tailieunhanh - Báo cáo hóa học: " Research Article Positive Solutions for Integral Boundary Value Problem with φ-Laplacian Operator Yonghong Ding"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Positive Solutions for Integral Boundary Value Problem with φ-Laplacian Operator Yonghong Ding | Hindawi Publishing Corporation Boundary Value Problems Volume 2o11 Article ID 827510 15 pages doi 2011 827510 Research Article Positive Solutions for Integral Boundary Value Problem with f-Laplacian Operator Yonghong Ding Department of Mathematics Northwest Normal University Lanzhou 730070 China Correspondence should be addressed to Yonghong Ding dyh198510@ 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. We consider the existence multiplicity of positive solutions for the integral boundary value problem with f-Laplacian f u t f t u t u t 0 t e 0 1 u 0 0 u r g r dr u 1 Jo u r h r dr where f 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. 1. Introduction We are interested in the existence of positive solutions for the integral boundary value problem f u W f t u t u t 0 t e 0 1 u 0 ufr g r dr u 1 u r h r dr o o where f f g and h satisfy the following conditions. H1 f is an odd increasing homeomorphism from R onto R and there exist two increasing homeomorphisms V1 and 2 of 0 to onto 0 to such that 1 u f v f uv 2 u f v Vu v 0. Moreover f f 1 e C1 R where f 1 denotes the inverse of f. 2 Boundary Value Problems H2 f 0 1 X 0 to X -TO to 0 to is continuous. g h e L1 0 1 are nonnegative and 0 J1 g t dt 1 0 J1 h t dt 1. The assumption H1 on the function ộ was first introduced by Wang 1 2 it covers two important cases ộ ù u and ộ ù u p-2u p 1. The existence of positive solutions for two above cases received wide attention see 3-10 . For