Existence of Positive Solutions for Semipositone Higher-Order BVPS on Time Scales
© Y. Lin and M. Pei. 2010
Received: 4 December 2009
Accepted: 29 March 2010
Published: 10 May 2010
Throughout this paper, let be a time scale, for any , the interval defined as . Analogous notations for open and half-open intervals will also be used in the paper. We also use the notation to denote the real interval . To understand further knowledge about dynamic equations on time scales, the reader may refer to [1–3] for an introduction to the subject.
Various cases of BVP (1.1) and (1.2) have attracted a lot of attention in the literature. When , BVP (1.1) and (1.2) has been studied by many specialists. For example, Agarwal et al.  have established the existence of positive solutions for continuous case of the semipositone Sturm-Liouville BVPs. Erbe and Peterson  andHao et al.dealt with Sturm-Liouville BVPs on time scale of positone nonlinear term. In addition, Agarwal and O'Regan  obtained positive solution of second-order right focal BVPs on time scale by using nonlinear alternative of Leray-Schauder type. In 2005, Chyan and Wong  obtained triple solutions of the same BVPs with . Recently,Sun and Li[9, 10] investigated semipositone Dirichlet BVPs on time scale. For higher-order BVPs, continuous case of BVP (1.1) and (1.2) have been investigated by Agarwal and Wong , Wong and Agarwal  and Wong . The discrete positone case of BVP (1.1) and (1.2) has been tackled by using a fixed point theorem for mappings that are decreasing with respect to a cone in . Especially, time-scale case of (1.1) with four-point boundary condition has been studied by Liu and Sang . Besides, BVP (1.1) and (1.2) of nonlinear positone term which satisfied Nagumo-type conditions have been dealt with in . Motivated by the works mentioned above, the purpose of this paper is to tackle semipositone BVP (1.1) and (1.2). In fact, BVPs appeared in [7–14] can be looked at as special case of BVP (1.1) and (1.2) in this paper. For other related works, we also refer to [17–19].
The paper is outlined as follows. In Section 2, we will present some notations and lemmas which will be used later. In Section 3, by using Krasnoselskii's fixed point theorem in a cone, we offer criteria for the existence of positive solution of BVP (1.1) and (1.2).
In this section, we offer some notations and lemmas, which will be used in main results. Throughout this paper, we always use the following notations:
Lemma 2.3 (see ).
3. Main Results
In this section, by using Lemma 2.3, we offer criteria for the existence of positive solution of BVP (1.1) and (1.2).
Then BVP (1.1) and (1.2) has a positive solution.
Then BVP (1.1) and (1.2) has a positive solution.
Therefore, from Theorem 3.3, BVP(1.1) and (1.2) has a positive solution.
Finally we present an example to illustrate our result.
In Example 3.6, because nonlinear term may attain negative value, the result in  is not applicable.
The authors thank the referee for valuable suggestions which led to improvement of the original manuscript.
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