# Positive almost periodic solutions of some convolution equations

Positive almost periodic solutions of some convolution equations Consider a Hausdorff σ-compact, locally compact abelian group G. We are looking for positive almost periodic solutions of the following functional equation: $${\displaystyle f(x)=M_y\left[(A\circ f)(xy^{-1})\mu(y)\right], \quad x\in G.}$$ In this context μ is a positive almost periodic measure on G, A is a uniformly continuous function on $${{\mathbb R}}$$ and M y [μ(y)] is the mean of μ. A more general equation which we investigate is the following $${\displaystyle f(x)=g(x)+\nu*f(x)+M_y\left[(A\circ f)(xy^{-1})\mu(y)\right], \quad x\in G,}$$ where g is a positive almost periodic function on G, μ a positive almost periodic measure, ν a positive bounded measure and A a Lipschitz function. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positive almost periodic solutions of some convolution equations

, Volume 14 (4) – Aug 11, 2010
14 pages

/lp/springer_journal/positive-almost-periodic-solutions-of-some-convolution-equations-ZP3DlYa1gS
Publisher
SP Birkhäuser Verlag Basel
Copyright © 2010 by Springer Basel AG
Subject
Mathematics; Econometrics; Calculus of Variations and Optimal Control; Optimization; Potential Theory; Operator Theory; Fourier Analysis
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-010-0081-9
Publisher site
See Article on Publisher Site

### Abstract

Consider a Hausdorff σ-compact, locally compact abelian group G. We are looking for positive almost periodic solutions of the following functional equation: $${\displaystyle f(x)=M_y\left[(A\circ f)(xy^{-1})\mu(y)\right], \quad x\in G.}$$ In this context μ is a positive almost periodic measure on G, A is a uniformly continuous function on $${{\mathbb R}}$$ and M y [μ(y)] is the mean of μ. A more general equation which we investigate is the following $${\displaystyle f(x)=g(x)+\nu*f(x)+M_y\left[(A\circ f)(xy^{-1})\mu(y)\right], \quad x\in G,}$$ where g is a positive almost periodic function on G, μ a positive almost periodic measure, ν a positive bounded measure and A a Lipschitz function.

### Journal

PositivitySpringer Journals

Published: Aug 11, 2010

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