Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 142, pp. 1-11.
Title: Periodicity and stability in neutral nonlinear
differential equations with functional delay
Authors: Youssef M. Dib (Univ. of Louisiana at Lafayette, LA, USA)
Mariette R. Maroun (Baylor Univ., Waco, TX, USA)
Youssef N. Raffoul (Univ. of Dayton, Dayton, OH, USA)
Abstract:
We study the existence and uniqueness of periodic
solutions and the stability of the zero solution of the nonlinear
neutral differential equation
$$
\frac{d}{dt}x(t) = -a(t)x(t)+ \frac{d}{dt}Q(t, x(t-g(t)))
+G(t,x(t), x(t-g(t))).
$$
In the process we use integrating factors and
convert the given neutral differential equation into an equivalent
integral equation. Then we construct appropriate mappings and
employ Krasnoselskii's fixed point theorem to show the existence
of a periodic solution of this neutral differential equation. We
also use the contraction mapping principle to show the existence
of a unique periodic solution and the asymptotic stability of the
zero solution provided that $Q(0,0)= G(t, 0,0) = 0$.
Submitted November 30, 2004. Published December 6, 2005.
Math Subject Classifications: 34K20, 45J05, 45D05.
Key Words: Krasnoselskii; contraction; neutral differential equation;
integral equation; periodic solution; asymptotic stability