On the existence and uniqueness of ( N , λ ) $(N,\lambda )$ -periodic solutions to a class of Volterra difference equations

Abstract

In this paper we introduce the class of (N, λ)-periodic vector-valued sequences and show several notable properties of this new class. This class includes periodic, anti-periodic, Bloch and unbounded sequences. Furthermore, we show the existence and uniqueness of (N, λ)-periodic solutions to the following class of Volterra difference equations with infinite delay: u(n + 1) = α n j=-∞ a(n-j)u(j) + f (n, u(n)), n ∈ Z, α ∈ C, where the kernel a and the nonlinear term f satisfy suitable conditions.

Existence of $$(N,\lambda )$$-Periodic Solutions for Abstract Fractional Difference Equations

“…We finish this section recalling the notion of (N, λ)-periodic sequences and their main properties. The notion of (N, λ)-periodic sequences was introduced in [6] as a discrete counterpart of the concept of (ω, c)-periodic functions defined in [10]. Definition 2.6 [6].…”

“…The notion of (N, λ)-periodic sequences was introduced in [6] as a discrete counterpart of the concept of (ω, c)-periodic functions defined in [10]. Definition 2.6 [6]. A vector-valued function f : Z → X is called (N, λ)periodic discrete function (or (N, λ)-periodic sequence) if there exist N ∈ N and λ ∈ C\ {0}, such that f (n + N ) = λf (n) for all n ∈ Z. N is called the λ-period of f .…”

“…In (1.1), A is a possibly unbounded operator defined on a Banach space X and f : Z × X → X is given. This class of (N, λ)-periodic functions was introduced in the reference [6] as the discrete counterpart of the notion of (ω, c)-periodic functions [10], a notion that has been studied by various authors, see, e.g., [8,9,[15][16][17][18][19]26] and [30]. It is worth noting that class of (N, λ)-periodic functions contains the classes of discrete periodic (λ = 1), discrete anti-periodic (λ = −1), discrete Bloch-periodic (λ = e ikN , k ∈ Z fixed), and unbounded functions.…”

“…It is worth noting that class of (N, λ)-periodic functions contains the classes of discrete periodic (λ = 1), discrete anti-periodic (λ = −1), discrete Bloch-periodic (λ = e ikN , k ∈ Z fixed), and unbounded functions. Existence and uniqueness of (N, λ)-periodic solutions for scalar models, such as Volterra difference equations with infinite delay, were recently investigated in [6]. Anticipating a growing theoretical and practical interest in this class of solutions, we study in this article the existence and uniqueness of solutions for the abstract Cauchy problem (1.1).…”

Hartman and Nirenberg type results for systems of delay differential equations under $ (\omega,Q) $-periodic conditions

“…Furthermore, various generalizations such as c-semiperiodic, c-almost periodic functions were considered in [12,13]. Many other extensions to impulsive, discrete or fractional differential equations have been investigated in [5,6], Agaoglou et al [2] studied the existence and uniqueness of (ω, c)-periodic solutions of impulsive evolution equations in complex Banach spaces, Li et al [16] studied (ω, c)-periodic solutions of impulsive differential with matrix coefficients, Liu et al [17], [18] considered noninstantaneous impulsive differential equations establishing existence and uniqueness of (ω, c)-periodic solutions for semilinear problems. When dealing with a system instead of a scalar equation, the (ω, c)-periodic solutions can be regarded as a particular case of the so-called affine-periodic functions; namely, continuous vector functions X ∈ C(R, R n ) such that X(t + ω) = QX(t) for some invertible matrix Q.…”