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Second‐order boundary value problems with nonhomogeneous boundary conditions (I)
Author(s) -
Kong Lingju,
Kong Qingkai
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410234
Subject(s) - mathematics , disjoint sets , boundary value problem , mathematical analysis , order (exchange) , nonlinear system , boundary (topology) , plane (geometry) , value (mathematics) , geometry , physics , finance , quantum mechanics , economics , statistics
We study the nonlinear boundary value problem with nonhomogeneous multi‐point boundary condition$$ u \prime \prime + f(t, u, u\prime) = 0\, , \quad t \in (0, 1) \, , $$$$ u(0) - \sum ^{m} _{i=1} a _{i} u(t_{i}) = \lambda _{1}\, , \quad u(1) - \sum ^{m} _{i=1} b_{i}u(t_{i}) = \lambda_{2} \, . $$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)