Second order ordinary differential equations with fully nonlinear two point boundary conditions II

Abstract

We establish existence results for two point boundary value problems for second order ordinary differential equations of the form y″ = f(x,y,y′), x ∈ [0,1], where f satisfies the Carathéodory measurability conditions and there exist lower and upper solutions. We consider boundary conditions of the form G((y/(0),y/(1));(y′(0),y′(1))) = 0 for fully nonlinear, continuous G and of the form (y(i),y′(i)) ∈ J(i), i = 0,1 for closed connected subsets J(i) of the plane. We obtain analogues of our results for continuous f. In particular we introduce compatibility conditions between the lower and upper solutions and : (i) G; (ii) the J(i), i = 0,1. Assuming those compatibility conditions hold and, in addition, f satisfies assumptions guarenteeing a'priori bounds on the derivatives of solutions we show that solutions exist. As an application we generalise some results of Palamides.