Existence and Comparison Results for Nonlinear Volterra Integral Equations in a Banach Space
Abstract
**Please note that the full text is embargoed** ABSTRACT: In this paper we prove the existence of solutions to nonlinear Volterra integral equations in a Banach Space. A comparison theorem and the existence of maximal solutions are also obtained using the notion of ordering with respect to a cone. As is known, in infinite dimensional Banach spaces compactness-type conditions are needed to prove existence, whereas in finite dimensional cases these assumptions are not necessary. The results of the paper generalize the corresponding results of Nohel [6]. See also Miller [5] and Lakshmikantham and Leela [2]. Throughout this paper, E will denote a Banach space, and
[to,to + a] = J C R. We also use Be(xo) to denote the ball of radius
centered at xo , i.e., [see pdf for notation]. The equation
under consideration is(1.1) [see pdf for notation] where ^ C E open, xo :J ^ ^ and K:J x J x ^ ^ E.