- Open Access
Fractional integral problems for fractional differential equations via Caputo derivative
© Tariboon et al.; licensee Springer 2014
- Received: 8 April 2014
- Accepted: 25 June 2014
- Published: 22 July 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.
- fractional differential equations
- nonlocal boundary conditions
- fixed point theorems
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.
where denotes the Caputo fractional derivative of order q, is a continuous function, , , for all , , , , and is the Riemann-Liouville fractional integral of order (, , , ).
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.
where denotes the integer part of the real number q.
provided the integral exists.
where , ().
for some , ().
Substituting constants and into (2.7), we obtain (2.6), as required. □
where and , , , .
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.
where a constant Φ is defined by (3.3).
which implies that .
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 , .
then the boundary value problem (1.1) has a unique solution.
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
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
Then the boundary value problem (1.1) has a unique solution.
Note that the function Ψ satisfies and for all .
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 .
This shows that . It is easy to see using (3.6) that is a contraction mapping.
Now we prove the compactness of the operator .
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 )
F has a fixed point in , or
there is a (the boundary of U in C) and with .
Theorem 3.6 Assume 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 .
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.
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.
Thus . Hence, by Theorem 3.1, the boundary value problem (4.1) has a unique solution on .
Hence, by Theorem 3.3, the boundary value problem (4.2) has a unique solution on .
Hence, by Theorem 3.4, the boundary value problem (4.3) has at least one solution on .
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.
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.
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