1. Introduction
Fractional differential equations arise in many engineering and scientific disciplines such as physics, aerodynamics, polymer rheology, regular variations in thermodynamics, biophysics, blood flow phenomena, electrical circuits, biology, etc. In fact, the tools of fractional calculus have considerably improved the mathematical modeling of many real world problems. For theoretical development and applications of the subject, we refer the reader to [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11] and the references cited therein.
The nonlocal boundary conditions are important in describing some peculiarities happening inside the domain of physical, chemical or other processes [
12], while the integral boundary conditions provide the means to assume an arbitrary shaped cross-section of blood vessels in computational fluid dynamics (CFD) studies of blood flow problems [
13,
14].
Non-local boundary value problems of nonlinear fractional order differential equations have recently been investigated by several researchers. The domain of study ranges from the theoretical aspects to the analytic and numerical methods for fractional differential equations.
Agarwal et al. [
4] discussed the existence of solutions for a boundary value problem of integro-differential equations of fractional order
with non-local three-point boundary conditions
, where
.
Motivated by the works mentioned, in this paper, we investigate the existence and uniqueness of solutions for the non-linear fractional integro-differential equation
with non-local boundary conditions
where
denote the Caputo fractional derivative of order
p and
q respectively,
is a given continuous function,
are real constants,
,
,
,
,
and
.
These boundary conditions are interesting and important from a physical point.
The rest of paper is arranged as follows: In
Section 2 we recall some basic definitions of fractional calculus and present an auxiliary lemma.
Section 3 contains main existence and uniqueness results. We also present some examples to to illustrate our results.
2. Preliminaries
First of all, we recall some basic definitions of fractional calculus [
7].
Definition 1. For a function , the Caputo derivative of fractional order p is defined aswhere denotes the integer part of real number p and is the gamma function, which is defined by . Definition 2. The Riemann-Liouvill fractional integral of order p is defined asprovided the integral exists. Now, we present an auxiliary lemma which plays a key role in the sequel.
Lemma 1. Let and . Then the unique solution of the boundary value problem with boundary conditions and is given bywhere , and . Proof. It is well known [
7] that the general solution of the equation
is given by
where
are arbitrary constants. By using the boundary conditions, we get
and
Substituting the values of
in (
3), completes the proof. ☐
3. The Main Results
For , let denote the Banach space of all continuous functions defined on into endowed with the norm .
Using Lemma 1, we define an operator
associated with the problem (
1)–(
2) as
Observe that the problem (
1)–(
2) has solutions if and only if the operator
T has fixed points.
Theorem 1 ([15]). Let X be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then T has a fixed point in X.
Theorem 2. Assume that there exists such that for with . Then the problem (1)–(2) has at least one solution on . Proof. As a first step, we show that the operator
T is completely continuous. Let
be a bounded set. Then for each
, we get
on the taking the norm for
, we obtain
We set
. In a similar manner, we find that
Put
Next, for each
and
, we have
Hence
In a similar manner, we get
The functions
are uniformaly continuous on
since
. Therefore, by Arzela-Ascoli Theorem, the sets
and
are relatively compact in
. Thus, we deduce that
is a relatively compact subset of
.
Now, we consider the set
and show that it is bounded. Let
, then
. For any
, it implies from
that
This shows that the set
B is bounded. Thus, by Theorem 1, we conclude that the operator
T has at least one fixed point. Consequently , the problem (
1)–(
2) has at least one solution on
. ☐
Now, we establish the uniqueness of solutions for problem (
1)–(
2) by means of classical contraction mapping principle.
Theorem 3. Assume that is a continuous function satisfying the conditionwith , where . Then, the problem (1)–(2) has a unique solution on . Proof. First, we show that
, where
T is the operator defined by (4),
with
, where
. For
, we have
Similarly, we obtain
Therefore, we get that
implies that
. Moreover, for
and for any
, we have
Analogously, we can obtain
Therefore, with the condition
, we deduce that the operator
T is a contraction. Hence, it follows Banach’s fixed point theorem that the problem (
1)–(
2) has a unique solution on
. ☐
Example 1. Consider the following fractional boundary value problem given bywhere , andClearly with . Hence, by Theorem 2, the problem (5) has at least one solution on . Example 2. Consider the following fractional boundary value problem given byHere , andUsing the given values of the parameters, we obtain and . It is easy to see thatWe have and . Therefore, by Theorem 3 we deduce that the problem (6) has a unique solution on . 4. Conclusions
In this paper, we have obtained some existence results for a non-linear fractional integro-differential problem with non-local boundary conditions by means of Schauder type fixed point theorem and contraction mapping principle. Our results are not only new in the given configuration but also correspond to some new situations associated with the specific values of the parameters involved in the given problem.