(ω,c)‐asymptotically periodic functions, first‐order Cauchy problem, and Lasota‐Wazewska model with unbounded oscillating production of red cells

Abstract

In this paper, we study a new class of functions, which we call ( , c)-asymptotically periodic functions. This collection includes asymptotically periodic, asymptotically antiperiodic, asymptotically Bloch-periodic, and unbounded functions. We prove that the set conformed by these functions is a Banach space with a suitable norm. Furthermore, we show several properties of this class of functions as the convolution invariance. We present some examples and a composition result. As an application, we prove the existence and uniqueness of ( , c)-asymptotically periodic mild solutions to the first-order abstract Cauchy problem on the real line. Also, we establish some sufficient conditions for the existence of positive ( , c)-asymptotically periodic solutions to the Lasota-Wazewska equation with unbounded oscillating production of red cells.

Multi-dimensional $ρ$-almost periodic type functions and applications

“…It is well known that a continuous function f : I → X is (ω, c)-periodic if and only if the function g(•) ≡ c −•/ω f (•) is periodic and g(t + ω) = g(t) for all t ∈ I; here, c −•/ω denotes the principal branch of the exponential function (cf. also [1] and [4]- [5]). In [29], the authors have recently analyzed various classes of (ω, c)-almost periodic type functions.…”

Almost Periodic Functions and Their Applications: A Survey of Results and Perspectives

“…e notions of antiperiodicity and Bloch periodicity are special cases of the notion of an (ω, c)-periodicity, which has also been analyzed in [110]. e authors of [109] analyzed the existence and uniqueness of mild (ω, c)-periodic solutions to abstract semilinear integrodifferential equation (10).…”

$(\omega ,c)$-Pseudo periodic functions, first order Cauchy problem and Lasota–Wazewska model with ergodic and unbounded oscillating production of red cells

“…When c = e ik/ω , we will call this set of functions pseudo Bloch-periodic functions. Also, it should be noted that the space of (ω, c)-periodic functions, asymptotically Bloch periodic functions (see [12,13]), and the space of (ω, c)-asymptotically periodic functions (which basically are sums of (ω, c)-periodic functions with continuous functions h such that c -t/ω h(t) goes to 0 as t goes to ∞, see [2,Definition 2.5]) are contained in the space of (ω, c)-pseudo periodic functions. For other works related to pseudo periodicity, see [10,14,25,26].…”