Abstract: In this paper，the following nonlinear Volterra equation with variable delays was studied by using the fixed point theory. Some new conditions were given to ensure that the zero solutions are globally asymptotically stable in C¹. Previously，almost all scholars used fixed point theory to study the asymptotic stability of zero solutions of nonlinear neutral differential equations with variable delays，it required τ quadratic differentiability and τ′≠1. Unlike most research methods，these conditions do not require a quadratic differentiability of delay τ and τ′≠1. The results obtained generalize the corresponding results in the literatures. An example is given to verify the validity of the conclusions.

Key words: nonlinear, Volterra equation, fixed point theorem, asymptotic stability, zero solutions