Wiener, Joseph; Aftabizadeh, A. R. Differential equations alternately of retarded and advanced type. (English) Zbl 0671.34063 J. Math. Anal. Appl. 129, No. 1, 243-255 (1988). The differential equation \(x'(t)=f(x(t),x(m[(t+k)/m]))\), where [\(\cdot]\) is the greatest integer function, is alternately of advanced and retarded type on intervals \([mn-k,m(n+1)-k]\), \(n=integer\). A theorem asserting the existence of a unique solution of the initial value problem \(x(0)=c_ 0\) is proved. The linear case \(f=ax(t)+a_ 0x(m[(t+k)/m])\) is studied in detail. Several results dealing with the asymptotic behavior of solutions are proved. For example, necessary and sufficient conditions are given for the asymptotic stability of the solution \(x=0\) when the coefficients a and \(a_ 0\) are constants. When these coefficients are functions of t, necessary and sufficient conditions for the non-oscillation of solutions and for the periodicity of solutions are given. Reviewer: J.M.Cushing Cited in 15 Documents MSC: 34K20 Stability theory of functional-differential equations 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C25 Periodic solutions to ordinary differential equations 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations Keywords:asymptotic stability; non-oscillation of solutions PDF BibTeX XML Cite \textit{J. Wiener} and \textit{A. R. Aftabizadeh}, J. Math. Anal. Appl. 129, No. 1, 243--255 (1988; Zbl 0671.34063) Full Text: DOI OpenURL References: [1] Aftabizadeh, A.R.; Wiener, J., Oscillatory properties of first order linear functional differential equations, Appl. anal., 20, 165-187, (1985) · Zbl 0553.34045 [2] {\scA. R. Aftabizadeh and J. Wiener}, Oscillatory and periodic solutions of an equation alternately of retarded and advanced type, to appear. · Zbl 0598.34059 [3] Busenberg, S.; Cooke, K.L., Models of vertically transmitted diseases with sequential-continuous dynamics, (), 179-187 [4] Cooke, K.L.; Wiener, J., Retarded differential equations with piecewise constant delays, J. math. anal. appl., 99, 265-297, (1984) · Zbl 0557.34059 [5] {\scK. L. Cooke and J. Wiener}, An Equation alternately of retarded and advanced type, to appear. · Zbl 0628.34074 [6] Shah, S.M.; Wiener, J., Advanced differential equations with piecewise constant argument deviations, Internat. J. math. math. sci., 6, No. 4, 671-703, (1983) · Zbl 0534.34067 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.