Operator Equations in Banach Spaces Relative to the Weak Topology

Abstract

A lot has been written on differentia] and integral equations in a Banach space relative to the strong topology over the last twenty years or so; see [11, 12] and their references (and also Chapter 8). However only a few results have been obtained for equations in a Banach space relative to the weak topology. The first paper [19] appeared in 1971. There Szep discussed in detail the abstract Cauchy problem

$$ \left\{ {\begin{array}{*{20}c} {y'{\rm{ = f(t'y) on [0}}{\rm{.T]}}} \\ {y(0) = y_0 } \\\end{array}} \right. $$

(10.1.1)

with: [0, T] × B → B a weakly-weakly continuous function and B a reflexive Banach space. The nonreflexive case was examined by Cramer, Lakshmikantham and Mitchell [5] and more recently by Kubiaczyk and Szufla [10] and Szufla [20]. Motivated by the above studies, our goal in this chapter is to present an existence theory for the general operator equation in Banach spaces relative to the weak topology. Using a well known fixed point theorem we will obtain a variety of existence principles for (10.1.2). Our general theory will include as particular cases the Cauchy problem (10.1.1) and the Volterra integral equation the integral in (10.1.3) is understood to be the Pettis integral. The theory presented in this chapter was adapted from O’Regan [13–16]. Also in this chapter we will discuss approximation of solutions to (10.1.2) using the notion of a set of collectively compact operators in a Banach space relative to the weak topology.