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

Abstract

In this paper we study a new class of functions, which we call (ω, c)-pseudo periodic functions. This collection includes pseudo periodic, pseudo anti-periodic, pseudo 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 (ω, c)-pseudo 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 (ω, c)-pseudo periodic solutions to the Lasota-Wazewska equation with unbounded oscillating production of red cells.

Generalized $c$-almost periodic type functions in ${\mathbb{R}}^{n}$

“…Concerning the regular solutions of the inhomogeneous wave equations given by the d'Alembert formula, we would like to note that the analysis carried out in the issues [35, 2.2-2.3, Section 4] can be also used to justify the introduction of notion in Definition 2.1 and Definition 2. Suppose that D is any unbounded set in the plane R 2 such that (g [1] (•) ≡…”

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

“…e definitions and basic properties of (ω, c)-periodic and (ω, c)-pseudo-periodic functions were introduced and analyzed by Alvarez, Gómez, and Pinto in [108,109], motivated by some known results regarding the qualitative properties of the solution to Mathieu's linear differential equation y″(t) + [a − 2q cos 2 t]y(t) � 0, arising in modeling of railroad rails and seasonally forced population dynamics (ω > 0 and c ∈ C\ 0 { }). e linear delayed equations can have (ω, c)-periodic solutions as well (see, e.g., [109], Example 2.5).…”

“…e authors of [109] analyzed the existence and uniqueness of mild (ω, c)-periodic solutions to abstract semilinear integrodifferential equation (10). Furthermore, Alvarez, Castillo, and Pinto analyzed in [108] the existence and uniqueness of mild (ω, c)-pseudo-periodic solutions to the abstract semilinear differential equation of the first order:…”

$ (\omega,\mathbb{T}) $-periodic solutions of impulsive evolution equations

“…Chang et al [9] presented the existence of rotating periodic solutions of second order dissipative dynamical systems. Alvarez et al [4] studied (ω, c)-pseudo periodic functions and establish some sufficient conditions for the existence of positive (ω, c)-pseudo periodic solutions to the Lasota-Wazewska equation with unbounded oscillating production of red cells. Alvarez et al [5] studied (ω, c)-asymptotically periodic functions and give several properties of this class of functions as the convolution invariance.…”