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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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