ℎ-Stability of Dynamic Equations on Time Scales with Nonregressivity

Abstract

We study the ℎ-stability of dynamic equations on time scales, without the regressivity condition on the right-hand side of dynamic equations. This means that we can include noninvertible difference equations into our results.

1. Introduction

The theory of calculus on time scales, which has recently received a lot of attention, was created by Hilger [1] in order to unify the theories of differential equations and of difference equations and in order to extend those theories to other kinds of the so-called “dynamic equations.” The two main features of the calculus on time scales are unification and extension of continuous and discrete analysis.

Time scale calculus is especially useful when studying systems with discrete and continuous elements in their domains, such as an insect population that is continuous for parts of the year and discrete for other parts of the year.

The calculus on time scales and dynamic equations on time scales have applications in any field that requires simultaneous modeling of continuous and discrete processes, because they bridge the divide between continuous and discrete aspects of processes. The applications include insect population models, epidemic models, neural networks, and heat transfer. Foundational definitions and results from the time scale calculus appear in an excellent introductory text by Bohner and Peterson [2].

In [3], we extended the concept of ℎ-stability introduced by Pinto [4] to dynamic equations on time scales. The notion of ℎ-stability is quite flexible because it includes the classical notions of uniform or exponential stability within one common framework. For the detailed results of ℎ-stability for differential and difference systems, see [4–7].

Regressivity plays a crucial role in developing the fundamental theory of linear dynamic equations. Consider the scalar dynamic equation on time scale 𝕋:𝑥^Δ(𝑡)=𝑎(𝑡)𝑥(𝑡). (1.1)Equation (1.1) is said to be regressive if 𝑎(𝑡) satisfies1+𝑎(𝑡)𝜇(𝑡)≠0 (1.2)for all 𝑡∈𝕋^𝜅. A system is nonregressive if it is not regressive. When we consider the generalized time scale exponential function, we need the concept of regressiveness since the exponential function 𝑒_𝑎(𝑡,𝑡₀) is defined only for 𝑎(𝑡) satisfying condition (1.2). The continuous dynamical systems (e.g., ordinary differential equations) are always regressive since 𝕋=ℝ has the graininess function 𝜇(𝑡)≡0. However, nonregressivity is always a possibility in discrete dynamical systems (e.g., difference equations), where the underlying domain consists of a mixture of discrete and continuous parts. In fact, if there is even one point in 𝕋 with nonzero graininess, then nonregressivity is possible [8].

In this paper, we investigate the ℎ-stability for dynamic equations on time scales, with the nonregressivity condition. Thus, we improve some results in [3].

2. Calculus on Time Scales

We mention without proof several foundational definitions and theorems, as well as a general introduction to the theory of time scales in an excellent introductory text by Bohner and Peterson [2].

Definition 2.1. A time scale 𝕋 is any nonempty closed subset of the real numbers ℝ.

One assumes throughout that 𝕋 has the topology that it inherits from the standard topology on ℝ.

It is also assumed throughout that in 𝕋 the interval [𝑎,𝑏] means the set {𝑡∈𝕋∶𝑎≤𝑡≤𝑏} for the points 𝑎<𝑏 in 𝕋. Since a time scale may or may not be connected, one needs the following concept of jump operators.

Definition 2.2. The functions 𝜎,𝜌∶𝕋→𝕋 defined by 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡},𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡} (2.1)are called the jump operators.

The jump operators 𝜎 and 𝜌 allow the classification of points in 𝕋 in the following way.

Definition 2.3. A nonmaximal element 𝑡∈𝕋 is said to be right-dense if 𝜎(𝑡)=𝑡, and right-scattered if 𝜎(𝑡)>𝑡. Also, a nonminimal element 𝑡∈𝕋 is called left-dense if 𝜌(𝑡)=𝑡, and left-scattered if 𝜌(𝑡)<𝑡.

Definition 2.4. The function 𝜇∶𝕋→ℝ₊ defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡 is called the graininess function.

