Existence and Compactness Results for a System of Fractional Differential Equations

Guendouz, Cheikh; Lazreg, Jamal Eddine; Nieto, Juan J.; Ouahab, Abdelghani

doi:https://doi.org/10.1155/2020/5735140

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Abstract Introduction Preliminaries Applications Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Volume 2020 | Article ID 5735140 | https://doi.org/10.1155/2020/5735140

Existence and Compactness Results for a System of Fractional Differential Equations

Cheikh Guendouz,¹Jamal Eddine Lazreg,¹Juan J. Nieto,²and Abdelghani Ouahab^1,3

Guest Editor: Lishan Liu

Received26 Jan 2020

Accepted23 May 2020

Published11 Jul 2020

Abstract

The existence and uniqueness, boundedness, and continuous dependence of solutions for fractional differential equations with Caputo fractional derivative is proven by Perov’s fixed point theorem in vector Banach spaces. We study the existence and compactness of solution sets and the u.s.c. of operator solutions.

1. Introduction

In the past twenty years, the fractional differential equation has aroused great consideration not only in its application in mathematics but also in other applications in physics, engineering, finance, fluid mechanics, viscoelastic mechanics, electroanalytical chemistry, and biological and other sciences [1–7].

In recent decades, the Riemann-Liouville, Caputo, and Hadamard fractional calculus are paid more attention; see the monographs [5, 8–13].

Applied problems requiring definitions of fractional derivatives are those that are physically interpretable for initial conditions containing , etc. The same requirements are true for boundary conditions. Caputo’s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both the Riemann-Liouville type and the Caputo type, see Podlubny [12] and Diethelm [14].

The theory of fractional differential equations and inclusions has been extensively studied and developed by many authors; see [15–21] and the references therein.

Perov in 1964 [22] and Perov and Kibenko [23] extended the classical Banach contraction principle for contractive maps on space endowed with a vector-valued metric. Later, they attempted to generalize the Perov fixed point theorem in several directions which has a number of applications in various fields of nonlinear analysis, semilinear differential equations, and system of ordinary differential equations.

In [24], Dezideriu and Precup studied the following system of semilinear equations where are linear operators and are nonlinear operators.

Precup, in [25], explained the advantage of vector-valued norms and the role of matrices that are convergent to zero in the study of semilinear operator systems.

Many authors studied the existence of solutions for a system of differential equations and impulsive differential equations by using the vector version fixed point theorem; their results are given in [26–30].

Our goal of this paper is to treat the systems of fractional differential equations. More precisely, we will consider the following problem: where and are the Caputo fractional derivatives, are given functions, and .

In the case where , the above system was used to analyze initial value problems and boundary value problems for nonlinear competitive or cooperative differential systems from mathematical biology [31] and mathematical economics [32] which can be set in the operator from ((2)).

The plan of this paper is as follows: in Section 2, we introduce all the background material used in this paper such as some properties of generalized Banach spaces, fixed point theory, and fractional calculus theory. In Section 3, we state and prove our main results by using Perov’s fixed point type theorem in generalized Banach spaces. By the Leray-Schauder fixed point in vector Banach space, we prove the existence and compactness of solution sets of the above problems.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Definition 1 [22]. Let be a nonempty set. The mapping which satisfies all the usual axioms of the metric is called a generalized metric in Perov’s sense and is called a generalized metric space.

In a generalized metric space in Perov’s sense, the concepts of Cauchy sequence, convergent sequence, completeness, and open and closed subsets are similarly defined as those for usual metric space.

If and , then by , we mean for each , and by , we mean for each . Also and . If , then means for each . Denote by the open ball centered in with radius , and the closed ball centered in with radius .

Definition 2. A square matrix of real numbers is said to be convergent to zero if and only if as .

Lemma 3 [33]. Let . The following statements are equivalent: (i) is a matrix convergent to zero(ii)The eigenvalues of are in open disc, i.e., , for every with (iii)The matrix is nonsingular and (iv)The matrix is nonsingular and has nonnegative elements(v) and as , for any

Example 4. Some examples of matrix convergent to zero are where and , where and , and where and .

