Abstract
In this paper we study a new class of functions, which we call (omega ,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 (omega ,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 (omega ,c)-pseudo periodic solutions to the Lasota–Wazewska equation with unbounded oscillating production of red cells.
Highlights
Note that when c = 1 we obtain the space of pseudo periodic functions defined in [28, Definition 2 p. 873], and when c = –1 we obtain the space of pseudo anti-periodic functions defined in [27]
We prove a convolution theorem and that the space of (ω, c)pseudo periodic functions is a Banach space with the norm · pωc defined below
We prove the existence of positive (ω, c)-pseudo periodic solutions to the Lasota–Wazewska equation with (ω, c)-pseudo periodic coefficients y (t) = –δy(t) + h(t)e–a(t)y(t–τ), t ≥ 0
Summary
Note that when c = 1 we obtain the space of pseudo periodic functions defined in [28, Definition 2 p. 873] (see [16]), and when c = –1 we obtain the space of pseudo anti-periodic functions defined in [27]. Definition 2.5 A function f ∈ C(R, X) is said to be (ω, c)-pseudo periodic if f = g +h where g ∈ Pωc(R, X) and h ∈ AA0,c(X). The collection of those functions (with the same c-period ω for the first component) will be denoted by PPωc(X). (a) is integrable, or (b) Lp-integrable for 1 < p < ∞, or (c) asymptotic at t in –∞ and ∞
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