Existence and Uniqueness of Solutions of Boundary Value Problems for Second Order Systems of Nonlinear Differential Equations

摘要

Questions of the existence and uniqueness of solutions of the two point boundary value problem y double prime + f)t,y,y') = O, y(a) = A, y(b) = B, where f(t,y,y') satisfies a Lipschitz condition, have a long history, agoing back to Picard, 1893. The problem is to determine, in terms of the Lipschitz constants, the best possible interval (a,b) on which there exists a unique solution of these equations. A second question of interest is, given that solutions of these equations are unique, i.e., given that there exists at most one solution, when does this imply that there exists at least one solution. This paper investigates these two types of questions for the second order system, x' = f sub 1 (t,x,y), y' = f sub 2 (t,x,y), subject to more general boundary conditions g sub 1 (x(a),y(a)) = c sub 1, g sub 2 (x(b),y(b)) = c sub 2. (Author)