Some General Theorems on Uniform Boundedness for Functional Differential Equations

Abstract

Consider the functional differential equation with bounded delay
X”² = F(t,X_t), X ∈ Rⁿ.

We discuss uniform boundedness and uniform ultimate boundedness by Liapunov‘s second method with conditions such as:

(i) W₁(|X(t)|) ≤ V(t, X_t) ≤ W₂(|X(t)| + 

(ii) V(t, φ) ≤ W₃(”–φ”–);

(iii) V”²₍₁₎ (t, X_t) ≤

where m(φ) = min_{−h≤s≤0}|φ(s)|.

The theorem discussed in this paper generalizes some results on uniform boundedness and uniform ultimate boundedness for functional differential equations with bounded delay. Some examples are also discussed in this paper.