Convergence for essentially strongly increasing discrete time semi-flows
The article deals with iterations of a monotone and \(\alpha\)-condensing mapping \(T: \;[p,q] \to [p,q]\), where \([p,q]\), \(p \ll q\), is an order interval \(\{x: a\leq x \leq b\}\) in a strongly ordered Banach space \(V\); it is assumed that a cone \(V_+ = \{x \in V: x\geq 0\}\) is solid and, probably, normal (the authors state that order bounded subsets of \(V\) are bounded). About \(T\) it is assumed that it is essentially strongly increasing, i.e. \(x \leq y\) implies that either \(Tx = Ty\) or \(Tx \ll Ty\), and \(x \ll y\) always implies \(Tx \ll Ty\). The main results are the following: if \(F(T) = \{p,q\}\) then \(\{x \in [p,q]:p \ll x \ll q\}\) is attracted by either \(p\) or \(q\); if \(F(T)\) is a singleton, then \(\omega(x) = F(T)\) for all \(x \in [p,q]\); if \(F(T)\) is not a singleton and each element \(F(T)\) is Lyapunov stable then \(F(T) = J([0,1])\) where \(J:[0,1] \to [p,q]\) is a strictly incresing continuous mapping and some others (in these statements \(\omega(x)\) is the set of limit points for \(\{T^nx: n\in {\mathbb N}\}\)). As applications the periodic solutions of delay differential equations are considered. In particular, the systems of type \[ \frac{dx_i(t)}{dt} = F_i(t,x_t), \qquad i = 1,2,\dots,n \] and \[ \begin{cases} dx^1 / dt = f(t,x^n_t) - \alpha_1(t)x^1(t), \\ dx^i / dt = x^{i-1}(t - r_{i-1}) - \alpha_i(t)x^i(t),\end{cases} \qquad 2 \leq i \leq n, \] are studied.
essentially strongly increasing discrete time semi-flow
periodic quasi-monotone delay differential equations
scientific article; zbMATH DE number 5566214
scientific article