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 sσ0, 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.

Keywords

• higher-order periodic problems
• lower and upper solutions
• Nagumo condition
• degree theory