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

Abstract

We investigate the solvability of a fully fourth-order periodic boundary value problem of the form $x^{(4)}=f(t,x,x^{\mathrm{\prime }},x^{\mathrm{\prime }\mathrm{\prime }},x^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}),$ $x^{(i)}(0)=x^{(i)}(T),\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }i=\mathrm{0,1},\mathrm{2,3},$ where $f:[0,T]\times {\mathbb{R}}^4\to \mathbb{R}$ satisfies Carathéodory conditions. By using the coincidence degree theory, the existence of nontrivial solutions is obtained. Meanwhile, as applications, some examples are given to illustrate our results.