Discrete order preserving semigroups and stability for periodic parabolic differential equations (Q912287)

For a discrete order preserving semigroup \(\{S^ n\}_{n\geq 1}\) in a closed subset D of a Banach lattice (X,\(\geq)\) with order-continuous norm, under natural assumptions on S and D, it is proved that for each \(x\in D\) with compact semiorbit, the sequence \(\{S^ nx\}_{n\geq 1}\) converges to a fixed point of S in a suitable norm. This result is then extended to asymptotically autonomous discrete dynamical processes, that is, \(T_ n=S_ n\circ...\circ S_ 1\), where \(S_ n\to S\) in a convenient way. Under weaker convergence assumptions in \(S_ n\to S\), one may obtain only that \(\omega\) (x)\(\subset E\), where \(E=\{x\in D\); \(S(x)=x\}.\) Further on, there are presented applications in the case of a parabolic equation with a periodic (respectively, asymptotically periodic) nonlinear term.