Differential equations alternately of retarded and advanced type (Q1119812)

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.