Existence of Solutions of Ordinary Differential Equations with Generalized Boundary Conditions

Abstract

An investigation of the existence of solutions of the nonlinear boundary value problem $x’ = f(t,x,y),y’ = g(t,x,y),AV(a,x(a),y(a)) + BW(a,x(a),y(a)) = {C_1},CV(b,x(b),y(b)) + DW(b,x(b),y(b)) = {C_2}$, is made. Here we assume $g,f:[a,b] \times {R^p} \times {R^q} \to {R^p}$ are continuous, and $V,W:[a,b] \times {R^p} \times {R^q} \to R$ are continuous and locally Lipschitz. The main techniques used are the theory of differential inequalities and Lyapunov functions.