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Monotone iterative method for dynamic systems on time scales. (English) Zbl 0783.34005
Employing notions and calculus developed in [B. Aulbach and S. Hilger, Nonlinear dynamics and quantum dynamical systems, Contrib. Int. Semin., ISAM-90, Gaussig/GDR 1990, Math. Res. 59, 9-20 (1990; Zbl 0719.34088)] and results from a preceding own paper [Existence and comparison results for dynamic systems on time scales. J. Math. Anal. Appl. (to appear)], the author extends the method of upper and lower solutions to dynamical systems on time scales, ${u}^{{\Delta }}=f\left(t,u\right)$, $u\left(0\right)={u}_{0}$, $f\in {C}_{rd}\left[{T}^{k}×ℝ,ℝ\right]$. He uses monotone iterative technique for initial value problems and periodic boundary value problems in order to obtain extremal solutions.
##### MSC:
 34A45 Theoretical approximation of solutions of ODE 37-99 Dynamic systems and ergodic theory (MSC2000) 34C11 Qualitative theory of solutions of ODE: growth, boundedness 34A34 Nonlinear ODE and systems, general 34C25 Periodic solutions of ODE