Definition 5 [34]. Let be a generalized metric space. An operator is said to be contractive if there exists a convergent to zero matrix such that

Notice now that the Banach fixed point theorem can be extended to generalized metric spaces in the sense of Perov.

Theorem 6 [22, 28]. Let be a complete generalized metric space and be a contractive operator with Lipschitz matrix . Then, has a unique fixed point , and for each , we have

We recall now the following Leary-Schauder type theorem.

Theorem 7 [28, 35]. Let be a generalized Banach space and let be a completely continuous operator. Then, either (i)the equation has at least one solution, or(ii)the set is unbounded

We will use the following notations. Let and be two metric spaces and .

Definition 8 [28, 36]. A multivalued map is called upper semicontinuous (u.s.c.) at a point provided that for every open subset with , there exists such that is called upper semicontinuous if it is u.s.c. at every point

The mapping is said to be completely continuous if it is u.s.c., and for every bounded subset , is relatively compact, i.e., there exists a relatively compact set such that

Also, is compact if is relatively compact, and it is called locally compact if for each , there exists an open set containing , such that is relatively compact.

Theorem 9 [36]. Let be a closed locally compact multifunction. Then, is u.s.c.

Now, we recall some notations and definitions of fractional calculus theory.

Definition 10 [5]. The Riemann-Liouville fractional integral of the function of order is defined by where is the Euler gamma function defined by .

Definition 11 [5]. For a function , the Caputo fractional-order derivative of order of is defined by where .

We recall Gronwall’s lemma for singular kernels, whose proof can be found in Lemma 7.1.1 of [37].

Lemma 12. Let be a real function; is a nonnegative, locally integrable function on ; is a nonnegative, nondecreasing continuous function defined on ; (constant); and suppose is nonnegative and locally integrable on . Assume such that then for every .

3. Existence, Uniqueness, and Bounded Solutions

In order to define a solution for problem (2), consider the following functional spaces. Let and be the space of all continuous functions from into .

is a Banach with norm

We need the following auxiliary result.

Lemma 13 [14]. Concerning the problem, where the function is continuous. The function is the unique solution of the problem (19) if and only if

Definition 14. A function is said to be a solution of (2) if and only if

In this section, we assume the following conditions.

(H1). There exists functions , , such that where where for ,

(H2). The functions are defined by satisfies

Now, we are in a position to prove our existence and uniqueness solution for the problem (2) using the Perov fixed point theorem and show that for each initial condition , the solution is bounded.

Theorem 15. Assume that (H1)-(H2) are satisfied. If the matrix converges to zero. Then, the problem (2) has a unique bounded solution.

Proof. Transform the problem (2) into a fixed point theorem of the operator defined by , where First, we show that the operator is well-defined. Let , then we have Then, Hence, the operator is well-defined.
Clearly, the fixed points of operator are solutions of problem (2). Now, we show that is a contraction. For all , we have Then, Similarly, we have Therefore, According to Theorem 6, we deduce that the operator has unique fixed point which is a solution of problem (2). Now, we will prove that the solution of problem (2) is bounded. For all , we have Therefore, Hence, where Then, From (H1) and (H2), we deduce that the solution () is bounded.

For the next result, we prove the continuous dependence of solutions on initial conditions.

Theorem 16. Assume that (H1) and (H2) hold. If , and the matrix defined in (27) converges to zero.
For every , we denoted by the solution of problem (2). Then, the map is continuous.

Proof. From Theorem 15, for each initial condition , there exists unique solution , , then we get Therefore, Hence,

4. Existence and Compactness of Solution Sets

For the existence and compactness result of problem (2), we consider the following Banach space: with norm

It is evident that is a Banach space. The following compactness criterion on unbounded domains is called Corduneanu compactness criterion in which the proof is easy and similar to the classical one in (see [38]).

Lemma 17. Let . Then, is relatively compact if the following conditions hold: (a) is uniformly bounded in (b)The functions belonging to are almost equicontinuous on , i.e., for all compact interval, for any , there exists such that for every with , we have for all (c)The functions from are equiconvergent, that is, given , there corresponds such that

In the sequel of this section, we will consider the following assumption.

