Advanced differential equations with piecewise constant argument deviations. (English) Zbl 0534.34067

Biological models often lead to functional differential equations which are reducible to systems of ordinary differential equations. In the present article functional differential equations with arguments having intervals of constancy are considered. These equations are similar in structure to those found in certain ”sequential-continuous” models of disease dynamics. Our equations are directly related to difference equations of a discrete argument, the theory of which has been very intensively developed in numerous works. Connections are also established with impulse and loaded equations. The main feature of equations with piecewise constant argument deviations is that it is natural to pose initial-value and boundary-value problems for them not on intervals, but at a number of individual points. The study is concentrated on equations of advanced type. The initial-value problem is posed at \(t=0\), and the solution is sought for \(t>0\). The existence and uniqueness of solution on \([0,\infty]\) and of its backward continuation on (-\(\infty,0]\) is proved for linear equations with constant coefficients. Necessary and sufficient conditions of stability and asymptotic stability of the trivial solution are determined explicitly via coefficients of the equations, and oscillatory properties of solutions are studied. For linear systems with continuous coefficients the existence and uniqueness of solution on \([0,\infty)\) is proved. A simple algorithm of computing the solution by means of continued fractions is indicated for a class of scalar equations. A general estimate of the solution’s growth as \(t\to +\infty\) is found. Special consideration is given to the problem of stability. An existence criterion of periodic solutions to linear equations with periodic coefficients is established. Some non-linear equations are also tackled. The work will be extended in the future to equations of neutral type.


34K10 Boundary value problems for functional-differential equations
34K05 General theory of functional-differential equations
39A10 Additive difference equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25 Periodic solutions to ordinary differential equations
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