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Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions
Advances in Difference Equations volume 2014, Article number: 315 (2014)
Abstract
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.
1 Introduction
In this paper, we concentrate on the study of existence and uniqueness of solution for the following nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions:
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.
The significance of studying problem (1.1) is that the nonlocal conditions are very general and include many conditions as special cases. In particular, if , for all , , then the nonlocal condition of (1.1) reduces to
and if , , , then (1.2) is reduced to
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 [14]. 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.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus [2, 3] and present preliminary results needed in our proofs later.
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 , ().
In the following, for the sake of convenience, we set constants
and .
Lemma 2.2 Let , , , , , , , , and . Then the nonlinear fractional Caputo-Langevin equation
subject to the nonlocal Riemann-Liouville fractional integral conditions
has a unique solution given by
Proof The general solution of equation (2.1) is expressed as the following integral equation:
where and are arbitrary constants. By taking the Riemann-Liouville fractional integral of order for (2.4), we get
In particular, for , we have
Repeating the above process for the Riemann-Liouville fractional integral of order , substituting and applying the nonlocal condition (2.2), we obtain the following system of linear equations:
Solving the linear system of equations in (2.5) for constants , , we have
Substituting and into (2.4), we obtain solution (2.3). □
3 Main results
Throughout this paper, for convenience, the expression means
Let denote the Banach space of all continuous functions from to ℝ endowed with the norm defined by . 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 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 .
If
where constants , are defined by
then problem (1.1) has a unique solution on .
Proof Problem (1.1) is equivalent to a fixed point problem by defining the operator as in (3.1), which yields . Using the Banach contraction mapping principle, we will show that problem (1.1) has a unique solution. Setting , we define a set ,
where
For any , we have
which implies that . Next, we need to show that is a contraction mapping. Let . Then, for , we have
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
Now we give another existence and uniqueness result for problem (1.1) by using Banach’s fixed point theorem and Hölder’s inequality. For , we set
Theorem 3.2 Let be a continuous function. In addition we assume that
(H2) for each , , where , .
Denote .
If
where and are defined by (3.4) and (3.6), respectively, then problem (1.1) has a unique solution.
Proof For and each , by Hölder’s inequality, we have
Therefore,
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 [25])
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 .
Then problem (1.1) has at least one solution on if
where is defined by (3.4).
Proof We define the operators and ℬ on by
where the ball is defined by for some suitable r such that
with and , and Φ are defined by (3.3), (3.4) and (3.5), respectively. To show that , we let . Then we have
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.
The continuity of f implies that the operator is continuous. For , we obtain
This means that the operator is uniformly bounded on . Next we show that is equicontinuous. We set , and consequently we get
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 [26])
Let E be a Banach space, C be a closed, convex subset of E, U be an open subset of C and . Suppose that is a continuous, compact (that is, is a relatively compact subset of C) map. Then either
-
(i)
has a fixed point in , or
-
(ii)
there is (the boundary of U in C) and with .
Theorem 3.5 Let be a continuous function. Assume that
(H4) there exists a continuous nondecreasing function denoted by and a function such that
(H5) there exists a constant such 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 .
Proof Let the operator be defined by (3.1). Firstly, we shall show that maps bounded sets (balls) into bounded sets in . For a number , let be a bounded ball in . Then, for , we have
and consequently,
Next, we will show that 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 x be a solution. Then, for , and following similar computations as in the first step, we have
which leads to
By (H5) there is 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 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
(H6) there exist constants and such that
where , are defined by (3.3) and (3.4), respectively.
Then problem (1.1) has at least one solution on .
Proof We define an operator as in (3.1) and consider the fixed point equation
We shall prove that there exists a fixed point satisfying (1.1).
Set a ball as
where a constant radius . Hence, we show that satisfies the condition
We define
As shown in Theorem 3.5, the operator is continuous, uniformly bounded and equicontinuous. Then, by the Arzelá-Ascoli theorem, a continuous map defined by is completely continuous. If (3.9) holds, then the following Leray-Schauder degrees are well defined, and by the homotopy invariance of topological degree, it follows that
where I denotes the unit operator. By the nonzero property of Leray-Schauder degree, for at least one . Let us assume that for some and for all so that
Taking norm , we get
Solving the above inequality for yields
If , then inequality (3.9) holds. This completes the proof. □
4 Examples
Example 4.1 Consider the following fractional Caputo-Langevin equation with Riemann-Liouville fractional integral conditions:
Here , , , , , , , , , , , , , , , , , , , , , , , , , and . Since , then (H1) is satisfied with . We can find that
Therefore, we have
Hence, by Theorem 3.1, problem (4.1) has a unique solution on .
Example 4.2 Consider the following fractional Caputo-Langevin equation with Riemann-Liouville fractional integral conditions:
Here , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and . Since , then (H2) is satisfied with such that . We can find that
and . Therefore, we have
Hence, by Theorem 3.2, problem (4.2) has a unique solution on .
Example 4.3 Consider the following fractional Caputo-Langevin equation with Riemann-Liouville fractional integral conditions:
Here , , , , , , , , , , , , , , , , , , , , , , , , , , , , and .
Since , then (H3) is satisfied. We can find that
This means that . By Theorem 3.3 problem (4.3) has as least one solution on .
Example 4.4 Consider the following fractional Caputo-Langevin equation with Riemann-Liouville fractional integral conditions:
Here , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and . Then we can find that
Clearly,
By choosing and , we can show that
which implies . By Theorem 3.5, problem (4.4) has at least one solution on .
Example 4.5 Consider the following fractional Caputo-Langevin equation with Riemann-Liouville fractional integral conditions:
Here , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , and . By a direct computation, we have
Choosing and , we can show that
which satisfies (H6). By Theorem 3.6, problem (4.5) has at least one solution on .
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Acknowledgements
The research of JT is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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Yukunthorn, W., Ntouyas, S.K. & Tariboon, J. Nonlinear fractional Caputo-Langevin equation with nonlocal Riemann-Liouville fractional integral conditions. Adv Differ Equ 2014, 315 (2014). https://doi.org/10.1186/1687-1847-2014-315
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DOI: https://doi.org/10.1186/1687-1847-2014-315