Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses

Abstract

We investigate the existence of mild solutions for fractional order differential equations with integral boundary conditions and not instantaneous impulses. By some fixed-point theorems, we establish sufficient conditions for the existence and uniqueness of solutions. Finally, two interesting examples are given to illustrate our theory results.

1. Introduction

Impulsive differential equations are used to describe many practical dynamical systems including evolutionary processes characterized by abrupt changes of the state at certain instants. Such processes are naturally seen in biology, physics, engineering, and so forth. Due to their significance, many authors have established the solvability of impulsive differential equations. Nowadays, the theory of impulsive differential equations has received great attention. Differential equations with instantaneous impulses have been treated in several works (see, e.g., the monographs [1–3], the works on time variable impulses problem [4–7], and the references therein).

However, in almost all the papers concerning impulsive differential equations, the impulses are all instantaneous impulses, and the classical models with instantaneous impulses cannot characterize many practical problems, for example, the dynamics of evolution processes in pharmacotherapy. Let us consider the hemodynamic equilibrium of a person. The introduction of the drugs in the bloodstream and the consequent absorption for the body are gradual and continuous processes. In fact, this situation should be characterized by a new case of impulsive action, which starts at an arbitrary fixed point and stays active on a finite time interval . To this end, Hernández and O’Regan [8] initially offered to study a new class of abstract semilinear impulsive differential equations with not instantaneous impulses in a PC-normed Banach space. In [8], the authors discussed the following problem: where is the generator of a -semigroup of bounded linear operators defined on a Banach space , are prefixed numbers, , for all , and is a suitable function. Meanwhile, Pierri et al. [9] continued the work in [8] in a -normed Banach space.

On the one hand, the absorption of drugs has a memory effect; thus, the new class of impulsive conditions introduced by [8] may not explain this phenomenon very well. On the other hand, fractional calculus provides a powerful tool for the description of hereditary properties of various materials and memory processes [10, 11]. Fractional differential equations have recently proved to be strong tools in the modeling of medical, physics, economics, and technical sciences. For more details on fractional calculus theory, one can see the monographs of Diethelm [12], Kilbas et al. [13], Lakshmikantham et al. [14], Miller and Ross [15], Podlubny [16], and Tarasov [17]. Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attention (see [10, 11, 18–22]).

The theory of boundary value problems (BVPs) with integral boundary conditions for differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For BVPs with integral boundary conditions and comments on their importance, we refer the readers to the papers by Gallardo [23], Karakostas and Tsamatos [24], Lomtatidze and Malaguti [25], and the references therein. For more information about the general theory of integral equations and their relation with BVPs, we refer to the books of Corduneanu [26] and Agarwal and O’Regan [27]. Moreover, BVPs with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal BVPs as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention. To identify a few, we refer the readers to [28–31] and references therein.

In [32], the authors consider the following problem: where is the Caputo fractional derivative of order with the lower limit zero, is a given function, , , ,  , and satisfy , and represent the right and left limits of at . Obviously, the impulses in (2) are instantaneous. Motivated by the work in [8, 9, 32], in this paper, we consider the following impulsive fractional differential equations with integral boundary conditions and not instantaneous impulses: where is the Caputo fractional derivative of order with the lower limit zero, are prefixed numbers, , for , , , , , and , .

The rest of this paper is organized as follows. In Section 2, some lemmas which are essential to prove our main results are stated. In Section 3, we give the main results. In Section 4, two interesting examples are given to illustrate our theory results.

2. Preliminaries

At first, we present the necessary definitions for the fractional calculus theory.

Definition 1 (see [13]). The Riemann-Liouville fractional integral of order of a suitable function is given by where the right side is pointwise defined on .

Definition 2 (see [13]). The Caputo fractional derivative of order of a suitable function is given by where , denotes the integer part of number and the right side is pointwise defined on .

Lemma 3 (see [13]). Let ; then the fractional differential equation has solutions where , , .

Lemma 4 (see [13]). Let , then one has where , , .

Lemma 5 (Krasnoselskii’s fixed point theorem [33]). Let be a closed convex and nonempty subset of a Banach space . Let and be two operators such that(1) whenever ;(2) is compact and continuous;(3) is a contraction mapping.Then there exists such that .
In order to study problem (3), we define , , and , exist with , .
It is easy to check that is a Banach space with the norm .

Let , ; then .

If satisfies problem (3), then for , , integrating the first equation of (3) from to by virtue of Definition 1, one can obtain From the second equation in (3), we know . Then, for , we have

For , integrating the first equation in (3) from to by virtue of Definition 1, one can obtain By the boundary conditions, we have Multiplying (11) with and integrating from 0 to , we have Multiplying (9) with and integrating from to , , we have

Multiplying the second equation of (3) with and integrating from to , , we can obtain Adding (12), (13), and (14), one has Hence So, for , we have

Then similar to Definition  2.1 in [9], we can define the mild solution for (3).

