Boundary Value Problems
Volume 2011 (2011), Article ID 867615, 17 pages
doi:10.1155/2011/867615
Research Article

Multiple Positive Solutions for Second-Order 𝑝 -Laplacian Dynamic Equations with Integral Boundary Conditions

Yongkun Li and Tianwei Zhang

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Received 13 July 2010; Revised 21 November 2010; Accepted 25 November 2010

Academic Editor: Gennaro Infante

Copyright © 2011 Yongkun Li and Tianwei Zhang. 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

We are concerned with the following second-order 𝑝 -Laplacian dynamic equations on time scales ( 𝜑 𝑝 ( 𝑥 Δ ( 𝑡 ) ) ) ∇ + 𝜆 𝑓 ( 𝑡 , 𝑥 ( 𝑡 ) , 𝑥 Δ ( 𝑡 ) ) = 0 , 𝑡 ∈ ( 0 , 𝑇 ) 𝕋 , with integral boundary conditions 𝑥 Δ ( 0 ) = 0 , ∫ 𝛼 𝑥 ( 𝑇 ) − 𝛽 𝑥 ( 0 ) = 𝑇 0 𝑔 ( 𝑠 ) 𝑥 ( 𝑠 ) ∇ 𝑠 . By using Legget-Williams fixed point theorem, some criteria for the existence of at least three positive solutions are established. An example is presented to illustrate the main result.