Fixed Points and Stability of a Class of Integrodifferential Equations

We study a class of integrodifferential functional differential equations x¨+f(t,x,x˙)x˙+∑j=1N∫t-rj(t)taj(t,s)gj(s,x(s))ds=0 with variable delay. By using the fixed point theory, we establish necessary and sufficient conditions ensuring that the zero solution of this equation is asymptotically stable