Existence and Uniqueness of Solutions to Impulsive Fractional Integro-Differential Equations with Nonlocal Conditions ()
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.