs with finite time delay. We first use the Leray–Schauder alternation theorem and the Banach fixed-point theorem to study the existence and uniqueness of solutions for this class of
-Caputo Fra-Diff-Equs. Then, we study the Hs-Um-St of the Fra-Diff-Equs (
1). With the aim of overcoming the difficulty in proving Hs-Um-St, some inequalities are obtained for delayed Fra-Diff-Equs. In addition, there are two examples listed to illustrate the conclusions. The findings of our study enhance and extend the results presented in [
23,
24].
Author Contributions
Conceptualization, Y.X.; Methodology, Q.D.; Formal analysis, A.B.E. and Q.D.; Investigation, Y.X.; Resources, A.B.E.; Data curation, Q.D.; Writing—original draft, Y.X.; writing—review and editing, Y.X., A.B.E. and Q.D. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported by the National Natural Science Foundations of China [Grant No. 12271469 and 12371140] and Jiangsu Students’ Innovation and Entrepreneurship Training Program [202411117164Y].
Data Availability Statement
The data presented in this study are openly available in [WOS] at [10.3390/math10071013].
Acknowledgments
We sincerely express my gratitude to every editor and reviewer for their dedication. Wish you all a happy life.
Conflicts of Interest
The authors declare no conflict of interest.
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