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, ![](https://www.scirp.org/html/1-7401455\ce1e4292-3222-4e5c-a629-70f40bcd100a.jpg)
![](https://www.scirp.org/html/1-7401455\be2e24d9-13b5-4a60-b74c-76c5dcc17b82.jpg)
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
![](https://www.scirp.org/html/1-7401455\27e7df65-b344-483a-8b4a-090a29c8e00a.jpg)
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
![](https://www.scirp.org/html/1-7401455\0ad7ffa1-74f8-4750-9053-c0892253e49d.jpg)
Definition 2.1 [5,8]: The fractional (arbitrary) order integral of the function
of order
is defined by
![](https://www.scirp.org/html/1-7401455\b1cd3eb2-6b19-452c-a889-4ec16219bd99.jpg)
where
is the gamma function, when ![](https://www.scirp.org/html/1-7401455\881d81dd-1cbc-45ed-b93a-8c14a438d7a3.jpg)
Definition 2.2 [5,8]: For a function
given on the interval
, Riemann-Liouville fractional-order derivative of order
of
, is defined by
![](https://www.scirp.org/html/1-7401455\622734c4-5f2b-4334-9025-8242f9523171.jpg)
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
![](https://www.scirp.org/html/1-7401455\50e34d26-e9f9-4989-9156-9c01a44e6928.jpg)
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
![](https://www.scirp.org/html/1-7401455\77cfc881-e8d1-4fcc-8d02-710e8d235ed4.jpg)
Definition 3.1: A function
whose
-derivative exists on
is said to be a solution of (1.1)-(1.3), if
satisfies the equation
![](https://www.scirp.org/html/1-7401455\c27930df-0c16-4e97-8611-7e4e4d0b2866.jpg)
on
and satisfies the conditions
![](https://www.scirp.org/html/1-7401455\d53538d1-5d22-4865-8a51-089a38a52937.jpg)
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
![](https://www.scirp.org/html/1-7401455\74b11e1a-2eb3-4a8a-ada4-b65d536b0639.jpg)
has solutions
![](https://www.scirp.org/html/1-7401455\00f3ef7e-1ac7-4584-af0a-b1d8489c7b14.jpg)
Lemma 3.3: Let
, then
![](https://www.scirp.org/html/1-7401455\a57d8eaa-0e94-4fcc-85c5-eb2d4e8fb0df.jpg)
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
![](https://www.scirp.org/html/1-7401455\83a3ce70-35d0-4047-9d19-692a81361704.jpg)
If
, by Lemma 3.3, it follows that
![](https://www.scirp.org/html/1-7401455\72774a76-71d4-4af3-9ad5-dbf46581fc12.jpg)
If
, then from Lemma 3.3 we get
![](https://www.scirp.org/html/1-7401455\af65d8da-75dd-4144-85dd-74ad138fe163.jpg)
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 ![](https://www.scirp.org/html/1-7401455\c989fba9-c7fc-4d73-9f5e-1e1ed3b82557.jpg)
Also, we can easily show that
![](https://www.scirp.org/html/1-7401455\6e63cc6f-f8b2-422b-a54d-c694fd2d57fd.jpg)
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
![](https://www.scirp.org/html/1-7401455\292d7f25-a811-4ca6-8ba5-b9a26193abf3.jpg)
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
![](https://www.scirp.org/html/1-7401455\c5593416-3530-4a75-ba59-58b47a65c0c3.jpg)
Since
is continuous function, we have
as
.
For each
,
![](https://www.scirp.org/html/1-7401455\afa99c5c-901f-45d0-8ac7-23d752178784.jpg)
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
![](https://www.scirp.org/html/1-7401455\45a4353d-472b-46ef-bad6-5c38eb4f634a.jpg)
For
, we have
![](https://www.scirp.org/html/1-7401455\0ea41ac1-0773-48a6-8552-6a87481f6ea2.jpg)
Let
![](https://www.scirp.org/html/1-7401455\765ab696-7074-4cd0-83ff-b17d7ad68cdd.jpg)
then ![](https://www.scirp.org/html/1-7401455\76ccca23-d61a-4804-87ec-112fc4a675e8.jpg)
Step 3:
maps bounded sets into equicontinuous sets of
.
Let
,
be a bounded set of
as in Step 2, and let
. For
, we have
![](https://www.scirp.org/html/1-7401455\fa05f5ef-3d15-4a83-858c-4ba2187cdc36.jpg)
For
, we have
![](https://www.scirp.org/html/1-7401455\6599e61c-3b29-4cbf-bf15-75b387e0c834.jpg)
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.