A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

Khaldi, Somia; Mecheraoui, Rachid; Mukheimer, Aiman

doi:https://doi.org/10.1155/2020/4237680

Journal of Function Spaces

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Research Article | Open Access

Volume 2020 | Article ID 4237680 | https://doi.org/10.1155/2020/4237680

A Nonlinear Fractional Problem with Mixed Volterra-Fredholm Integro-Differential Equation: Existence, Uniqueness, H-U-R Stability, and Regularity of Solutions

Somia Khaldi,¹Rachid Mecheraoui,¹and Aiman Mukheimer²

Academic Editor: Adrian Petrusel

Received26 Jun 2020

Revised19 Oct 2020

Accepted25 Oct 2020

Published23 Nov 2020

Abstract

This paper considers nonlinear fractional mixed Volterra-Fredholm integro-differential equation with a nonlocal initial condition. We propose a fixed-point approach to investigate the existence, uniqueness, and Hyers-Ulam-Rassias stability of solutions. Results of this paper are based on nonstandard assumptions and hypothesis and provide a supplementary result concerning the regularity of solutions. We show and illustrate the wide validity field of our findings by an example of problem with nonlocal neutral pantograph equation, involving functional derivative and -Caputo fractional derivative.

1. Introduction

Over the last few decades, there has been significant development in the area of ordinary and partial fractional differential equations with boundary integral conditions. This is reflected by various state-of-the-art papers, e.g., [1–15]. Problems with initial integral conditions have many important applications. An example is when a direct measurement quantity is impossible but their mean values are known. The fractional derivatives and integrals appear to be a very efficient tool to model various physical phenomena: kinetic theories, statistical mechanics, dynamics in complex media, control theory, signal processing, bioengineering and biomedical applications, and many others. We refer the reader to [16–18].

In an attempt to formulate different problems, distinct definitions of the fractional derivative and fractional integral are available in the literature. We mention some of them.

Let , an integrable function defined on , and an increasing function with for all . (i)The Riemann–Liouville fractional derivative:(ii)The Caputo fractional derivative:(iii)The -Riemann–Liouville fractional integral:(iv)The -Riemann–Liouville fractional derivative:

where

Very recently, the author introduced in [10] the so-called -Caputo-type fractional integral and derivative with respect to another function as follows:

Definition 1 (-Caputo fractional derivative). Let , , and be an increasing function with for all . The -Caputo fractional derivative operator of order of a function is defined by where . In particular, when , we have For some special cases of , we obtain the Caputo fractional derivative, the Caputo–Hadamard fractional derivative, and the Caputo–Erdélyi–Kober fractional derivative.
The relationship between the -Caputo and the -Riemann–Liouville integrals can be written as follows:

Lemma 2 [10]. Let and . Then, we have In particular, given , we have In this paper, we state and prove a weak form of Banach contraction principal. Together with the Schauder’s fixed-point theorem, we discuss the existence, uniqueness, Hyers-Ulam-Rassias stability, and regularity of solutions for the following nonlinear fractional mixed Volterra-Fredholm integro-differential equation with nonlocal initial condition: where is the -Caputo derivative of order , , , , , , is a continuous function on with values in Banach space , , , and are continuous valued functions. For the sake of simplicity, we denote

The main contributions in this paper reside in the following: (i)This work extends, improves, and generalizes several recent state-of-the-art results, including [19–24](ii)Assumptions (A2)-(A6) are not standard; they have been considerably weakened. In fact, they are not supposed to hold on the overall space, but only on a subspace. Example in last section shows the importance of this new form of assumptions(iii)Results in Section 2 are obtained on the basis of a weak form of the Banach contraction principal, which is stated and proved in this paper(iv)We discuss the Hyers-Ulam-Rassias stability of problems (9) and (10) with respect to a function . Generally, in the existing literature, function is supposed to be bounded, continuous, and verifies . In our result, function is supposed to be neither continuous nor bounded nor positive. It is just an integrable locally bounded function(v)In addition to the existence, uniqueness, and stability of solutions, the main theorem in Section 3 provides a supplementary result concerning the regularity of solutions(vi)We give an Hyers-Ulam-Rassias stability result on unbounded intervals(vii)Last, in this paper, we prove the existence, uniqueness, Ulam-Hyers-Rassias stability, and regularity of solutions for an example of problems with nonlocal neutral pantograph equation to show the large validity field of our results

