ASYMPTOTIC PROPERTY FOR PERTURBED NONLINEAR FUNCTIONAL DIFFERENTIAL SYSTEMS

Abstract

This paper shows that the solutions to the perturbed nonlinear functional differential system

1. Introduction

Elaydi and Farran [8] 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. Pachpatte [13] investigated the stability and asymptotic behavior of solutions of the functional differential equation. Gonzalez and Pinto [9] proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems. Choi et al. [6,7] examined Lipschitz and exponential asymptotic stability for nonlinear functional systems. Also, Goo [11] and Choi and Goo [2,4] investigated Lipschitz and asymptotic stability for perturbed differential systems.

In this paper we will obtain some results on asymptotic property for nonlinear perturbed differential systems. We will employ the theory of integral inequalities to study asymptotic property for solutions of the nonlinear differential systems. 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.

 

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, we consider the perturbed differential system of (1)

where g ∈ C(ℝ+ × ℝn × ℝn, ℝn), g(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 Φ(t, t0, x0) of (4) is given by

and Φ(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 [8].

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,

(AS)asymptotically stable if it is stable and if there exists δ = δ(t0) > 0 such that if |x0| < δ , then |x(t)| → 0 as t → ∞,

(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,

(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 ([9]). 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 ≥ t0 such that x(t, t0, y0) ∈ ℝn, y(t, t0, y0) ∈ ℝn,

Lemma 2.2 (Bihari-type inequality). Let u, λ ∈ C(ℝ+), w ∈ C((0,∞)) and w(u) be nondecreasing in u. Suppose that, for some c > 0,

Then

where t0 ≤ t < b1, , W−1(u) is the inverse of W(u), and

Lemma 2.3 ([10]). Let u, λ1, λ2, λ3, λ4 ∈ C(ℝ+), w ∈ C((0,∞)), and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0 and 0 ≤ t0 ≤ t,

Then

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

Lemma 2.4 ([3]). Let u, λ1, λ2, λ3, λ4, λ5, λ6, λ7 ∈ C(ℝ+), w ∈ C((0,∞)), and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0 and 0 ≤ t0 ≤ t,

Then

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

Proof. Define a function v(t) by the right member of (6). Then, we have v(t0) = c and

t ≥ t0, since v(t) is nondecreasing, u ≤ w(u), and u(t) ≤ v(t). Now, by integrating the above inequality on [t0, t] and v(t0) = c, we have

It follows from Lemma 2.2 that (8) yields the estimate (7). □

For the proof we need the following corollary from Lemma 2.4.

Corollary 2.5. Let u, λ1, λ2, λ3, λ4, λ5, λ6 ∈ C(ℝ+), w ∈ C((0,∞)), and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0 and 0 ≤ t0 ≤ t,

Then

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

Lemma 2.6 ([5]). Let u, λ1, λ2, λ3, λ4, λ5, λ6 ∈ C(ℝ+), w ∈ C((0,∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0,

Then

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

Proof. Define a function z(t) by the right member of (9). Then, we have z(t0) = c and

since z(t) and w(u) are nondecreasing, u ≤ w(u), and u(t) ≤ z(t). Therefore, by integrating on [t0, t], the function z satisfies

It follows from Lemma 2.2 that (11) yields the estimate (10). □

We prepare two corollaries from Lemma 2.6 that are used in proving the theorems.

Corollary 2.7. Let u, λ1, λ2, λ3 ∈ C(ℝ+), w ∈ C((0,∞)) and w(u) be nonde-creasing in u, u ≤ w(u). Suppose that for some c > 0,

Then

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

Corollary 2.8. Let u, λ1, λ2, λ3, λ4 ∈ C(ℝ+), w ∈ C((0,∞)) and w(u) be non-decreasing in u, u ≤ w(u). Suppose that for some c > 0,

Then

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

 

3. Main results

In this section, we investigate asymptotic property for solutions of perturbed nonlinear functional differential systems.

Theorem 3.1. Let the solution x = 0 of ( 1) be EASV. Suppose that the per-turbing term g(t, y, Ty) satisfies

and

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

where t ≥ t0 and c = |y0|Meαt0, then all solutions of (2) approach 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, (12), and (13), we have

Then, we obtain

since w is nondecreasing. Set u(t) = |y(t)|eαt. By Lemma 2.3 and (14) we have

where c = M|y0|eαt0. The above estimation yields the desired result. □

Remark 3.1. Letting b(t) = 0 in Theorem 3.1, we obtain the similar result as that of Theorem 3.5 in [4].

Theorem 3.2. Let the solution x = 0 of (1) be EASV. Suppose that the per-turbing term g(t, y, Ty) satisfies

and

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

where t ≥ t0 and c = |y0|Meαt0, then all solutions of (2) approach 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. Applying Lemma 2.1, (15), and (16), we have

Since w is nondecreasing, we obtain

Set u(t) = |y(t)|eαt. By Corollary 2.5 and (17), we have

where c = M|y0|eαt0. From the above estimation, we obtain the desired result. □

Remark 3.2. Letting w(u) = u, b(t) = c(t) = 0 in Theorem 3.2, we obtain the similar result as that of Corollary 3.6 in [4].

Theorem 3.3. Let the solution x = 0 of (1) be EASV. Suppose that the perturbed term g(t, y, Ty) satisfies

and

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

where b1 = ∞ and c = M|y0|eαt0, then all solutions of (2) approach 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 conditions, Lemma 2.1, (18), and (19), we have

Since w is nondecreasing, we have

Then, it follows from Corollary 2.7 with u(t) = |y(t)|eαt and (20) that

where c = M|y0|eαt0. Hence, all solutions of (2) approach zero as t → ∞ , and so the proof is complete. □

Remark 3.3. Letting b(t) = 0 in Theorem 3.3, we obtain the similar result as that of Theorem 3.7 in [4].

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

and

where α > 0, a, b, c, k, w ∈ C(ℝ+), a, b, c, k, w ∈ L1(ℝ+) and w(u) is nonde-creasing in u, u ≤ w(u). If

where b1 = ∞ and c = M|y0|eαt0, then all solutions of (2) approach 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 means of Lemma 2.1, (21), and (22), we have

Since w is nondecreasing, we have

Set u(t) = |y(t)|eαt. Then, it follows from Corollary 2.8 and (23) that

where c = M|y0|eαt0. From the above inequality, we obtain the desired result. □

Remark 3.4. Letting w(u) = u, b(t) = c(t) = 0 in Theorem 3.4, we obtain the similar result as that of Corolary 3.8 in [4].