On a Volterra's Integral Equation of Functions with Values in Banach Space.

In this paper we shall consider an integral equation of functions with values in Banach spaces over t-he real field R briefly rB-spaces that iswhere U(e)A(xt) and B(x) are functiens taklng values in rB-spaceshere the variables xt are real And in the rBrmspace considered here the integral of functioRs is based on the theory of Bochner integyals 1) The purpose of this paper is to show that under suitable conditions the solutioxx ef (A) is analytic Next we shall apply the above result to a differential equation of some type 1PreliariesIn this sectiolt we summalize a few definitions notations and results in the ferm that we shall need We commence with Definition A series(xn)of elements of a normed space X is saldo be absolutey convergent if and only i f the series (ll xn fi) is coRvergent From this follows that alt absolutely convergent series is convergent Definition-2 Let A be any set and F an rB ralgebra (real Banach 31gebra) A mapping f of A into F is bounded if sup (il f(O liteA) is finite Let BE(A) be the set of all bounded mappings of A into F lf addition scaiar multiplication and multiplication ik this set are defined by the equations for anyrespectively and if in addition the topology is givelt by norm such as YfiiBsup(Ii f(t)gt Egi A) then it can be proved that B p(A) becomes an rB-algebra REMARK ln what follows the symbol ll r jiB denotes the supnorm on the ftmction space whose elemeRts are bounded functions in a given domain That the norm in BF(A) is a supnorm implies that absolute convergence of a sequence of elements of BF(A) means uniform and absolute convergence of the