Open access

DOI: 10.24193/fpt-ro.2019.1.06

Authors: T.A. Burton

Abstract: In this note we show a simple way to obtain a unique solution on [0, ∞) of a scalar integral equation



where x, y ∈ ℜ 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 ∈ ℜ 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 → ∞.

Key Words and Phrases: Progressive contractions, integral equations, existence, uniqueness, fixed points.

2010 Mathematics Subject Classification: 45D05, 45G05, 47H09, 47H10.

Published on-line: February 1st, 2019.

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