Positive almost periodic solutions of some convolution equations

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.