Positive solutions of Riemann-Stieltjes integral boundary problems for the nonlinear coupling system involving fractional-order differential
© Zhao and Gong; licensee Springer. 2014
Received: 4 July 2014
Accepted: 5 September 2014
Published: 29 September 2014
In this article, we study Riemann-Stieltjes integral boundary value problems of nonlinear fractional functional differential coupling system involving higher-order Caputo fractional derivatives. Some sufficient criteria are obtained for the existence, multiplicity, and nonexistence of positive solutions by applying fixed-point theorems on a convex cone. As applications, some examples are provided to illustrate our main results.
where , . , are the Caputo fractional derivatives of order , . , are continuous functions. The integrals from (1.2) are Riemann-Stieltjes integrals. are the function of bounded variation with and . To the best of our knowledge, the study of existence of positive solutions of nonlinear fractional differential system (1.1)-(1.2) has not been done.
The rest of this paper is organized as follows. In Section 2, we recall some useful definitions and properties, and present the properties of the Green’s functions. In Section 3, we give some sufficient conditions for the existence and nonexistence of positive solutions for boundary value problem (1.1)-(1.2). Some examples are also provided to illustrate our main results in Section 4.
For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the recent literature.
provided that the right-hand side is pointwise defined on .
where , provided that the right-hand side is pointwise defined on .
Lemma 2.1 (see )
for some, , where n is the smallest integer greater than or equal to α.
Here we introduce the following useful fixed-point theorems.
Lemma 2.2 (see )
, and, , or
holds. Then T has at least one fixed point in.
Let E be a real Banach space with a cone . Define a partial order ≺ in E as if . For , the order interval is defined as .
Lemma 2.3 (see )
Let P be a normal cone in a real Banach space E, andbe an increasing operator. If T is completely continuous, then T has a fixed point.
Now we present the Green’s functions for system associated with BVPs (1.1)-(1.2).
where and are defined by (2.2) and (2.3).
Similar to the above argument, we get , that is , which mean that the solution for BVPs (2.1) is unique. The proof is complete. □
, for all.
, for all, where, .
- (3)for all, and for every, we have
Clearly, for which indicates is increasing with respect to . Therefore, for .
- (2)From (2.2), we have
- (3)From (2.4), for , we have . Thus, is decreasing with respect to . Therefore, for , we have
The proof of Lemma 2.5 is complete. □
From Lemma 2.5, we have the following lemma.
Lemma 2.6 Ifis a nondecreasing function and, then the Green’s functions, of BVPs (2.1) are continuous onand satisfyfor all. Moreover, ifsatisfiesfor all, then the unique solutionof BVPs (2.1) satisfies, , for alland.
3 Existence and nonexistence of positive solutions
In this section, we will discuss the existence and nonexistence of positive solutions to the BVPs (1.1)-(1.2) under various assumptions on f and g.
We present the assumptions that we shall use in the sequel.
(H1) are nondecreasing functions, , .
(H2) The functions , are continuous and for all .
Theorem 3.1 Assume that (H1)-(H2) hold. Assume, and. Then BVPs (1.1)-(1.2) have at least a pair of positive solutions.
Let . Define the operator the same as (3.3). We shall prove the theorem through two steps.
Thus, we show that and is uniformly bounded.
By (3.7), (3.11), and condition (i) of Lemma 2.2, we know that T has at least one fixed point . Consequently, BVPs (1.1)-(1.2) have at least a pair of positive solution , here , . The proof is complete. □
Similarly, we can get the following theorem.
Theorem 3.2 Assume that (H1)-(H2) hold. Assumeand. Then BVPs (1.1)-(1.2) have at least a pair of positive solution.
Then BVPs (1.1)-(1.2) have at least two pairs of positive solutions.
By (3.17), (3.19), and condition (ii) of Lemma 2.2, we know that T has at least a fixed point in , that is, . Equations (3.18) and (3.19) together with condition (i) of Lemma 2.2 imply that T has at least one fixed point , namely, . It is worth noting that , and (3.19) is a strict inequality, that is to say, the operator T has not the fixed point on the boundary . So we conclude that BVPs (1.1)-(1.2) have at least two pairs of positive solutions and with the properties of and (). The proof is complete. □
Similarly, we get the following theorem.
Then BVPs (1.1)-(1.2) have at least two pairs of positive solutions.
Theorem 3.5 Assume that (H1)-(H2) hold. Further suppose thatandare nondecreasing functions with respect to each variable x, y, z, w for each, and there exist, satisfying, for, , . Then BVPs (1.1)-(1.2) have at least a pair of positive solutionsuch that, .
Proof Define the normal cone as (3.2) and the operator as (3.3). By the definition of T, it is easy to show that T is continuous. For any bounded subset of P, similar to the proof of (3.6) in Theorem 3.1, we know that which implies that P is relatively compact set in E. Hence is completely continuous.
Hence T is an increasing operator. By the assumptions , , we have . Since is completely continuous, by Lemma 2.3, T has one fixed point . Thus BVPs (1.1)-(1.2) have at least a pair of positive solution such that , . The proof is complete. □
Theorem 3.6 Assume that (H1)-(H2) hold. Assumeandfor, . Then BVPs (1.1)-(1.2) have no monotone positive solution.
Proof Define the cone as (3.2), the operator as (3.3) and the partial order ≤ on P as the proof of Theorem 3.5. By the definition of T, it is easy to show that T is continuous. For any bounded subset of P, similar to the proof of (3.6) in Theorem 3.1, we know that , which implies that P is relatively compact set in E. Hence is completely continuous.
which is a contradiction. Then BVPs (1.1)-(1.2) have no monotone positive solution. The proof is complete. □
Similarly, we obtain the following theorem.
Theorem 3.7 Assume that (H1)-(H2) hold. Ifandfor, . Then BVPs (1.1)-(1.2) have no monotone positive solution.
4 Illustrative examples
Clearly, . By a simple computation, we get , , , and , which implies that and . Hence BVPs (4.1)-(4.2) have at least a pair of positive solutions by Theorem 3.1.
Clearly, . By a simple computation, we obtain , , and , which shows that , , , , and .
Hence BVPs (4.1)-(4.2) have at least two pairs of positive solutions by Theorem 3.3.
The author would like to thank the anonymous referees for their useful and valuable suggestions. This work is supported by the National Natural Sciences Foundation of Peoples Republic of China under Grant (No. 11161025), Yunnan Province natural scientific research fund project (No. 2011FZ058).
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