If 𝕋 has a left-scattered maximum 𝑚, then 𝕋^𝜅=𝕋−{𝑚}. Otherwise 𝕋^𝜅=𝕋.

Definition 2.5. A function 𝑓∶𝕋→ℝ is called differentiable at 𝑡∈𝕋^𝜅 with (delta) derivative 𝑓^Δ(𝑡)∈ℝ if given 𝜀>0 there exists a neighborhood 𝑈 of 𝑡 such that, for all 𝑠∈𝑈, |𝑓^𝜎(𝑡)−𝑓(𝑠)−𝑓^Δ(𝑡)[𝜎(𝑡)−𝑠]|≤𝜀|𝜎(𝑡)−𝑠|, (2.2)where 𝑓^𝜎=𝑓∘𝜎. If 𝑓 is delta differentiable for every 𝑡∈𝕋^𝜅, then 𝑓∶𝕋→ℝ is called delta differentiable on 𝕋^𝜅.

Some basic properties of delta derivatives are as follows [9, 10].

(i) If 𝑓 is differentiable at 𝑡∈𝕋^𝜅, then𝑓^𝜎(𝑡)=𝑓(𝑡)+𝜇(𝑡)𝑓^Δ(𝑡). (2.3)(ii) If both 𝑓 and 𝑔 are differentiable at 𝑡∈𝕋^𝜅, then the product 𝑓𝑔 is also differentiable at 𝑡∈𝕋^𝜅 with(𝑓𝑔)^Δ(𝑡)=𝑓^Δ(𝑡)𝑔(𝑡)+𝑓^𝜎(𝑡)𝑔^Δ(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑓^Δ(𝑡)𝑔^𝜎(𝑡). (2.4)

Definition 2.6. The function 𝑓∶𝕋→ℝ is said to be rd-continuous (denoted by 𝑓∈𝐶_rd(𝕋,ℝ)) if
(i)𝑓 is continuous at every right-dense point 𝑡∈𝕋,(ii)lim_{𝑠→𝑡−}𝑓(𝑠) exists and is finite at every left-dense point 𝑡∈𝕋.

Definition 2.7. Let 𝑓∈𝐶_rd(𝕋,ℝ). The function 𝑔∶𝕋→ℝ is called the antiderivative of 𝑓 on 𝕋 if it is differentiable on 𝕋 and satisfies 𝑔^Δ(𝑡)=𝑓(𝑡) for 𝑡∈𝕋^𝜅. In this case, one defines ^𝑡_𝑎𝑓(𝑠)Δ𝑠=𝑔(𝑡)−𝑔(𝑎),𝑡∈𝕋. (2.5) Some basic properties of delta integral are as follows [2, 9, 10].
Let 𝑓,𝑔∶𝕋→ℝ be rd-continuous.
(i) If 𝑟,𝑠∈𝕋,𝛼,𝛽∈ℝ, then^𝑠_𝑟[𝛼𝑓(𝑡)+𝛽𝑔(𝑡)]Δ𝑡=𝛼^𝑠_𝑟𝑓(𝑡)Δ𝑡+𝛽^𝑠_𝑟𝑔(𝑡)Δ𝑡. (2.6)(ii) If 𝑡∈𝕋^𝜅, then_𝑡^𝜎(𝑡)𝑓(𝜏)Δ𝜏=𝜇(𝑡)𝑓(𝑡). (2.7)(iii) If 𝕋=ℝ and 𝑎,𝑏∈𝕋, then^𝑏_𝑎𝑓(𝑡)Δ𝑡=^𝑏_𝑎𝑓(𝑡)𝑑𝑡, (2.8)where the integral on the right is the usual Riemann integral from calculus.(iv) If [𝑎,𝑏] consists of only isolated points, then^𝑏_𝑎𝑓(𝑡)Δ𝑡=^{𝑡∈[𝑎,𝑏)}𝜇(𝑡)𝑓(𝑡)if0𝑎<𝑏,if𝑎=𝑏,−_{𝑡∈[𝑏,𝑎)}𝜇(𝑡)𝑓(𝑡)if𝑎>𝑏. (2.9)(v) If 𝕋=𝛿ℤ, where 𝛿>0, then^𝑏_𝑎𝑓(𝑡)Δ𝑡=^{𝑏/𝛿−1𝑘=𝑎/𝛿}𝑓(𝑘𝛿)𝛿if0𝑎<𝑏,if−𝑎=𝑏,_{𝑎/𝛿−1𝑘=𝑏/𝛿}𝑓(𝑘𝛿)𝛿if𝑎>𝑏. (2.10)

