Second-order boundary value problems with nonhomogeneous boundary conditions (I)

✍ Scribed by Lingju Kong; Qingkai Kong

• Publisher: John Wiley and Sons
• Year: 2005
• Tongue: English
• Weight: 250 KB
• Volume: 278
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410234

✦ Synopsis

Abstract

We study the nonlinear boundary value problem with nonhomogeneous multi‐point boundary condition

Sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relation. Using this existence result, under some assumptions, we obtain explicit ranges of values of λ~1~ and λ~2~ with which the problem has a solution, has a positive solution, and has no solution, respectively. Furthermore, we prove that the whole plane for λ~1~ and λ~2~ can be divided into two disjoint connected regions Λ^E^ and Λ^N^ such that the problem has a solution for (λ~1~, λ~2~) ∈ Λ^E^ and has no solution for (λ~1~, λ~2~) ∈ Λ^N^. We also show that under different assumptions, the problem has a solution for all (λ~1~, λ~2~) ∈ ℝ^2^. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)