Existence of solutions of ordinary differential equations with generalized boundary conditions

Bernfeld, Stephen R.; Lakshmikantham, V.

doi:10.1090/S0002-9947-1973-0326043-2

Existence of solutions of ordinary differential equations with generalized boundary conditions
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by Stephen R. Bernfeld and V. Lakshmikantham PDF

Trans. Amer. Math. Soc. 182 (1973), 261-274 Request permission

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.

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