Asymptotic stability and exponential stability for a class of impulsive neutral differential equations with discrete and distributed delays

TL;DR

This work analyzes the asymptotic and exponential stability of a class of impulsive neutral differential equations with discrete and distributed delays using fixed-point theory rather than Lyapunov methods. By formulating a vector-matrix model with delays $\tau(t)$, $\delta(t)$, $r(t)$ and impulses at times $t_k$, it derives sufficient conditions encoded in a contraction constant $\rho<1$ that combine Lipschitz bounds, impulse terms, and delay magnitudes. The results establish global asymptotic and exponential stability (via weighted spaces with $e^{\lambda t}$) without requiring differentiability of delays or fixed-sign coefficient functions, thereby extending and unifying prior Lyapunov- and LMI-based approaches. Two numerical examples demonstrate practical verification of the contraction condition, underscoring applicability to multi-dimensional impulsive neutral delay systems.

Abstract

In this paper, we present sufficient conditions for asymptotic stability and exponential stability of a class of impulsive neutral differential equations with discrete and distributed delays. Our approaches are based on the method using fixed point theory, which do not resort to any Lyapunov functions or Lyapunov functionals. Our conditions do not require the differentiability of delays, nor do they ask for a fixed sign on the coefficient functions. Our results improve some previous ones in the literature. Examples are given to illustrate our main results.

Figures (2)

• Figure 1: The solution of Example 5.1
• Figure 2: The solution of Example 5.2

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4
• Lemma 2.5
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• Remark 3.3