tailieunhanh - báo cáo hóa học:" Research Article Existence and Multiplicity of Positive Solutions of a Boundary-Value Problem for Sixth-Order ODE with Three Parameters"

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 Existence and Multiplicity of Positive Solutions of a Boundary-Value Problem for Sixth-Order ODE with Three Parameters | Hindawi Publishing Corporation Boundary Value Problems Volume 2o10 Article ID 878131 13 pages doi 2010 878131 Research Article Existence and Multiplicity of Positive Solutions of a Boundary-Value Problem for Sixth-Order ODE with Three Parameters Liyuan Zhang and Yukun An Nanjing University of Aeronautics and Astronautics 29 Yudao st. Nanjing 210016 China Correspondence should be addressed to Liyuan Zhang binghaiyiyuan1@ Received 13 May 2010 Accepted 14 August 2010 Academic Editor Kanishka Perera Copyright 2010 L. Zhang and Y. An. 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 study the existence and multiplicity of positive solutions of the following boundary-value problem -u 6 - yuAi fu - au f t u 0 t 1 u 0 u 1 m 0 m 1 uW 0 uW 1 0 where f 0 1 X R R is continuous a f and Y e R satisfy some suitable assumptions. 1. Introduction The following boundary-value problem u 6 Au 4 Bu Cu - f x Ù 0 0 x L u 0 u L u 0 u L u 4 0 u 4 L 0 where A B and C are some given real constants and f x ù is a continuous function on R2 is motivated by the study for stationary solutions of the sixth-order parabolic differential equations du d6u d4u nd2u . . 6 A 4 B 2 f tx ú - dt dx6 dx4 dx2 This equation arose in the formation of the spatial periodic patterns in bistable systems and is also a model for describing the behaviour of phase fronts in materials that are undergoing a 2 Boundary Value Problems transition between the liquid and solid state. When f x Ù u-u3 it was studied by Gardner and Jones 1 as well as by Caginalp and Fife 2 . If f is an even 2L-periodic function with respect to x and odd with respect to u in order to get the 2L-stationary spatial periodic solutions of one turns to study the two points boundary-value problem . The 2L-periodic extension u of the odd extension of the .