© F. Xu and Z. Meng. 2009
Received: 25 February 2009
Accepted: 2 June 2009
Published: 5 July 2009
We study the following third-order -Laplacian -point boundary value problems on time scales , , , , , where is -Laplacian operator, that is, , , , . We obtain the existence of positive solutions by using fixed-point theorem in cones. In particular, the nonlinear term is allowed to change sign. The conclusions in this paper essentially extend and improve the known results.
The theory of time scales was initiated by Hilger  as a mean of unifying and extending theories from differential and difference equations. The study of time scales has lead to several important applications in the study of insect population models, neural networks, heat transfer, and epidemic models, see, for example [2–6]. Recently, the boundary value problems with -Laplacian operator have also been discussed extensively in literature; for example, see [7–18]. However, to the best of our knowledge, there are not many results concerning the higher-order -Laplacian mutilpoint boundary value problem on time scales.
A time scale is a nonempty closed subset of . We make the blanket assumption that are points in . By an interval , we always mean the intersection of the real interval with the given time scale; that is .
author studied the existence of solutions for the nonlinear boundary value problem by using Krasnoselskii's fixed point theorem and Leggett and Williams fixed point theorem, respectively.
They established a corresponding iterative scheme for the problem by using the monotone iterative technique.
All the above works were done under the assumption that the nonlinear term is nonnegative. The key conditions used in the above papers ensure that positive solution is concave down. If the nonlinearity is negative somewhere, then the solution needs no longer to be concave down. As a result, it is difficult to find positive solutions of the -Laplacian equation when the nonlinearity changes sign. In particular, little work has been done on the existence of positive solutions for higher order -Laplacian -point boundary value problems with nonlinearity being nonnegative on time scales. Therefore, it is a natural problem to consider the existence of positive solution for higher order -Laplacian equations with sign changing nonlinearity on time scales. This paper attempts to fill this gap in literature.
2. Preliminaries and Lemmas
for all . If , is said to be right scattered, if , is said to be left scattered; if , is said to be right dense, and if , is said to be left dense. If has a right scattered minimum , define ; otherwise set . If has a left scattered maximum , define ; otherwise set .
By caculating, we can easily get (2.7). So we omit it.
The proof is completed.
This completes the proof.
Lemma 2.11 (see ).
3. Main Results
We now give our results on the existence of positive solutions of BVP (1.6).
Then, the BVP (1.6) has at least one positive solution.
Then, the BVP (1.6) has at least two positive solutions.
4. An Example
In the section, we present some simple examples to explain our results.
This project was supported by the National Natural Science Foundation of China (10471075, 10771117).
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