Pham Ky Anh; Nguyen Huu Du; Le Cong Loi Singular difference equations: an overview. (English) Zbl 1148.39009 Vietnam J. Math. 35, No. 4, 339-372 (2007). 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. Reviewer: Christian Pötzsche (München) Cited in 8 Documents 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 Keywords:implicit difference equation; stochastic difference equation; Lyapunov functions; stability; robust stability; Floquet theory; multiplicative ergodic theorem; Furstenberg-Kifer decomposition PDFBibTeX XMLCite \textit{Pham Ky Anh} et al., Vietnam J. Math. 35, No. 4, 339--372 (2007; Zbl 1148.39009)