In this study, we establish some conditions for existence and uniqueness of the solutions to semilinear fractional impulsive integro-differential evolution equations with non-local conditions by using Schauder’s fixed point theorem and the contraction mapping principle.
MSC: 26A33, 34A37.
Keywords:boundary value problem; Caputo type fractional derivative; existence and uniqueness; fixed point theorem; impulsive integro-differential equation; nonlocal condition
The topic of fractional differential equations has received a great deal of attention from many scientists and researchers during the past decades; see, for instance, [1-7]. This is mostly due to the fact that fractional calculus provides an efficient and excellent instrument to describe many practical dynamical phenomena which arise in engineering and science such as physics, chemistry, biology, economy, viscoelasticity, electrochemistry, electromagnetic, control, porous media; see [8-13]. Moreover, many researchers study the existence of solutions for fractional differential equations; see [14-16] and the references therein.
In particular, several authors have considered a nonlocal Cauchy problem for abstract evolution differential equations having fractional order. Indeed, the nonlocal Cauchy problem for abstract evolution differential equations was studied by Byszewski [17,18] initially. Afterwards, many authors [19-21] discussed the problem for different kinds of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces. Balachandran et al.[22,23] established the existence of solutions of quasilinear integrodifferential equations with nonlocal conditions. N’Guérékata  and Balachandran and Park  researched the existence of solutions of fractional abstract differential equations with a nonlocal initial condition. Ahmad  obtained some existence results for boundary value problems of fractional semilinear evolution equations. Recently, Balachandran and Trujillo  have investigated the nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces.
On the other hand, the theory of impulsive differential equations for integer order has emerged in mathematical modeling of phenomena and practical situations in both physical and social sciences in recent years. One can see a significant development in impulsive theory. We refer the readers to [28-31] for the general theory and applications of impulsive differential equations. Besides, some researchers (see [32-35] and the references therein) have addressed the theory of boundary value problems for impulsive fractional differential equations.
However, only a few studies were concerned with the Cauchy problem for impulsive evolution differential equations of fractional order; see [36-38]. Further, in , Balachandran et al. studied the existence of solutions for fractional impulsive integrodifferential equations of the following type:
where and , by using the contraction mapping principle.
Motivated by the aforementioned works, in this paper, we deal with the existence and uniqueness of solutions for a boundary value problem (BVP), for the following impulsive fractional semilinear integro-differential equation with nonlocal conditions:
where , is the Caputo fractional derivative, is a bounded linear operator on a Banach space X, , , , ( will be defined in the next section),
and has a similar meaning for , , and . Here . For brevity, let us take .
Meanwhile, nonlinear functions f of this type with the integral term k occur in mathematical problems that are concerned with the heat flow in materials having memory and viscoelastic problems; see . Also, as indicated in [40,41], nonlocal conditions can be more useful than standard conditions to describe physical phenomena. For example, in , the author described the diffusion phenomenon of a small amount of gas in a transparent tube by using the formula
where , are given constants and .
Note that in this work, to the best of our knowledge, it is the first time that a general boundary value problem for impulsive semilinear evolution integrodifferential equations of fractional order with nonlocal conditions has been considered.
The rest of this paper is organized as follows. In Section 2, we present some notations and preliminary results about fractional calculus and differential equations to be used in the following sections. In Section 3, we discuss some existence and uniqueness results for solutions of BVP (1.1). Namely, the first result is based on Schauder’s fixed point theorem and the second one is based on Banach’s fixed point theorem. Finally, we shall give an illustrative example for our results.
In order to model the real world application, the fractional differential equations are considered by using the fractional derivatives. There are many different starting points for the discussion of classical fractional calculus; see, for example, . One can begin with a generalization of repeated integration. If is absolutely integrable on , it can be found [42,43]
where and . On writing , an immediate generalization in the form of the operation defined for is
where is the gamma function and is called the convolution product of and . Now Eq. (2.1) is known as a fractional integral of order α for the function .
Let , , …, , , and , then we define the set of functions as follows:
which is a Banach space with the norm
Now, denotes the Banach space of bounded linear operators from X into X with the norm .
The fractional (arbitrary) order integral of the function of order is defined by
where is the Euler gamma function.
For a function h given on the interval J, the Caputo-type fractional derivative of order is defined by
where the function has absolutely continuous derivatives up to order .
Let , then the differential equation
has the following solution:
Let , then
for some , , .
Now, by using the Kronecker convolution product, see , the fractional integral becomes
Thus, if can be integrated, then expanded in block pulse functions, the fractional integral is solved via the block pulse functions operational matrix as follows:
Now, we need the following lemma for our study.
