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Existence results for fractional differential equations with three-point boundary conditions
Advances in Difference Equations volume 2013, Article number: 257 (2013)
Abstract
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.
1 Introduction
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.
In recent years, there has been a great deal of research on the questions of existence and uniqueness of solutions to boundary value problems for differential equations of fractional order. For example, Ahmad and Nieto [8] investigated the existence and uniqueness of solutions for an anti-periodic fractional boundary value problem
where denotes the Caputo fractional derivative of order α, f is a given continuous function.
In [16], the author discussed the existence of solutions for the following nonlinear fractional differential equations with anti-periodic-type fractional boundary conditions
where denotes the Caputo fractional derivative of order q, , , , are real constants, and f is a given continuous function.
Fractional differential equations with three-point integral boundary conditions of the following form were considered in [15] by Ahmad et al.
where denotes the Caputo fractional derivative of order α, f is a given continuous function, and with .
By a simple computation, we observed that in equations (1) and (2). This implies that the boundary conditions in (1) and in (2) actually are equivalent to the boundary conditions and , respectively.
Motivated by the papers above, in this article, firstly, we study fractional differential equations with the three-point boundary conditions in the following form
where denotes the Caputo fractional derivative of order q, , , , are real constants such that , , and f is a given continuous function.
Then we consider the fractional differential equations with three-point integral boundary conditions
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. [8]).
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.
2 Preliminaries
Definition 2.1 [17]
The Riemann-Liouville fractional integral of order q for a continuous function is defined as
provided the integral exists.
Definition 2.2 [17]
For times absolutely continuous function , the Caputo derivative of fractional order q is defined as
where denotes the integer part of the real number q.
Lemma 2.1 [12]
Let , then the differential equation
has solutions and
here , , .
The following are two standard fixed point theorems, which will be used in Sections 3 and 4 (see [26]).
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)
Lemma 3.1 For any , the unique solution of the three-point boundary value problem
is given by
Proof For , by Lemma 2.1, we know that the general solution of the equation can be written as
where are arbitrary constants. Since , , , we have
Using the boundary conditions, we obtain
Therefore, we have
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).
Let and be the Banach space of all continuous real functions from J into ℝ equipped with the norm . In view of Lemma 3.1, we define an operator as follows
Note that problem (4) has solutions if and only if the operator has fixed points. We denote by , where
Here the constants and are given by
Now, we are in a position to present our main results.
Theorem 3.1 Suppose that is continuous and satisfies
for , , and . If
then problem (4) has a unique solution, where
Proof Denote . For any and each , we have
Therefore, we have
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. □
Corollary 3.1 Suppose that is continuous and satisfies
for , , and . Then problem (4) has a unique solution, provided
Theorem 3.2 Let be a continuous function. Assume that
for each , , , and . Then problem (4) has at least one solution.
Proof Let , , where
Observe that is a closed, bounded convex subset of the Banach space .
Firstly, we prove that . For any , we have
Hence, we have
This implies that .
Secondly, we prove that ℱ maps bounded sets into equicontinuous sets. Let B be any bounded set of . Notice that f is continuous on J, therefore, without loss of generality, we can assume that there is an N such that
for any and . Now, we let . Then for each , we have
and
Hence, we have
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.
Theorem 3.3 Let be a continuous function. Assume that
-
(1)
there exists a function and a non-decreasing function such that
where , ;
-
(2)
there exists a constant such that
where
Then problem (4) has at least one solution.
Proof Firstly, we prove that ℱ maps bounded sets into bounded sets in . Let B be a bounded subset of and for any . As in the proof of Theorem 3.2, we have
Hence,
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.
Finally, we let for some . Then for each , we have
This implies that
According to the assumptions, we know that there exists K such that . Let
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)
Lemma 4.1 For any , the unique solution of the three-point boundary value problem
is given by
Proof For and some constants , the general solution of the equation can be written as
From , it follows that . Using the integral boundary conditions of (9), we obtain
Therefore, we have
Substituting the values of , , we obtain the result. This completes the proof. □
Define the space endowed with the norm . Obviously, is a Banach space. In order to obtain the existence results of problem (5), by Lemma 4.1, we define an operator as follows
where
Since f is continuous, it is easy to see that
here k is a constant given by
Theorem 4.1 Let be a continuous function satisfying that
for , , and , . Then problem (5) has a unique solution provided that , where Δ, Λ are given by
Proof Let and . Then for each , we have
By the Hölder inequality, we have
Similarly, we have
From the inequalities above, we can deduce that
By the contraction principle, we know that problem (5) has a unique solution. □
Theorem 4.2 Assume that
-
(1)
there exist two non-decreasing functions and a function with such that
for and ;
-
(2)
there exists a constant such that
where and
Then problem (5) has at least one solution on .
Proof The proof consists of the following steps.
Firstly, we show that the operator maps bounded sets into bounded sets. Let be a bounded set in X. Then for each , we have
By using the Hölder inequality, we have
Similarly, we can obtain that
Therefore, we have
That is to say, we have
Secondly, by a discussion similar to that of Theorem 3.2, we can get
as . This implies that
Finally, we let for . Then for each , we have
That is to say,
By the assumptions and a discussion similar to the one in the proof of Theorem 3.3, we can deduce that has a fixed point in X. So the proof of this theorem is completed. □
5 Examples
In this section, we give two examples to illustrate the main results.
Example 1 Consider the boundary value problem
Here , , , , , , , , and
Since
let , and . Thus, by Theorem 3.2, problem (11) has at least one solution on .
Example 2 Consider the following fractional differential equation
In this case , , , , , , and
Since
let , we have
By Theorem 4.1, we know that problem (12) has at least one solution.
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Acknowledgements
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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Fu, X. Existence results for fractional differential equations with three-point boundary conditions. Adv Differ Equ 2013, 257 (2013). https://doi.org/10.1186/1687-1847-2013-257
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DOI: https://doi.org/10.1186/1687-1847-2013-257