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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SynopsisWe study the planar delay differential equation x′(t) = −x(t) + αF(x(t − 1)), for α > 0. An existence theorem for nonconstant periodic solutions is achieved for a certain class of maps F, for α > some α0. Besides a condition of nondegeneracy at x = 0, we assume F is bounded and satisfies a kind of planar negative feedback condition. The nonconstant periodic solutions are associated with nontrivial fixed points of a certain operator defined by the flow in the plase space C([−l, 0], R2). In our approach, the existence of such fixed points depends on the ejectivity of O ϵ C([−1, 0], R2) with respect to that operator. Relaxing the boundedness condition on F, we show the existence of a sequence of values of α, α0 < α1 <…, where a Hopf bifurcation occurs.