Periodic solution and stability in nonlinear neutral system with infinite delay

Abstract

We study the existence and uniqueness of periodic solutions and the stability of the zero solution of the nonlinear neutral differential equation d dt x(t) = −a(t)x(t) + d dt Q(t, x(t − g(t))) + t −∞ D(t, s)f (x(s))ds 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) = f (0) = 0.

Stability in nonlinear neutral differential equations with infinite delay

“…An asymptotic stability theorem with a necessary and sufficient condition is proved. The results presented in this paper improve and generalize the main results in [1,6,17].…”

“…Nevertheless, the application of this method to problems of stability in differential equations with delay has encountered serious difficulties if the delay is unbounded or if the equation has unbounded terms [4][5][6]. Recently, investigators such as Burton, Zhang, Raffoul and others have noticed that some of these difficulties vanish or might be overcome by means of fixed point theory see ( [1]- [15], [17]). The fixed point theory does not only solve the problem on stability but has a significant advantage over Lyapunov's direct method.…”

“…On the other hand, Althubiti, Makhzoum, Raffoul in [1] considered the following nonlinear neutral differential equation…”

“…Theorem C (Althubiti, Makhzoum, Raffoul [1]). Suppose (2)- (5) hold, and there exists a constant α ∈ (0, 1) such that for t ≥ 0,…”

“…Our purpose here is to give, by using the contraction mapping principle, asymptotic stability results of a nonlinear neutral differential equation with infinite delay (1). An asymptotic stability theorem with a necessary and sufficient condition is proved.…”