BOUNDEDNESS IN THE FUNCTIONAL NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS

Abstract

Alexseev's formula generalizes the variation of constants formula and permits the study of a nonlinear perturbation of a system with certain stability properties. In this paper, we investigate bounds for solutions of the functional nonlinear perturbed differential systems using the two notion of h-stability and $t\infty$-similarity.

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 : the use of integral inequalities, the method of variation of constants formula, and Lyapunov’s second 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. Also, he studied some general results about asymptotic integration and gave some important examples in [15]. Choi and Koo [2], Choi and Ryu [3], and Choi et al. [4,5,6] investigated h-stability and bounds of solutions for the perturbed functional differential systems. Also, Goo [8,9,10] studied the boundedness of solutions for the perturbed differential systems.

The main conclusion to be drawn from this paper is that the use of inequalities provides a powerful tool for obtaining bounds for solutions.

 

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 functional nonlinear perturbed differential systems of (2.1)

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 (2.1) with x(t0, t0, x0) = x0, existing on [t0, ∞). Then we can consider the associated variational systems around the zero solution of (2.1) and around x(t), respectively,

and

The fundamental matrix Φ(t, t0, x0) of (2.4) is given by

and Φ(t, t0, 0) is the fundamental matrix of (2.3).

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

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

x(t)| ≤ c |x0| h(t) h(t0)−1

for t ≥ t0 ≥ 0 and |x0| ≤ δ (here ).

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 [7].

Definition 2.2. 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 .

The notion of t∞-similarity is an equivalence relation in the set of all n × n continuous matrices on ℝ+, and it preserves some stability concepts [4, 12].

In this paper, we investigate bounds for solutions of the nonlinear differential systems using the notion of t∞-similarity.

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

Lemma 2.3 ([16]). The linear system

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

for t ≥ t0 ≥ 0, where φ(t, t0) is a fundamental matrix of (2.6).

We need Alekseev formula to compare between the solutions of (2.1) 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 (2.8) 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 2.4. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.8), respectively. If y0 ∈ ℝn, then for all t such that x(t, t0, y0) ∈ ℝn,

Theorem 2.5 ([3]). If the zero solution of (2.1) is hS, then the zero solution of (2.3) is hS.

Theorem 2.6 ([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 (2.3) is hS, then the solution z = 0 of (2.4) is hS.

Lemma 2.7 ([6). (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 , W−1 (u) is the inverse of W(u) and

Lemma 2.8 ([5]). 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 2.7, and

Lemma 2.9 ([9]). 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 W, W−1 are the same functions as in Lemma 2.7, and

 

3. MAIN RESULTS

In this section, we investigate boundedness for solutions of the functional nonlinear perturbed differential systems via t∞-similarity.

Theorem 3.1. Let a, b, c, k, u, w ∈ C(ℝ+), w(u) be nondecreasing in u such that 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 (2.1) is hS with the increasing function h, and g in (2.2) satisfies

and

where and Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0,∞) and it satisfies

where c = c1|y0| h(t0)−1, W, W−1 are the same functions as in Lemma 2.7, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. By Theorem 2.5, since the solution x = 0 of (2.1) is hS, the solution v = 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution z = 0 of (2.4) is hS. Applying Lemmma 2.3, Lemma 2.4, the increasing property of the function h, (3.1), and (3.2), we have

Defining u(t) = |y(t)||h(t)|−1, then, by Lemma 2.8, we have

t0 ≤ t < b1, where c = c1|y0| h(t0)−1. Thus, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0, ∞). This completes the proof. □

Remark 3.2. Letting c(t) = 0 in Theorem 3.1, we obtain the same result as that of Theorem 3.2 in [10].

Theorem 3.3. Let a, b, c, k, u, w ∈ C(ℝ+), w(u) be nondecreasing in u such that 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 (2.1) is hS with the increasing function h, and g in (2.2) satisfies

and

where and

Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0,∞) and it satisfies

where c = c1|y0| h(t0)−1, W, W−1 are the same functions as in Lemma 2.7, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. By Theorem 2.5, since the solution x = 0 of (2.1) is hS, the solution v = 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution z = 0 of (2.4) is hS. Using the nonlinear variation of constants formula and the hS condition of x = 0 of (2.1), (3.3), and (3.4), we have

Set u(t) = |y(t)||h(t)|−1. Then, an application of Lemma 2.9 yields

t0 ≤ t < b1, where c = c1|y0| h(t0)−1. The above estimation yields the desired result since the function h is bounded. Thus, the theorem is proved. □

Remark 3.4. Letting w(u) = u and b(s) = c(s) = 0 in Theorem 3.3, we obtain the same result as that of Theorem 3.3 in [11].

Lemma 3.5. Let u, λ1, λ2, λ3, λ4, λ5 ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u. Suppose that for some c > 0 and 0 ≤ t0 ≤ t,

Then

where W, W−1 are the same functions as in Lemma 2.7, and

Proof. Setting

then we have z(t0) = c and

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

It follows from Lemma 2.7 that (3.6) yields the estimate (3.5). □

Theorem 3.6. Let a, b, c, k, u, w ∈ C(ℝ+), w(u) be nondecreasing in u such that 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 (2.1) is hS with the increasing function h, and g in (2.2) satisfies

and

where and Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0,∞) and

t0 ≤ t < b1, where c = c1|y0| h(t0)−1, W, W−1 are the same functions as in Lemma 2.7, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. By Theorem 2.5, since the solution x = 0 of (2.1) is hS, the solution v = 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution z = 0 of (2.4) is hS. By Lemma 2.3, Lemma 2.4, the increasing property of the function h, (3.7), and (3.8), we have

Set u(t) = |y(t)||h(t)|−1. Then, Lemma 3.5, we obtain

t0 ≤ t < b1, where c = c1|y0| h(t0)−1. Thus, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0, ∞). This completes the proof. □

Lemma 3.7. Let u, λ1, λ2, λ3, λ4, λ5 ∈ C(ℝ+), w ∈ C((0, ∞)) and w(u) be nondecreasing in u, u ≤ w(u). Suppose that for some c > 0 and t0 ≤ t < b1,

Then

where W, W−1 are the same functions as in Lemma 2.7, and

Proof. Define a function v(t) by the right member of (3.9) . Then

which implies

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

Then, by the well-known Bihari-type inequality, (3.11) yields the estimate (3.10). □

Theorem 3.8. Let a, b, c, k, u, w ∈ C(ℝ+), w(u) be nondecreasing in u such that 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 (2.1) is hS with the increasing function h, and g in (2.2) satisfies

and

where and Then, any solution y(t) = y(t, t0, y0) of (2.2) is bounded on [t0,∞) and

t0 ≤ t < b1, where c = c1|y0| h(t0)−1, W, W−1 are the same functions as in Lemma 2.7, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (2.1) and (2.2), respectively. By Theorem 2.5, since the solution x = 0 of (2.1) is hS, the solution v = 0 of (2.3) is hS. Therefore, by Theorem 2.6, the solution z = 0 of (2.4) is hS. Using two Lemma 2.3, Lemma 2.4, the hS condition of x = 0 of (2.1), (3.12), and (3.13), we have

Using Lemma 3.7 with u(t) = |y(t)||h(t)|−1, we have

t0 ≤ t < b1, where c = c1|y0| h(t0)−1. The above estimation implies the boundedness of y(t), and so the proof is complete. □

Remark 3.9. Letting b(t)w(u(t)) = b(t)u(t) in Theorem 3.8, we obtain the same result as that of Theorem 2.4 in [9].