Boundary Value Problems
Volume 2011 (2011), Article ID 827510, 15 pages
doi:10.1155/2011/827510
Research Article

Positive Solutions for Integral Boundary Value Problem with ϕ-Laplacian Operator

Yonghong Ding

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 20 September 2010; Revised 31 December 2010; Accepted 19 January 2011

Academic Editor: Gary Lieberman

Copyright © 2011 Yonghong Ding. 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 consider the existence, multiplicity of positive solutions for the integral boundary value problem with 𝜙 -Laplacian ( 𝜙 ( 𝑢 ( 𝑡 ) ) ) + 𝑓 ( 𝑡 , 𝑢 ( 𝑡 ) , 𝑢 ( 𝑡 ) ) = 0 , 𝑡 ∈ [ 0 , 1 ] , ∫ 𝑢 ( 0 ) = 1 0 𝑢 ( 𝑟 ) 𝑔 ( 𝑟 ) d 𝑟 , ∫ 𝑢 ( 1 ) = 1 0 𝑢 ( 𝑟 ) ℎ ( 𝑟 ) d 𝑟 , where 𝜙 is an odd, increasing homeomorphism from ℝ onto ℝ . We show that it has at least one, two, or three positive solutions under some assumptions by applying fixed point theorems. The interesting point is that the nonlinear term 𝑓 is involved with the first-order derivative explicitly.