3. ℎ-Stability

Let 𝑀_𝑛(ℝ) be the set of all 𝑛×𝑛 matrices over ℝ and the class of all rd-continuous operators 𝐴∶𝕋→𝑀_𝑛(ℝ) denoted by𝐶_rd(𝕋,𝑀_𝑛(ℝ)). (3.1)We consider the linear homogeneous dynamic system𝑥^Δ=𝐴(𝑡)𝑥,𝑥(𝑡₀)=𝑥₀, (3.2)where 𝐴∈𝐶_rd(𝕋,𝑀_𝑛(ℝ)).

Definition 3.1 (See [8]). System (3.2) is said to be regressive if det[𝐼+𝜇(𝑡)𝐴(𝑡)]≠0 (3.3)for all 𝑡∈𝕋^𝜅, where 𝐼 denotes the 𝑛×𝑛 identity matrix.

It turns out that condition (3.3) is equivalent to having all of the eigenvalues of 𝐴(𝑡) regressive in the sense of (1.2) [8].

The norm of an 𝑛×𝑛 matrix 𝐴 is defined to be|𝐴|=max_{1≤𝑗≤𝑛}|𝐴^𝑗|, (3.4)where 𝐴^𝑗 is the 𝑗th column of 𝐴.

We recall the notion of the transition matrix of the linear dynamic systems without regressivity.

Definition 3.2 (See [11, Definition 1.3.5]). Let 𝜏∈𝕋 and 𝐴∈𝐶_rd(𝕋,𝑀_𝑛(ℝ)), and assume that Φ_𝐴∶{(𝑡,𝜏)∈𝕋×𝕋∶𝜏≤𝑡}→𝑀_𝑛(ℝ) is an operator with Φ_𝐴(𝜏,𝜏)=𝐼 for all 𝜏∈𝕋. Then, the unique solution Φ_𝐴 of the IVP 𝑋^Δ(𝑡)=𝐴(𝑡)𝑋(𝑡),𝑋(𝜏)=𝐼 (3.5)is called the transition matrix of (3.2).

Definition 3.3 (See [2, Definition 5.18]). Let 𝜏∈𝕋 and assume that 𝐴∈𝐶_rd(𝕋,𝑀_𝑛(ℝ)) is regressive. The unique solution Φ_𝐴(𝑡,𝜏) of IVP (3.5) is called the matrix exponential function of (3.2).

Note that the solution of (3.2) through (𝑡₀,𝑥₀) can be represented as 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥₀)=Φ_𝐴(𝑡,𝑡₀)𝑥(𝑡₀). The transition operator has the following properties.

