1. Introduction

The notion of uniformly Lipschitz stability (ULS) was introduced by Dannan and Elaydi [9]. This notion of ULS lies somewhere between uniformly stability on one side and the notions of asymptotic stability in variation of Brauer [4] and uniformly stability in variation of Brauer and Strauss [3] on the other side. An important feature of ULS is that for linear systems, the notion of uniformly Lipschitz stability and that of uniformly stability are equivalent. However, for nonlinear systems, the two notions are quite distinct. Also, Elaydi and Farran [10] introduced the notion of exponential asymptotic stability(EAS) which is a stronger notion than that of ULS. They investigated some analytic criteria for an autonomous differential system and its perturbed systems to be EAS. Gonzalez and Pinto [11] proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems.

In this paper, we investigate Lipschitz and asymptotic stability for solutions of the functional differential systems. To do this we need some integral inequalities. The method incorporating integral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov’s method.

2. Preliminaries

We consider the nonlinear nonautonomous differential system

where f ∈ C(ℝ+ × ℝn, ℝn), ℝ+ = [0, ∞) and ℝn is the Euclidean n-space. We assume that the Jacobian matrix fx = ∂f /∂x exists and is continuous on ℝ+ × ℝn and f (t, 0) = 0. Also, consider the perturbed differential system of (1)

where g ∈ C(ℝ+ × ℝn, ℝn), h ∈ C[ℝ+ × ℝn × ℝn, ℝn] , g(t, 0) = 0, h(t, 0, 0) = 0, and T : C(ℝ+, ℝn) → C(ℝ+, ℝn) is a continuous operator .

For x ∈ ℝn, let For an n × n matrix A, define the norm |A| of A by |A| = sup|x|≤1|Ax|.

Let x(t, t0, x0) denote the unique solution of (1) with x(t0, t0, x0) = x0, existing on [t0, ∞). Then we can consider the associated variational systems around the zero solution of (1) and around x(t), respectively,

and

The fundamental matrix Փ(t, t0, x0) of (4) is given by

and Փ(t, t0, 0) is the fundamental matrix of (3).

Before giving further details, we give some of the main definitions that we need in the sequel [9].

Definition 2.1. The system (1) (the zero solution x = 0 of (1)) is called

(S)stable if for any ϵ > 0 and t0 ≥ 0, there exists δ = δ(t0, ϵ) > 0 such that if |x0| < δ , then |x(t)| < ϵ for all t ≥ t0 ≥ 0,

(US)uniformly stable if the δ in (S) is independent of the time t0,

(ULS) uniformly Lipschitz stable if there exist M > 0 and δ > 0 such that |x(t)| ≤ M|x0| whenever |x0| ≤ δ and t ≥ t0 ≥ 0

(ULSV) uniformly Lipschitz stable in variation if there exist M > 0 and δ > 0 such that |Փ(t, t0, x0) ≤ M for |x0| ≤ δ and t ≥ t0 ≥ 0,

(EAS) exponentially asymptotically stable if there exist constants K > 0 , c > 0, and δ > 0 such that

provided that |x0| < δ,

(EASV) exponentially asymptotically stable in variation if there exist constants K > 0 and c > 0 such that

provided that |x0| < ∞.

Remark 2.1 ([11]). The last definition implies that for |x0| ≤ δ

We give some related properties that we need in the sequel. We need Alekseev formula to compare between the solutions of (1) and the solutions of perturbed nonlinear system

where g ∈ C(ℝ+ × ℝn, ℝn) and g(t, 0) = 0. Let y(t) = y(t, t0, y0) denote the solution of (5) passing through the point (t0, y0) in ℝ+ × ℝn.

The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1].

Lemma 2.1. Let x and y be a solution of (1) and (5), respectively. If y0 ∈ ℝn, then for all t such that x(t, t0, y0) ∈ ℝn,

Lemma 2.2 ([8]). Let u, λ1, λ1, w ∈ C(ℝ+), w(u) be nondecreasing in u and for some v > 0. If, for some c> 0,

then

where is the inverse of W (u), and

Lemma 2.3 ([15]). Let u, p, q, w, and r ∈ C(ℝ+) and suppose that, for some c ≥ 0, we have

Then

Lemma 2.4 ([13]). Let u, λ1, λ2, λ3 ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u ,u ≤ w(u). Suppose that for some c > 0,

Then

where W , W−1 are the same functions as in Lemma 2.2, and

Lemma 2.5 ([13]). Let u, p, q, w, r ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c ≥ 0,

Then

where W , W−1 are the same functions as in Lemma 2.2, and

Lemma 2.6 ([6]). Let u, λ1, λ2, λ3 ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u. Suppose that for some c > 0,

Then

where W , W−1 are the same functions as in Lemma 2.2, and

Lemma 2.7 ([5]). Let u, λ1, λ2, λ3, λ4, w ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c ≥ 0,

for t ≥ t0 ≥ 0 and for some c ≥ 0. Then

for t0 ≤ t < b1, where W, W−1 are the same functions as in Lemma 2.2, and

3. Main results

In this section, we investigate Lipschitz and asymptotic stability for solutions of the perturbed functional differential systems.

