On periodic asymptotic equilibria of systems of nonlinear finite-difference equations
Consider \(N\) a positive integer, \(t\in \mathbb{R}\), \(x= (x_1,\dots, x_n): \mathbb{R}\to\mathbb{R}^n\), \(|x|= \max_{1\leq i\leq n}|x_i|\), \(w: \mathbb{R}\to\mathbb{R}^+\) an \(N\)-periodic function and suppose: {\parindent7.5mm \begin{itemize}\item[(i)] \(f= (f_1,\dots, f_n): \mathbb{R}\times\mathbb{R}^n\to \mathbb{R}^n\) is a continuous function, \(N\)-periodic with respect to \(t\in\mathbb{R}\) and \(|f|= \max_{1\leq i\leq n}|f_i|\); \item[(ii)] \((\forall(t,x,y)\in \mathbb{R}\times\mathbb{R}^n\times\mathbb{R}^n)(|f(t,x)- f(t,y)|\leq w(t)|x-y|)\); \item[(iii)] \((\exists\theta\in (0,1))(\forall t\in\mathbb{R})(w(t)w(t+ 1)\cdots w(t+ N-1)\leq \theta)\). \end{itemize}} The author considers the system \[ x(t+ 1)= f(t,x(t)),\quad (t,x(t))\in\mathbb{R}\times \mathbb{R}^n\tag{1} \] of nonlinear finite-difference equations. The system (1) has an \(N\)-periodic asymptotic equilibrium \(\gamma\) continuous for \(t\geq 0\) if: (a) \(x\) is an arbitrary continuous solution of (1), bounded for \(t\in\mathbb{R}^+\), is a continuous \(N\)-periodic solution of (1) and the relation \[ x(t)= \gamma(t)+ o(1)\tag{2} \] holds as \(t\to+\infty\), (b) there exists a family of continuous solutions \(x\) of (1), bounded for \(t\in\mathbb{R}^+\) such that (2) is satisfied as \(t\to+\infty\). The author proves that there exists a unique continuous \(N\)-periodic solution \(\gamma\) such that an arbitrary continuous solution \(x\) of (1), bounded for \(t\in\mathbb{R}^+\), satisfies (2), that there exists a family of continuous solutions \(x\) of (1), bounded for \(t\in\mathbb{R}^+\), such that (2) holds as \(t\to+\infty\), and that the system (1) has a continuous \(N\)-periodic asymptotic equilibrium \(\gamma\). Also, if \(T> 0\) and \(f\) is \(T\)-periodic in \(t\in\mathbb{R}\) and satisfies the Lipschitz condition: \[ (\exists L\in (0,1))(\forall(t,x,y)\in \mathbb{R}\times \mathbb{R}^n\times\mathbb{R}^n)(|f(t,x)- f(t,y)|\leq L|x- y|), \] then the system (1) has a unique continuous \(T\)-periodic solution \(v\) such that any continuous solution \(x\) of (1), bounded for \(t\in\mathbb{R}^+\), satisfyies \[ \lim_{t\to+\infty} |x(t)- v(t)|= 0\tag{3} \] and for an arbitrary continuous vector function \(\varphi\), there exists a family \(x(t)= x(t,\varphi(t))\) of continuous solutions of (1), bounded for \(t\geq 0\) and satisfying (3). The author studies the special case \(x(t+1)= f(x(t))\).
system of nonlinear finite-difference equations
scientific article; zbMATH DE number 5824110