h-STABILITY AND BOUNDEDNESS IN FUNCTIONAL PERTURBED DIFFERENTIAL SYSTEMS
h-STABILITY AND BOUNDEDNESS IN FUNCTIONAL PERTURBED DIFFERENTIAL SYSTEMS
The Pure and Applied Mathematics. 2015. May, 22(2): 145-158
• Received : November 27, 2014
• Accepted : January 27, 2015
• Published : May 31, 2015 PDF e-PUB PubReader PPT Export by style
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YOON HOE GOO

Abstract
In this paper, we investigate h -stability and boundedness for solutions of the functional perturbed differential systems using the notion of t -similarity.
Keywords
1. INTRODUCTION AND PRELIMINARIES
We consider the nonlinear nonautonomous differential system PPT Slide
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where f C (ℝ + × ℝ n ,ℝ n ), ℝ + = [0, ∞) and ℝ n is the Euclidean n -space. We assume that the Jacobian matrix f x = ∂f / ∂x exists and is continuous on ℝ + × ℝ n and f ( t , 0) = 0. Also, consider the functional perturbed differential systems of (1.1) PPT Slide
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where g C (ℝ + × ℝ n , ℝ n ), h C (ℝ + × ℝ n × ℝ n , ℝ n ), g ( t , 0) = 0, h ( t , 0, 0) = 0, and T : C (ℝ + ,ℝ n ) → C (ℝ + ,ℝ n ) is a continuous operator .
For x ∈ ℝ n , let PPT Slide
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For an n × n matrix A , define the norm | A | of A by | A | = sup |x|≤1 | Ax |.
Let x ( t , t 0 , x 0 ) denote the unique solution of (1.1) with x ( t 0 , t 0 , x 0 ) = x 0 , existing on [ t 0 , ∞). Then we can consider the associated variational systems around the zero solution of (1.1) and around x ( t ), respectively PPT Slide
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and PPT Slide
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The fundamental matrix Φ( t , t 0 , x 0 ) of (1.4) is given by PPT Slide
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and Φ( t , t 0 , 0) is the fundamental matrix of (1.3).
We recall some notions of h -stability  .
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 | x 0 | h ( t ) h ( t 0 ) −1
for t t 0 ≥ 0 and | x 0 | small enough (here PPT Slide
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).
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 | x 0 | ≤ 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  and Choi, Koo, and Ryu  investigated bounds of solutions for nonlinear perturbed systems. Also, Goo [7 , 8 , 9] and Goo et al.  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 C 1 with the property that S ( t ) and S −1 ( t ) are bounded. The notion of t -similarity in M was introduced by Conti  .
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., PPT Slide
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such that PPT Slide
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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 (  ). The linear system PPT Slide
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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 + such that PPT Slide
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for t t 0 ≥ 0, where ϕ ( t , t 0 ) 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 PPT Slide
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where g C (ℝ + × ℝ n , ℝ n ) and g ( t , 0) = 0. Let y ( t ) = y ( t , t 0 , y 0 ) denote the solution of (1.8) passing through the point ( t 0 , y 0 ) in ℝ + × ℝ n .
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev  .
Lemma 1.5. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (1.1) and (1.8), respectively. If y 0 ∈ ℝ n , then for all t such that x ( t , t 0 , y 0 ) ∈ ℝ n , PPT Slide
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Theorem 1.6 (  ). If the zero solution of (1.1) is hS, then the zero solution of (1.3) is hS.
Theorem 1.7 (  ). Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ 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 PPT Slide
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Then PPT Slide
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Where PPT Slide
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W −1 ( u ) is the inverse of W ( u ), and PPT Slide
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Lemma 1.9 ( [10) . ] Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t , PPT Slide
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Then PPT Slide
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where W , W −1 are the same functions as in Lemma 1.8, and PPT Slide
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Lemma 1.10 (  ). Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t , PPT Slide
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Then PPT Slide
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Where W , W −1 are the same functions as in Lemma 1.8, and PPT Slide
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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 , λ 1 , λ 2 , λ 3 , λ 4 , λ 5 C [ℝ + , ℝ + ] and suppose that, for some c ≥ 0 and t t 0 , we have PPT Slide
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Then PPT Slide
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Proof. Define a function v ( t ) by the right member of (2.1). Then, we have v ( t 0 ) = c and PPT Slide
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since v ( t ) is nondecreasing and u ( t ) ≤ v ( t ). Now, by integrating the above inequality on [ t 0 , t ] and v ( t 0 ) = c , we have PPT Slide
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Thus (2.3) yields the estimate (2.2). ☐
Theorem 2.2. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ 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 PPT Slide
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and PPT Slide
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where a, b, c, k, q C ( PPT Slide
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), PPT Slide
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and PPT Slide
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Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is hS.