Lemma 3Let and be continuous. A function is a solution of the fractional integral equation
if and only if is a solution of the fractional BVP
Proof Let u be the solution of (2.4). If , then Lemma 2 implies that
for some .
Applying the boundary condition for , we find that
If , then Lemma 2 implies that
for some . Thus, we have
In the view of
By repeating the process, for , we have
Now, applying the boundary condition
we find that
Substituting the value of in (2.5) and (2.6), we obtain Eq. 2.3.
Conversely, if we assume that u satisfies the impulsive fractional integral equation (2.3), then by direct computation, we can easily see that the solution given by (2.3) satisfies (2.4). Thus, the proof of Lemma 3 is complete. □
3 Main results
Definition 3 A function with its q-derivative existing on is said to be a solution of (1.1) if u satisfies the equation
on and satisfies the conditions
Now, we define the operator by
Clearly, the fixed points of the operator T are the solutions of problem (1.1). To begin with, we need the following assumptions to prove the existence and uniqueness of a solution of the integral equation (2.3) which satisfies BVP (1.1):
(A1) is a continuous bounded linear operator and there exists a constant such that for all ;
(A2) The function is continuous and there exists a constant such that ;
(A3) are continuous and there exist constants and such that for each and ;
(A4) There exist constants and are continuous functions such that , ;
(A5) There exists a constant such that
, and ;
(A6) is continuous and there exists a constant such that
for all ;
(A7) There exist constants , such that , for each and ;
(A8) There exist constants such that , .
The following are the main results of this paper. Our first result relies on Schauder’s fixed point theorem which gives an existence result for solutions of BVP (1.1).
Theorem 1Assume that the assumptions (A1)-(A4) hold. Then BVP (1.1) has at least one solution onJ.
Proof In order to show the existence of a solution of BVP (1.1), we need to transform BVP (1.1) to a fixed point problem by using the operator T in (3.1). Now, we shall use Schauder’s fixed point theorem to prove T has a fixed point which is then a solution of BVP (1.1). First, let us define for any . Then it is clear that the set is a closed, bounded and convex. The proof will be given in several steps.
Step 1: T is continuous.
Let be a sequence such that in . Then
Since A is a continuous operator and f, g, I, are continuous functions, we have as .
Step 2: T maps bounded sets into bounded sets.
Now, it is enough to show that there exists a positive constant l such that for each . Then we have, for each ,
Then it follows that .
Step 3: T maps bounded sets into equicontinuous sets.
Let be a bounded set of as in Step 2, and let . Then, letting with , , we have
Hence, is equicontinuous on all the subintervals , . Then we can deduce that is completely continuous as a result of the Arzela-Ascoli theorem together with Steps 1 to 3.
As a consequence of Schauder’s fixed point theorem, we conclude that T has a fixed point. That is, BVP (1.1) has at least one solution. The proof is complete. □
Our second result is about the uniqueness of the solution of BVP (1.1). And it depends on Banach’s fixed point theorem.
Theorem 2Assume that (A1)-(A8) hold with
Proof First, we show that . Indeed, in order to do this, it is adequate to replace l with r in Step 2 in Theorem 1. Thus, T maps into itself. Now, define the mapping . Then, for each , we have
Observing the inequality
which implies that
Therefore, by (3.2), the operator T is a contraction. As a consequence of Banach’s fixed point theorem, we deduce that T has a fixed point which is a unique solution of BVP (1.1). □
Example 1 Consider the following boundary value problem for impulsive integrodifferential evolution equation of fractional order:
where , , and , are given positive constants with and .
Here, , , , . Obviously, , , , , , , and by (2.5), it can be found that
Therefore, due to the fact that all the assumptions of Theorem 2 hold, BVP (3.3) has a unique solution. Besides, one can easily check the result of Theorem (1) for BVP (3.3).
In the literature, the authors consider impulsive fractional semilinear evolution integro-differential equations of order in different aspects as mentioned above. Besides, either impulsive fractional semilinear equations of order or impulsive fractional integro-differential equations of order are studied by different authors (see, for instance, [44,45]). But, to the best of our knowledge, no study considering both cases has been carried out. Thus, in this article, we consider a general boundary value problem for impulsive fractional semilinear evolution integro-differential equations of order with nonlocal conditions. Therefore, the present results are new and complementary to previously known literature.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final draft.
The authors express their sincere thanks to the referees for the careful and noteworthy reading of the manuscript and very helpful suggestions that improved the manuscript substantially. The second author gratefully acknowledges that this research was partially supported by the University Putra Malaysia under the ERGS Grant Scheme (project No. 5527068).
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