Theorem 3.1. For the perturbed (2), we asssume that

and

where a, b, c, k ∈ C(ℝ+), a, b, c, k ∈ L1 (ℝ+), w ∈ C((0, ∞), and w(u) is nondecreasing in

where M (t0) < ∞ and b1 = ∞. Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1) and (2), respectively. Since x = 0 of (1) is ULSV, it is ULS ([9],Theorem 3.3). Applying Lemma 2.1, condition (6), and condition (7), we have

Set u(t) = |y(t)||y0|−1. Now an application of Lemma 2.4 yields

By condition (8), we have |y(t)| ≤ M (t0)|y0| for some M (t0) > 0 whenever |y0| < δ. This completes the proof. □

Remark 3.1. Letting c(t) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.6 in [12].

Theorem 3.2. For the perturbed (2), we asssume that

and

where a, b, c, k ∈ C(ℝ+), a, b, c, k ∈ L1 (ℝ+), w ∈ C((0, ∞), and w(u) is nondecreasing in

where M (t0) < ∞ and b1 = ∞. Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1) and (2), respectively. Since x = 0 of (1) is ULSV, it is ULS . Using the nonlinear variation of constants formula , condition (9), and condition (10), we have

Set u(t) = |y(t)||y0|−1. Then, it follows from Lemma 2.7 that

By condition (11), we have |y(t)| ≤ M (t0)|y0| for some M (t0) > 0 whenever |y0| < δ, and so the proof is complete. □

Remark 3.2. Letting c(t) = 0 in Theorem 3.2, we obtain the same result as that of Theorem 3.7 in [12].

Theorem 3.3. For the perturbed (2), we asssume that

and

where a, b, c, k ∈ C(ℝ+), a, b, c, k ∈ L1 (ℝ+), w ∈ C((0, ∞), and w(u) is nondecreasing in

where M (t0) < ∞ and b1 = ∞. Then the zero solution of (2) is ULS whenever the zero solution of (1) is ULSV.

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1) and (2), respectively. Since x = 0 of (1) is ULSV, it is ULS. Using the nonlinear variation of constants formula, condition (12), and condition (13), we have

Set u(t) = |y(t)||y0|−1. Now an application of Lemma 2.5 yields

t ≥ t0. From condition (14) we get |y(t)| ≤ M (t0)|y0| for some M (t0) > 0 whenever |y0| < δ, and so the proof is complete. □

Remark 3.3. Letting c(s) = 0 in Theorem 3.3, we obtain the same result as that of Theorem 3.7 in [12].

Theorem 3.4. Let the solution x = 0 of (1) be EASV. Suppose that the perturbing term g(t, y) satisfies

and

where α > 0, a, b, c, k ∈ C(ℝ+), a, b, c, k ∈ L1 (ℝ+), w(u) is nondecreasing in u. If

where c = |y0|Meαt0, then all solutions of (2) approch zero as t → ∞

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1) and (2), respectively. Since the solution x = 0 of (1) is EASV, by remark 2.1, it is EAS. Using Lemma 2.1, condition (15), and condition (16), we have

Set u(t) = |y(t)|eαt. By Lemma 2.3, we obtain

where c = M |y0|eαt0. By condition (17), we have |y(t)| ≤ ce−αt M (t0). This estimation yields the desired result. □

Remark 3.4. Letting c(s) = 0 in Theorem 3.4, we obtain the same result as that of Theorem 3.8 in [12].

Theorem 3.5. Let the solution x = 0 of (1) be EASV. Suppose that the perturbed term g(t, y) satisfies

and

where α > 0, a, b, c, k, w ∈ C(ℝ+), a, b, c, k ∈ L1 (ℝ+), w(u) is nondecreasing in u, and

where c = M |y0|eαt0,then all solutions of (2) approch zero as t → ∞

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1) and (2), respectively. Since the solution x = 0 of (1) is EASV, it is EAS by remark 2.1. Using Lemma 2.1, condition (18), and condition (19), we have

Set u(t) = |y(t)|eαt. Since w(u) is nondecreasing, an application of Lemma 2.6 obtains

where c = M |y0|eαt0. By condition (20), we have |y(t)| ≤ e−αt M (t0). From this estimation, we obtain the desired result. □

Remark 3.5. Letting c(t) = 0 in Theorem 3.5, we obtain the same result as that of Theorem 3.9 in [12].