Existence, non-existence and multiplicity results for a third order eigenvalue three-point boundary value problem

Abstract

This paper provides sufficient conditions to guarantee the existence, non-existence and multiplicity of solutions for a third order eigenvalue fully differential equation, coupled with three point boundary value conditions. Although the change of sign, some bounds for the second derivative of the Green's function are obtained, which allow to define a different kind of cone that, as far as we know, has not been previously used in the literature. The main arguments are based on the fixed point index theory for bounded and unbounded sets. Some examples are presented in order to show that the different existence theorems proved are not comparable.

Existence and multiplicity results for some generalized Hammerstein equations with a parameter

“…In [5], the authors consider a third order three-point boundary value problem, whose solutions are the fixed points of the integral operator where G(t, s) is an explicit Green's function, verifying some adequate properties such that G(t, s) and ∂ G ∂ t (t, s) are bounded and non-negative in the square [0, 1] × [0, 1], but ∂ 2 G ∂ t 2 (t, s) could change sign, being non-negative in a subset of the square.…”

“…In [5], the authors consider a third order three-point boundary value problem, whose solutions are the fixed points of the integral operator…”