On Stability of Linear Delay Differential Equations under Perron's Condition

and Applied Analysis 3 where A and B are n × n matrix functions defined and continuous on 0,∞ and g is a continuously differentiable increasing function defined on 0,∞ satisfying g t < t and g ′ t ≤ 1. We set h : g−1. Obviously, h ∈ C1 0,∞ and increases on 0,∞ and h t > t. Perron’s condition takes the following form. Definition 1.5. System 1.4 is said to satisfy Perron’s condition P if, for any given vector function f ∈ CB 0,∞ , the solution x t of x′ t A t x t B t x ( g t ) f t 1.5 satisfying x t 0, t ≤ 0 is bounded. A natural question is whether the zero solution of 1.4 is uniformly asymptotically stable under Perron’s condition P . It turns out that the answer depends on the delay function g. The paper is organized as follows. In Section 2, we only state our results; the proofs are included in Section 5. We define an adjoint system and give a variation of parameters formula in Section 3 to be needed in proving the main results. Section 4 contains also some lemmas concerning Perron’s condition and a relation useful for changing the order of integration. 2. Stability Theorems The conclusion obtained by Bellman and Halanay for systems L1 and L2 , respectively, is quite strong. We are only able to prove the stability of the zero solution for more general equation 1.4 under Perron’s condition. To get uniform stability or asymptotic stability or uniform asymptotic stability, we impose restrictions on the delay function. For our purpose, we denote h∗ t : h t − t, t ≥ 0, g∗ t, t0 : sup r∈ h t0 ,t { r − g r , t, t0 ≥ 0. 2.1 Theorem 2.1. Let P hold. If there are positive numbers M1 andM2 such that |A t | ≤ M1, |B t | ≤ M2 ∀t ≥ 0, 2.2 then the zero solution of 1.4 is stable. Theorem 2.2. Let P hold. If 2.2 is satisfied and if there exists a positive real number M3 such that h∗ t ≤ M3 ∀t ≥ 0, 2.3 then the zero solution of 1.4 is uniformly stable. 4 Abstract and Applied Analysis Theorem 2.3. Let P hold. If 2.2 and lim sup t→∞ g∗ t, t0 t − t0 0 for each t0 ≥ 0 2.4 are satisfied, then the zero solution of 1.4 is asymptotically stable. Theorem 2.4. Let P hold. If 2.2 , 2.3 , and lim sup t→∞ g∗ t, t0 t − t0 0 uniformly for t0 ≥ 0 2.5 are satisfied, then the zero solution of 1.4 is uniformly asymptotically stable. Remark 2.5. Note that if g t t − τ , then h t t τ and hence the conditions 2.3 , 2.4 , and 2.5 are automatically satisfied. In this case, all theorems become equivalent, that is, the zero solution is uniformly asymptotically stable. Thus, the results obtained by Bellman and Halanay are recovered. 3. Variation of Parameters Formula To establish a variation of parameters formula to represent the solutions of 1.5 , one needs an adjoint system. The following lemma helps to define the adjoint of 1.4 . Lemma 3.1. Let x t be a solution of 1.4 . If y t is a solution of y′ t −AT t y t − B h t y h t h′ t , 3.1


Introduction
We begin with a classical result for the linear system where A is an n × n matrix function defined and continuous on 0, ∞ . By C B 0, ∞ , we will denote the set of bounded functions defined and continuous on 0, ∞ and by |·| the Euclidean norm.
In 1930, Perron first formulated the following definition being named after him. The proof is accomplished by making use of the basic properties of a fundamental matrix, the Banach-Steinhaus theorem, and the adjoint system It is shown by an example in 3 that Theorem 1.2 may not be valid if the function f appearing in N1 is replaced by a constant vector. However, such a theorem is later obtained in 4 under a Perron-like condition. Theorem 1.2 is extended by Halanay 5 to linear delay systems of the form where A, B are n × n matrix functions defined and continuous on 0, ∞ and τ is a positive real number. Definition 1.3. System L2 is said to satisfy Perron's condition P2 if for any given vector The method used to prove Theorem 1.4 is similar to Bellman's except that the adjoint system is not constructed with respect to an inner product but the functional For some extensions to impulsive differential equations, we refer the reader in particular to 6, 7 .
In this paper, we consider the more general linear delay system Abstract and Applied Analysis 3 where A and B are n × n matrix functions defined and continuous on 0, ∞ and g is a continuously differentiable increasing function defined on 0, ∞ satisfying g t < t and g t ≤ 1. We set h : g −1 . Obviously, h ∈ C 1 0, ∞ and increases on 0, ∞ and h t > t. Perron's condition takes the following form.
Definition 1.5. System 1.4 is said to satisfy Perron's condition P if, for any given vector A natural question is whether the zero solution of 1.4 is uniformly asymptotically stable under Perron's condition P . It turns out that the answer depends on the delay function g.
The paper is organized as follows. In Section 2, we only state our results; the proofs are included in Section 5. We define an adjoint system and give a variation of parameters formula in Section 3 to be needed in proving the main results. Section 4 contains also some lemmas concerning Perron's condition and a relation useful for changing the order of integration.

Stability Theorems
The conclusion obtained by Bellman and Halanay for systems L1 and L2 , respectively, is quite strong. We are only able to prove the stability of the zero solution for more general equation 1.4 under Perron's condition. To get uniform stability or asymptotic stability or uniform asymptotic stability, we impose restrictions on the delay function.
For our purpose, we denote

Variation of Parameters Formula
To establish a variation of parameters formula to represent the solutions of 1.5 , one needs an adjoint system. The following lemma helps to define the adjoint of 1.4 .

Lemma 3.1. Let x t be a solution of 1.4 . If y t is a solution of y t −A T t y t − B T h t y h t h t ,
Proof. Verify directly.
Definition 3.2. The system 3.1 is said to be adjoint to system 1.4 .
It is easy to see that the adjoint of system 3.1 is system 1.4 ; thus the systems are mutually adjoint to each other.

Lemma 3.3. Let Y t, s be a matrix solution of 3.1 for t < s satisfying Y s, s I and Y t, s 0 for t > s. Then x t is a solution of 1.5 if and only if
Proof. Replacing t by β in 1.5 and then integrating the resulting equation multiplied by 3.5 Comparing both sides and using Proof. The proof follows as in 5 . We provide only the steps for the reader's convenience. Define for each rational number t k , k ∈ AE.
In view of P , the family of continuous linear operators {S k } from C B 0, ∞ to C B 0, ∞ is pointwise-bounded. For the space of bounded continuous functions C B 0, ∞ , the usual sup norm · is used.
By the Banach-Steinhaus theorem, the family is uniformly bounded. Thus, there is a positive number M such that S k f ≤ M f for every f ∈ C B 0, ∞ .
As the rational numbers are dense in the real numbers, for each t there is t k such that t k → t as k → ∞ and so t 0 X t, β f β dβ ≤ M f ∀f ∈ C B 0, ∞ .

4.3
The final step involves choosing a sequence of functions and using a limiting process.