The Method of Upper, Lower Solutions and Volterra Integral Equations

Abstract

**Please note that the full text is embargoed** ABSTRACT: In employing the method of upper and lower solutions to dynamical systems, one is required to impose a certain monotone property on the given system [5,6,11] When the given system does not possess such a monotonic property, stronger forms of upper and lower solutions have to be assumed in order to obtain similar results [4,6,9]. Furthermore, if the system enjoys a mixed monotone property the method of quasi-upper and lower solutions, which is recently introduced, is most useful [7]. In this paper we shall extend these ideas to Volterra integral equations. We want to note that Volterra integral equations and inequalities which can not be reduced to differential ones were first considered in [1,10], We shall first consider various aspects of Muller's type result and then develop monotone iterative technique to establish the existence of coupled quasi-minimal and maximal solutions.