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.