n-th Order Functional Problems with Resonance of Dimension One

Abstract

We consider the nonlinear n-th order boundary value problem Lu=u(n)=f(t,u(t),u′(t),…,u(n−1)(t))=Nu given arbitrary bounded linear functional conditions Bi(u)=0, i=1,…,n and develop a method that allows us to study all such resonance problems of order one, as well as implementing a more general constructive method for deriving existence criteria in the framework of the coincidence degree method of Mawhin. We demonstrate applicability of the formalism by giving an example for n=4.