#### 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