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Pullback attractors in nonautonomous difference equations. (English) Zbl 0961.39007
The author considers the nonautonomous difference equation $x_{n+1}= f_n(x_n), \quad x_n\in\mathbb{R}^d,\;n\in\mathbb{Z}, \tag{1}$ where $$\mathbb{R}^d$$ is the Euclidean state space and $$f_n$$ are continuous mappings from $$\mathbb{R}^d$$ to $$\mathbb{R}^d$$. Consider the forwards nonautonomous solution mapping $$\varphi$$ of (1), defined through iteration by $x_n=\varphi (n,n_0,x_0)= (f_{n-1}\circ \cdots \circ f_{n_0})(x_0),$ $$n>n_0$$, $$n_0\in\mathbb{Z}$$, $$x_0\in\mathbb{R}^d$$, for the initial value $$\varphi(n_0,n_0,x_0)=x_0$$. The author expresses this solution as a mapping $$\widetilde\varphi:\mathbb{Z}^+ \times\mathbb{Z} \times\mathbb{R}^d \to\mathbb{R}^d$$ defined by $\widetilde\varphi (j,n_0,x_0)= \varphi(n_0+j, n_0,x_0), \quad j\in\mathbb{Z}^+,\;n_0\in \mathbb{Z},\;x_0\in \mathbb{R}^d.$ Nonautonomous difference equations are formulated as cocycles which generalize semigroups corresponding to autonomous difference equations of the form $$x_{n+1}= f(x_n)$$, $$x_n\in\mathbb{R}^d$$, $$n\in\mathbb{Z}$$.
The solution $$\widetilde\varphi$$ of (1) satisfies the cocycle property if $\widetilde\varphi (i+j,n_0,x_0)= \widetilde\varphi \bigl(i,n_0+j,\;\widetilde\varphi (j,n_0,x_0)\bigr) \tag{2}$ for all $$i,j\in \mathbb{Z}^+$$, $$n_0\in \mathbb{Z}$$, $$x_0\in \mathbb{R}^d$$, and as well the initial condition $\widetilde\varphi (0, n_0,x_0)=x_0. \tag{3}$ A mapping $$\widetilde\varphi$$ satisfying (2) and (3) is called a difference cocycle.
The author defines the pullback attractors as an appropriate generalization of autonomous attractors to cocycles. A solution $$\varphi^*$$ of a difference cocycle $$\widetilde\varphi$$ is said to be (globally) Lyapunov asymptotically stable if it is Lyapunov stable and satisfies the forwards running convergence $\bigl\|\widetilde\varphi (j,n_0,x_0)- \varphi^*(n_0+j) \bigr\|\to 0\quad\text{as }j\to+ \infty$ for all $$x_0\in\mathbb{R}^d$$ and $$n_0\in Z$$. If $\bigl\|\widetilde\varphi(j,n_0-j, x_0)-\varphi^* (n_0)\bigr\|\to 0\quad \text{as }j\to+ \infty$ for all $$x_0 \in\mathbb{R}^d$$ and $$n_0\in\mathbb{Z}$$, then we say that the solution $$\varphi^*$$ satisfies the pullback convergence.
A family $$\widehat A=\{A_n;n \in\mathbb{Z}\}$$ of nonempty subsets of $$\mathbb{R}^d$$ is said to be $$\widetilde\varphi$$-invariant if $$\varphi^*(j,n_0, A_{n_0})= A_{n_0+j}$$ for all $$j\in\mathbb{Z}^+$$, $$n_0\in\mathbb{Z}$$; this family is called pullback attracting if it is $$\widetilde\varphi$$-invariant and $$H^*(\widetilde\varphi (j,n_0-j,x_0), A_{n_0})\to 0$$ as $$j\to+\infty$$ for all $$n_0\in\mathbb{Z}$$ and $$x_0\in\mathbb{R}^d$$, where $$H^*(A,B)$$ is the Hausdorff separation of nonempty compact subsets $$A,B$$ of $$\mathbb{R}^d$$. Such a pullback attracting family is called a pullback attractor.
A family $$\widehat B=\{ B_n;n \in\mathbb{Z}\}$$ of nonempty compact subsets of $$\mathbb{R}^d$$ is called a pullback absorbing set family for a difference cocycle $$\widetilde\varphi$$ on $$\mathbb{R}^d$$ if for each $$n_0\in\mathbb{Z}$$ and every bounded subset $$D$$ of $$\mathbb{R}^d$$ there exists an $$N_{n_0,D} \in\mathbb{Z}^+$$ such that $$\widetilde\varphi (j,n_0-j,D)\subset B_{n_0}$$ for all $$j\geq N_{n_0,D}$$ and $$n_0\in\mathbb{Z}$$.
The existence of a pullback attractor follows when the difference equation cocycle has a pullback absorbing set. The paper includes also the construction of a Lyapunov function characterizing pullback attraction and gives several examples.
Reviewer: D.M.Bors (Iaşi)

##### MSC:
 39A12 Discrete version of topics in analysis 39A11 Stability of difference equations (MSC2000) 37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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