A product integral representation for a Gronwall inequality

Abstract

Introduction.This paper shows that if G and H are functions from RXR to R such that 1-G^c>0, G and H are integrable and have bounded variation on [a, b], f is bounded and f(x)^k + (LR)fxa(fH+fG) for xE [a, b] then (1) if G^O and H^O, then f(x) k JH (1+P)(1-G)-1 and ^ITa (1+PT)(1 -G)_1 is a solution of the inequality, (2) ifl-|G| ^c>Oand/^0,then/(x)^/fen*(l + |P|)(l-|G|)-1, and (3) if k^O, the requirement 1-G^c>0 cannot be relaxed.Also a Gronwall-type inequality is stated and proved for functions/, G and H which have ranges in a normed ring.The Main Theorem of Schmaedeke and Sell [3] is a special case of Theorem 4 of this paper.The linear function J(f) defined by Herod [2] is more general than the function J(f) = (LR)f(fH+fG) defined above; however, there are linear functions (LR)Jl(fH+fG) which will satisfy the hypothesis of Theorem 4 but will not satisfy the hypothesis of Herod's theorem.2. Definitions and preliminary theorems.For detailed definitions, see [l, p. 299].All sum and product integrals (represented by the symbol YLa G) are subdivision-refinement-type limits of appropriate