Optimal existence theory for single and multiple positive periodic solutions to functional difference equations
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).
scientific article; zbMATH DE number 2136197
scientific article