BOUNDEDNESS IN THE NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS VIA t_∞-SIMILARITY

Abstract

This paper shows that the solutions to the nonlinear perturbed differential system $y{\prime}=f(t,y)+\int_{t_0}^{t}g(s,y(s),T_1y(s))ds+h(t,y(t),T_2y(t))$, have the bounded property by imposing conditions on the perturbed part $\int_{t_0}^{t}g(s,y(s),T_1y(s))ds,h(t,y(t),T_2y(t))$, and on the fundamental matrix of the unperturbed system y′ = f(t, y) using the notion of h-stability.

1. INTRODUCTION AND PRELIMINARIES

We are interested in the relations between the solutions of the unperturbed nonlinear nonautonomous differential system

and the solutions of the perturbed differential system of (1.1) including two operators T1, T2 such that

where f ∈ C(ℝ+ × ℝn, ℝn), g, h ∈ C(ℝ+ × ℝn × ℝn, ℝn), ℝ+ = [0, ∞) , f(t, 0) = 0, g(t, 0, 0) = h(t, 0, 0) = 0, and T1, T2 : C(ℝ+, ℝn) → C(ℝ+, ℝn) are a continuous operator and ℝn is an n-dimensional Euclidean space. We always assume that the Jacobian matrix fx = ∂f /∂x exists and is continuous on ℝ+ × ℝn. The symbol | · | will be used to denote any convenient vector norm in ℝn.

Let x(t, t0, x0) denote the unique solution of (1.1) with x(t0, t0, x0) = x0, existing on [t0, ∞). Then we can consider the associated variational systems around the zero solution of (1.1) and around x(t), respectively,

and

The fundamental matrix Φ(t, t0, x0) of (1.4) is given by

and Φ(t, t0, 0) is the fundamental matrix of (1.3).

We recall some notions of h-stability [16].

Definition 1.1. The system (1.1) (the zero solution x = 0 of (1.1)) 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

Definition 1.2. The system (1.1) (the zero solution x = 0 of (1.1)) is called (hS)h-stable if there exists δ > 0 such that (1.1) is an h-system for |x0| ≤ δ and h is bounded.

Pachpatte[14, 15] investigated the stability, boundedness, and the asymptotic behavior of the solutions of perturbed nonlinear systems under some suitable conditions on the perturbation term g and on the operator T. The purpose of this paper is to investigate bounds for solutions of the nonlinear differential systems

The notion of h-stability (hS) was introduced by Pinto [16,17] 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. Choi, Ryu [5] and Choi, Koo, and Ryu [6] investigated bounds of solutions for nonlinear perturbed systems. Also, Goo [8,9,10] and Goo et al. [3,4] studied the boundedness of solutions for the perturbed differential systems.

Let denote the set of all n×n continuous matrices A(t) defined on ℝ+ and be the subset of 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 was introduced by Conti [7].

Definition 1.3. A matrix is t∞-similar to a matrix if there exists an n × n matrix F(t) absolutely integrable over ℝ+, i.e.,

such that

for some .

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 [7, 12].

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

Lemma 1.4 ([17]). 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 (1.6).

We need Alekseev formula to compare between the solutions of (1.1) 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 (1.8) 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 1.5 ([2]). Let x and y be a solution of (1.1) and (1.8), respectively. If y0 ∈ ℝn, then for all t ≥ t0 such that x(t, t0, y0) ∈ ℝn, y(t, t0, y0) ∈ ℝn,

Theorem 1.6 ([5]). If the zero solution of (1.1) is hS, then the zero solution of (1.3) is hS.

Theorem 1.7 ([6]). 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 (1.3) is hS, then the solution z = 0 of (1.4) is hS.

Lemma 1.8. (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 is the inverse of W(u) and

Lemma 1.9 ([11]). Let u, λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8 ∈ 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 1.8, and

For the proof we prepare the following lemma.

Corollary 1.10. 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 1.8, and

Lemma 1.11 ([3]). 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 1.8, and

 

2. MAIN RESULTS

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

We need the lemma to prove the following theorem.

