Existence results for fractional differential equations with three-point boundary conditions
© Fu; licensee Springer. 2013
Received: 21 April 2013
Accepted: 8 August 2013
Published: 22 August 2013
In this paper, we study three-point boundary value problems of nonlinear fractional differential equations. Existence and uniqueness results are obtained by using standard fixed point theorems. Some examples are given to illustrate the results.
MSC:34A60, 26A33, 34B15.
Fractional differential equations have gained much importance and attention due to the fact that they have been proved to be valuable tools in the modeling of many phenomena in engineering and sciences such as physics, mechanics, economics and biology, etc. [1–3]. For some developments on the existence results of fractional differential equations, we can refer to [4–25] and the references therein.
where denotes the Caputo fractional derivative of order α, f is a given continuous function.
where denotes the Caputo fractional derivative of order q, , , , are real constants, and f is a given continuous function.
where denotes the Caputo fractional derivative of order α, f is a given continuous function, and with .
where denotes the Caputo fractional derivative of order q, , , , are real constants such that , , and f is a given continuous function.
where denotes the Caputo fractional derivative of order q, the Riemann-Liouville fractional integral of order γ, f is a given continuous function, and a, b, c are real constants with .
We remark that when , and , problem (3) reduces to the anti-periodic fractional boundary value problem (1) (cf. ).
The paper is organized as follows: in Section 2 we present the notations, definitions and give some preliminary results that we need in the sequel, Sections 3 and 4 are dedicated to the existence results of problems (4) and (5), respectively, in the final Section 5, two examples are given to illustrate the results.
Definition 2.1 
provided the integral exists.
Definition 2.2 
where denotes the integer part of the real number q.
Lemma 2.1 
here , , .
The following are two standard fixed point theorems, which will be used in Sections 3 and 4 (see ).
Theorem 2.1 Let X be a Banach space, let B be a nonempty closed convex subset of X. Suppose that is a continuous compact map. Then F has a fixed point in B.
Theorem 2.2 (Nonlinear alternative for single-valued maps)
Let X be a Banach space, let B be a closed, convex subset of X, let U be an open subset of B and . Suppose that is a continuous and compact map. Then either (a) P has a fixed point in , or (b) there exist an (the boundary of U) and with .
3 Existence results for problem (4)
Substituting the values of , in (7), we obtain the result. This completes the proof. □
From the proof of Lemma 3.1, we note that when , , that is to say, the non-separateness feature in (4) is more expressed than those in (1).
Now, we are in a position to present our main results.
This together with (8) implies that ℱ is a contraction mapping. The contraction mapping principle yields that ℱ has a unique fixed point, which is the unique solution of problem (4). This completes the proof. □
for each , , , and . Then problem (4) has at least one solution.
This implies that .
and the limit is independent of . Therefore, the operator is equicontinuous and uniformly bounded. The Arzela-Ascoli theorem implies that is relatively compact in . By Theorem 2.1, we know that problem (4) has at least one solution. The proof is completed. □
Corollary 3.2 Assume that for , with . Then problem (4) has at least one solution.
- (1)there exists a function and a non-decreasing function such that
- (2)there exists a constant such that
Then problem (4) has at least one solution.
Secondly, we claim that ℱ is equicontinuous. The proof of this claim is the same as the one in the proof of Theorem 3.2.
The operator is continuous and completely continuous. Combining the choice of O and Theorem 2.2, we can deduce that ℱ has a fixed point in , which is a solution of problem (4). □
4 Existence results for problem (5)
Substituting the values of , , we obtain the result. This completes the proof. □
By the contraction principle, we know that problem (5) has a unique solution. □
- (1)there exist two non-decreasing functions and a function with such that
- (2)there exists a constant such that
Then problem (5) has at least one solution on .
Proof The proof consists of the following steps.
In this section, we give two examples to illustrate the main results.
let , and . Thus, by Theorem 3.2, problem (11) has at least one solution on .
By Theorem 4.1, we know that problem (12) has at least one solution.
The author would like to express his thanks to the referees for their helpful suggestions. This work is partially supported by Shaoxing University (No. 20125009).
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