Abstract. A variety of linear generalizations of Gronwall's inequality, including recent multivariable results of D. R. Snow and E. C. Young, are subsumed and extended by simple arguments involving the resolvent kernel of the integral operator."Everyone knows" that Gronwall's1 inequality [5] is but one example of an inequality for a monotone operator % in which the exact solution of w = a + %w provides an upper bound on all solutions of u < a + %u. Nevertheless, this idea is often neglected in deriving new variants of this classical inequality.Here Gronwall's inequality is generalized to systems of n linear inequalities in m variables by arguments that amount to manipulation of the resolvent kernel equation for %. These results encompass work of Chu and Metcalf [4], Snow [9], [10

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