Throughout this paper, besides τ being a periodic function with period T, we suppose in addition that (tauin C^{1}( {mathbb {R},mathbb{R}})) with (tau'(t)<1), (forall tin[0,T]).
Let (mathbb{T}) be a periodic function under the matched space ((mathbb{T},Pi,F,delta)) and f be a Δ-periodic function with period (tau in Pi^).
Since f is an almost periodic function, (W_{psi _{k+frac{nu}{2}}}f) is an almost periodic function in b as well [27].
f ( t ) = g ( t ) + α ( t ) in R +, and g ( t ) is an almost periodic function in ℝ.
In this section, we consider the almost periodic system frac{{d} x t)}{{d}t}=fbigl t,x t),x t-tau bigr), (2) where (f:Rtimes Srightarrow R^{n}) is an almost periodic function in t uniformly with respect to (x t-tau bigrR^{n}).
By Theorems 3 and 4, it is natural to ask: Can we provide a difference analogue of Theorem 3? Or, can we delete the condition that '(a z)) is a periodic entire function with period c' in Theorem 4?
For example, in [2], Zhang studied the following Liénard equation with a singularity: x"(t)+fbigl(x t bigr)x'(t)+g bigl t,x t bigr)=0, (1.1) where (f: {R}rightarrow{R}), (g: {R}times 0,+infty)rightarrow {R}) is an (L^{2} -Carathéodory function, (g(t,x)) is a T-periodic function in the first argument and can be singuL^{2} -Carathéodory (g(t,x)) can be unbounded as (xrightarrow0^).
In the following we shall use the notation begin{aligned}& l=min bigl{ [ 0,infty ) capmathbb{T} bigr},qquad I_{omega}= [ l,l+omega ] capmathbb{T}, & f^{M}= sup_{tin mathbb{T}}f ( t ),qquad f^{m}=inf _{tin mathbb{T}}f ( t ),qquad overline{f}=frac{1}{omega}int_{I_{omega}}f ( s ) Delta s, end{aligned} where f is a periodic rd-continuous function with period ω in (mathbb{T}).
Noting that (psi(t)) is an ω-periodic function in terms of the periodicity of ((u(t),v(t))) and (tau(t)), we choose (zetain [0,omega]) such that (psi zeta)=psi(theta-tau(theta))).
Let T be a time scale that is periodic in shifts δ ± with the period P, and let f be a Δ-periodic function in shifts δ ± with the period ω ∈ [ P, ∞ ) T ∗. Suppose that f ∈ C rd ( T ), then ∫ t 0 t f ( s ) Δ s = ∫ δ ± ω ( t 0 ) δ ± ω ( t ) f ( s ) Δ s. Lemma 2.5 [10].
