Equations with almost periodic functions

Denote by AP(G) (respectively ap(G)) the space of all almost periodic functions (respectively almost periodic measures) defined on a Hausdorff locally compact abelian group G and by M(µ) or Mx[µ(x)], the mean of the almost periodic measure µ. In this paper we study the solvability of the nonlinear problem ϕ ∈ AP(G), F ϕ(x), Mx ϕ(xy−1 )f(y)µ(y)
= H(x), x ∈ G. Using the theory of nearness, the theory of Fourier-Bohr series for almost periodic measures, and the Banach fixed point theorem, we establish an existence and uniqueness result for the above equation.