Forced Oscillations in Nonlinear Neutral Differential Equations

It is known that if a periodic neutral differential equation of certain type (which includes equations like $( {d / dt} )[ {x( t ) - q \times ( {t - r} )} ] = f( {x( t ),x( {t - r} ) + p( t ),| q | < 1,p( t )} )$ periodic) is uniform ultimately bounded, then there is a periodic solution. By generalizing the so-called Lyapunov-Razumikhin techniques, we give sufficient conditions for uniform ultimate boundedness. Stability results are obtained as particular cases. Two applications in equations arising in the theory of transmission lines are given.