Bifurcation results for periodic third-order Ambrosetti-Prodi-type problems

Abstract

This paper presents sufficient conditions for the existence of a bifurcation point for nonlinear periodic third-order fully differential equations. In short, the main discussion on the parameter s about the existence, non-existence, or the multiplicity of solutions, states that there are some critical numbers σ0 and σ1 such that the problem has no solution, at least one or at least two solutions if s<σ0, s=σ0 or σ0>s>σ1, respectively, or with reversed inequalities. The main tool is the different speed of variation between the variables, together with a new type of (strict) lower and upper solutions, not necessarily ordered. The arguments are based in the Leray–Schauder’s topological degree theory. An example suggests a technique to estimate for the critical values σ0 and σ1 of the parameter.