Definition 6. A function is a mild solution of problems (3) if for all and for all , .

3. Main Results

This section deals with the existence of mild solutions for (3). Before stating and proving the main results, we make the following hypotheses.  is jointly continuous. There exists a function such that  for all , for all .  is continuous and there exists a function such that   and there exist , such that

Let

Now we are in the position to establish the main results. Our first theorem is based on contraction mapping principle.

Theorem 7. Let hold and ; then the problem (3) has a unique mild solution, where

Proof. Let be the map defined by for and for , . Clearly, is well defined.
Next we show that is contraction on .
Fix ; we consider three cases.
Case 1. If , , by the assumptions and the property , we have
Case 2. If , by , , one can obtain Then, by a similar argument, we can get
Case 3. If , , from the assumption , we get
Therefore, , for all , which implies that is a contraction mapping. Then, there exists a unique mild solution of (3).
In order to get the second main result, we give assumption .The function is jointly continuous and strongly measurable on . There exist and a nondecreasing function such that
Our second result is based on Krasnoselskii’s fixed point theorem.

Theorem 8. Assume that hold; if and there exists a constant such that where , then the problem (3) has at least a mild solution.

Proof. Let be the map introduced in the proof of Theorem 7. We consider the decomposition , where
Let . We divide our proof into three steps.
Step 1. First we show that whenever .
From , we know that is continuous, and then is bounded, for . Let .
Let ; if , we have By the definition of , the integral mean value theorem, , and the property , we have where .
By a similar argument, let , if ; we have From the condition , , we can get
For the other cases, from the definition of , one can get .
From the proof above, let ; then for all , we have
Let ; if , , we can obtain
Let ; if , we can get
Proceeding as above, we obtain that , , for , .
Then, for all , we have that
Step 2. We show that is a contraction mapping.
From the definition of , , and , we can easily get which implies that is a contraction mapping.
Step 3. Next we will prove that is compact and continuous.
We also divide the proof into 3 steps.
(I) We show that is continuous.
Let be a sequence such that in . Then for all , by the definition of , , we have
From , , we can get the continuity of and . Then one has which shows that the operator is continuous.
(II) We show that maps bounded sets into bounded sets in .
Indeed, it is enough to show that, for any , there exists a such that, for each , we have .
For all , from the definition of , , and , , one can obtain
Then we conclude that maps bounded sets into bounded sets in .
(III) At last, we prove that maps bounded sets into equicontinuous sets in .
For interval , , , , by definition of and , we have which is independent of . As , the right-hand side of the above inequality tends to zero. Therefore is equicontinuous on interval , .
Proceeding as above, we can also prove that is equicontinuous for the time interval . From the definition of , it is easy to see that is equicontinuous for the other cases.
By Arzela-Ascoli Theorem, is continuous and compact.

As a consequence of Lemma 5, we deduce that the operator has at least a fixed point on which means that problem (3) has at least a mild solution.

4. Examples

Consider the following impulsive system of fractional differential equations.

Example 1. Consider where are pre-fixed numbers, , , and .
We prove that Example 1 satisfies all the assumptions of Theorem 7.
In Example 1, set
It is easy to see that is jointly continuous. We can also check that and are satisfied with and .
For , , and , then we know with , so is also satisfied.
From (47), we can get , , , , , , , and then
So all the conditions of Theorem 7 are satisfied. As a consequence of Theorem 7, Example 1 has a unique mild solution.

Example 2. Consider where are prefixed numbers, , , and .
It is easy to see that is bounded. Set
Then we have with , being nondecreasing. So is satisfied. Similarly to the proof of Example 1, we know that and are satisfied.
From (51), we can obtain , , , , , , , , , and , and then the inequality (32) becomes . Hence, inequality (32) holds for all .
Thus, all the assumptions in Theorem 8 are satisfied, and our results can be applied to Example 2. So Example 2 has at least one mild solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China (nos. 11001274, 11101126, and 11261010), China Postdoctoral Science Foundation (no. 20110491249), Key Scientific and Technological Research Project of Department of Education of Henan Province (no. 12B110006), Youth Science Foundation of Henan University of Science and Technology (no. 2012QN010), and Innovative Natural Science Foundation of Henan University of Science and Technology (no. 2013ZCX020). The authors thank the referees for their careful reading of the paper and insightful comments, which help to improve the quality of the paper. The authors would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contribute to the perfection of the paper.