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0 A(t− s)f(s, x(s))ds where x, y ∈ < and t ≥ 0 imply that |g(t, x) − g(t, y)| ≤ α|x − y|, 0 < α < 1, and for each E > 0 there is a K > 0 so that x, y ∈ < and 0 ≤ t ≤ E imply |f(t, x) − f(t, y)| ≤ K|x − y|. We introduce a progressive contraction. The constant K is a function of E and, hence, may tend to infinity as E → ∞. The conclusion is that there is a single function ξ(t) satisfying the equation on [0,∞) without resorting to any of the classical translations and extensions of solutions which, in fact, must invoke Zorn’s Lemma and which can encounter difficulties as K →∞.

A note on existence and uniqueness for integral equations with sum of two operators: progressive contractions