A Lipschitz boundary value problem for nonlinear ordinary differential equations on non compact intervals

Abstract

(L(T) is called the Lipschitz constant of T.) Many authors have been interested in BVP for ordinary differential equations mainly when J is a real compact interval: Conti [3] deals with boundary conditions of the form x E Q c C(J, Rd) for a differential equation as x’ = A(t)x +f(t) (I) or, more generally, x’ = A(t)x +f(t, x). (If) Opial [7] considers systems of differential equations as x’ = F(t, x) for which Lx = y is a linear boundary condition. For many other cases, often for the two points problem for a second order differential equation, wider references may be found in the expository papers of Conti [3], Opial[7], Jackson [S] and in the book of Bernfeld and Lakshmikantham [2]. We want also to recall the papers of Furi, Martelli and Vignoli [4], Anichini [l], Tiba [lo], for the case of a nonlinear boundary condition, and the paper of Kartsatos [6] for the case in which .I is a noncompact real interval.