- Open Access
Positive solution of singular fractional differential system with nonlocal boundary conditions
© Wu et al.; licensee Springer. 2014
Received: 20 August 2014
Accepted: 4 December 2014
Published: 22 December 2014
In this paper, we consider the existence of positive solutions for a singular fractional differential system involving a nonlocal boundary condition which is given by a linear functional on with a signed measure. By looking for the upper and lower solutions of the system, the sufficient condition of the existence of positive solutions is established; some further cases are discussed. This is proved in the case of strong singularity and with a signed measure.
where , and are the standard Riemann-Liouville derivatives, and denote the Riemann-Stieltjes integral, where A, B are functions of bounded variation. are continuous and may be singular at and .
In system (1.1), the boundary condition is given by a nonlocal condition involving a Stieltjes integral type linear functional on with a signed measure, but it does not need to be a positive functional. In particular, if or , then the BVP (1.1) reduces to an integral boundary value problem, and thus it also includes the multi-point boundary value problem as a special case. So the problem with Stieltjes integral boundary condition contains various boundary value problems (see ).
where , , , , and are nonnegative. is allowed to be singular at , and f may be singular at . By using the fixed point index theorem, the existence of positive solutions for the BVP (1.3) is established.
where , , , is the Hadamard fractional derivative of fractional order, is the Hadamard fractional integral of order γ and are continuous functions. The existence of solutions for the system (1.4) is derived from Leray-Schauder’s alternative, whereas the uniqueness of the solution is established by the Banach contraction principle. More recently, Ahmad et al.  studied the existence of solutions for a system of coupled hybrid fractional differential equations with Dirichlet boundary conditions. By using the standard tools of the fixed point theory, the existence and uniqueness results were established.
Motivated by the above work, we consider the existence of positive solutions for the singular fractional differential system with nonlocal Stieltjes integral boundary conditions when f, g can be singular at and . It is well known from linear elastic fracture mechanics that the stress near the crack tip exhibits a power singularity of , where r is the distance measured from the crack tip, and this classical singularity also exists in nonlocal nonlinear problems. But due to the singularity of f, g at , we cannot handle the system (1.1) like in [4, 5]. Thus, this work we shall devote to finding the upper and lower solution of the system (1.1), and by means of the Schauder fixed point theorem to establish the criterion of the existence of positive solutions for the system (1.1). To the best of our knowledge, there has been no work done for the singular fractional differential system with the Riemann-Stieltjes integral boundary conditions, and this work aims to contribute in this field. Our work also extends the results of [4–6, 9] to fractional systems with which f, g can be singular at and .
2 Preliminaries and lemmas
for any . By a positive solution of problem (1.1), we mean a pair of functions satisfying (1.1) with , for all and .
Now we begin our work based on theory of fractional calculus; for details of the definitions and semigroup properties of Riemann-Liouville fractional calculus, one refers to [15–17]. In what follows, we give the definitions of the lower and upper solution of the system (1.1).
Remark 2.1 Normally, it is difficult to find the lower solution and upper solution of the system (1.1). In Theorem 3.1 of this paper, we will give a general strategy to find the lower solution and upper solution of the system (1.1) through a series of integral calculations form the initial value .
According to the strategy of , we can get easily the Green functions of the corresponding linear boundary value problem for the system (1.1).
, for all .
- (2)There exist two constants λ, μ such that(2.7)
Proof (1) is obvious. We only prove the first inequality of (2.7), the proof of second one is similar to those of the first one.
The proof is completed. □
Lemmas 2.1 and 2.2 lead to the following maximum principle.
Lemma 2.4 (Schauder fixed point theorem)
is bounded. Then T has a fixed point.
3 Main results
We make the following assumptions throughout this paper:
(H0) A and B are functions of bounded variation satisfying for and ;
Remark 3.1 The conditions (H1)-(H2) imply that f, g have a powder singularity at , and some typical functions are , , with and , , .
In particular, if , then is positive solution of the system (1.1).
We claim that T is well defined and .
It follows from (3.12) and (3.14)-(3.18) that , are lower and upper solutions of the system (1.1), and .
Obviously, a fixed point of the operator is a solution of the BVP (3.21).
So , which implies that is uniformly bounded. In addition, it follows from the continuity of , and the uniform continuity of , , and (H1) that is continuous.
Let be bounded, by standard discuss and the Arzela-Ascoli theorem, we easily know is equicontinuous. Thus is completely continuous, and by using Schauder fixed point theorem, has at least a fixed point such that .
i.e., on , which contradicts . Thus we have on . In the same way, on . Consequently, (3.22) is satisfied; then is a positive solution of the problem (1.1).
The proof is completed. □
4 Further results
In this section, we discuss some special case for system (1.1) and obtain some further results. We firstly discuss that f, g have no singularity at , but can be singular at .
Theorem 4.1 Suppose (H0) holds, and f, g satisfies
Clearly, is well defined.
Thus (4.5) and (4.7)-(4.10) imply that , are lower and upper solutions of the system (1.1), and .
Thus the rest of proof is similar to those of Theorem 3.1. □
Next, if f, g have no singularity at and , we copy the proof of Theorem 4.1, and we have the following interesting result.
Clearly, for also hold.
By Theorem 3.1, the system (5.1) with boundary condition (5.2) has at least a positive solution .
Thus by Theorem 4.1, the system (5.3) with boundary condition (5.2) has at least a positive solution .
Remark 5.1 In this work, the monotone assumption of f and g is an essential condition. In particular, for nonsingular case, the result is interesting since only monotone assumption is requested, which meets a large classes of functions.
The authors were supported financially by the National Natural Science Foundation of China (11371221).
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