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Positive solution of singular fractional differential system with nonlocal boundary conditions
Advances in Difference Equations volume 2014, Article number: 323 (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.
In this paper, we consider the existence of positive solutions for a singular nonlinear fractional differential system with nonlocal boundary conditions,
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 ).
Since the nonlocal boundary value problems can describe a class of very interesting and important phenomena arising from heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics, this type of problem has attracted much attention of many researchers (see [2–14] and the references therein). Especially, based on the fixed point theory of a strict set of contraction operators in a cone, Feng et al.  investigated the existence and nonexistence of positive solutions of the following second order BVPs with integral boundary conditions in Banach space:
Subsequently, Liu et al.  studied a singular integral boundary value problem,
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.
By means of a monotone iterative technique, Zhang and Han  established the existence and uniqueness of the positive solutions for a class of higher conjugate-type fractional differential equation with one nonlocal term,
where , is the standard Riemann-Liouville derivative, A is a function of bounded variation, denotes the Riemann-Stieltjes integral of u with respect to A, dA can be a signed measure. Recently, some work on systems of nonlinear fractional differential equations was developed [7–9]. In , Ahmad and Ntouyas studied the existence and uniqueness of solutions for a system of Hadamard type fractional differential equations with integral boundary conditions
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
The basic space used in this paper is , where ℝ is a real number set. Obviously, the space E is a Banach space if it is endowed with the norm as follows:
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).
Definition 2.1 A pair of functions is called a lower solution of the system (1.1), if it satisfies
Definition 2.2 A pair of functions is called an upper solution of the system (1.1), if it satisfies
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).
Lemma 2.1 Given and , then the following boundary value problems:
have the unique solution
where , are the Green functions of the BVP (2.4), respectively, and
Lemma 2.2 Let and for , then the Green functions defined by (2.6) satisfy
, for all .
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.
Since for any , we have
On the other hand, from (2.1), obviously,
then we have
The proof is completed. □
Lemmas 2.1 and 2.2 lead to the following maximum principle.
Lemma 2.3 If satisfies
and , for any . Then
Lemma 2.4 (Schauder fixed point theorem)
Let T be a continuous and compact mapping of a Banach space E into itself, such that the set
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 ;
(H1) are decreasing in second and third variables and such that
(H2) for all , there exist constants such that, for any ,
Remark 3.1 The conditions (H1)-(H2) imply that f, g have a powder singularity at , and some typical functions are , , with and , , .
Theorem 3.1 Suppose (H0)-(H2) hold. Then the system (1.1) has at least a positive solution , which satisfies
In particular, if , then is positive solution of the system (1.1).
Proof Define a cone
then P is nonempty since . Now let us denote an operator T by
We claim that T is well defined and .
In fact, for any , we have
So from Lemma 2.1 and (H1)-(H2), one gets
On the other hand, by Lemma 2.1 and (H1)-(H2), we also have
Thus it follows from (3.3)-(3.6) that T is well defined and . Moreover, by Lemma 2.2, we have
since , , we have
then by (3.8)-(3.11) and (H2), we have
Consequently, it follows from (3.7) and (3.10)-(3.13) that
It follows from (3.12) and (3.14)-(3.18) that , are lower and upper solutions of the system (1.1), and .
Define the functions , , and the operator in E by
It follows from the assumption that and are continuous. Consider the following boundary value problem:
Obviously, a fixed point of the operator is a solution of the BVP (3.21).
For all , by (3.19)-(3.20), we have
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 .
Now we prove
We firstly prove . Otherwise, suppose . According to the definition of , , we have
On the other hand, as is an upper solution of (1.1), we have
Let , , (3.23)-(3.24) imply that
On the other hand, since is an upper solution of the BVP (1.1) and is a fixed point of , we know
It follows from Lemma 2.3 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).
It follows from and (3.22) that
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
(H∗1) are decreasing in second and third variables and such that
Then the system (1.1) has at least a positive solution , which satisfies
Proof Similar to the proof of Theorem 3.1, we take the cone
Clearly, is well defined.
then by (4.1)-(4.4) and (H∗1), we have
Consequently, it follows from (4.5) and (4.6) that
Thus (4.5) and (4.7)-(4.10) imply that , are lower and upper solutions of the system (1.1), and .
On the other hand, by Lemma 2.2,
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.
Theorem 4.2 Suppose (H0) holds, , , , and are decreasing in second and third variables. Then the system (1.1) has at least a positive solution , which satisfies
Take functions of bounded variation,
Example 5.1 Suppose that , and , , . We consider the following singular fractional differential system:
subject to the nonlocal boundary condition
By a simple calculation, the system (5.1) with boundary condition (5.2) is equivalent to the following system with coefficients of both signs:
Clearly, for also hold.
Let , , then f, g are decreasing in x and y, and
Moreover, for all and , we have
By Theorem 3.1, the system (5.1) with boundary condition (5.2) has at least a positive solution .
Example 5.2 Consider the singular fractional differential system
subject to nonlocal boundary condition (5.2). Let
then f, g are decreasing in x and y, and
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.
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The authors were supported financially by the National Natural Science Foundation of China (11371221).
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between all authors. JW and XZ completed the main part of this paper, LL and YW corrected the main theorems and polished the manuscript. All authors read and approved the final manuscript.