- Open Access
Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions
© Yukunthorn et al.; licensee Springer. 2014
Received: 1 September 2014
Accepted: 26 November 2014
Published: 12 December 2014
In this paper, we study the existence and uniqueness of solution for a problem consisting of a sequential nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions. A variety of fixed point theorems, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder degree theory, are used. Examples illustrating the obtained results are also presented.
MSC:26A33, 34A08, 34B10.
where , , and are the Caputo fractional derivatives of order q and p, respectively, is the Riemann-Liouville fractional integral of order ϕ, where , are given points, , , , and is a continuous function.
where and . Note that the nonlocal conditions (1.2) and (1.3) do not contain values of an unknown function x on the left-hand side and the right-hand side of boundary points and , respectively.
Fractional differential equations have been shown to be very useful in the study of models of many phenomena in various fields of science and engineering, such as physics, chemistry, biology, signal and image processing, biophysics, blood flow phenomena, control theory, economics, aerodynamics and fitting of experimental data. For examples and recent development of the topic, see [1–13] and the references cited therein.
The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments . For some new developments on the fractional Langevin equation, see, for example, [15–24].
In the present paper several new existence and uniqueness results are proved by using a variety of fixed point theorems (such as Banach’s contraction principle, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder’s degree theory).
The rest of the paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel. In Section 3 we present our existence and uniqueness results. Examples illustrating the obtained results are presented in Section 4.
where denotes the integer part of the real number q.
provided the integral exists.
where , ().
for some , ().
Substituting and into (2.4), we obtain solution (2.3). □
3 Main results
It should be noticed that problem (1.1) has solutions if and only if the operator has fixed points.
In the following subsections, we prove existence, as well as existence and uniqueness results, for 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 Let be a continuous function. Assume that
(H1) there exists a constant such that for each and .
then problem (1.1) has a unique solution on .
which leads to . Since , is a contraction mapping. Therefore has only one fixed point, which implies that problem (1.1) has a unique solution. □
3.2 Existence and uniqueness result via Banach’s fixed point theorem and Hölder’s inequality
Theorem 3.2 Let be a continuous function. In addition we assume that
(H2) for each , , where , .
where and are defined by (3.4) and (3.6), respectively, then problem (1.1) has a unique solution.
Hence, from (3.7), is a contraction mapping. Banach’s fixed point theorem implies that has a unique fixed point, which is the unique solution of problem (1.1). This completes the proof. □
3.3 Existence result via Krasnoselskii’s fixed point theorem
Lemma 3.1 (Krasnoselskii’s fixed point theorem )
Let M be a closed, bounded, convex and nonempty subset of a Banach space X. Let A, B be operators such that (a) whenever ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists such that .
Theorem 3.3 Let be a continuous function. Moreover, we assume that
(H3) , and .
where is defined by (3.4).
It follows that , and thus condition (a) of Lemma 3.1 is satisfied. For , we have . Since , the operator ℬ is a contraction mapping. Therefore, condition (c) of Lemma 3.1 is satisfied.
which is independent of x and tends to zero as . Then is equicontinuous. So is relatively compact on , and by the Arzelá-Ascoli theorem, is compact on . Thus condition (b) of Lemma 3.1 is satisfied. Hence the operators and ℬ satisfy the hypotheses of Krasnoselskii’s fixed point theorem; and consequently, problem (1.1) has at least one solution on . □
3.4 Existence result via Leray-Schauder’s nonlinear alternative
Theorem 3.4 (Nonlinear alternative for single-valued maps )
has a fixed point in , or
there is (the boundary of U in C) and with .
Theorem 3.5 Let be a continuous function. Assume that
where , and Φ are defined by (3.3), (3.4) and (3.5), respectively.
Then 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 has a fixed point which is a solution of problem (1.1). This completes the proof. □
3.5 Existence result via Leray-Schauder’s degree theory
Theorem 3.6 Let be a continuous function. Suppose that
where , are defined by (3.3) and (3.4), respectively.
Then problem (1.1) has at least one solution on .
We shall prove that there exists a fixed point satisfying (1.1).
If , then inequality (3.9) holds. This completes the proof. □
Hence, by Theorem 3.1, problem (4.1) has a unique solution on .
Hence, by Theorem 3.2, problem (4.2) has a unique solution on .
Here , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
This means that . By Theorem 3.3 problem (4.3) has as least one solution on .
which implies . By Theorem 3.5, problem (4.4) has at least one solution on .
which satisfies (H6). By Theorem 3.6, problem (4.5) has at least one solution on .
The research of JT is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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