Almost Periodic and Periodic Solutions of Difference Equations

I t is easy to see that for every point (y, N) in WXI there is a solution (n) of (1) that satisfies <1>(N) =y. This solution is defined and unique on some set N^nKN*» where N<» is maximal. (That is, either iV» = °o or ^ i V ^ — l) (£W.) The solution may or may not be continuable for nƒ> rc)> 0^w(y,f, 0) = y. One can view the f unction ƒ defined on WXI as the restriction of a continuous function defined on WXR, where R denotes the real numbers. This viewpoint will be assumed in the sequel. Our problem is to find sufficient conditions such that Equation (1) has a periodic or almost periodic solution. We shall consider only functions ƒ (y, t) tha t are uniformly almost periodic. This means that ƒ is uniformly continuous on each set KXRy where K is compact in W, and for each x, ƒ(#, t) is Bohr almost periodic in t. Our principal result states that under certain stability conditions Equation (1) has an almost periodic solution. If ƒ is periodic in /, with integral period, then we are able to prove the existence of a periodic solution. Let ƒ be a uniformly almost periodic function defined on W XI and let {ƒ&: kÇzl] be the space of translates of/, where ƒk (y, n) =f(y, k+n). The hull of/, H (J), is defined to be the closure of {fk: & £ / } in the topology of uniform convergence on sets of the form KXI, where K is compact in W. (For the uniformly almost periodic functions this is the same as the closure of {ƒ&: & £ / } in the compact-open topology [8].) In addition to Equation (1) we shall be interested in the solutions of