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Abstract
This paper is concerned with positive almost periodic type solutions to a class of nonlinear difference equation with delay. By using a fixed point theorem in partially ordered Banach spaces, we establish several theorems about the existence and uniqueness of positive almost periodic type solutions to the addressed difference equation. In addition, in order to prove our main results, some basic and important properties
Highlights
Introduction and preliminariesIn this paper, we consider the following nonlinear difference equation with delay:n x(n) = h(x(n)) +f (j, x(j)), n ∈ Z, j=n−k(n) (1.1)where h : R+ → R+, k : Z → Z+, and f : Z × R+ → R+.Equation (1.1) can be regarded as a discrete analogue of the integral equation t x(t) =f (s, x(s))ds, t ∈ R, t−τ (t) (1.2)which arise in the spread of some infectious disease
F (s, x(s))ds, t ∈ R, t−τ (t) which arise in the spread of some infectious disease
Since the work of Fink and Gatica [2], the existence of positive almost periodic type and positive almost automorphic type solutions to equation (1.2) and its variants has been of great interest for many authors
Summary
A function f : Z → R is called almost periodic if ∀ε, ∃N (ε) ∈ N such that among any N (ε) consecutive integers there exists an integer p with the property that Let Ω ⊂ R and f be a function from Z × Ω to R such that f (n, ·) is continuous for each n ∈ Z. A function f : Z × Ω → R is called asymptotically almost periodic in n uniformly for x ∈ Ω if it admits a decomposition f = g + h, where g ∈ AP (Z × Ω) and h ∈ C0(Z × Ω).
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