- Open Access
Existence results of fractional integro-differential equations with m-point multi-term fractional order integral boundary conditions
© Sudsutad and Tariboon; licensee Springer 2012
- Received: 19 April 2012
- Accepted: 9 August 2012
- Published: 30 August 2012
In this article, we present some new existence and uniqueness results for nonlinear fractional integro-differential equations with m-point multi-term fractional order integral boundary conditions. Our results are based on the Banach contraction principle and Krasnoselskii’s fixed point theorem.
- Caputo fractional derivative
- Riemann-Liouville fractional integral
- integral boundary value problem
Differential equations of fractional order arise in several research areas of science and engineering, such as physics, chemistry, aerodynamics, electro-dynamics of complex medium, polymer rheology, electrical networks, control of dynamical systems, etc. Fractional differential equations provide an excellent tool for description of memory and hereditary properties of various materials and processes. Some recent contributions to the theory of fractional differential equations and its applications can be seen in [1–5].
Recently, many researchers have given attention to the existence of solutions of the initial and boundary value problems for fractional differential equations. There are some papers that have studied the existence of solutions to boundary value problems with two-point, three-point, multi-point or integral boundary conditions. See for examples [6–30]. However, to the best of the authors’ knowledge, there are only a few papers that consider nonlinear fractional differential equations with nonlocal fractional order integral boundary conditions, see [31–33].
and is such that . Let be a Banach space and denote the Banach space of all continuous functions from endowed with a topology of uniform convergence with the norm denoted by .
In this case, the boundary condition corresponds to intervals of area under the curve of solution from to for .
provided that exists, where denotes the integer part of the real number q.
provided that such integral exists.
provided the right-hand side is pointwise defined on .
Lemma 2.1 (see )
where , ().
for some , ().
We state a result due to Krasnoselskii  which is needed to prove the existence of at least one solution of the problem (1.1)-(1.2).
A is compact and continuous,
B is a contraction mapping.
Then there exists such that .
for some constants .
Substituting the values of and in (2.5), we obtain the solution (2.4). □
Now we are in the position to establish the main result. Our first result is based on Banach’s fixed point theorem.
Theorem 3.1 Assume that is jointly continuous and maps bounded subsets of in to relatively compact subsets of X, and is continuous with . In addition, suppose that there exist positive constants , such that
() , for all , ,
() , where Ω is defined by (2.7).
Then the problem (1.1)-(1.2) has a unique solution.
By (), , therefore, F is a contraction. Hence, by the Banach fixed point theorem, we get that F has a fixed point which is a unique solution of the problem (1.1)-(1.2). □
Our second result is based on Krasnoselskii’s fixed point theorem.
for all , where .
Then the BVP (1.1)-(1.2) has at least one solution on .
Hence, U is a contraction mapping.
Next, we show that S is compact and continuous.
which is independent of x. As , the right-hand side of the above inequality tends to zero. So, S is relatively compact on . Hence, by Arzelá-Ascoli theorem, S is compact on . Thus, all the assumptions of Theorem 2.2 are satisfied. As a consequence of Theorem 2.2, we have that the boundary value problem (1.1)-(1.2) has at least one solution on . This completes the proof. □
Hence, by Theorem 3.1, the boundary value problem (4.1)-(4.2) has a unique solution on .
This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
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