The remainder of this paper is structured as follows. In Section 2, we briefly recall some basic definitions and some preliminary concepts about fractional calculus and auxiliary results used in the following sections. In Section 3, we discuss the existence and uniqueness of mild solutions for problems (9) and (10), using a weak form of Banach contraction principal and Schauder’s fixed-point theorem. Section 4 is devoted to present an Ulam-Hyers-Rassias stability result for problems (9) and (10), and we give a regularity result of solutions for this problem. Finally, we supply in Section 5 a well-suited example to illustrate the application and the validity of our results. In this example, we discuss the existence, uniqueness, Ulam-Hyers-Rassias stability, and regularity of solutions for a problem with nonlocal neutral pantograph equation involving a functional derivative and -Caputo fractional derivative.

2. Preliminaries

Throughout this paper, we will use the following notations:

For the sake of simplicity, we used the same symbol, , for all norms. The closure of a set will be denoted by . are given functions (problems (9) and (10)), and denote the constants:

Define operators and by

For any , we set: where denotes composed with itself times.

Now, we present some important theorems and lemmas in obtaining the main results.

Theorem 3 (Weak form of the Banach contraction principal). Let and be two complete metric spaces, a nonempty subset of , and such that . Let (, resp.) denotes the induced metric on (, resp.). If is continuous and is a contraction mapping, i.e., there exists such that for all . Then, has a unique fixed point in the closure of . Further, the sequence converges to for all .

Proof. For all and , using (16) and the triangle inequality, we get on one hand that and on the other that Now, for all and , by virtue of inequalities (17) and (18), we obtain This means that is a Cauchy sequence in the complete metric space . Then, it converges to a point of . Reusing (19), we obtain Letting tend to infinity, taking into account continuity of mapping , we deduce that is a fixed point of . If is a fixed point of , there exits a sequence in that converges to . By (16), we get for all and , Letting tends to infinity, by the continuity of in , we deduce that . But are two fixed points of so . This achieves the proof.

Theorem 4 (Schauder’s fixed-point theorem). Let be a closed convex set in a Banach space and assume that is a continuous mapping such that is a relatively compact subset of . Then, has a fixed point.

Theorem 5 (Arzela-Ascoli theorem). Assume that is a compact set in , . Then, a set is relatively compact in if the functions in are uniformly bounded and equicontinuous on .

Lemma 6. Let be a nondecreasing function and an increasing function with for all , then (i)Function is Riemann integrable and bounded(ii)There exist positive constants and such thatfor all .

Proof. is a nondecreasing function on ; then, it is Riemann integrable, and it attains its maximum and minimum at points and , respectively. Let , and clearly, This achieves the proof.

Lemma 7. Let a continuous function, an integrable function, and two positive constants. Suppose that (i)There exists (ii)Inequality holds for all (iii)Inequality holds for all Then, for all .

Proof. Let , and by items (5) and (9), we have . Then, a.a. . So, there exits a sequence converges to and verifies , for all . But, by item (10), we deduce that . Keeping in mind that is continuous, it yields , and this concludes the proof.

Lemma 8. Let . The following statements hold true: (i)Operators are continuous(ii)Find a solution of (9) and (10) is equivalent to find a fixed point of , that is, an element such that(i)If is a solution of problems (9) and (10), then for all function , we have

Proof. To prove that is continuous, let be a sequence such that in . For each , we have But are the continuous functions on the compact ; hence, there exists verifying the following: So, by (26), Therefore, since is continuous, operator is continuous. Similarly, we prove that is continuous.
For point , let be a solution of -fractional integro-differential problems (9) and (10). Applying the fractional integral operator on both sides of (9), we get On the other hand, keeping in mind initial condition (10), we have by Dirichlet’s formula, and it follows that Substituting this latter in (29) and putting , we obtain Conversely, let in verifying . We have
Now, reusing the Dirichlet’s formula, (24) gives and this achieves the proof of point 2. For point 3, let , and we have Therefore, Similarly, we get QED.
Let , and the following assumptions are also used:
There exists a positive constant such that There exists a positive constant such that, for all , , and : There exists a positive constant such that, for all and : There exists a positive constant such that, for all and : With , we have For all , if , then .

