Solvability for a nonlinear three-point boundary value problem withp-Laplacian-like operator at resonance

M García-Huidobro,R Manásevich,C P Gupta

doi:10.1155/s1085337501000550

Abstract

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 Caratheodory 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)

Highlights

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 ), (1.1)
We are interested in the case when problem (1.1) is at resonance, meaning by this that the associated three-point boundary value problem φ u (t) = 0 a < t < b, (1.2)
We introduce here a technical condition for the homeomorphism φ which will be used in Section 4 in order to guarantee some properties of the time-mapping for non-homogeneous operators

Summary

Introduction

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 ), (1.1). Our main purpose here is to obtain existence results with one-sided growth restrictions for the three-point boundary value problem. Conditions of this type have been considered by Schmitt [20], Mawhin and Ward [18], and Fernandes and Zanolin [4] for the periodic case and the second-order linear differential operator, by de Figueiredo and Ruf in [1] for the second-order linear differential operator and Neumann boundary conditions, and by Manásevich and Zanolin in [16] for the one-dimensional p-Laplacian and Dirichlet boundary value conditions.

Abstract formulation and a deformation lemma

First existence results