Fractional Differential EquationsView this Special Issue
Positive Solution to Nonzero Boundary Values Problem for a Coupled System of Nonlinear Fractional Differential Equations
We consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of fractional differential equations. The differential operator is taken in the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, the existence and uniqueness of positive solution are obtained. Two examples are given to demonstrate the feasibility of the obtained results.
Fractional differential equation can describe many phenomena in various fields of science and engineering such as control, porous media, electrochemistry, viscoelasticity, and electromagnetic. There are many papers dealing with the existence and uniqueness of solution for nonlinear fractional differential equation; see, for example, [1–5]. In , the authors investigated a singular coupled system with initial value problems of fractional order. In , Su discussed a boundary value problem of coupled system with zero boundary values. By means of Schauder fixed point theorem, the existence of the solution is obtained. The nonzero boundary values problem of nonlinear fractional differential equations is more difficult and complicated. No contributions exist, as far as we know, concerning the existence of positive solution for coupled system of nonlinear fractional differential equations with nonzero boundary values.
In this paper, we consider the existence and uniqueness of positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations: where , are given functions, and is the standard Riemann-Liouville differentiation. By using Banach fixed point theorem and nonlinear differentiation of Leray-Schauder type, some sufficient conditions for the existence and uniqueness of positive solution to the above coupled boundary values problem are obtained.
The rest of the paper is organized as follows. In Section 2, we introduce some basic definitions and preliminaries used in later. In Section 3, the existence and uniqueness of positive solution for the coupled boundary values problem (1.1) will be discussed, and examples are given to demonstrate the feasibility of the obtained results.
2. Basic Definitions and Preliminaries
In this section, we introduce some basic definitions and lemmas which are used throughout this paper.
Remark 2.3 (see ). The following properties are useful for our discussion:(1), , ,(2)(3)
Lemma 2.4 (the nonlinear alternative of Leray and Schauder type ). Let be a Banach space with closed and convex. Let be a relatively open subset of with and let be a continuous and compact mapping. Then either(a)the mapping has a fixed point in , or(b)there exist and with . Consider then one has the following lemma.
Lemma 2.5. Let and , then is a solution of BVP (2.3) if and only if is a solution of the integral equation: where
Proof. Assume that is a solution of BVP (2.3), then by Remark 2.3, we have
By (2.3), we have
Therefore, we obtain
Conversely, if is a solution of integral equation (2.4), using the relation where is the smallest integer greater than or equal to [3, Remark 2.1], we have A simple computation showed . The proof is complete.
Let we call Green's function of the boundary value problem (1.1).
Lemma 2.6. Let , then the function is continuous and satisfies(1)(2)
Proof. It is easy to prove that is continuous on , here we omit it. Now we prove . Let
We only need to prove Since
set , we have
Then is decreasing on . Meanwhile,
Therefore, for Clearly so It is easy to show that , Hence, .
Similarly, The proof of is completed.
Let then, therefore, So, is decreasing with respect to . Similarly, is decreasing with respect to . Also and are increasing with respect to . We obtain that is decreasing with respect to for and increasing with respect to for .
With the use of the monotonicity of , we have Similarly, The proof of is completed.
3. Main Result
In this section, we will discuss the existence and uniqueness of positive solution for boundary value problem (1.1).
We define the space endowed with , endowed with .
For , let .
Define , then the cone .
Lemma 3.1. Suppose that and are continuous, then is a solution of BVP (1.1) if and only if is a solution of the integral equations Let be the operator defined as then by Lemma 3.1, the fixed point of operator coincides with the solution of system (1.1).
Lemma 3.2. Let and be continuous on , then defined by (3.2) is completely continuous.
Proof. Let , in view of nonnegativeness and continuity of functions , , and , we conclude that is continuous.
Let be bounded, that is, there exists a positive constant such that for all .
Let then we have Hence, . is uniformly bounded.
Since is continuous on , it is uniformly continuous on . Thus, for fixed and for any , there exists a constant , such that any and
Similarly, For the Euclidean distance on , we have that if are such that , then That is to say, is equicontinuous. By the means of the Arzela-Ascoli theorem, we have is completely continuous. The proof is completed.
Theorem 3.3. Assume that and are continuous on , and there exist two positive functions that satisfy
Then system (1.1) has a unique positive solution if
Proof. For all , by the nonnegativeness of and , we have . Hence, Similarly, We have, From Lemma 3.2, is completely continuous, by Banach fixed point theorem, the operator has a unique fixed point in , which is the unique positive solution of system (1.1). This completes the proof.
Theorem 3.4. Assume that and are continuous on and satisfy
Then the system (1.1) has at least one positive solution in
Proof. Let with , define the operator as (3.2).
Let , that is, . Then
Similarly, , so , . From Lemma 3.2 is completely continuous.
Consider the eigenvalue problem
Under the assumption that is a solution of (3.15) for a , one obtains Similarly, , so , which shows that . By Lemma 2.4, has a fixed point in . We complete the proof of Theorem 3.4.
This work was jointly supported by the Natural Science Foundation of Hunan Provincial Education Department under Grants 07A066 and 07C700, the Construct Program of the Key Discipline in Hunan Province, Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Foundation of Xiangnan University.
S. Q. Zhang, “Positive of solution for boundary-value problems of nonlinear fractional differential equations,” Electronic Journal of Differential Equations, vol. 36, pp. 1–12, 2006.View at: Google Scholar