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Singular difference equations: an overview. (English) Zbl 1148.39009

This nicely written paper surveys the recent development in the theory of linear and nonlinear singular (i.e., implicit) difference equations (SDEs), as well as singular stochastic difference equations (SSDEs). It focusses on the author’s work, where the following aspects are covered:
(1) Index-\(1\) tractable SDEs: For linear and specific nonlinear problems, the property of being index-\(1\) tractable is preserved under explicit Euler discretization with sufficiently small stepsize. Moreover, a variety of conditions is given for the existence of forward solutions to initial values problems for (nonlinear) SDEs. Finally, also the solvability of multipoint boundary value problems for linear SDEs is studied, as well as their persistence under discretization.
(2) Qualitative behavior of SDEs: A Floquet theory for index-\(1\) problems is presented. Moreover, using Lyapunov functions the authors deduce conditions for (asymptotic) stability of solutions for semilinear SDEs.
(3) The final section deals with SSDEs, and in particular features a multiplicative ergodic theorem, as well as a Furstenberg-Kifer decomposition.

MSC:

39A11 Stability of difference equations (MSC2000)
34A09 Implicit ordinary differential equations, differential-algebraic equations
37H15 Random dynamical systems aspects of multiplicative ergodic theory, Lyapunov exponents
39A12 Discrete version of topics in analysis
37H10 Generation, random and stochastic difference and differential equations
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