Monotonicity of solutions of Volterra integral equations in Banach space

Abstract

for almost all t. In this definition it is assumed, of course, that the integral on the right-hand side of (1.2) is in the domain of A. In a recent paper [6], Friedman and Shinbrot have studied the equation (1.1) even in the more general case where xo and A depend on t and -, respectively. They proved theorems of existence, uniqueness, differentiability and asymptotic behavior of solutions. They also constructed fundamental solutions and derived asymptotic bounds for them. We recall [6] that if xo is in the domain of A4, for some ,u>0, then the solution of (1.1) is given by S(t)xo. The purpose of the present paper is to derive monotonicity theorems for solutions of (1.1). We shall generalize some of the monotonicity theorems of Friedman [2] (see also [4]) from the case X= R1 (R1 the one-dimensional Euclidean space) to the case where X is any Banach space. In ?2 we give some auxiliary results. These results are concerned with Volterra equations in one-dimension (i.e., X=R1). In particular, we study the behavior of the solutions with respect to a certain parameter. In ?3 we give an integral formula for S(t) in case A is a bounded operator. For A unbounded, we construct a fundamental solution as a limit of fundamental solutions corresponding to the bounded operators A(I+ A/n) -1. We prove that the fundamental solution coincides with the fundamental solution of [6, Chapter 1]