Abstract
In this paper, we study a class of (omega ,c)-periodic time varying impulsive differential equations and establish the existence and uniqueness results for (omega ,c)-periodic solutions of homogeneous problem as well as nonhomogeneous problem.
Highlights
1 Introduction It is well known that the concept of (ω, c)-periodic functions is the same of “affine-periodic functions” or “periodic of second kind”, which were introduced by Floquet [1] and have been studied in the past decades
We introduce a Banach space PC(R, R) = {x : R → R : x ∈ C((ti, ti+1], R), and x(ti–) = x(ti), x(ti+) exists ∀i ∈ N} endowed with the norm x = supt∈R |x(t)|
Lemma 2.5 Assume that the following conditions hold: (A1) a(·) is ω-periodic, i.e., a(t + ω) = a(t), ∀t ∈ R. (A2) Set t0 = 0 and ti < ti+1, i ∈ N
Summary
It is well known that the concept of (ω, c)-periodic functions is the same of “affine-periodic functions” or “periodic of second kind”, which were introduced by Floquet [1] and have been studied in the past decades. Alvarez et al [2] introduced a new concept of (ω, c)-periodic function by considering Mathieu’s equation z + [α – 2β cos(2t)]z = 0, and its solution satisfies z(t + ω) = cz(t), c ∈ C. (ω, c)-periodic functions become the standard ω-periodic functions when c = 1 and ω-antiperiodic functions when c = –1 For these particular cases, we refer readers to [3,4,5,6]. Alvarez et al [7] transferred the same idea to study (N, λ)-periodic discrete functions and established the existence and uniqueness of (N, λ)-periodic solutions to a class of Volterra difference equations with infinite delay. Li et al [9] transferred the similar idea to consider (ω, c)-periodic solutions impulsive differential systems. The main purpose of this paper is to derive existence and uniqueness results for (ω, c)periodic solutions of nonhomogeneous linear problem as well as homogeneous linear problem
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