Abstract
This paper is devoted to generalize Halanay's inequality which plays an important rule in study of stability of differential equations. By applying the generalized Halanay inequality, the stability results of nonlinear neutral functional differential equations (NFDEs) and nonlinear neutral delay integrodifferential equations (NDIDEs) are obtained.
Highlights
In 1966, in order to discuss the stability of the zero solution of u t −Au t Bu t − τ∗, τ∗ > 0, 1.1
1.4 y t φ t, t ≤ t0, φ bounded and continuous for t ≤ t0, Baker and Tang 2 give a generalization of Halanay inequality as Lemma 1.2 which can be used for discussing the stability of solutions of some general Volterra functional differential equations
Neutral functional differential equations NFDEs are frequently encountered in many fields of science and engineering, including communication network, manufacturing systems, biology, electrodynamics, number theory, and other areas see, e.g., 8–11
Summary
In 1966, in order to discuss the stability of the zero solution of u t −Au t Bu t − τ∗ , τ∗ > 0, Halanay used the inequality as follows. If v t ≤ −Av t B sup v s , for t ≥ t0, 1.2 t−τ ≤s≤t where A > B > 0, there exist c > 0 and κ > 0 such that v t ≤ ce−κ t−t0 , for t ≥ t0, 1.3 and v t → 0 as t → ∞. 1.4 y t φ t , t ≤ t0, φ bounded and continuous for t ≤ t0, Baker and Tang 2 give a generalization of Halanay inequality as Lemma 1.2 which can be used for discussing the stability of solutions of some general Volterra functional differential equations. The Halanay inequality has been extended to more general type and used for investigating the stability and dissipativity of various functional differential equations by several researchers see, e.g., 3–7.
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