By John E Littlewood

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5 Let f i E L~ (I) and let ki E L+ (1 • 1), and let ui be the unique L2 solution of t f~ ui(t) -- f i(t) "Jr-/ ki(t, s)ui(s) ds, 0 where i -- 1, 2 . . . n. e. on I. i=1 E] P r o o f : Since f i E L+(I) and ki E L+(I x I), then F ~ L2(I) and k u(t) exists. e. on I is by induction on n. The theorem is obvious for n - 1. Assume its truth for i - 1, 2 , . . , n - 1. Let n-1 t n-1 v~t~ = ~ f i(t~ + f }~ ki(t, s~u(~l~ . i=1 0 i=1 Therefore n-1 t Un(t) q- v(t) = f n (t) + Z f i(t) + [ kn(t, s)u. (s) ds ,!

5) and splitting we get f(t)

F i(t) > O, pi(t) > O, i = 1, 2..... n, f Pk-l(Sk-1) "-' "ilpk(sk)fk(Sk) 0 0 • dsk dsk_l . . , ds2 (iS1. o LINEAR INTEGRAL INEQUALITIES 54 In the following three theorems some basic inequalities given by Pachpatte (1988a, in press j) are presented. 1 Let u > O, h >_ O, rj(t) > O, j - l , 2 . . , n - l , g(t) > 0 be continuous functions defined on R+ and uo > 0 be a constant. 8) for t ~ R+, then for t ~ R+. (a2) If for t ~ R+, then for t ~ R+. (a3) If for t ~ R+, then for t ~ R+. (a4) If for t ~ R+, then for t ~ R+.