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April 10, 2015 By

Non-negative solutions of a convolution equation

C. R. Math. Rep. Acad. Sci. Canada Vol. 28, (1), 2006 pp. 1–5

March 30, 2006

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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