Abstract: In this paper, the authors study the first order neutral differential equation [x(t)-cx(t-h)-c^*x(t+h^*)]'=p(t)x(g(t)) by the Krasnoselskii fixed point theorem, and obtain some new sufficient conditions for existence of asymptotically decaying positive solutions of the first order neutral differential equation.

Key words: Mixed neutral differential equation, Asymptotically Decaying positive solution, Krasnoselskii fixed point theorem, Completely continuous