1. INTRODUCTION AND 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 perturbed differential systems of (1.1)
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 (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
|x(t)| ≤ c |x0| h(t) h(t0)−1
for t ≥ t0 ≥ 0 and |x0| small enough (here ).
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.
Integral inequalities play a vital role in the study of boundedness and other qualitative properties of solutions of differential equations. In particular, Bihari’s integral inequality continuous to be an effective tool to study sophisticated problems such as stability, boundedness, and uniqueness of solutions. 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. Choi, Ryu [2] and Choi, Koo, and Ryu [3] investigated bounds of solutions for nonlinear perturbed systems. Also, Goo [7,8,9] and Goo et al. [11] investigated boundedness of solutions for nonlinear perturbed systems.
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 [5].
Definition 1.3. 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 .
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 [5, 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 1.4 ([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 (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. Let x(t) = x(t, t0, y0) and y(t) = y(t, t0, y0) be solutions of (1.1) and (1.8), respectively. If y0 ∈ ℝn, then for all t such that x(t,t0,y0) ∈ ℝn ,
Theorem 1.6 ([2]). If the zero solution of (1.1) is hS, then the zero solution of (1.3) is hS.
Theorem 1.7 ([3]). 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 ([4]). (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 1.9 ([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 W, W−1 are the same functions as in Lemma 1.8, and
Lemma 1.10 ([8]). 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 1.8, and
2. MAIN RESULTS
In this section, we investigate hS and boundedness for solutions of the functional perturbed differential systems via t∞-similarity.
Lemma 2.1. Let u, λ1, λ2, λ3, λ4, λ5 ∈ C[ℝ+, ℝ+] and suppose that, for some c ≥ 0 and t ≥ t0, we have
Then
Proof. Define a function v(t) by the right member of (2.1). Then, we have v(t0) = c and
since v(t) is nondecreasing and u(t) ≤ v(t). Now, by integrating the above inequality on [t0, t] and v(t0) = c, we have
Thus (2.3) yields the estimate (2.2). ☐
Theorem 2.2. 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 (1.1) is hS with the increasing function h, and g in (1.2) satisfies
and
where a, b, c, k, q ∈ C(), and
Then, any solution y(t) = y(t, t0, y0) of (1.2) is hS.
Proof. Using the nonlinear variation of constants formula of Alekseev [1], any solution y(t) = y(t, t0, y0) passing through (t0, y0) is given by
By Theorem 1.6, since the solution x = 0 of (1.1) is hS, the solution v = 0 of (1.3) is hS. Therefore, by Theorem 1.7, the solution z = 0 of (1.4) is hS. In view of Lemma 1.4, the hS condition of x = 0 of (1.1), (2.4),(2.5), and (2.6), we have
Set u(t) = |y(t)||h(t)|−1. Now an application of Lemma 2.1 yields
Where Thus, any solution y(t) = y(t, t0, y0) of (1.2) is hS, and so the proof is complete. ☐
Theorem 2.3. 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 (1.1) is hS with the increasing function h, and g in (1.2) satisfies
and
Where and . Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0, ∞) and it satisfies
where 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, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula (2.6), the hS condition of x = 0 of (1.1), (2.7), and (2.8), we have
Defining u(t) = |y(t)||h(t)|−1, then, by Lemma 1.10, we have
Thus, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0, ∞). This completes the proof. ☐
Remark 2.4. Letting c(t) = 0 in Theorem 2.3, we obtain the same result as that of Theorem 3.2 in [7].
Theorem 2.5. 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 (1.1) is hS with the increasing function h, and g in (1.2) satisfies
and
Where and . Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0, ∞) and it satisfies
where 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, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Applying Lemma 1.4, the hS condition of x = 0 of (1.1), (2.6), (2.9), and (2.10), we have
Set u(t) = |y(t)||h(t)|−1. Then, by Lemma 1.9, 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, ∞). Hence, the proof is complete. ☐
Remark 2.6. Letting c(t) = 0 and w(u) = u in Theorem 2.5, we obtain the same result as that of Theorem 3.1 in [6].
Lemma 2.7. ([8]. ] 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 1.8, and
Proof. Define a function v(t) by the right member of (2.11). 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, (2.13) yields the estimate (2.12). ☐
Theorem 2.8. 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 (1.1) is hS with the increasing function h, and g in (1.2) satisfies
and
Where and . Then, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0, ∞) and
where 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, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula (2.6), the hS condition of x = 0 of (1.1), (2.14), and (2.15), we have
Set u(t) = |y(t)||h(t)|−1. Then, it follows from Lemma 2.7 that we have
where c = c1|y0| h(t0)−1. From the above estimation, we obtain the desired result. Thus, the theorem is proved. ☐
Lemma 2.9. 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
t0 ≤ t < b1, where W, W−1 are the same functions as in Lemma 1.8, and
Proof. Define a function v(t) by the right member of (2.16) . 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, (2.18) yields the estimate (2.17). ☐
Theorem 2.10. 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 (1.1) is hS with the increasing function h, and g in (1.2) satisfies
and
Where and . Then, any solution y = 0 of (1.2) is bounded on [t0, ∞) and it satisfies
t0 ≤ t < b1, where c = c1|y0| h(t0)−1, 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, by Theorem 1.7, the solution z = 0 of (1.4) is hS. Using the nonlinear variation of constants formula (2.6), the hS condition of x = 0 of (1.1), (2.19), and (2.20), we have
Set u(t) = |y(t)||h(t)|−1 with c = c|y0| h(t0)−1. Then, an application of Lemma 2.9 yields
where t0 ≤ t < b1. Thus, any solution y(t) = y(t, t0, y0) of (1.2) is bounded on [t0, ∞), and so the proof is complete. ☐
Remark 2.11. Letting c(t) = 0 and b(t) = a(t) in Theorem 2.10, we obtain the similar result as that of Theorem 3.3 in [11].
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