Stability in nonlinear neutral integro-differential equations with variable delay using fixed point theory

摘要

The nonlinear neutral integro-differential equation (d)/(dt)x(t)=?∫_(t?τ(t))~t a(t, s)g(x(s))ds + (d)/(dt)G(t,x(t ? τ(t)), with variable delay τ(t) ≥ 0 is investigated.We find suitable conditions for τ, a, g and Gso that for a given continuous initial function ψ amapping P for the above equation can be defined on a carefully chosen complete metric space S_ψ~0 in which P possesses a unique fixed point. The final result is an asymptotic stability theorem for the zero solution with a necessary and sufficient condition. The obtained theorem improves and generalizes previous results due to Burton (Proc. Am. Math. Soc. 132:3679– 3687, 2004), Becker and Burton (Proc. R. Soc. Edinb., A 136:245–275, 2006) and Jin and Luo (Comput. Math. Appl. 57:1080–1088, 2009).