BOUNDEDNESS IN NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS

Abstract

In this paper, we investigate bounds for solutions of nonlinear perturbed differential systems.

1. Introduction

The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are three useful methods for showing the qualitative behavior of the solutions of perturbed nonlinear system : Lyapunov’s second method, the use of integral inequalities, and the method of variation of constants formula. 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.

The notion of h-stability (hS) was introduced by Pinto [15,16] with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some perturbations. That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called h-systems. Using this notion, Choi and Ryu [3,5] investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo et al. [8] studied the boundedness of solutions for nonlinear perturbed systems.

In this paper, we obtain some results on boundedness of solutions of nonlinear perturbed differential systems under suitable conditions on perturbed term. To do this we need some integral inequalities.

 

2. Preliminaries

We are interested in the relations of the unperturbed system

and the solutions of the perturbed system

where x,y,f and g are elements of ℝn, an n-dimensional real Euclidean space.

We assume that f, g ∈ C(ℝ+ × ℝn,ℝn), ℝ+ = [0,∞), and that f is continuously differentiable with respect to the components of x on ℝ+ × ℝn, f(t, 0) = 0 for all t ∈ ℝ+. The symbol ｜·｜ will be used to denote arbitrary vector norm in ℝn.

Let x(t, t0, x0) denote the unique solutions of (1) and (2), satisfying the initial conditions x(t0, t0, x0) = x0, and y(t0, t0, y0) = y0, existing on [t0,∞), respectively. Then we can consider the associated variational systems around the zero solution of (1) and around x(t), respectively,

and

Here, fx(t, x) is the matrix whose element in the ith row, jth column is the partial derivative of the ith component of f with respect to the jth component of x. The fundamental matrix Փ(t, t0, x0) of (4) is given by

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

We recall some notions of h-stability [15].

Definition 2.1. The system (1)(the zero solution x = 0 of (1)) is called an h-system if there exist a constant c ≥ 1, and a positive continuous function h on ℝ+ such that

for t ≥ t0 ≥ 0 and |x0| small enough(here

Definition 2.2. The system (1) (the zero solution x = 0 of (1)) is called h-stable(hS) if there exists δ > 0 such that (1) is an h-system for |x0| ≤ δ and h is bounded.

Let M denote the set of all n × n continuous matrices A(t) defined on ℝ+ and N be the subset of M consisting of those nonsingular matrices S(t) that are of class C1 with the property that S(t) and S-1(t) are bounded. The notion of t∞-similarity in M was introduced by Conti [6].

Definition 2.3. A matrix A(t) ∈ M is t∞-similar to a matrix B(t) ∈ M if there exists an n × n matrix F(t) absolutely integrable over ℝ+, i.e.,

such that

for some S(t) ∈ N.

We give some related properties that we need in the sequel.

Lemma 2.1 ([16]). The linear system

where A(t) is an n×n continuous matrix, is an h-system( h-stable, respectively) if and only if there exist c ≥ 1 and a positive continuous( bounded, repectively) function h defined on ℝ+ such that

for t × t0 × 0, where ϕ(t, t0) is a fundamental matrix of (6).

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

Lemma 2.2. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1) and (2), respectively. If y0 ∈ ℝn, then for all t such that x(t, t0, y0) ∈ ℝn,

Theorem 2.3 ([3]). If the zero solution of (1) is hS, then the zero solution of (3) is hS.

Theorem 2.4 ([4]). Suppose that fx(t, 0) is t∞-similar to fx(t, x(t, t0, x0)) for t ≥ t0 ≥ 0 and |x0| ≤ δ for some constant δ > 0. If the solution v = 0 of (3) is hS, then the solution z = 0 of (4) is hS.

Lemma 2.5 ([13]). Let u, f, g ∈ C(ℝ+), for which the inequality

holds, where u0 is a nonnegative constant. Then,

Lemma 2.6 ([5]). Let u, λ1, λ2,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 2.7 ([11]). 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 is the inverse of W (u) and

 

3. Main results

In this section, we investigate bounds for the nonlinear differential systems. Also, we examine the bounded property for the perturbed system of (1)

where g ∈ C(ℝ+ × ℝn,ℝn) and g(t, 0) = 0.

