Non-negative solutions of a convolution equation

C. R. Math. Rep. Acad. Sci. Canada Vol. 28, (1), 2006 pp. 1–5
(Received: 2005-08-16 , Revised: 2005-09-02)

Karol Baron, Instytut Matematyki, Uniwersytet Slaski, ul. Bankowa 14, PL–40–007 Katowice, Poland

Witold Jarczyk, Instytut Matematyki, Uniwersytet Slaski ul. Bankowa 14, PL–40–007 Katowice, Poland and Wydzia l Matematyki, Informatyki i Ekonometrii, Uniwersytet Zielonogorski, ul. Szafrana 4a, PL–65–516 Zielona Gora, Poland

Abstract/Résumé:

We show that any Lebesgue measurable function \(f \colon \mathbb{R} \to [0,\infty)\) satisfying \( f(x) = \int_0^{\infty} f(x+y) f(y) \,dy \) has the form \( f(x) = 2 \lambda e^{-\lambda x} \) with a \(\lambda \in [0,\infty)\).

Nous démontrons que toute fonction mesurable au sens de Lebesgue \(f \colon \mathbb{R} \to [0,\infty)\) satisfaisant à \( f(x) = \int_0^{\infty} f(x+y) f(y) \,dy \) est de la forme \( f(x) = 2 \lambda e^{-\lambda x} \) avec un \(\lambda \in [0,\infty)\).

Keywords: Lebesgue measurable and non-negative solution, convolution equation, integrated Cauchy equation, nonlinear integral equation

AMS Subject Classification: Other nonlinear integral equations 45G10

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