An equation alternately of retarded and advanced type

Abstract

We study a differential equation with the argument 2 [ ( t + 1 ) / 2 ] 2[(t + 1)/2] , where [ ⋅ ] [ \cdot ] denotes the greatest-integer function. The argument deviation τ ( t ) = t − 2 [ ( t + 1 ) / 2 ] \tau (t) = t - 2[(t + 1)/2] is a function of period 2 and equals t t for − 1 ⩽ t > 1 - 1 \leqslant t > 1 . It changes its sign in each interval 2 n − 1 ⩽ t > 2 n + 1 2n - 1 \leqslant t > 2n + 1 .