Abstract
In this study, we establish some conditions for existence and uniqueness of the solutions to semilinear fractional impulsive integrodifferential evolution equations with nonlocal 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 integrodifferential equation; nonlocal condition1 Introduction
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, [17]. 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 [813]. Moreover, many researchers study the existence of solutions for fractional differential equations; see [1416] 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 [1921] 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 [24] and Balachandran and Park [25] researched the existence of solutions of fractional abstract differential equations with a nonlocal initial condition. Ahmad [26] obtained some existence results for boundary value problems of fractional semilinear evolution equations. Recently, Balachandran and Trujillo [27] 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 [2831] for the general theory and applications of impulsive differential equations. Besides, some researchers (see [3235] 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 [3638]. Further, in [38], Balachandran et al. studied the existence of solutions for fractional impulsive integrodifferential equations of the following type:
where
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 integrodifferential equation with nonlocal conditions:
where
with
and
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 [39]. Also, as indicated in [40,41], nonlocal conditions can be more useful than standard conditions to describe physical phenomena. For example, in [41], the author described the diffusion phenomenon of a small amount of gas in a transparent tube by using the formula
where
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
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.
2 Preliminaries
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, [42]. One can begin with a generalization of repeated integration. If
where
where
Next, we give some basic definitions and properties of fractional calculus theory used in this paper; see [1,4,28,31,32].
Let
Now,
The fractional (arbitrary) order integral of the function
where
For a function h given on the interval J, the Caputotype fractional derivative of order
where the function
Lemma 1[1]
Let
has the following solution:
Lemma 2[14]
Let
for some
Now, by using the Kronecker convolution product, see [7], the fractional integral becomes
Thus, if
where
for
see [7].
Now, we need the following lemma for our study.
Lemma 3Let
if and only if
where
Proof Let u be the solution of (2.4). If
for some
Applying the boundary condition
If
for some
In the view of
we have
Hence,
By repeating the process, for
Now, applying the boundary condition
we find that
Substituting the value of
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
on
Now, we define the operator
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)
(A2) The function
(A3)
(A4) There exist constants
(A5) There exists a constant
(A6) is continuous and there exists a constant
for all
(A7) There exist constants
(A8) There exist constants
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
Step 1: T is continuous.
Let
Since A is a continuous operator and f, g, I,
Step 2: T maps bounded sets into bounded sets.
Now, it is enough to show that there exists a positive constant l such that
Thus,
Then it follows that
Step 3: T maps bounded sets into equicontinuous sets.
Let
where
Hence,
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
Observing the inequality
we have
Thus,
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
Here,
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).
Conclusion
In the literature, the authors consider impulsive fractional semilinear evolution
integrodifferential equations of order
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final draft.
Acknowledgements
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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