Successive approximations in ordered vector spaces and global solutions of nonlinear Volterra integral equations

Abstract

Conditions are found under which a nonlinear operator in an ordered topological vector space will have a fixed point. This result is applied to study a nonlinear Volterra integral operator in the space of continuous, real valued functions on [0, oo) equipped with the topology of uniform convergence on compact subsets. Two theorems on the global existence of solutions to the related Volterra integral equation as limits of successive approximations are proved in this manner. Introduction. In the first section of this paper, we establish conditions under which an operator equation Fx = x in a partially ordered topological vector space, X, will have a solution, unique in a certain class, as a limit of successive approximations. The conditions involve a related operator, G, for which certain inequalities hold, guaranteeing F is contractive in an appropriate space. The second and third sections are given over to two examples in which this general method is applied to establish existence of global solutions for the scalar nonlinear Volterra integral equation rt (0.1) x(t) = y(t)+J k(t, s)f(s, x(s)) ds, where X will be the space of continuous functions on [0, oo) under its natural ordering. In the example of ?1, the related operator, G, is effectively linear, whilef of (0.1) is required to satisfy a loose local Lipschitz condition similar to the type considered by Miller and Sell [7]. The example to which ?2 is devoted involves a nonlinear G, and the Lipschitz condition is replaced by a condition of the form ff(t, XI)-f(t, X2)J 0. The work in this article relates to some earlier work as follows. The underlying method is related in concept to the works of Corduneanu [2], [3], Lakshmikantham Received by the editors March 1, 1971. AMS 1970 subject classifications. Primary 47H15, 47H10, 45D05, 45G99.