Existence and Attractivity Results for Coupled Systems of Nonlinear Volterra–Stieltjes Multidelay Fractional Partial Integral Equations

Abstract

We are concerned with some existence and attractivity results of a coupled fractional Riemann–Liouville–Volterra–Stieltjes multidelay partial integral system. We prove the existence of solutions using Schauder’s fixed point theorem; then we show that the solutions are uniformly globally attractive.

1. Introduction

Fractional integral and fractional differential equations are among the most fast growing field in mathematics. They are used to describe many phenomena, especially the ones with long memory. Examples include but are not limited to viscoelasticity, viscoplasticity, biochemistry, control theory, mathematical psychology, mechanics, modeling in complex media (porous, etc.), and electromagnetism [1–4]. In recent years, there has been a significant development in ordinary and partial fractional integral equations; see, for instance, the monographs of Abbas et al. [5–7], Agarwal et al. [8], Kilbas et al. [9], Miller and Ross [10], Podlubny [11], Samko et al. [12], and the papers [13–18] and the references therein.

In this paper we study the existence and attractivity of solutions to the following coupled system of nonlinear fractional Riemann–Liouville–Volterra–Stieltjes quadratic multidelay partial integral equations:where , , , are given continuous functions, , , are bounded, , , are continuous and bounded functions with for each and , for each , and is the (Euler’s) Gamma function defined by

2. Preliminaries

In this section, we recall some notations, definitions, and preliminary facts which will be used in this paper. , will denote the space of all Lebesgue-integrable functions equipped with the norm will denote the usual Banach space of all bounded and continuous functions from into equipped with the standard norm It is clear that the product space turns out to be a Banach space if equipped with the norm

Definition 1 (see [19]). Let and The left-sided mixed Riemann–Liouville integral of order of is defined by provided the integral exists.

Example 2. Let and , then

If is a real-valued function defined on the interval , then we will use the symbol to denote the variation of on We say that is of bounded variation on the interval whenever is finite. If , then the symbol indicates the variation of the function on the interval , where is arbitrarily fixed in the interval Analogously we define For more details on the properties of functions of bounded variation we refer the reader to [20].

If and are two real-valued functions defined on the interval , then under some appropriate conditions (see [20]) we can define the Stieltjes integral (in the Riemann–Stieltjes sense) of the function with respect to In this case we say that is Stieltjes integrable on with respect to Several conditions are known to ensure Stieltjes integrability [20]. One of the most frequently used requires that is continuous and is of bounded variation on

Now we recall a few properties of the Stieltjes integral included in the lemmas below.

Lemma 3 (see [20, 21]). If is Stieltjes integrable on the interval with respect to a function of bounded variation, then

Lemma 4 (see [20, 21]). Let and be Stieltjes integrable functions on the interval with respect to a nondecreasing function such that for Then

From now on, we will also consider Stieltjes integrals of the form and Riemann–Liouville–Stieltjes integrals of fractional order of the form where and the symbol indicates the integration with respect to

Let , and let , and consider the solutions of equationIn light of the definition of the attractivity of solutions of integral equations (for instance, [15]), we will introduce the following concept of attractivity of solutions for (14).

Definition 5. A solutions of (14) is said to be locally attractive if there exists a ball in the space such that, for arbitrary solutions and of (14) belonging to , we have that, for each ,When the limit (15) is uniform with respect to , solutions of (14) are said to be uniformly locally attractive (or equivalently that solutions of (14) are locally asymptotically stable).

Definition 6 (see [15]). The solution of (14) is said to be globally attractive if (15) holds for each solution of (14). If condition (15) is satisfied uniformly with respect to the set , solutions of (14) are said to be globally asymptotically stable (or uniformly globally attractive).

Lemma 7 (see [22], p. 62). Let Then is relatively compact in if the following conditions hold:(a) is uniformly bounded in (b)The functions belonging to are almost equicontinuous on , i.e., equicontinuous on every compact subset of (c)The functions from are equiconvergent; that is, given , there corresponds such that for any and

3. Existence and Attractivity Results

Definition 8. By a solution to problem (1)-(2), we mean every coupled functions such that satisfies (1) on and (2) on

We will use the following assumptions in the sequel:There exist positive functions such that For all such that , the function is nondecreasing on .The function is nondecreasing on .The functions and are continuous on for each fixed or , respectively.There exist continuous functions ; , such that  for , ; Moreover, assume that

Remark 9. Set , for and From the above assumptions, we infer that are finite.

Theorem 10. Assume that hypotheses hold. Then problem (1)-(2) has at least one solution in the space Moreover, solutions to problem (1)-(2) are uniformly globally attractive.

Proof. Define the operators ; byand consider the operator such that, for any ,From the hypotheses above, we deduce that is continuous on Now let us prove that for any For arbitrarily fixed , we have and for all and each , we have Thus,HenceTherefore The problem of finding the solutions of the coupled system (1)-(2) is reduced to finding the solutions of the operator equation From (25), we infer that transforms the ball into itself. Now we will show that satisfies the Schauder’s fixed point theorem [23]. The proof will be presented in several steps and cases.
Step 1 ( is continuous). Let be a sequence such that and in Then, for each , we haveCase 1. Assume that , then, since as and are continuous, (26) implies Case 2. Let , then from and (26) we obtainSince , then (28) gives Let us show that is continuous in the same way as continuity of .
Step 2 ( is uniformly bounded). This fact is obvious because and is a bounded set.
Step 3 ( is equicontinuous on every compact subset of , ). Let , , and let Without loss of generality, let us assume that Then we obtain Thus Using continuity of the functions , and since and , the right-hand side of the above inequality tends to zero. The equicontinuity of for the cases , and , is immediate.
We can also prove that Hence Step 4 ( is equiconvergent). Let and , then we get Now, since as , we conclude that, for each , we obtain Also, for each , we get Then, for each , we get Hence, In view of Steps 1 to 4, along with the Lemma 7, we deduce that is continuous and compact. From an application of Schauder’s theorem [23], we conclude that has a fixed point which is a solution of the coupled system (1)-(2).
Step 5 (the uniform global attractivity of solutions). Now let us study the stability of solutions of the coupled system (1)-(2). Let and be two solutions of (1)-(2). Then, for each , we obtain ThusHenceBy using (41) and , we obtain Therefore, all solutions of the coupled system (1)-(2) are uniformly globally attractive.
Let (product space) be the Banach space equipped with the following norm:

From the above theorem, we deduce the following consequence.

Corollary 11. Consider the system of nonlinear fractional Riemann–Liouville–Volterra–Stieltjes quadratic multidelay partial integral equations of the formwhere , ; , , , , , , are given continuous functions, ; are bounded, , ; are continuous and bounded functions with for each and ; for each .
Suppose that and the following assumptions are verified: There exist positive functions , , such that There exist continuous functions , , , such that  for , , , Moreover, assume that Then problem (44)-(45) has at least one solution in the space In addition, the solutions are uniformly globally attractive.

4. An Example

To illustrate our results, we consider the following coupled system of nonlinear fractional order Riemann–Liouville–Volterra–Stieltjes quadratic multidelay partial integral equationwhere , , , ; , , First, we can see that and Next, the assumption is satisfied with and consequently Also, it is clear that the function satisfies assumptions

Finally, the functions , , satisfy the assumption Indeed, are continuous and satisfy the inequality , Also, we have and For , we have also andConsequently, Theorem 10 implies that the coupled system (49)-(50) has a solution defined on ; moreover solutions of this system are uniformly globally attractive.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.