Global asymptotic stability of nonlinear neutral differential equation

In this paper, the following nonlinear neutral differential equation x ′ ( t ) = - a ( t ) x ( t ) + c ( t ) x ′ ( t - τ 1 ( t ) ) + q ( t , x ( t ) , x ( t - τ 2 ( t ) ) ) is considered. By using fixed point theory, we give some new conditions to ensure that the zero solution is globally asymptotically stable in C 1 . By weakening the assumptions on the neutral coefficient c and the delay τ 1 and by obtaining more manageable criteria, some existing results are improved and generalized.