Lemma 9. Let , and if and are satisfied, then the following statements hold true. for any , and Proof. Immediate.

3. Existence and Uniqueness Results

Our first result is based on the weak form of the Banach contraction principle.

Theorem 10. Let . If assumptions are satisfied, then the fractional integro-differential problems (9)-(10) have a unique solution continuous on . Furthermore, the sequence converges to in .

Proof. Let where assumptions hold. Clearly, . We shall prove that is a contraction on . For this, let . By (13), we get By means of assumption , we obtain and by virtue of Lemma 9, it follows that On the other hand, a simple calculation shows that Substituting estimates (45) and (46) in relation (44), we deduce that or, keeping in mind condition , operator is a contraction on and consequently continuous on . Therefore, by the weak form of the Banach contraction principal, has a unique fixed point in the closure of , which is a solution of (24) and hence, by item 2 in Lemma 8, a solution of problems (9) and (10). Note that if is a solution of (9) and (10), by (9), we get . Assumption () implies that ; then, , so . By (47), we have and this leads to the uniqueness of solutions. As a recap, (i)Problems (9) and (10) have a unique solution (ii)The sequence converges to This achieves the proof.

In the next theorem, we shall use Schauder’s fixed-point theorem to establish the existence of solutions for problems (9) and (10), with less conditions.

Theorem 11. Under assumption (A1), problems (9) and (10) have at least one solution on .

Proof. Let denotes the closed ball in of radius : , with and defines the operator on the Banach space by Clearly, whenever , i.e., , and item 2 in Lemma 8 assures that is continuous. Now, we shall prove that is an equicontinuous set of . Let and , we have Taking into account assumption (A1), we obtain But, Therefore, and consequently, as , which means that is an equicontinuous set of .
But , so is uniformly bounded, and the Arzela-Ascoli theorem implies that is relatively compact. Therefore, is continuous on the closed convex set , and is relatively compact. According to the Schauder’s fixed-point theorem, mapping has at least a fixed point. Item 2 in Lemma 8 achieves the proof.

4. Ulam-Hyers-Rassias Stability

In this section, we discuss the Ulam-Hyers-Rassias stability of problems (9) and (10). First, we introduce a basic definition and some notations and hypotheses for this section.

Definition 12 (Ulam-Hyers-Rassias stability). If for each function satisfying where is a nonnegative function, there exists a solution of the fractional differential problems (9) and (10) and a constant independent of and such that for all ; then, we say that problems (9) and (10) are the Hyers-Ulam-Rassias stability.

Let (H) denotes the following hypothesis:

We denote by the set of all functions verifying is locally bounded, , and there exist two positive constants such that, for all

Let , , and a function.

Let be a continuously differential function. Assume that assumptions hold together with one of the following: (i)For all we have(ii)For all and we have

Finally, we denote by the constant and we assume that

Now, we state and prove the main result of this section.

Theorem 13. If the function satisfies for all . Then, there exists a unique function solution of problems (9) and (10), with

Remark 14. Here, we give some important remarks: (i)If function is member of we getfor all (ii)Using Lemma 6, we can substitute space by the space of nondecreasing functions(iii)The case is obvious, so we have omitted this case, supposing that

