- Research Article
- Open Access
Existence Results for Higher-Order Boundary Value Problems on Time Scales
© J. Liu and Y. Sang. 2009
- Received: 22 March 2009
- Accepted: 16 June 2009
- Published: 29 July 2009
- Boundary Value Problem
- Small Positive Number
- Singular Boundary
- Jump Operator
- Nonempty Closed Subset
Time scales and time-scale notationare introduced well in the fundamental texts by Bohner and Peterson [1, 2], respectively, as important corollaries. In, the recent years, many authors have paid much attention to the study of boundary value problems on time scales (see, e.g., [3–17]). In particular, we would like to mention some results of Anderson et al. [3, 5, 6, 14, 16], DaCunha et al. , and Agarwal and O'Regan , which motivate us to consider our problem.
with the same boundary conditions where is a positive parameter. They obtained some results for the existence of positive solutions by using the Krasnoselskii, the Schauder, and the Avery-Henderson fixed-point theorem.
and got many excellent results.
In this paper, by constructing one integral equation which is equivalent to the BVP (1.6) and (1.7), we study the existence of positive solutions. Our main tool of this paper is the following fixed-point index theorem.
Theorem 1.1 ().
The outline of the paper is as follows. In Section 2, for the convenience of the reader we give some definitions and theorems which can be found in the references, and we present some lemmas in order to prove our main results. Section 3 is developed in order to present and prove our main results. In Section 4 we present some examples to illustrate our results.
Throughout we assume that are points in , and define the time-scale interval . In this paper, we also need the the following theorem which can be found in .
We will discuss it from three perspectives.
Sufficiency. Suppose that
For convenience, we set
Proof of Theorem 3.1.
Proof of Theorem 3.2.
Proof of Theorem 3.3.
In this section, in order to illustrate our results, we consider the following examples.
By Theorem 2.1, we have
By simple calculations, we have
The authors would like to thank the anonymous referee for his/her valuable suggestions, which have greatly improved this paper.
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