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: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\displaystyle f(x)=M_y\left[(A\circ f)(xy^{-1})\mu(y)\right], \quad x\in G.}$$\end{document}In this context μ is a positive almost periodic measure on G, A is a uniformly continuous function on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb R}}$$\end{document} and My[μ(y)] is the mean of μ. A more general equation which we investigate is the following \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\displaystyle f(x)=g(x)+\nu*f(x)+M_y\left[(A\circ f)(xy^{-1})\mu(y)\right], \quad x\in G,}$$\end{document}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.