Proof. Using the nonlinear variation of constants formula of Alekseev  , any solution y ( t ) = y ( t , t 0 , y 0 ) passing through ( t 0 , y 0 ) is given by PPT Slide
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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 PPT Slide
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Set u ( t ) = | y ( t )|| h ( t )| −1 . Now an application of Lemma 2.1 yields PPT Slide
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Where PPT Slide
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Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) 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 PPT Slide
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for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ 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 PPT Slide
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and PPT Slide
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Where PPT Slide
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and PPT Slide
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. Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞) and it satisfies PPT Slide
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where W, W −1 are the same functions as in Lemma 1.8, and PPT Slide
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Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) 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 PPT Slide
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Defining u ( t ) = | y ( t )|| h ( t )| −1 , then, by Lemma 1.10, we have PPT Slide
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Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞). 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  .
Theorem 2.5. Let a, b, c, k, u, w C (ℝ + ), w ( u ) be nondecreasing in u such that u w ( u ) and PPT Slide
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for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ 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 PPT Slide
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and PPT Slide
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Where PPT Slide
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and PPT Slide
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. Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞) and it satisfies PPT Slide
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where W , W −1 are the same functions as in Lemma 1.8, and PPT Slide
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Proof . Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) 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 PPT Slide
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Set u ( t ) = | y ( t )| |h ( t )| −1 . Then, by Lemma 1.9, we have PPT Slide
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where c = c 1 | y 0 | h ( t 0 ) −1 . Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞). 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  .
Lemma 2.7. (  . ] Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t , PPT Slide
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Then PPT Slide
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where W, W −1 are the same functions as in Lemma 1.8, and PPT Slide
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Proof. Define a function v ( t ) by the right member of (2.11). Then PPT Slide
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which implies PPT Slide
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since v and w are nondecreasing, u w ( u ), and u ( t ) ≤ v ( t ) . Now, by integrating the above inequality on [ t 0 , t ] and v ( t 0 ) = c , we have PPT Slide
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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 PPT Slide
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for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ 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 PPT Slide
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and PPT Slide
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Where PPT Slide
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and PPT Slide
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. Then, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞) and PPT Slide
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where W , W −1 are the same functions as in Lemma 1.8, and PPT Slide
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Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) 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 PPT Slide
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Set u ( t ) = | y ( t )|| h ( t )| −1 . Then, it follows from Lemma 2.7 that we have PPT Slide
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where c = c 1 | y 0 | h ( t 0 ) −1 . From the above estimation, we obtain the desired result. Thus, the theorem is proved. ☐
Lemma 2.9. Let u, λ 1 , λ 2 , λ 3 , λ 4 C (ℝ + ), w C ((0, ∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0 and 0 ≤ t 0 t , PPT Slide
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Then PPT Slide
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t 0 t < b 1 , where W , W −1 are the same functions as in Lemma 1.8, and PPT Slide
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Proof . Define a function v ( t ) by the right member of (2.16) . Then PPT Slide
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which implies PPT Slide
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since v and w are nondecreasing, u w ( u ), and u ( t ) ≤ v ( t ) . Now, by integrating the above inequality on [ t 0 , t] and v ( t 0 ) = c , we have PPT Slide
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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 PPT Slide
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for some v > 0. Suppose that fx ( t , 0) is t - similar to fx ( t , x ( t , t 0 , x 0 )) for t t 0 ≥ 0 and | x 0 | ≤ 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 PPT Slide
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and PPT Slide
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Where PPT Slide
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and PPT Slide
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. Then, any solution y = 0 of (1.2) is bounded on [ t 0 , ∞) and it satisfies PPT Slide
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t 0 t < b 1 , where c = c 1 | y 0 | h ( t 0 ) −1 , W, W −1 are the same functions as in Lemma 1.8, and PPT Slide
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Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) 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 PPT Slide
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Set u ( t ) = | y ( t )|| h ( t )| −1 with c = c | y 0 | h ( t 0 ) −1 . Then, an application of Lemma 2.9 yields PPT Slide
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where t 0 t < b 1 . Thus, any solution y ( t ) = y ( t , t 0 , y 0 ) of (1.2) is bounded on [ t 0 , ∞), 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  .
Acknowledgements
The author is very grateful for the referee’s valuable comments.
References