Positive solutions to boundary value problems of fractional difference equation with nonlocal conditions
© Kang et al.; licensee Springer. 2014
Received: 29 July 2013
Accepted: 11 December 2013
Published: 7 January 2014
In this paper, we will use the Krasnosel’skii fixed point theorem to investigate a discrete fractional boundary value problem of the form , , , where , , is a continuous function, , are given functionals, where Ψ, Φ are linear functionals, and λ is a positive parameter.
MSC:26A33, 39A05, 39A12.
where , , , for each . are given functionals, and is continuous for each admissible i.
Extensive literature exists on boundary value problems of fractional difference equations [9–13]. Ferreiraa  provided sufficient conditions for the existence and uniqueness of solution to some discrete fractional boundary value problems of order less than 1. Goodrich [11–13] studied a ν order () discrete fractional three-point boundary value problem and semipositone discrete fractional boundary value problems.
where , , is a continuous function. , are given linear functionals and λ is a positive parameter. The boundary conditions (2)-(3) are generally called nonlocal conditions. Our analysis relies on the Krasnosel’kill fixed-point theorem to get the main results of problem (1)-(3).
The paper will be organized as follows. In Section 2, we will present basic definitions and demonstrate some lemmas in order to prove our main results. In Section 3, we establish some results for the existence of solutions to problem (1)-(3), and we provide an example to illustrate our main results.
Let us first recall some basic lemmas which plays an important role in our discussions.
Definition 2.1 
for any t and ν for which the right-hand side is defined. We also appeal to the convention that if is a pole of the Gamma function and is not a pole, then .
Definition 2.2 
Lemma 2.1 
Lemma 2.2 
for some , with .
Lemma 2.3 
Lemma 2.4 
, for each ;
for each ; and
- (iii)there exists a number such that
and is as given in Lemma 2.3.
Consequently, we see that is a solution of (1)-(3) if and only if is a fixed point of (5), as desired. □
Lemma 2.5 The function is strictly decreasing in t, for . In addition, , and . On the other hand, the function is strictly increasing in t, for . In addition, , and .
In a similar way, it may be shown that satisfies the properties given in the statement of this lemma. We omit the details. □
Corollary 2.1 Let . There are constants such that , , where is the usual maximum norm.
Lemma 2.6 
, , , ; or
, , , .
Then S has a fixed point in .
3 Main results
(S1) , .
(S2) , .
(S3) , .
(S4) , .
(G1) The functionals Ψ, Φ are linear. In particular, we assume that
(G2) We have both and , for each , and
(G3) Each of , , , is nonnegative.
Lemma 3.1 Assume that (G1)-(G3) hold, and let T be the operator defined in (5). Then .
It also shows that .
So, we conclude that , and the proof is complete. □
Then, problem (1)-(3) has at least one positive solution such that .
that is, for .
that is, for . By means of Lemma 2.6, there exists such that . □
An application of Lemma 3.2 leads to two distinct solutions of (1)-(3) which satisfy . □
The proof is similar to Theorem 3.1 and hence is omitted.
equations (1)-(3) have a positive solution.
That is, for and .
that is, for and .
, and y is defined on the time scale , f and ψ, ϕ satisfy conditions of Theorem 3.1.
A computation shows that . Then, for every , problem (18)-(20) has at least two positive solutions.
The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to an improvement of this paper. Project supported by the National Natural Science Foundation of China (Grant No. 11271235) and Shanxi Province (2008011002-1) and Shanxi Datong University Institute (2009-Y-15, 2010-B-01, 2013K5) and the Development Foundation of Higher Education Department of Shanxi Province (20101109, 20111020).
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