We characterize the weak almost periodicity of a vector-valued, bounded, continuous function. We show that if the range of the function is relatively weakly compact, then the relative weak compactness of its right orbit is equivalent to that of its left orbit. At the same time, we give the function some other equivalent properties. 1. Introduction. Let S be a semitopological semigroup, let ᐄ be...

Given a topological group G let C(G) denote the Banach space of bounded, continous real valued function on G. Eberlein [1] defined a function f ∈ C(G) to be weakly almost periodic if the weak closure of all of its translates is compact in the weak topology on C(G) — in other words, if fx(y) is defined to be f(yx−1) then the weak closure of {fx | x ∈ G} is weakly compact. The set of weakly almos...

In particular, (6) implies that the functions/i, • • • , /„ are all regular in the domain (7) a > 0, i wi\ < r, • • • ,| to„| < r. In these notations, the following theorem will be proved: Theorem. In the domain (7), let fx, • ■ • ,fnbe regular functions of the form (1) with uniformly almost periodic coefficient functions defined on (2) having Fourier expansions of the type (3), satisfying (4)....

We show that a completely multiplicative automatic function, which dos not have 0 as value, is almost periodic.

This paper studies some sufficient conditions for the existence and uniqueness of a pseudo almost periodic mild solution to abstract hyperbolic differential equations with infinite delay. Applications include the existence and uniqueness of a pseudo almost periodic mild solution to the heat equation.

L is a continuous function from R× into X and f is a continuous function from R to X . The theory of functional differential equations with infinite delay has been developed by several authors. The fundamental task of that kind of equations is the choice of phase space . Since the history function t→ ut is not continuous in general if u : (−∞,T] → X is continuous (where T > 0). The paper [9] co...

Given two finite sets of integers S ⊆ N \ {0} and D ⊆ N \ {0, 1}, the impartial combinatorial game i-Mark(S,D) is played on a heap of tokens. From a heap of n tokens, each player can move either to a heap of n − s tokens for some s ∈ S, or to a heap of n/d tokens for some d ∈ D if d divides n. Such games can be considered as an integral variant of Mark-type games, introduced by Elwyn Berlekamp ...