(H3). For every , the functions are uniformly continuous on the sets uniformly with respect to , i.e., and satisfied the following condition: for all , , there exists such that for all and for all with , , we have

(H4). There exist , such that

Theorem 18. Assume that (H3) and (H4) hold. Then, the problem (2) has at least one bounded solution. Moreover, the solution set is compact and the multivalued map is u.s.c.

Proof. Let is defined in the proof of Theorem 15.

Step 1. is well defined. Let , then

Hence,

Step 2. is continuous.

Let in . Then, there exists such that for any and , we have

By (H3), for every , there exists such that for all and for all with , we have

Since converge to , then there exists such that for all ,

Hence,

Thus

Step 3. We will show that maps bounded sets into bounded sets in .

Let , where and if , then we obtain

Similarly, we have

Hence, where

Step 4. Now, we prove that maps bounded sets in into almost equicontinuous sets of .

Let . Then, for all ,

Thus,

Similarly, we have

Step 5. The set is equiconvergent, i.e., for every , there exists such that , for every and each .

It is clear that

Then, for any , there exists such that for all , we have

Step 6. The set is defined as follows:
is bounded. Let , then

Similarly, we get

Thus, where

Hence,

By the Gronwall-Bellman Lemma 12, we have

Then, for every , we have where is the Mittag-Leffler function defined by

Then,

According to Theorem 7, problem (2) has at least one solution.

Now, we show that the set is compact. It is clear that . From (75), we deduced that is bounded sets in . Since is compact, then is compact if and only if is closed. Let be a sequence converge to . Thus,

Similarly to Step 2, we can prove that

This implies that is compact.

The solution operator is u.s.c. (a) has a closed graph:

To see this, first note that the graph of is the set

Let be a sequence in , and let .

Since , then we have

Let

As in Step 2, we can prove that (b)Using the same method as Steps 3 to 4, we find that for each bounded set , we can show that is compact.

From (a) and (b), we concluded that, is u.s.c.

5. Applications

In this section, we show the applicability of our main result. We begin by illustrating by Theorem 15.

Example 19. Consider the problem: where It is clear that Then, Observe that every and , we have Hence, the condition (H1) holds.
We see that Hence, the condition (H2) holds. Assume that . For this, we have . If we add that or k₃ and . The . By Theorem 15, it follows that problem (83) has a unique solution.

In this last example, we illustrate the applicability of Theorem 18.

Lemma 20. Let us consider the function defined by where satisfy the following assumptions:
(S1) .
(S2) The function is continuous and bounded. Then, the function satisfied the condition (H3).

Proof. Since is bounded, then there exists such that Let , then for any , we obtain Hence, It is clear that form (S2) is uniformly continuous on the . Thus, for any there exists such that for all with then, This implies that for every such that for all with , , we get

Theorem 21. Let be functions that satisfy the conditions (S1) and (S2).
Assume
(S3) There exists such that Then, the following problem where

Proof. From Lemma 20, we deduce that and satisfies (H3). By (S1) and (S3), there exist such that Therefore, all the conditions of Theorem 18 hold. Then, problem (98) has at least one solution in .

Remark 22. For another application, we can replace the condition (S2) with (S4). The functions and are uniformly continuous and bounded.

6. Conclusion

In this paper, we investigated a system of fractional differential equations under various assumptions on the right-hand-side nonlinearity and we obtain a number of results regarding the existence and uniqueness of solutions in an appropriate space of continuous functions. In this paper, we have focused on the dependence continuity of a solution, compactness of solution sets, and upper semicontinuity of operator solutions. We hope this paper can provide some contribution to the questions of existence and topological structure for the system of fractional differential equations on unbounded domains.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

The research of Juan J. Nieto has been partially supported by the Agencia Estatal de Investigacin (AEI) of Spain, cofinanced by the European Fund for Regional Development (FEDER) corresponding to the 2014-2020 multiyear financial framework, project MTM2016-75140-P, and by Xunta de Galicia under grant ED431C 2019/02.

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Copyright © 2020 Cheikh Guendouz 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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