×

Stability of dynamic systems on the time scales. (English) Zbl 1027.34059

Here, the author surveys some results on the stability of dynamic equations on time scales. In particular, one can find theorems on local existence and uniqueness of solutions in forward time (of Peano and Perron type), dynamic inequalities and existence of extremal solutions, comparison results, the linear and nonlinear variation of parameters, as well as results on various stability and strict stability notions using Lyapunov functions and generalized derivatives in terms of the monograph V. Lakshmikantham, S. Sivasundaram and B. Kaymakçalan [Dynamic systems on measure chains. Mathematics and its Applications. 370. Dordrecht: Kluwer Academic Publishers (1996; Zbl 0869.34039)]. Proofs are largely omitted, and the paper is not self-contained in the sense that, for example, certain notions like nondegenerate points, measure chains or dynamic triples are not introduced.
The paper should be read with care, since most of the results are copied from the monograph mentioned above, including their obvious errors. E.g., one has to replace \(\mu^\ast(t,s)\) by \(\mu(t,s)\) in Definition 2.3, the rd-continuity of \(v,w\) in Theorems 4.1, 4.2 is redundant, the integrals in Theorem 8.1(i),(ii) are Riemann and not time scale-integrals (Cauchy integrals), or one has to consider the one-sided limit \(h\to\mu^\ast(t)+\) in (9.3) and assume that \(u+\mu^\ast(t)g(t,u)\) is nondecreasing in Theorem 9.2. Moreover, the author uses the book of B. Aulbach [Continuous and discrete dynamics near manifolds of equilibria. Lecture Notes in Mathematics. 1058. Berlin etc.: Springer-Verlag (1984; Zbl 0535.34002)] as one of the standard references on the time scales calculus. This book, however, is not concerned with time scales at all.

MSC:

34D20 Stability of solutions to ordinary differential equations
39A13 Difference equations, scaling (\(q\)-differences)
39A14 Partial difference equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
PDFBibTeX XMLCite