Boundary Value Problems
Volume 2011 (2011), Article ID 279752, 11 pages
doi:10.1155/2011/279752
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

Received 31 March 2010; Revised 8 December 2010; Accepted 9 December 2010

Academic Editor: Daniel Franco

Copyright © 2011 Changxiu Song and Xuejun 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.

Abstract

The authors study the boundary value problems for a 𝑝 -Laplacian functional dynamic equation on a time scale, [ 𝜑 𝑝 ( 𝑥 Δ ∇ ( 𝑡 ) ) ] ∇ + 𝑎 ( 𝑡 ) 𝑓 ( 𝑥 ( 𝑡 ) , 𝑥 ( 𝜇 ( 𝑡 ) ) ) = 0 , 𝑡 ∈ ( 0 , 𝑇 ) , 𝑥 0 ( 𝑡 ) = 𝜓 ( 𝑡 ) , 𝑡 ∈ [ − 𝑟 , 0 ] , 𝑥 Δ ( 0 ) = 𝑥 Δ ∇ ( 0 ) = 0 , 𝑥 ( 𝑇 ) + 𝐵 0 ( 𝑥 Δ ( 𝜂 ) ) = 0 . By using the twin fixed-point theorem, sufficient conditions are established for the existence of twin positive solutions.