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Optimal existence theory for single and multiple positive periodic solutions to functional difference equations. (English) Zbl 1068.39009
For any $$(a,b)\in \mathbb{Z}^2$$, $$a< b$$, denote $\begin{gathered} Z[a, b]= \{a,a+1,\dots, b\},\quad Z(a, b)= \{a-1,\dots, b-1\},\\ Z[a, b)= \{a,a+1,\dots, b-1\},\quad Z(a, b]= \{a-1,\dots, b\},\\ Z[a,+\infty)= \{a,a+1,\dots\},\quad Z(-\infty, a]= \{\dots,a-1,a\},\\ Z(a,+\infty)= \{a+1,\dots\},\quad Z(-\infty, a)= \{\dots, a-1\},\end{gathered}$ therefore $$Z[1,+\infty)= N$$ and $$Z(-\infty,+\infty)= Z$$, let $$T\in N$$.
The authors consider the general periodic logistic difference equation $\Delta y(i)= y(i)(a(i)- f(i,y(i- \tau_1(i)),\dots, y(i-\tau_n(i)))),\quad i\in \mathbb{Z},\tag{1}$ where $$y: Z\to [0,+\infty)$$, $$\Delta y(i)= y(i+ 1)- y(i)$$, $$i\in \mathbb{Z}$$, the function $$f: \mathbb{Z}\times [0,+\infty)^n\to [0,+\infty)$$ is continuous, the function $$a: \mathbb{Z}\to (0,+\infty)$$ is continuous, $$a(i)= a(i+ T)$$, $$i\in \mathbb{Z}$$, the functions $$\tau_m: \mathbb{Z}\to \mathbb{Z}$$, $$1\leq m\leq n$$, satisfying $$\tau_m(i)= \tau_m(i+ t)$$, $$i\in \mathbb{Z}$$, $$1\leq m\leq n$$, and $f(i,u_1,\dots, u_n)= f(i+ T,u_1,\dots, u_n),\quad i\in \mathbb{Z},\quad (u_1,\dots, u_n)\in [0,+\infty)^n.$ They also consider the general periodic functional difference equation $\Delta y(i)= -a(i)y(i)+ g(i,y(i-\tau(i))),\quad i\in \mathbb{Z},\tag{2}$ where $$a\in C(\mathbb{Z},(0,1))$$, $$g\in C(\mathbb{Z}\times [0,+\infty),[0,+\infty))$$, $$\tau\in C(\mathbb{Z},\mathbb{Z})$$, $$a(i)= a(i+ T)$$, $$i\in\mathbb{Z}$$, $$\tau(i)= \tau(i+ T)$$, $$i\in\mathbb{Z}$$ and $$g(i,u)= g(i+ T,u)$$, $$i\in\mathbb{Z}$$, $$u\in [0,+\infty)$$.
Let $X= \{y;\,y\in C(\mathbb{Z},\mathbb{R}),\;y(i)= y(i+ T),\;i\in\mathbb{Z}\}$ be the Banach space endowed with the norm $\| y\|= \sup_{i\in\mathbb{Z}[0,T-1]}\,|y(i)|,\quad y\in X$ and the cone in $$X$$ defined by $K= \{y\in X;\;(\forall i\in \mathbb{Z})(y(i)\geq 0\wedge y(i)\geq \sigma\| y\|\},$ where $\sigma= \Biggl(\prod^{T-1}_{i=0} (1+ a(i))\Biggr)^{-1}.$ Using a fixed point theorem in cones the authors establish the existence of positive $$T$$-periodic solutions $$y$$ to equation (1) and the existence for single and multiple positive $$T$$-periodic solutions $$y$$ to equation (2).
Reviewer: D. M. Bors (Iaşi)

##### MSC:
 39A11 Stability of difference equations (MSC2000)
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