Investigating a Class of Generalized Caputo-Type Fractional Integro-Differential Equations

Department of Basic Engineering Sciences, College of Engineering, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, Saudi Arabia Department of Mathematics and General Sciences, Prince Sultan University, Riyadh, Saudi Arabia Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan Department of Mathematics, Hashemite University, Zarqa, Jordan Department of Mathematics, Hodeidah University, Al-Hudaydah, Yemen Department of Computer Science, Cihan University-Erbil, Kurdistan Region, Iraq


Introduction
It is notable that fractional calculus was and still is a new tool that uses fractional differential and integral equations to construct more modern mathematical models that can precisely describe complex frameworks. There are many definitions of fractional integrals (FIs) and fractional derivatives (FDs) accessible in the literature, for instance, the Riemann-Liouville and Caputo definitions that assumed a significant part in the advancement of the theory of fractional analysis. Referring to all books and papers in this field will be extremely many. In this regard, here, we refer to the most important of main references, e.g., Samko et al. [1] gave a broad comprehensive mathematical handling of fractional derivatives and integrals. Podlubny [2] and Kilbas et al. [3] have been introduced many useful results related to fractional differential equations (FDEs). Several applications have been implemented recently by a wide range of works on this subject, see [4][5][6][7][8].
However, the currently common operator is the generalized FD regarding another function, see [1,3]. Agrawal [7] studied further various properties for generalized fractional derivatives and integrals. More recently, Almeida [9] inspired an idea of that generalization by projecting this generalization onto the definition of the Caputo fractional derivative with respect to another function, so-called ψ-Caputo, and introduced many interesting properties, which are more general than the classical Caputo FD. Jarad and Abdeljawad [10] provided interesting properties for generalized FDs and Laplace transform. Specifically, ψ-Caputo type FDEs with initial, boundary, and nonlocal conditions have been investigated by many researchers using fixed-point theories, see Almeida et al. [11,12], Abdo et al. [13], and Wahash et al. [14]. A recent survey on ψ-Caputo type FDEs can be found in [15][16][17][18]. For more results in this direction, we refer to interesting works provided by Zhang et al. [19], Zhao et al. [20], Baitiche et al. [21], Benchohra et al. [22], Ravichandran et al. [23], Trujillo et al. [24], and Furati et al. [25].
Li in [17,18] investigated some interesting results of the integral equations and the integro-differential equations involving Hadamard-type. In this work, our goal is to intend to address a general extension of these studies. Precisely, we consider the following ψ-Caputo type fractional integrodifferential equation (FIDE) where (i) 0 < ϱ 1 < ϱ < 1,ϱ 2 > 0, and a 1 , denotes the generalized Caputo FD of order σ ∈ fϱ, ϱ 1 g (iii) The notation I ψ,ϱ 2 a means the generalized Riemann-Liouville FI of order ϱ 2 and D ψ,σ a exist and are both continuous in ½a, b Observe that the considered system (1) covers the previous standard cases of nonlinear FIDEs by defining the kernel, i.e., if ψðxÞ = x, ψðxÞ = log ðxÞ, and ψðxÞ = x ρ , then, the problem (1) reduces to the Caputo type FIDE, Caputo-Hadamard type FIDE, and Caputo-Katugampola type FIDE, respectively.
The aim of this work is to develop the nonlinear FIDEs. In particular, we investigate the uniqueness and Ulam-Hyers stability of solution for the problem (1) by Banach's fixed point theorem and Babenko's technique [26]. Note that the presentation and structuring of the arguments for our problem are new, and our results generalize and cover some of the known results in the literature. In addition, the obtained results here are valid when the left hand side of the considered problem (1) involves many FDs and FIs. For more details, see Remark 22. The remainder of this paper is organized as follows: in Section 2, we present some important tools related the fractional calculus and the functional spaces, in which we aim to determine our analysis strategies. Section 3 gives the main results and their illustrative examples. Finally, our brief conclusion is included in Section 4.

Preliminaries
In this section, we present some properties, lemmas, definitions, and important estimations needed in the proof of our result.

