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

Abstract

The nonlinear neutral integro-differential equation $$\frac{d}{dt}x ( t ) =-\int_{t-\tau ( t ) }^{t}a ( t,s ) g \bigl( x ( s ) \bigr) ds+\frac{d}{dt}G \bigl( t,x \bigl( t-\tau ( t ) \bigr) \bigr) , $$ with variable delay τ(t)≥0 is investigated. We find suitable conditions for τ, a, g and G so that for a given continuous initial function ψ a mapping P for the above equation can be defined on a carefully chosen complete metric space \(S_{\psi }^{0}\) 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).