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Fractional integral problems for fractional differential equations via Caputo derivative
Advances in Difference Equations volume 2014, Article number: 181 (2014)
In this paper, we study the existence and uniqueness of solutions for fractional boundary value problems involving nonlocal fractional integral boundary conditions, by means of standard fixed point theorems. Some illustrative examples are also presented.
MSC:26A33, 34A08, 34B15.
Differential equations with fractional order have recently proved to be valuable tools for the description of hereditary properties of various materials and systems. Many phenomena in engineering, physics, continuum mechanics, signal processing, electro-magnetics, economics, and science describes efficiently by fractional order differential equations. For a reader interested in the systematic development of the topic, we refer the books [1–5]. Many researchers have studied the existence theory for nonlinear fractional differential equations with a variety of boundary conditions; for instance, see the papers [6–21], and the references therein.
In this paper, we study the existence and uniqueness of solutions for the following boundary value problem for the fractional differential equation with nonlocal fractional integral boundary conditions
where denotes the Caputo fractional derivative of order q, is a continuous function, , , for all , , , , and is the Riemann-Liouville fractional integral of order (, , , ).
The significance of studying problem (1.1) is that the boundary conditions are very general and include many conditions as special cases. In particular, if , for all , , then the boundary conditions reduce to
Note that the condition (1.2) does not contain the values of the unknown function u at the left side and right side of the boundary points and , respectively.
We develop some existence and uniqueness results for the boundary value problem (1.1) by using standard techniques from fixed point theory. The paper is organized as follows: in Section 2, we recall some preliminary facts that we need in the sequel and Section 3 contains our main results. Finally, Section 4 provides some examples for the illustration of the main results.
Definition 2.1 For an at least n-times differentiable function , the Caputo derivative of fractional order q is defined as
where denotes the integer part of the real number q.
Definition 2.2 The Riemann-Liouville fractional integral of order q is defined as
provided the integral exists.
Lemma 2.1 For , the general solution of the fractional differential equation is given by
where , ().
In view of Lemma 2.1, it follows that
for some , ().
For convenience we set
Lemma 2.2 Let , , , for , , and . Then the problem
has a unique solution given by
Proof Using Lemma 2.1, (2.4) can be expressed as an equivalent integral equation,
Taking the Riemann-Liouville fractional integral of order for (2.7), we have
From the first condition of (2.5) and (2.8) with , it follows that
According to the above process, the second condition of (2.5) and (2.8) with and imply that
Solving the system of linear equations for constants , , we have
Substituting constants and into (2.7), we obtain (2.6), as required. □
3 Main results
Let denote the Banach space of all continuous functions from to ℝ endowed with the norm defined by . Throughout this paper, for convenience, the expression means
where and , , , .
As in Lemma 2.2, we define an operator by
It should be noticed that problem (1.1) has solutions if and only if the operator ℱ has fixed points.
In the following subsections we prove existence, as well as existence and uniqueness results, for the boundary value problem (1.1) by using a variety of fixed point theorems.
3.1 Existence and uniqueness result via Banach’s fixed point theorem
Theorem 3.1 Assume that
(H1) there exists a constant such that , for each and .
where Λ is defined by (3.2), then the boundary value problem (1.1) has a unique solution on .
Proof We transform the problem (1.1) into a fixed point problem, , where the operator ℱ is defined as in (3.1). Observe that the fixed points of the operator ℱ are solutions of problem (1.1). Applying Banach’s contraction mapping principle, we shall show that ℱ has a unique fixed point.
We let and choose
where a constant Φ is defined by (3.3).
Now, we show that , where . For any , we have
which implies that .
Next, we let . Then for , we have
which implies that . As , ℱ is a contraction. Therefore, we deduce, by Banach’s contraction mapping principle, that ℱ has a fixed point which is the unique solution of problem (1.1). The proof is completed. □
3.2 Existence and uniqueness result via Banach’s fixed point theorem and Hölder inequality
Theorem 3.2 Suppose that is a continuous function satisfying the following assumption:
(H2) , for , and , .
Denote . If
then the boundary value problem (1.1) has a unique solution.
