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Retarded differential equations with piecewise constant delays. (English) Zbl 0557.34059
Functional differential equations with piecewise constant delays are studied. They are closely related to impulse, loaded and, especially, to difference equations, and have the structure of continuous dynamical systems within intervals of unit length. Continuity of a solution at a point joining any two consecutive intervals implies recursion relations for the solution at such points. The equations are thus similar in structure to those found in certain ”sequential-continuous” models of disease dynamics. The main feature of equations with piecewise constant delays is that it is natural to pose initial-value and boundary-value problems for them not on intervals but at a certain number of individual points. Existence and uniqueness theorems are established for equations with bounded and unbounded operators. A general estimate of the solutions’ growth as $t\to +\infty$ is found. Special consideration is given to the problem of stability, and oscillatory properties of solutions are studied, too. The instrument of continued fractions plays an important role in the computation of solutions and in the analysis of their asymptotic behavior. Equations with unbounded delay arise in cases of several argument deviations. In such problems, the initial function is prescribed on (-$\infty ,0\right]$ and the solution is sought for $t>0$.

##### MSC:
 34K05 General theory of functional-differential equations 34K20 Stability theory of functional-differential equations 34K30 Functional-differential equations in abstract spaces 39A10 Additive difference equations