SOLVABILITY FOR A NONLINEAR THREE-POINT BOUNDARY VALUE PROBLEM WITH p-LAPLACIAN-LIKE OPERATOR AT RESONANCE

Let φ be an odd increasing homeomorphism from R onto R which satisfies φ(0)= 0 and let f : [a,b]×R×R → R be a function satisfying Carathéodory conditions. Separated two-point and periodic boundary value problems containing the nonlinear operator (φ(u′))′, or its more particular form, the so-called p-Laplace operator, have received a lot of attention lately (cf. [6, 7, 8, 14, 15] and the references therein). On the other hand, three-point (or m-point) boundary value problems for the case when (φ(u′))′ = u′′, that is, the linear operator, have been considered by many authors (cf. [3, 9, 10, 12, 13]). The purpose of this paper is to study the following three-point boundary value problem which contains the nonlinear operator (φ(u′))′, ( φ(u′) )′ = f (t,u,u′), u′(a)= 0, u(η)= u(b), (1.1)