This paper is devoted to the study of the class of continuous and bounded functions f : [0, ∞) → X for which exists ω > 0 such that lim t→∞ (f (t + ω) − f (t)) = 0 (in the sequel called S-asymptotically ω-periodic functions). We discuss qualitative properties and establish some relationships between this type of functions and the class of asymptotically ω-periodic functions. We also study the existence of S-asymptotically ω-periodic mild solutions of the first-order abstract Cauchy problem in Banach spaces.

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