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A survey of differential equations with piecewise continuous arguments. (English) Zbl 0737.34045
Delay differential equations and dynamical systems, Proc. Conf., Claremont/CA (USA) 1990, Lect. Notes Math. 1475, 1-15 (1991).

[For the entire collection see Zbl 0727.00007.]

The article is a survey of the present status of the theory of differential equations with piecewise continuous arguments (EPCA). A typical EPCA is of the form ${x}^{\text{'}}\left(t\right)=f\left(t,x\left(t\right),x\left(h\left(t\right)\right)$, where the argument $h\left(t\right)$ has intervals of constancy. For example: $h\left(t\right)=\left[t\right]$, $\left[t-n\right]$, $t-n\left[t\right]$, where $\left[\phantom{\rule{4pt}{0ex}}\right]$ denotes the greatest integer function. The main topics are: existence, uniqueness, representation and stability of the solutions; stability as a function of the delay; oscillatory and periodic solutions; approximation of equations with discrete delay; equations of alternating type and chaotic behaviour of solutions.

##### MSC:
 34K05 General theory of functional-differential equations 34K99 Functional-differential equations 34-02 Research monographs (ordinary differential equations)