On higher order fully periodic boundary value problems

Abstract

a b s t r a c tIn this paper we present sufficient conditions for the existence of periodic solutions of the higher order fully differential equation→ R a continuous function verifying a Nagumotype growth condition.A new type of lower and upper solutions, eventually non-ordered, allows us to obtain, not only the existence, but also some qualitative properties on the solution. The last section contains two examples to stress the application to both cases of n odd and n even.

Bifurcation Results for Periodic Third-Order Ambrosetti-Prodi-Type Problems

“…The study of the periodic orbits of differential equations is an important line of research, namely: to obtain sufficient conditions for the non-existence and multiplicity for strongly nonlinear differential equations [6]; the existence of periodic orbits as limit cycles [7], or as solutions of the φ-Laplacian generalized Liénard equations [8]; solvability of higher-order periodic problems with fully differential equations [9], and singular third order problems via cones theory [10]; equations with asymptotically sign-changed nonlinearities [11], or with anti-periodic boundary conditions [12]; oscillations of nonlinear even order differential equations [13].…”

Nontrivial Solutions of a Fully Fourth-Order Periodic Boundary Value Problem

“…However, most of the works in the above-mentioned references allow only having , or , , in the right-hand side nonlinear function ; see [2-11, 13, 15-18, 20-30]. The works on the fully nonlinear cases of which contains explicitly and every derivative of up to order three have been quite rarely seen; see [1,12,14,19].…”