Existence and Compactness Results for a System of Fractional Differential Equations

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 yð0Þ, y′ð0Þ, 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


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][2][3][4][5][6][7].
Applied problems requiring definitions of fractional derivatives are those that are physically interpretable for initial conditions containing yð0Þ, y ′ ð0Þ, 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][16][17][18][19][20][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 A 1 , A 2 : DðAÞ ⊂ X ⟶ X are linear operators and F 1 , F 2 : J × X × X ⟶ X 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][27][28][29][30].
Our goal of this paper is to treat the systems of fractional differential equations. More precisely, we will consider the following problem: where c D α and c D β are the Caputo fractional derivatives, α, β ∈ ð0, 1, J = ½0, ∞Þ, f , g : J × ℝ × ℝ ⟶ ℝ are given functions, and x 0 , y 0 ∈ ℝ.
In the case where α = β = 1, 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.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
Definition 1 [22]. Let X be a nonempty set. The mapping d : X × X ⟶ ℝ m + which satisfies all the usual axioms of the metric is called a generalized metric in Perov's sense and ðX, dÞ 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 v, r ∈ ℝ m , v ≔ ðv 1 , v 2 , ⋯, v m Þ and r ≔ ðr 1 , r 2 , ⋯, r m Þ, then by v ≤ r, we mean v i ≤ r i for each i ∈ f1, ⋯, mg, and by v < r, we mean v i < r i for each i ∈ f1, ⋯, mg. Also |v | ≔ ð|v 1 |, ⋯, |v m | Þ and max ðu, vÞ ≔ ðmax ðu 1 the open ball centered in x 0 with radius r, and the closed ball centered in x 0 with radius r.

Definition 2.
A square matrix M of real numbers is said to be convergent to zero if and only if M n ⟶ 0 as n ⟶ ∞.
Definition 5 [34]. Let ðX, dÞ be a generalized metric space. An operator N : X ⟶ X is said to be contractive if there exists a convergent to zero matrix M 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 ðX, dÞ be a complete generalized metric space and N : X ⟶ X be a contractive operator with Lipschitz matrix M. Then, N has a unique fixed point x * , and for each x 0 ∈ X, we have We recall now the following Leary-Schauder type theorem.

2
Journal of Function Spaces Theorem 7 [28,35]. Let X be a generalized Banach space and let N : X ⟶ X be a completely continuous operator. Then, either (i) the equation NðxÞ = x has at least one solution, or (ii) the set M = fx ∈ X | μNðxÞ = x, μ ∈ ð0, 1Þg is unbounded We will use the following notations. Let ðX, dÞ and ðY, ρÞ be two metric spaces and G : X ⟶ P ðYÞ.
Definition 8 [28,36]. A multivalued map φ : The mapping G is said to be completely continuous if it is u.s.c., and for every bounded subset C ⊆ X, GðCÞ is relatively compact, i.e., there exists a relatively compact set K = K ⊂ X such that Also, G is compact if GðXÞ is relatively compact, and it is called locally compact if for each x ∈ X, there exists an open set W containing x, such that GðWÞ is relatively compact.
Theorem 9 [36]. Let G : X ⟶ P cp ðYÞ be a closed locally compact multifunction. Then, G 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 h ∈ L 1 ð½0, T, ℝ + Þ of order α ∈ ℝ + is defined by where Γ is the Euler gamma function defined by ΓðaÞ = Definition 11 [5]. For a function h ∈ AC n ðJ, ℝÞ, the Caputo fractional-order derivative of order α of h is defined by where n = ½α + 1.
We recall Gronwall's lemma for singular kernels, whose proof can be found in Lemma 7.1.1 of [37].

Existence, Uniqueness, and Bounded Solutions
In order to define a solution for problem (2), consider the following functional spaces. Let J = ½0, ∞Þ and CðJ, ℝÞ be the space of all continuous functions from J into ℝ.
We need the following auxiliary result.
Lemma 13 [14]. Concerning the problem, where the function f : ℝ + × ℝ ⟶ ℝ is continuous. The function x : ℝ + × ℝ ⟶ ℝ is the unique solution of the problem (19) if and only if 3 Journal of Function Spaces Definition 14. A function x, y ∈ C b is said to be a solution of (2) if and only if In this section, we assume the following conditions. where where for γ = α, β, 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 ðx 0 , y 0 Þ, the solution is bounded.
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 N : First, we show that the operator N is well-defined. Let ðx, yÞ ∈ C b × C b and t ∈ ½0, ∞Þ, then we have Then, Hence, the operator N is well-defined. Clearly, the fixed points of operator N are solutions of problem (2). Now, we show that N is a contraction. For all ðx, yÞ, ð x, yÞ ∈ C b × C b , we have 4 Journal of Function Spaces Then, Similarly, we have Therefore, According to Theorem 6, we deduce that the operator N has unique fixed point which is a solution of problem (2). Now, we will prove that the solution ðx, yÞ of problem (2) is bounded. For all t ∈ ½0, ∞Þ, we have Therefore, Hence, where Then, From (H1) and (H2), we deduce that the solution (x, y) 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 f ðt, 0, 0Þ = gðt, 0, 0Þ = 0, t ∈ J and the matrix M defined in (27) converges to zero.

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 BC b ðJ, ℝÞ 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 C b ðℝ + , ℝÞ (see [38]).
(c) The functions from H are equiconvergent, that is, given ϵ > 0, there corresponds TðϵÞ > 0 such that In the sequel of this section, we will consider the following assumption.
Proof. Let N = ðN 1 , N 2 Þ is defined in the proof of Theorem 15.

Conclusion
In this paper, we investigated a system of fractional differential equations under various assumptions on the righthand-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.