Invariants and Liapunov functions for nonautonomous systems (Q1600384)

Let \(T_n:M\to M\) \((M\subset \mathbb{R}^k)\) be a sequence of continuous self-mappings. If there exists an unbounded function \(U:\mathbb{N}\times C\to \mathbb{R}\) such that for every \(x\in M\) and some \(\alpha(x)\in\mathbb{R}\) \[ T_{n-1}\circ\cdots\circ T_0x\in C\Rightarrow U(n,T_{n-1}\circ\cdots\circ T_0x)\leq \alpha(x), \] then every solution of the recurrence \[ x_{n+1}=T_nx_n, n\in \mathbb{N}, \] eventually contained in \(C\), is boudned. Moreover, under some additional assumptions every such solution has the \(\omega\)-limit: \(\omega(x_0,T_n)\subset H\), where \(H\) is an appropriate set. The proofs are based on the concept of Lyapunov functions for nonautonomous discrete dynamical systems. Some illustrative examples are included.