Lemma 2.1. Let u, λ1, λ2, λ3, λ4, λ5, λ6, λ7, λ8 ∈ 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 1.8, and

Proof. Define a function v(t) by the right member of (2.1) and let us differentiate v(t) to obtain

This reduces to

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

By view of Lemma 1.8, (2.3) yields the estimate (2.2).      □

To obtain the bounded result, the following assumptions are needed:

(H1) 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.

(H2) The solution x = 0 of (1.1) is hS with the increasing function h.

(H3) w(u) be nondecreasing in u such that u ≤ w(u) and for some v > 0.

Theorem 2.2. Let a, b, c, k, q ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (1.2) satisfies

and

where a, b, c, k, q, w ∈ L1(ℝ+), w ∈ C((0, ∞)), T1, T2 are a continuous operator. Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0,∞) and it satisfies

where t0 ≤ t < b1, W, W−1 are the same functions as in Lemma 1.8, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1.1) and (1.2), respectively. By Theorem 1.6, since the solution x = 0 of (1.1) is hS, the solution v = 0 of (1.3) is hS. Therefore, from (H1), by Theorem 1.7, the solution z = 0 of (1.4) is hS. Applying the nonlinear variation of constants formula Lemmma 1.5, together with (2.4) and (2.5), we have

By the assumptions (H2) and (H3), we obtain

Define u(t) = |y(t)||h(t)|−1. Then, by Lemma 2.1, we have

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

Remark 2.3. Letting c(t) = 0 in Theorem 2.2, we obtain the same result as that of Theorem 3.1 in [10].

Theorem 2.4. Let a, b, c, d, k, q ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (1.2) satisfies

and

where a, b, c, d, k, q, w ∈ L1(ℝ+), w ∈ C((0, ∞)), T1, T2 are a continuous operator. Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0,∞) and it satisfies

where t0 ≤ t < b1, W, W−1 are the same functions as in Lemma 1.8, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1.1) and (1.2), respectively. By the same argument as in the proof in Theorem 2.2, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula Lemma 1.5, together with (2.6) and (2.7), we have

It follows from (H2) and (H3) that

Set u(t) = |y(t)||h(t)|−1. Then, by Lemma 1.11, we have

where c = c1|y0| h(t0)−1. Thus, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0, ∞), and so the proof is complete.      □

Remark 2.5. Letting c(t) = d(t) = 0 in Theorem 2.4, we obtain the same result as that of Theorem 3.7 in [10].

Theorem 2.6. Let a, b, c, d, k ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (1.2) satisfies

and

where a, b, c, d, k, w ∈ L1(ℝ+), w ∈ C((0, ∞)), T1, T2 are a continuous operator. Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0,∞) and it satisfies

where t0 ≤ t < b1, W, W−1 are the same functions as in Lemma 1.8, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1.1) and (1.2), respectively. By the same argument as in the proof in Theorem 2.2, the solution z = 0 of (1.4) is hS. By Lemma 1.4, Lemma 1.5, together with (2.8) and (2.9), we have

Using the assumptions (H2) and (H3), we obtain

Let u(t) = |y(t)||h(t)|−1. Then, it follows from Corollary 1.10 that we have

where c = c1|y0| h(t0)−1. From the above estimation, we obtain the desired result. Thus, the theorem is proved.      □

Remark 2.7. Letting c(t) = d(t) = 0 in Theorem 2.6, we obtain the same result as that of Theorem 3.5 in [10].

Theorem 2.8. Let a, b, c, k, q ∈ C(ℝ+). Suppose that (H1), (H2), (H3), and g in (1.2) satisfies

and

where a, b, c, k, q, w ∈ L1(ℝ+), w ∈ C((0, ∞)), T1, T2 are a continuous operator. Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0,∞) and it satisfies

where t0 ≤ t < b1, W, W−1 are the same functions as in Lemma 1.8, and

Proof. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1.1) and (1.2), respectively. By the same argument as in the proof in Theorem 2.2, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula Lemma 1.5,

together with (2.10) and (2.11), we have

Using (H2) and (H3), we obtain

Put u(t) = |y(t)||h(t)|−1.Then, an application of Lemma 1.11 yields

where c = c1|y0| h(t0)−1. Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0, ∞), and so the proof is complete.      □

Remark 2.9. Letting c(t) = 0 in Theorem 2.8, we obtain the same result as that of Theorem 3.7 in [10].