BOUNDEDNESS IN PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS

Abstract

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

1. Introduction

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. 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 investigating the qualitative behavior of the solutions of perturbed nonlinear system of differential systems: the method of variation of constants formula, Lyapunov’s second method, and the use of integral inequalities. 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 [13,14] 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. He obtained a general variational h-stability and some properties about asymptotic behavior of solutions of differential systems called h-systems. Also, he studied some general results about asymptotic integration and gave some important examples in [13]. Choi and Ryu [3], Choi, Koo [5], and Choi et al. [4] investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo [9,10] studied the boundedness of solutions for nonlinear functional perturbed systems.

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

 

2. Preliminaries

We consider the nonlinear functional differential equation

where t ∈ ℝ+ = [0,∞), x ∈ ℝn,f ∈ C(ℝ+ × ℝn, ℝn), f(t, 0) = 0, the derivative fx ∈ C(ℝ+ × ℝn, ℝn), g ∈ C((ℝ+ × ℝn, ℝn), g(t, 0, 0) = 0 and T is a continuous operator mapping from C(ℝ+,ℝn) into C(ℝ+,ℝn). The symbol | · | will be used to denote arbitrary vector norm in ℝn. We assume that for any two continuous functions u, v × C(I) where I is the closed interval, the operator T satisfies the following property:

implies Tu(t) ≤ Tv(t), 0 ≤ t ≤ t1, and |Tu| ≤ T|u|.

Equation (1) can be considered as the perturbed equation of

Let x(t, t0, x0) be denoted by the unique solution of (2) passing through the point (t0, x0) ∈ ℝ+ × ℝn such that x(t0, t0, x0) = x0. Also, we can consider the associated variational systems around the zero solution of (2) 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).

We recall some notions of h-stability [13].

Definition 2.1. The system (2) (the zero solution x = 0 of (2)) 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

Definition 2.2. The system (2) (the zero solution x = 0 of (2)) is called (hS) h-stable if there exists δ > 0 such that (2) 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 sequal.

Lemma 2.1 ([14]). The linear system

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

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

We need Alekseev formula to compare between the solutions of (2) 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 (8) 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 2.2. If y0 ∈ ℝn, then for all t such that x(t, t0, y0) ∈ ℝn,

Theorem 2.3 ([3]). If the zero solution of (2) 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 ([5]). Let u, λ1, λ2, w ∈ C(ℝ+) and w(u) be nondecreasing in u such that for some v > 0. If ,for some c > 0,

then

where is the inverse of W(u), and

Lemma 2.6 ([2]). 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 2.5, and

Lemma 2.7 ([10]). Let u, p, q, w, r ∈ 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 2.5, and

 

3. Main results

In this section, we investigate the bounded property for the nonlinear functional differential systems.

Theorem 3.1. Let a, c, u, w ∈ C(ℝ+), w(u) be nondecreasing in 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 (2) is hS with the increasing function h, and g in (1) satisfies

and

where Then, any solution y(t) = y(t, t0, y0) of (1) is bounded on [t0,∞) and it satisfies

where W, W−1 are the same functions as in Lemma 2.5 , β(t) = c2a(t) , k is a positive constant, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2) and (1), respectively. By Theorem 2.3, since the solution x = 0 of (2) 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, Lemma 2.2 and the increasing property of the function h, we have

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

where k = c1|y0|h(t0)−1 and β(t) = c2a(t). This completes the proof.

Remark 3.1. Letting c(τ) = 0 in Theorem 3.1, we have the similar result as that of Theorem 3.3 in [7].

Theorem 3.2. Let a, b, c, u, w ∈ C(ℝ+), w(u) be nondecreasing in u and for some v > 0. Suppose that the solution x = 0 of (2) is hS with a non-decreasing function h and the perturbed term g in (1) satisfies

and

where Then any solution y(t) = y(t, t0, y0) of (1) is bounded on [t0,∞) and it satisfies

where W, W−1 are the same functions as in Lemma 2.5, k is a positive constant, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2) and (1), respectively. By Theorem 2.2, we obtain

since h is nondecreasing. Set u(t) = |y(t)|h(t)−1. Then, by Lemma 2.6, we have

where k = c1|y0|h(t0)−1. Therefore, we obtain the result.

Remark 3.2. Letting c(τ ) = 0 in Theorem 3.2, we have the similar result as that of Theorem 3.1 in [8].

Theorem 3.3. Let a, b, c, u, w ∈ C(ℝ+), w(u) be nondecreasing in 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 (2) is hS with the increasing function h, and g in (1) satisfies

and

where Then any solution y(t) = y(t, t0, y0) of (1) is bounded on [t0,∞) and it satisfies

where W, W−1 are the same functions as in Lemma 2.5, k is a positive constant, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2) and (1), respectively. By Theorem 2.3, since the solution x = 0 of (2) 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, Lemma 2.2 and the increasing property of the function h, we have

Set u(t) = |y(t)|h(t)−1. Then, by Lemma 2.6, we obtain

where k = c1|y0|h(t0)−1. Hence, the proof is complete.

Remark 3.3. Letting c(τ ) = 0 in Theorem 3.3, we have the similar result as that of Theorem 3.2 in [8].

Theorem 3.4. Let b, c, u, w ∈ C(ℝ+), w(u) be nondecreasing in 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, If the solution x = 0 of (2) is an h-system with a positive continuous function h and g in (1) satisfies

and

where a : ℝ+ → ℝ+ is continuous with

for all t0 ≥ 0, then any solution y(t) = y(t, t0, y0) of (1) satisfies

, t0 ≤ t < b1, where W, W−1 are the same functions as in Lemma 2.5, k is a positive constant, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2) and (1), respectively. By Theorem 2.3, since the solution x = 0 of (2) is an h-system, the solution v = 0 of (3) is an h-system. Therefore, by Theorem 2.4, the solution z = 0 of (4) is an h-system. By Lemma 2.2, we have

Setting u(t) = |y(t)|h(t)−1 and using Lemma 2.7, we obtain

, t0 ≤ t < b1, where k = c1|y0|h(t0)−1. Hence, the proof is complete.

Remark 3.4. Letting c(τ ) = 0 in Theorem 3.4, we have the similar result as that of Theorem 3.5 in [8].