Journal of Function Spaces
Proof. Let ω ∈ AC 0 ½a, b: Then by Definition 1 and Property 4, we get Lemma 7. Let ϱ > 0: Then, I ψ,ϱ a is bounded from AC 0 ½a, b into itself, and Proof. Let ω ∈ AC 0 ½a, b: Then By virtue of Definition 1, we get Taking advantage of the Dirichlet's formula, we have Now, we will provide and prove the next lemma: Proof. Using Definition 1, we have Performing the substitution s = ψðxÞ − ψðζÞ, we get From Definition 2, we obtain

Main Results
a solution in the space Proof. Applying the operator I ψ,ϱ a to both sides of Eq. (24), we obtain According to Lemma 5, we find that Observe that φðaÞ = 0 and 0 < ϱ 1 < ϱ < 1: It follows that In view of Babenko approach, we have By using the multinomial theorem and Property 4, we 3 Journal of Function Spaces As x ⟶ a, we get φðaÞ = 0: Now, we need to prove the series is absolutely continuous on ½a, b and converges in the space AC 0 ½a, b: Indeed, by Lemma 7, we have It follows that where which is the value at v 1 = ja 1 jðψðbÞ − ψðaÞÞ ϱ−ϱ 1 , v 2 = ja 2 j ðψðbÞ − ψðaÞÞ ϱ+ϱ 2 of the multivariate Mittag-Leffler function E ðϱ−ϱ 1 ,ϱ+ϱ 2 ,ϱ+1Þ ðv 1 , v 2 Þ given in [3]. So, we conclude that the series to the right of Eq. (25) is convergent. Obviously, φðxÞ ∈ AC½a, b due to h ∈ AC½a, b: To affirm that the obtained series could be a solution, we must see that it fulfills Eq. (24), i.e., by the cancellation. Notice that each series is absolutely convergent and also the term arrangements are possibly cancellated. In fact, The remainder terms cancel each other similarly. Plainly, the uniqueness follows promptly from the fact that the FIDE only has solution zero due to the Babenko approach.
Remark 13. Notice that the solution of Eq. (24) in AC 0 ½a, b is stable, if for all ε > 0, there exists δ > 0 such that kφk 0 < ε 4 Journal of Function Spaces with khk 0 < δ: Taking advantage of the following inequality we conclude that φ is stable.
has the solution in AC 0 ½a, b, that is So, according to Theorem 11, we have From Property 3, we get Þ κ+0:2ℓ 1 +1:3ℓ 2 +0:9 : ð42Þ Example 2. The following ψ-Caputo type FIDE has the solution in AC 0 ½a, b described as By virtue of Lemma 9, we obtain The uniqueness result of Eq. (1) will be proved through the following theorem.

Theorem 14.
Let G : ½a, b × ℝ ⟶ ℝ be a continuous function, and assume that there exists a constant C such that If then, the problem (1) has a unique solution in AC 0 ½a, b: Proof. Consider the operator I on AC 0 ½a, b defined by For φ ∈ AC 0 ½a, b, we have Ð x a Gðζ, φ ′ ðζÞÞdζ ∈ AC 0 ½a, b, since φ ′ ðζÞ ∈ Lða, bÞ and Gðζ, φ ′ ðζÞÞ ∈ Lða, bÞ: Hence, Using the inequality (38), we obtain Besides, IðφÞ is absolutely continuous on ½a, b via Theorem 11. So, I : AC 0 ½a, b ⟶ AC 0 ½a, b. Now, we just have 5 Journal of Function Spaces to show that I is a contraction mapping. Let φ, φ * ∈ AC 0 ½a , b: Then Since we obtain Inequality (48) leads us to that I is contraction mapping.

Ulam-Hyers Stability (UHS).
The first results about this type of stability emerged in 1940 by Ulam [28,29]. From that point forward, the UHS is studied via several researchers. With the vast development of fractional calculus, the studying of stability for FDEs also attracted the numerous authors, see [30][31][32].
In this regard, we investigate some recent results on the UHS and generalized UHS of (1). For ε > 0,x ∈ ½a, b, and φ 1 ∈ AC 0 ½a, b, the following inequality is satisfied.

Remark 22.
All previous results can be generalized in which the left-hand side of (1) may contain several FDs and FIs. For example, Theorem 11 can be generalized as follows:

Conclusions
ψ-Caputo FD, a general fractional operator, is of great use because of its wide freedom to cover many classical fractional operators. In this work, we have studied the uniqueness of solution for the nonlinear ψ-Caputo type FIDE (1) by using the Banach space AC 0 ½a, b, Banach's fixed point theorem, and Babenko's method. Moreover, the UH stability results to the proposed problem have been discussed. Also, some pertinent examples have been provided to justify the main results. The obtained results in this study extended and developed the current results introduced by [17,18]. We have already concluded that our results are valid when the left-hand side of the considered problem (1) involves many FDs and IDs as shown in Remark 22. Furthermore, problem (1) covers previous standard cases of nonlinear FDEs and FIDEs by selecting the suitable standard kernel in the studied problem. More specifically, our results generalize some known results in literature like those that include Hadamard and Katugampola FDs.
For future research, we will consider a class of nonlinear FIDEs with the fuzzy initial conditions in a fractional case. It would also be interesting to study the same results for our current problem under the ψ-Hilfer operator [33] or Atangana-Baleanu operator [8].

Data Availability
The data of this study were used to support the findings of this study are available from the corresponding author upon request.