Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020View this Special Issue
Existence and Compactness Results for a System of Fractional Differential Equations
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.
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].
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  and Diethelm .
Perov in 1964  and Perov and Kibenko  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 , Dezideriu and Precup studied the following system of semilinear equations where are linear operators and are nonlinear operators.
Precup, in , 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  and mathematical economics  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.
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Definition 1 . 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 . 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 . 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.
We recall now the following Leary-Schauder type theorem.
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 . 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 . The Riemann-Liouville fractional integral of the function of order is defined by where is the Euler gamma function defined by .
Definition 11 . 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 .
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.
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.
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
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.
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 ).
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
Proof. Let is defined in the proof of Theorem 15.
Step 1. is well defined. Let , then
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 ,
Step 3. We will show that maps bounded sets into bounded sets in .
Let , where and if , then we obtain
Similarly, we have
Step 4. Now, we prove that maps bounded sets in into almost equicontinuous sets of .
Let . Then, for all ,
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