Необходимые и достаточные условия для колебательных функциональных дифференциальных уравнений (ҚазҰУ Хабаршысы. Математика, механика, информатика сериясы, 2018)
In this survey, necessary and sufficient conditions for the oscillation of solutions of retarded, advancedand neutral differential equations of first and higher order with one or several constantcoefficients and constant arguments, in terms of the characteristic equation, are presented. Explicit(in terms of the constant coefficient and constant argument only) necessary and sufficientconditions are also presented in the case of one argument. In the case of nth order equations necessaryand sufficient conditions for the oscillation of all solutions are presented when n is odd, whilenecessary and sufficient conditions for the oscillation of all bounded solutions are presented whenn is even. In this case explicit sufficient conditions for the oscillation of all solutions are presentedwhen n is odd, while explicit sufficient conditions for the oscillation of all bounded solutions forretarded equations and of all unbounded solutions for advanced equations are presented when nis even. In the case of several arguments explicit but sufficient conditions only are given and theresults are also extended to equations with several variable coefficients.
Keywords: Колебание; запаздывание; необходимые и достаточные условия; характеристическое уравн; разностные уравнения