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By means of Krasnoselskii's fixed point theorem we obtain boundedness and stability results of a neutral nonlinear differential equation with variable delays. A stability theorem with a necessary and sufficient condition is given. The results obtained here extend and improve the work of C. In the end we provide an example to illustrate our claim.

In this article we study the existence of periodic solutions of the second order nonlinear neutral differential equation with functional delay d 2 dt 2 x (t) + p (t) d dt x (t) + q (t) x 3 (t) = d dt g (t, x (t − τ (t))) + f ` t, x 3 (t) , x 3 (t − τ (t)) ´. The main tool employed here is the Burton-Krasnoselskii's hybrid fixed point theorem dealing with a… (More)

In this paper we use the contraction mapping theorem to obtain asymptotic stability results of the zero solution of a nonlinear neutral Volterra integro-differential equation with variable delays. Some conditions which allow the coefficient functions to change sign and do not ask the boundedness of delays are given. An asymptotic stability theorem with a… (More)

Let T be a periodic time scale. We use a fixed point theorem due to Krasnoselskii to show that the nonlinear neutral dynamic equation with infinite delay x ∆ (t) = −a(t)x σ (t) + (Q(t, x(t − g(t))))) ∆ + t −∞ D (t, u) f (x(u)) ∆u, t ∈ T, has a periodic solution. Under a slightly more stringent inequality we show that the periodic solution is unique using… (More)

We study the existence of periodic solutions of the second order nonlinear neutral differential equation with variable delay x ′′ (t) + p (t) x ′ (t) + q (t) h (x (t)) = c (t) x ′ (t − τ (t)) + f (t, x (t − τ (t))). We invert the given equation to obtain an integral, but equivalent, equation from which we define a fixed point mapping written as a sum of a… (More)

The purpose of this paper is to use a fixed point approach to obtain asymptotic stability results of a nonlinear neutral differential equation with variable delays. An asymptotic stability theorem with a necessary and sufficient condition is proved. In our consideration we allow the coefficient functions to change sign and do not require bounded delays. The… (More)

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