The generalization of a function h’s condition and the strong condition of a function g in Theorem 3.1 [10] are the following result.

Theorem 3.1. Suppose that fx(t, 0) is t∞-similar to fx(t, x(t, t0, x0)) for t ≥ t0 ≥ 0 and |x0| ≤ δ for some constant δ > 0, the solution x = 0 of (1) is hS with a positive continuous function h, and g in (9) satisfies

where Then, the solution y = 0 of (9) is hS.

Proof. Using the nonlinear variation of Alekseev[1], any solution y(t) = y(t, t0, y0) of (9) passing through (t0, y0) is given by

By Theorem 2.3, since the solution x = 0 of (1) is hS, the solution v = 0 of (3) is hS. Therefore, by Theorem 2.4, the solution z = 0 of (4) is hS. By Lemma 2.1 and (10) , we have

Set u(t) = |y(t)|h(t)−1. Then, by Lemma 2.5, we obtain

It follows that y = 0 of (9) is hS. Hence, the proof is complete.

Remark 3.1. In the linear case, we can obtain that if the zero solution x = 0 of (6) is hS, then the perturbed system

is also hS under the same hypotheses in Theorem 3.1 except the condition of t∞-similarity.

Remark 3.2. Letting k(t) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.3 in [9].

The weak condition of a function h and the strong condition of a function g in Theorem 3.3 [8] are the following result.

Theorem 3.2. Let a, b, k, u, w ∈ C(ℝ+), w(u) be nondecreasing in u, u ≤ w(u) and for some v > 0. Suppose that fx(t, 0) is t∞-similar to fx(t, x (t, t0, x0)) for t ≥ t0 ≥ 0 and |x0| ≤ δ for some constant δ > 0, the solution x = 0 of (1) is hS with a positive continuous function h, and g in (9) satisfies

where Then, any solution y(t) = y(t, t0, y0) of (9) is bounded on [t0,∞) and it satisfies

where c = c1|y0|h(t0)−1 and W, W−1 are the same functions as in Lemma 2.6 and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1) and (9), respectively. By Theorem 2.3, since the solution x = 0 of (1) is hS, the solution v = 0 of (3) is hS. Therefore, by Theorem 2.4, the solution z = 0 of (4) is hS. Using Lemma 2.1 and (10), we have

Set u(t) = |y(t)|h(t)−1. Now an application of Lemma 2.6 yields

where c = c1|y0|h(t0) −1. The above estimation yields the desired result since the function h is bounded, and the theorem is proved.

The generalization of a function h’s condition and a slight modification of a function g’s condition in Theorem 3.4[11] are the following result.

Theorem 3.3. Let a, b, k, u, w ∈ C(ℝ+), w(u) be nondecreasing in u, u ≤ w(u) and for some v > 0. Suppose that fx(t, 0) is t∞-similar to fx(t, x(t, t0, x0)) for t ≥ t0 ≥ 0 and |x0| ≤ δ for some constant δ > 0, the solution x = 0 of (1) is hS with the positive continuous function h, and g in (9) satisfies

where Then, any solution y(t) = y(t, t0, y0) of (9) is bounded on [t0,∞) and it satisfies

where W, W-1 are the same functions as in Lemma 2.6 and

Proof. It is known that the solution of (9) is represented by the integral equation(10). By Theorem 2.3, since the solution x = 0 of (1) is hS, the solution v = 0 of (3) is hS. Therefore, by Theorem 2.4, the solution z = 0 of (4) is hS. Using Lemma 2.1 and (10), we have

Set u(t) = |y(t)|h(t)−1. Now an application of Lemma 2.7 yields

where c = c1|y0|h(t0)−1. The above estimation implies the boundedness of y(t), and the proof is complete.

Remark 3.3. Letting k(t) = 0 in Theorem 3.5 and adding the increasing condition of the function h, we obtain the same result as that of Theorem 3.2 in [8].