Approximation of Solutions of Operator Equations on the Half Line

Abstract

In this chapter we discuss existence and approximation for the nonlinear operator equation on the half line $$ y(t){\rm{ = F y(t) on [0,}}\infty ). $$ (9.1.1) Solutions will be sought in C([0, ∞]), R k ), k ∈ N + = {1, 2, …}. A particular example of (9.1.1) will be the nonlinear integral equation $$ y(t){\rm{ = h(t) + }}\smallint _0^\infty {\rm{ k(t,s,y(s)) ds for t }} \in {\rm{ [0,}}\infty ). $$ (9.1.2) Finite section approximations for (9.1.2) are given by $$ y(t){\rm{ = h(t) + }}\smallint _0^n {\rm{ k(t,s,y(s)) ds for t }} \in {\rm{ [0,}}\infty ). $$ (901.3) for n ∈ N +. Note that (9.1.3) n , for fixed n ∈ N +, determines y(t) for t > n in terms of y(x) for x ∈ [0, n] so in fact the finite section approximations reduce to integral equations on bounded intervals (we note as well that various discretization techniques, such as numerical integration, are available for the approximate solution of (9.1.3) n , n ∈ N + fixed). The technique which we present in this chapter to establish existence and approximation of solutions to (9.1.2) (or more generally (9.1.1)) involves using a new fixed point approach for equations on the half line (see [7, 10, 11, 12]) together with the well known notion of strict convergence (see [2, 3, 4]). The ideas presented were adapted from [2, 13].