Lemma 3.4 (See [11, Theorem 1.3.9]). If 𝐴∈𝐶_rd(𝕋^𝜅,𝑀_𝑛(ℝ)), then there exists the transition matrix Φ_𝐴(𝑡,𝜏) of (3.2) which satisfies the following properties:
(i)Φ_𝐴∶{(𝑡,𝜏)∈𝕋×𝕋∶𝜏≤𝑡}→𝑀_𝑛(ℝ) with Φ_𝐴(𝜏,𝜏)=𝐼 for all 𝜏∈𝕋;(ii)Φ_𝐴(𝑡,𝜏)=Φ_𝐴(𝑡,𝑠)Φ_𝐴(𝑠,𝜏) for all 𝜏≤𝑠≤𝑡;(iii)Φ^Δ𝑡_𝐴(𝑡,𝜏)=𝐴(𝑡)Φ_𝐴Φ(𝑡,𝜏)∀𝜏≤𝑡,_Δ𝜏𝐴(𝑡,𝜏)=−Φ_𝐴(𝑡,𝜎(𝜏))𝐴(𝜏)∀𝜎(𝜏)≤𝑡; (3.6)(iv)Φ_𝐴(𝜎(𝜏),𝜏)=𝐼+𝜇(𝜏)𝐴(𝜏)∀𝜏∈𝕋^𝜅; (3.7)if 𝐴∈𝐶_rd(𝕋^𝜅,𝑀_𝑛(ℝ)) is regressive, that is, (𝐼+𝜇(𝑡)𝐴(𝑡)) is invertible for all 𝑡∈𝕋^𝜅, then Φ_𝐴∶𝕋×𝕋→𝑀_𝑛(ℝ) satisfies (ii) and (iii) for all 𝑡,𝜏∈𝕋;(v)Φ_𝐴(𝑡,𝜏) is invertible in 𝑀_𝑛(ℝ) with Φ_𝐴(𝑡,𝜏)⁻¹=Φ_𝐴(𝜏,𝑡) for all 𝑡,𝜏∈𝕋.

Assume throughout that 𝕋 is unbounded above.

We recall the definitions about the various types of stability for the solutions of (3.2).

Definition 3.5. The solution 𝑥=0 of (3.2) is said to be stable if to any pair of numbers 𝑡₀, 𝜀>0, there exists a 𝛿=𝛿(𝑡₀,𝜀)>0 such that, for any solution 𝑥(𝑡,𝑡₀,𝑥₀) of (3.2), the inequality |𝑥₀|≤𝛿 implies |𝑥(𝑡)|<𝜀 for all 𝑡≥𝑡₀∈𝕋. A system is said to be stable if all of its solutions are stable.

Definition 3.6. The solution 𝑥=0 of (3.2) is said to be uniformly stable if it is stable and 𝛿 does not depend on 𝑡₀.

Definition 3.7. The solution 𝑥=0 of (3.2) is said to be asymptotically stable if it is stable and if there exists a 𝛿₀>0 such that |𝑥₀|≤𝛿₀ implies |𝑥(𝑡)|→0 as 𝑡→∞.

Pinto [4] introduced the notion of ℎ-stability which is an extension of the notions of exponential stability and uniform Lipschitz stability.

Definition 3.8. System (3.2) is said to be
(i)an ℎ-system if there exist a positive function ℎ∶𝕋→ℝ and a constant 𝑐≥1 such that|𝑥(𝑡,𝑡₀,𝑥₀)|≤𝑐|𝑥₀|ℎ(𝑡)ℎ(𝑡₀)⁻¹,𝑡≥𝑡₀, (3.8)for |𝑥₀| small enough (here ℎ(𝑡)⁻¹=1/ℎ(𝑡));(ii)ℎ-stable if system (3.2) is an ℎ-system and the function ℎ is bounded.

Remark 3.9. If ℎ(𝑡)=𝑒^−𝑡, then ℎ-stability coincides with exponential stability, and if ℎ(𝑡) is constant, then we have uniform Lipschitz stability.

Example 3.10. A linear dynamic system on time scale 𝕋 with 𝜇(𝑡)≤1/2,𝑥^Δ(𝑡)=−21−1−sin𝑡−2𝑥(𝑡), (3.9)is ℎ-stable [3].

For the various definitions of stability, we refer to [12] and we obtain the following possible implications for system (3.2) among the various types of stability:ℎ-stability⇒uniformexponentialstability⇒uniformLipschitzstability⇒uniformstability (3.10)as in [6]. The above implications can be proved by the characterization due to Pinto [7, Lemma 1] for the case 𝕋=ℝ, in terms of the transition matrix for system (3.2).