Proof. For all , we denote by the set Easily, we can prove that defined by is a generalized metric on verifying the following assumptions: (i)If there exists a sequence in such that , then for all satisfying (ii)For all , we havefor any
We claim that is complete. In fact, let be a Cauchy sequence in , and having in mind definition (67), we have Since is complete, (69) implies that converges for each . Let us show that function defined by is continuous or belongs to . Passing to the limit with respect to in the latter inequality, it follows that But, must be bounded because it is locally bounded on the compact . So, Hence, converges uniformly to , and consequently, is continuous. Further, if we consider (67) and (69), we may conclude that This means, the Cauchy sequence converges to in ; thus, is complete. On the other hand, Theorem 10 assures the existence of a unique function solution of problems (9) and (10), verifying . We define an operator by , i.e., for all , and we discuss its contractivity. Note that for all . Let be a continuously differential function satisfies (62) and (63). By virtue of Lemma 7 and relation (62), we obtain and by (5), we deduce that Thus, keeping assumption , we deduce that , and consequently, by item 3 in Lemma 8, where . By definition, . So, taking into account assumption , for all , we obtain By virtue of assumption and relations (57) and (68), we immediately get and by relation (76), assumption , and hypotheses (H), we obtain Substituting estimates (78) and (79) in (77), it yields Or, with and using the continuity of , for all . Obviously, , and by induction, we get Keeping in mind assumption , from (81) we get By the two last inequalities, these yield for all . Applying the fractional integration operator to both sides of (62), taking into account the fact that , we obtain for all . This implies that , and consequently, as and tend to infinity. Space is complete; then, this Cauchy sequence converges to a point of . By Lemma 8, operator is continuous; so, using inequality (84), we deduce that is a fixed point of . But is a fixed point for , and inequality (81) assures that is none other than . Letting tends to infinity in (84), with , we get and consequently, for all . This achieves the proof.

Corollary 15. Assume that is an unbounded interval. If the function satisfies for all . Then, there exists a unique function solution of problems (9) and (10), with for all

Proof. Let be a continuously differential function satisfies (88) and (89) and a positive integer. For all , by Theorem 13., there exists a unique continuous function satisfying for all . By virtue of the uniqueness of the solution , we deduce that for all and all . We define as and the function by Easily, by (91)-(94), we can prove that This achieves the proof.

5. Example

We consider the following nonlocal problem with neutral pantograph equation involving functional derivative and -Caputo fractional derivative: where and such that .

We claim that problem (96) has a unique solution verifying the following: for all

We shall prove that if a continuously differential function satisfies for all . Then, there exists a unique continuous function solution of problem (96) with for all

Let verifying (98) and (99), and we put . It is seen that problem (96) is equivalent to the following: where for all , , and . Further, we have the following:

() is uniformly bounded, in fact

For all , if , then .

satisfies

In fact, for all , and , we have

On the other hand, we have

But, using , for all , gets

Therefore, remembering that , we obtain

Substituting (106) and (109) in (105), we get, immediately, the desired inequality.

is continuous and satisfies the Lipschitz condition: for all

is continuous and satisfies the Lipschitz condition: for all and . verifying is bounded, , and for all , we have , , and , with , and .

With , , , , , , , , , and , we have

By assumptions and Theorems 10 and 13., we deduce that problem (96) has, exactly, one solution and if a continuously differential function satisfies for all . Then,

With function , we get and

Therefore, by (109), for all .

6. Conclusions

In this paper, using a fixed-point approach and by means of the -Caputo fractional derivative, we have studied the stability of solutions of a nonlinear fractional mixed Volterra-Fredholm integro-differential equation with an integral initial condition.

In the first part of this paper, a weak form of the Banach contraction principal and some useful lemmas are stated and proved. In the second part, we established conditions that assure the existence and uniqueness of solutions for the considered problem, basing on the weak form of the Banach contraction principal and the Schauder’s fixed-point theorem. The study of the Ulam-Hyers-Rassias stability is with respect to a function. When we impose more restrictions on, it will make the result less interesting in the reality world and vice versa. In the third part of this work, we have discussed of Ulam-Hyers-Rassias with respect to a function with very less restrictions. Two tough restrictions were omitted, and the continuity and that must be positive. Results in this paper provide a supplementary result concerning the regularity of solutions. Finally, the whole analysis has been demonstrated by a suitable example.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Acknowledgments

The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

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Copyright © 2020 Somia Khaldi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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