Multiple solutions for systems of differential equations with nonlinear boundary conditions

Abstract

We establish the existence of three solutions in admissible bounding sets for systems of nonlinear differential equations of the form $y''=f(x,y,y')$, $x \in [0,1]$ satisfying the fully nonlinear boundary conditions $g((y(0),y(1));(y'(0),y'(1)))=0.$ We assume that $f$ and $g$ are continuous, that $g$ is compatible with the admissible bounding sets, and that $f$ satisfies a Nagumo-type condition that guarantees a priori bounds on the derivatives of solutions. We use Leray-Schauder degree theory in novel spaces.