Existence of positive solutions for a discrete fractional boundary value problem
© Wang et al.; licensee Springer. 2014
Received: 18 April 2014
Accepted: 9 September 2014
Published: 25 September 2014
This paper is concerned with the existence of positive solutions to a discrete fractional boundary value problem. By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established. Additionally, examples are provided to illustrate the effectiveness of the main results.
MSC:26A33, 39A10, 47H07.
where , denotes the Riemann-Liouville fractional difference operator, and for any number and each interval I of R, . We appeal to the convention that for any , where y is a function defined on . By using the Krasnosel’skii and Schaefer fixed point theorems, the existence results are established and two examples are also provided to illustrate the effectiveness of the main results.
The rest of the paper is organized as follows. In Section 2, we introduce some lemmas and definitions which will be used later. In Section 3, the existence of positive solutions for the boundary value problem (1.1) is investigated. In Section 4, two examples are provided to illustrate the effectiveness of the main results.
2 Basic definitions and preliminaries
Firstly we present here some necessary definitions and lemmas which are used throughout this paper.
Define for any t and v for which the right-hand side is defined. If is a pole of the gamma function and is not a pole, then .
Definition 2.2 
for . Define the v th fractional difference for by , and satisfies .
Lemma 2.3 
Let t and v be any numbers for which and are defined. Then .
Lemma 2.4 
for some , with .
Lemma 2.5 (The nonlinear alternative of Leray and Schauder )
the mapping T has a fixed point in ; or
there exist and with .
Lemma 2.6 
for and for ; or
for and for ;
then T has at least one fixed point in .
We state next the structural assumptions that we impose on (1.1).
(H1) Assume that the nonlinearity function is continuous.
(H2) Assume that there exist nonnegative continuous functions , , such that , , .
(H3) Assume that uniformly for .
(H4) Assume that uniformly for .
3 Existence results
In this section, we will establish the existence of at least one positive solution for problem (1.1). At first, we state and prove some preliminary lemmas.
By and Definition 2.1, we obtain .
Proof First of all, (3.3) implies that . Note that , we know .
The proof of Lemma 3.2 is completed. □
Let B be the collection of all functions with the norm .
In view of the continuity of f, it is easy to know that T is continuous. Furthermore, it is not difficult to verify that T maps bounded sets into bounded sets and equi-continuous sets. Therefore, in the light of the well-known Arzelá-Ascoli theorem, we know that T is a compact operator (see [11, 12]).
We have the following theorem.
which shows that .
It shows that . By Lemma 2.5, T has a fixed point in . The proof is completed. □
Lemma 3.4 Let T be the operator defined in (3.9) and K be the cone defined in (3.13). Then .
The conclusion of Lemma 3.4 holds. □
Theorem 3.5 Suppose that conditions (H1), (H3) and (H4) hold. Then problem (1.1) has at least one positive solution.
Proof We have already shown in Lemma 3.4. By condition (H3), we can select sufficiently small so that both and hold for all and , where .
It implies that T is a cone contraction on .
whenever , so that T is a cone expansion on .
In summary, we may invoke Lemma 2.6 to deduce the existence of a function such that , where is a positive solution to problem (1.1). The proof is completed. □
By a simple computation, we can obtain , , . Therefore, . The conditions of Theorem 3.3 hold, the boundary value problem (4.1) has at least one positive solution.
The conditions of Theorem 3.5 hold, the boundary value problem (4.2) has at least one positive solution.
This work was jointly supported by the Natural Science Foundation of China under Grants 11471278, the Natural Science Foundation of Hunan Province under Grants 13JJ3120 and 14JJ2133, and the Construct Program of the Key Discipline in Hunan Province. We would like to show our great thanks to the anonymous referee for his/her valuable suggestions and comments, which improve the former version of this paper and make us rewrite the paper in a more clear way.
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