W. ORLICZ [3] proved that the set of continuous €unctions f : (0, a] x R" -Rn, for which the differential equation x'=f(t, x) does not have uniqueness of all solutions, is a set of first category. The purpose of the paper is to prove similar theorems for VOLTERRA integral equations in BANACH spaces. Assume that hi is the set of positive integers, J=[O, a ] , E is a RANACH space with the norm 11 -11, T = ( ( t , s ) : O s s~t s i a ) , and m is a function from T into [ l , W ) such that 1 0 for any t c J the function m(t, -) is integrable on [ O , t ] and t sup J m(t, s) cis ec0 , t € J 0 and r 2" lim Denote by &I the set of all functions f : IT X E --E such that, 30 for any fixed t C J and xE E the function s -f(t, s , x) is strongly L-measurable m(z, s ) ds = 0 for fixed z or t . T -t -o f t on LO, tl; 5 0 sup Ilf(t, s, x)li ~m ( t , s ) for ( t , s)E T; 60 there exists a function (z, t , s) -+r(z, t , s) ( O s s s t t z z u ) such that the X E B I function s --r(t, t , s) is integrable on [ O , t ] for any fixed t , 7, lim ~( t , t , s ) ds = 0 r-t-.O+ 0 for fixed t or z, and sup llf(z, s, x) -f(t, .s, x)ll s r ( z , t , s) xEE Let D = T x E . For any f , gEN put It can be easily proved that ( M , d> is a complete metric space. Denote by L the set of all f C M such that for any zE E there exist a constant k: 2 0 and a neigh-20 Math. Nachr. Bd. 93
