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 ) , A V ( a , x ( a ) , y ( a ) ) + B W ( a , x ( a ) , y ( a ) ) = C 1 , C V ( b , x ( b ) , y ( b ) ) + D W ( b , x ( b ) , y ( b ) ) = C 2 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 ] × R p × R q → R p g,f:[a,b] \times {R^p} \times {R^q} \to {R^p} are continuous, and V , W : [ a , b ] × R p × R q → R 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.