Investigation of a Mild Solution to Coupled Systems of Impulsive Hybrid Fractional Differential Equations

Abstract

The study of coupled systems of hybrid fractional differential equations requires the attention of scientists for the exploration of their different important aspects. Our aim in this paper is to study the existence and uniqueness of the solution for impulsive hybrid fractional differential equations. The novelty of this work is the study of a coupled system of impulsive hybrid fractional differential equations with initial and boundary hybrid conditions. We used the classical fixed-point theorems such as the Banach fixed-point theorem and Leray–Schauder alternative fixed-point theorem for existence results. We also give an example of the main results.

1. Introduction

Fractional differential equations appear naturally in a number of fields such as physics, engineering, biophysics, blood flow phenomena, aerodynamics, electron-analytical chemistry, biology, and control theory. An excellent account of fractional differential equations is given in this study. Undergoing abrupt changes at certain moments of time like earthquake, harvesting, and shock, these perturbations can be well approximated as instantaneous change of state or impulses. Furthermore, these processes are modelled by impulsive differential equations.

On the other hand, impulsive differential equations appear as a natural description of many evolutionary phenomena in the real world. The majority of processes in applied sciences are represented by differential equations. However, the situation is different in certain physical phenomena undergoing abrupt changes during their evolution as mechanical systems with impact, biological systems (heartbeat, blood flow, and so on), the dynamics of populations, natural disasters, etc. These changes are often of very short duration and are therefore produced instantly in the form of pulses. The modeling of such phenomena requires the use of forms that explicitly and simultaneously involve the continuous evolution of the phenomenon as well as instantaneous changes.

Such models are said to be “impulsive;” they are evolutionary of continuous processes governed by differential equations combined with difference equations representing the effect impulsive has undergone.

For some recent developments on the topic, see [1–5] and the references therein.

Hybrid fractional differential equations have also been studied by several researchers. This class of equations involves the fractional derivative of an unknown hybrid function with the nonlinearity depending on it. Some recent results on hybrid differential equations can be found in a series of papers [6–11].

By applying the Banach fixed-point theorem and Kransnoselskii fixed-point theorem, the existence results for the solution are obtained. Shah et al. [12] studied the coupled system of fractional impulsive boundary problems:where , , , , , , and are continuous functions, , , , and are fixed continuous functionals, and , and , .

Motivated by some recent studies to the boundary value problem of a class of impulsive hybrid fractional differential, we consider the problem of coupled hybrid fractional differential equations: and stand for Caputo fractional derivative of order and , respectively; , , , and are continuous functions defined by and , where for , ; and and and represent the right and left limits of at , .

We assume that and .

This paper is arranged as follows. In Section 2, we recall some concepts and some fractional calculation laws and establish preparation results. In Section 3, we present the main results. Section 4 is devoted to an example of the main results.

2. Preliminaries

In this section, we recall some basic definitions and properties of the fractional calculus theory and preparation results.

Throughout, this paper denotes , , … , , , and we introduce the spaces: for such that and , define the space .

Then, clearly is the Banach space under the norm .

Similarly, for such that and , define the space .

Then, clearly is the Banach space under the norm .

Consequently, the product is a Banach space under the norms and .

Definition 1 (see [13]). The fractional integral of the function of order is defined bywhere is the gamma function.

Definition 2 (see [13]). For a function given on the interval , the Riemann–Liouville fractional-order derivative of is defined bywhere and denotes the integer part of .

Definition 3 (see [13]). For a function given on the interval , the Caputo fractional-order derivative of is defined bywhere and denotes the integer part of .

3. Main Results

In this section, we will prove the existence of a mild solution for (2).

To obtain the existence of a mild solution, we will need the following assumptions: : the function is increasing in for every  : the function is increasing in for every  : the functions are continuous and bounded; that is, there exist positive numbers such that for all  : for all for all there exist positive numbers , such that : there exist constants such that for all  : for any , there exist constants , such that : for any , there exist constants and , such that : there exist constants , such that and  : there exist constants , such that , , , and  : there exist constants and , such thatfor all .

For brevity, let us set

Lemma 1. Let and be continuous. A function is a solution to the fractional integral equation:if and only if is a solution to the following fractional Cauchy problems:

Lemma 2. Let us assume that hypotheses and hold. Let and be continuous. A function is a solution to the fractional integral equation:whereif and only if is a solution of the following impulsive problem:

Proof 1. Let us assume that satisfies (16). If , thenApplying on both sides of (17), we can obtainthenIf , thenAccording to Lemma 1 and the continuity of , we haveSincethere existssoIf , then we haveForwe havesoIf , using the same method, we getConversely, assume that satisfies (14). If , then we haveThen, divided by and applying on both sides of (32), (17) is satisfied.
Again, substituting in (32), we have . Since is increasing in for , the map is injective in . Then, we get (18).
If , then we haveThen, divided by and applying on both sides of (33), (19) is satisfied. Again by , substituting in (32) and taking the limit of (33), (33) minus (32) gives (22).
If , similarly we getThis completes the proof.

Lemma 3. Let are continuous, then is a solution of (2) if and only if is the solution of the integral equations:whereWe define an operator bywhereNow, we are in a position to present our first result that deals with the existence and uniqueness of solutions for the problem (2). This result is based on Banach’s contraction mapping principle.

Theorem 1. Suppose that the condition holds and that are continuous functions. In addition, there exist positive constants , , , such thatIf , and are given by (11), then the impulsive coupled system (2) has a unique mild solution.

Proof 2. Let us set and and define a closed ball: , whereThen, we show that . For , we obtainHence,Working in a similar manner, one can find thatFrom (42) and (43), it follows that .
Next, for , and for any , we havewhich implies thatSimilarly, we can show thatFrom (45) and (46), we deduce that

In view of this condition , it follows that is a contraction. So, Banach’s fixed point theorem applies, and hence the operator has a unique fixed point. This, in turn, implies that the problem (2) has a unique solution on . This completes the proof.

In our second result, we discuss the existence of solutions for the problem (2) by means of Leray–Schauder alternative.

For brevity, let us set

Lemma 4 (Leray–Schauder alternative, see [14]). Let be a completely continuous operator (i.e., a map that is restricted to any bounded set in is compact). Let . Then either the set is unbounded or has at least one fixed point.

Theorem 2. Let us assume that conditions and hold. Furthermore, it is assumed that and , where and are given by (48). Then, the boundary value problem (2) has at least one solution.

Proof 3. We will show that the operator satisfies all the assumptions of Lemma 4.
In the first step, we will prove that the operator is completely continuous.
Clearly, it follows by the continuity of functions , and that the operator is continuous.
Let bounded. Then, we can find positive constants and such that .
Thus, for any , we can getwhich yieldsIn a similar manner, one can show thatFrom the inequalities (51) and (52), we deduce that the operator is uniformly bounded.
Now, we show that the operator is equicontinuous.
We take with and obtainwhich tend to 0 independently of . This implies that the operator is equicontinuous. Thus, by the abovementioned findings, the operator is completely continuous.
In the next step, it will be established that the set is bounded.
Let . Then, we have . Thus, for any , we can writethenwhich implies thatIn consequence, we havewhich in view of (49), can be expressed asThis shows that the set is bounded. Hence, all the conditions of Lemma 4 are satisfied and consequently the operator has at least one fixed point, which corresponds to a solution of the system (2). This completes the proof.

4. Example

Consider the following coupled system of hybrid fractional differential equations:where , , , and .

Note that

Thus, all the assumptions in Theorem 2 are satisfied, and our results can be applied to the problem (59).

Data Availability

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Conflicts of Interest

The authors declare that they have no conflicts of interest.