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

摘要

In this note we show a simple way to obtain a unique solution on [0,?∞) of a scalar integral equation where x ,? y ?∈?? and t ?≥?0 imply that ∣ g ( t ,? x )??? g ( t ,? y )∣?≤? α ∣ x ??? y ∣,?0??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 ?→?∞ .