tailieunhanh - Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 279752, 11 pages

Hindawi Publishing Corporation Boundary Value Problems Volume 2011, Article ID 279752, 11 pages doi: Research Article Positive Solutions for Third-Order p-Laplacian Functional Dynamic Equations on Time Scales Changxiu Song and Xuejun Gao School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, China Correspondence should be addressed to Changxiu Song, scx168@ Received 31 March 2010; Revised 8 December 2010; Accepted 9 December 2010 Academic Editor: Daniel Franco Copyright q 2011 C. Song and X. Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is. | Hindawi Publishing Corporation Boundary Value Problems Volume 2011 Article ID 279752 11 pages doi 2011 279752 Research Article Positive Solutions for Third-Order p-Laplacian Functional Dynamic Equations on Time Scales Changxiu Song and Xuejun Gao School of Applied Mathematics Guangdong University of Technology Guangzhou 510006 China Correspondence should be addressed to Changxiu Song scx168@ Received 31 March 2010 Revised 8 December 2010 Accepted 9 December 2010 Academic Editor Daniel Franco Copyright 2011 C. Song and X. Gao. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. The authors study the boundary value problems for a p-Laplacian functional dynamic equation on a time scale ỷp xAV t v a f f x f x p f 0 t e 0 T x0 t y t t e -r 0 xA 0 xAV 0 0 x T B0 xA n 0. By using the twin fixed-point theorem sufficient conditions are established for the existence of twin positive solutions. 1. Introduction Let T be a closed nonempty subset of R and let T have the subspace topology inherited from the Euclidean topology on R. In some of the current literature T is called a time scale or measure chain . For notation we shall use the convention that for each interval of J of R J will denote time scales interval that is J J n T. In this paper let T be a time scale such that -r 0 T e T. We are concerned with the existence of positive solutions of the p-Laplacian dynamic equation on a time scale p xAV t a t f x t x p t Ỵ 0 t e 0 T Xo t y t t e -r 0 xA 0 xAV 0 0 x T B0 xA n 0 where ộpfù is the p-Laplacian operator that is ộpfù u p 2u p 1 ộp 1 u q u where 1 p 1 q 1 n e 0 p T and H1 the function f R 2 R is continuous H2 the function a T R is left dense continuous . a e Cld T R and does not vanish identically on any closed subinterval of 0 T . Here Cld T R denotes the set of all left dense continuous .