Proof For and for each , by Hölder’s inequality, we have
It follows that ℱ is contraction mapping. Hence Banach’s fixed point theorem implies that ℱ has a unique fixed point, which is the unique solution of the problem (1.1). The proof is completed. □
3.3 Existence and uniqueness result via nonlinear contractions
Definition 3.1 Let E be a Banach space and let be a mapping. F is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the property
Lemma 3.1 (Boyd and Wong )
Let E be a Banach space and let be a nonlinear contraction. Then F has a unique fixed point in E.
Theorem 3.3 Let be a continuous function satisfying the assumption
(H3) , , , where is continuous and a constant defined by
Then the boundary value problem (1.1) has a unique solution.
Proof We define the operator as (3.1) and a continuous nondecreasing function by
Note that the function Ψ satisfies and for all .
For any and for each , we have
This implies that . Therefore ℱ is a nonlinear contraction. Hence, by Lemma 3.1 the operator ℱ has a unique fixed point which is the unique solution of the boundary value problem (1.1). This completes the proof. □
3.4 Existence result via Krasnoselskii’s fixed point theorem
Lemma 3.2 (Krasnoselskii’s fixed point theorem )
Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) whenever ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists such that .
Theorem 3.4 Let be a continuous function satisfying (H1). In addition we assume that
(H4) , , and .
Then the boundary value problem (1.1) has at least one solution on provided
Proof Setting and choosing
(where Λ and Φ are defined by (3.2) and (3.3), respectively), we consider . We define the operators and on by
For any , we have
This shows that . It is easy to see using (3.6) that is a contraction mapping.
Continuity of f implies that the operator is continuous. Also, is uniformly bounded on as
Now we prove the compactness of the operator .
We define , and consequently we have
which is independent of u and tends to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus all the assumptions of Lemma 3.2 are satisfied. So the conclusion of Lemma 3.2 implies that the boundary value problem (1.1) has at least one solution on . □
3.5 Existence result via Leray-Schauder’s nonlinear alternative
Theorem 3.5 (Nonlinear alternative for single valued maps )
Let E be a Banach space, C a closed, convex subset of E, U an open subset of C, and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
F has a fixed point in , or
there is a (the boundary of U in C) and with .
Theorem 3.6 Assume that
(H5) there exist a continuous nondecreasing function and a function such that
(H6) there exists a constant such that
where Λ and Φ are defined by (3.2) and (3.3), respectively.
Then the boundary value problem (1.1) has at least one solution on .
Proof Let the operator ℱ be defined by (3.1). Firstly, we shall show that ℱ maps bounded sets (balls) into bounded sets in . For a number , let be a bounded ball in . Then for we have
Next we will show that ℱ maps bounded sets into equicontinuous sets of . Let with and . Then we have
As , the right-hand side of the above inequality tends to zero independently of . Therefore by the Arzelá-Ascoli theorem the operator is completely continuous.
Let u be a solution. Then, for , and following similar computations to those in the first step, we have
which leads to
In view of (H6), there exists M such that . Let us set
We see that the operator is continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type, we deduce that ℱ has a fixed point which is a solution of the boundary value problem (1.1). This completes the proof. □
In this section, we present some examples to illustrate our results.
Example 4.1 Consider the following fractional integral boundary value problem:
Here , , , , , , , , , , , , , , , , , , , and . Since , (H1) is satisfied with . We can show that
Thus . Hence, by Theorem 3.1, the boundary value problem (4.1) has a unique solution on .
Example 4.2 Consider the following fractional integral boundary value problem:
Here , , , , , , , , , , , , , , , , , , , , , , and . We choose and we obtain
Hence, by Theorem 3.3, the boundary value problem (4.2) has a unique solution on .
Example 4.3 Consider the following fractional integral boundary value problem:
Here , , , , , , , , , , , , , , , , , , , , , , and . Since , (H1) is satisfied with . We find that
Hence, by Theorem 3.4, the boundary value problem (4.3) has at least one solution on .
Example 4.4 Consider the following fractional integral boundary value problem:
Here , , , , , , , , , , , , , , , , , , , , , and . Then we get
Choosing and , we can show that
which implies that . Hence, by Theorem 3.6, the boundary value problem (4.4) has at least one solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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The research of J Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. This research of W Sudsutad is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.