ABSTRACT

In this article, by using Schaefer fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for a class of impulsive integro-differential equations with nonlocal conditions involving the Caputo fractional derivative.

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, control theory, etc., An excellent account in the study of fractional differential equations can be found in [1-11] and references therein. Undergoing abrupt changes at certain moment of times like earthquake, harvesting, shock etc, these perturbations can be well-approximated as instantaneous change of state or impulses. Furthermore, these processes are modeled by impulsive differential equations. In 1960, Milman and Myshkis introduced impulsive differential equations in their papers [12]. Based on their work, several monographs have been published by many authors like Semoilenko and Perestyuk [13], Lak-shmikantham et al. [14], Bainov and Semoinov [15,16], Bainov and Covachev [17] and Benchohra et al. [18]. Impulsive fractional differential equations represent a real framework for mathematical modelling to real world problems. Significant progress has been made in the theory of impulsive fractional differential equations [19-21].

We consider a class of impulsive fractional integrodifferential equations with nonlocal conditions of the form

(1.1)

(1.2)

(1.3)

Where is the Caputo fractional derivative, the function is continuous and the function is continuous,

and represent the right and left limits of at, and is a continuous function,.

Nonlocal conditions were initiated by Byszewski [22] who proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems. As remarked by Byszewski [23,24], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena. For example, may be given by

where are given constants and .

In this article, our aim is to show sufficient conditions for the existence and uniqueness of solutions of solutions to impulsive fractional integro-differential equations with nonlocal conditions.

2. Preliminaries

In this section, we introduce some notations, definitions and preliminary facts which are used throughout this paper. By we denote the Banach space of all continuous functions from into with the norm

Definition 2.1 [5,8]: The fractional (arbitrary) order integral of the function of order is defined by

where is the gamma function, when

Definition 2.2 [5,8]: For a function given on the interval, Riemann-Liouville fractional-order derivative of order of, is defined by

here and denotes the integer part of

, when.

Definition 2.3 [14]: For a function given on the interval, the Caputo fractional-order derivative of order of, is defined by

where.

Lemma 2.4 [25]: (Schaefer’s fixed point theorem). Let be a Banach space and be a completely continuous operator. If the set

is bounded, then has at least a fixed point in X.

3. Existence of Solutions

Consider the set of functions

Definition 3.1: A function whose -derivative exists on is said to be a solution of (1.1)-(1.3), if satisfies the equation

on and satisfies the conditions

where.

To prove the existence of solutions to (1.1)-(1.3), we need the following auxiliary lemmas.

Lemma 3.2: Let, then the equation

has solutions

Lemma 3.3: Let, then

for some.

As a consequence of Lemma 3.2 and Lemma 3.3, we have the following result Lemma 3.4: Let, and let be continuous. A function is a solution of the fractional integral equation

(3.1)

if and only if is a solution of the fractional nonlocal BVP

(3.2)

(3.3)

(3.4)

Proof Assume satisfies (3.2)-(3.4).

If then.

Lemma 3.3 implies

If, by Lemma 3.3, it follows that

If, then from Lemma 3.3 we get

If, then again from we have (3.1).

Conversely, assume that satisfies the impulsive fractional integral equation (3.1). If, then and using the fact that is the left inverse of, we get.

If and using the fact that, where is a constant, we conclude that

Also, we can easily show that

Theorem: Assume that:

(H₁) There exists a constant such that

for each and each;

(H₂) There exists a constant such that

, for each and;

(H₃) There exists a constant such that

, for each, then the problem

(1.1)-(1.3) has at least one solution on.

Proof Consider the operator

defined by

Clearly, the fixed points of the operator are solution of the problem (1.1)-(1.3).

We shall use Schaefer’s fixed point theorem to prove that has a fixed point. The proof will be given in several steps.

Step 1: is continuous.

Let be a sequence such that in. Then for each

Since is continuous function, we have

as.

For each,

Since and are continuous functions, we have as.

Therefore, is continuous.

Step 2: maps bounded sets into bounded sets in.

Indeed, it is enough to show that for any, there exists a positive constant such that for each

, we have

. By (H₁), (H₂) and (H₃), for each, we have

For, we have

Let

then

Step 3: maps bounded sets into equicontinuous sets of.

Let, be a bounded set of as in Step 2, and let. For

, we have

For, we have

As, the right-hand side of the above inequality tends to zero. As a consequence of Steps 1 to 3 together with the Arzel’a-Ascoli theorem, we can conclude that is completely continuous.

As a consequence of Lemma 2.4 (Schaefer’s fixed point theorem), we deduce that has a fixed point which is a solution of the problem (1.1)-(1.3).

4. Acknowledgements

This work was supported by the natural science foundation of Hunan Province (13JJ6068, 12JJ9001), Hunan provincial science and technology department of science and tech-neology project (2012SK3117), Science foundation of Hengyang normal university of China (No. 12B35) and Construct program of the key discipline in Hunan Province.

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