Now, we consider the linear dynamic system (3.2) without the regressivity condition.

Firstly, we show that stability for solutions of (3.2) is equivalent to boundedness of solutions.

Theorem 3.11. All solutions of (3.2) are stable if and only if they are bounded for all 𝑡≥𝑡₀∈𝕋.

Proof. Suppose that the solution 𝑥=0 of (3.2) is stable. Then, given any 𝜀>0, there exists a 𝛿>0 such that |𝑥₀|<𝛿 implies |𝑥(𝑡,𝑡₀,𝑥₀)|<𝜀. However, |𝑥(𝑡,𝑡₀,𝑥₀)|=|Φ_𝐴(𝑡,𝑡₀)𝑥₀|<𝜀 for all 𝑡≥𝑡₀∈𝕋. Now, let 𝑥₀ be a vector (𝛿/2)𝑒^𝑗 for 𝑗=1,2,…,𝑛. Then,|Φ_𝐴(𝑡,𝑡₀)𝑥₀|=|𝑥^𝑗(𝑡,𝑡₀𝛿)|2<𝜀,𝑡≥𝑡₀,𝑗=1,…,𝑛, (3.11)where 𝑥^𝑗(𝑡,𝑡₀) is the 𝑗th column of Φ_𝐴(𝑡,𝑡₀). Thus,|Φ_𝐴(𝑡,𝑡₀)|=max_{1≤𝑗≤𝑛}|𝑥^𝑗(𝑡,𝑡₀)|<2𝜀𝛿,𝑡≥𝑡₀. (3.12)Consequently, for any solution 𝑥(𝑡,𝑡₀,𝑥₀) of (3.2),|𝑥(𝑡,𝑡₀,𝑥₀)|=|Φ_𝐴(𝑡,𝑡₀)𝑥₀|<2𝜀𝛿|𝑥₀|=𝑀|𝑥₀|,𝑡≥𝑡₀, (3.13)where 𝑀=2𝜀/𝛿. That is, all solutions of (3.2) are bounded.
Similarly, we can prove the converse.

In [12, Theorem 2.1], DaCunha obtained the characterization of uniform stability for the regressive system (3.2). Also, we have the same characterization for the nonregressive system (3.2) in what follows.

Theorem 3.12. Equation (3.2) is uniformly stable if and only if there exists a constant 𝛾>0 such that|Φ_𝐴(𝑡,𝑡₀)|≤𝛾,𝑡≥𝑡₀∈𝕋, (3.14)where Φ_𝐴(𝑡,𝑡₀) is a transition matrix of (3.2).

Proof. Suppose that (3.2) is uniformly stable. Then, for any given 𝜀>0, there exists a 𝛿=𝛿(𝜀)>0 such that 𝑡₀≤𝑡₁∈𝕋 and |𝑥(𝑡₁)|<𝛿 imply |𝑥(𝑡)|<𝜀 for all 𝑡≥𝑡₁∈𝕋. Thus,|Φ_𝐴(𝑡,𝑡₁)𝑥(𝑡₁)|<𝜀∀𝑡≥𝑡₁≥𝑡₀∈𝕋. (3.15)Since 𝑥(𝑡₁)=Φ_𝐴(𝑡₁,𝑡₀)𝑥₀ can be selected for any 𝑡₀ and 𝑡₁≥𝑡₀, let 𝑥(𝑡₀) be a vector (𝛿/2)𝑒^𝑗 for each 𝑗=1,2,…,𝑛. Then,|Φ_𝐴(𝑡,𝑡₁)Φ_𝐴(𝑡₁,𝑡₀)𝑥₀|=|Φ_𝐴(𝑡,𝑡₀)𝑥₀|=|𝑥^𝑗(𝑡,𝑡₀𝛿)|2<𝜀,𝑡≥𝑡₁≥𝑡₀, (3.16)where 𝑥^𝑗(𝑡,𝑡₀) is the 𝑗th column of Φ_𝐴(𝑡,𝑡₀). Thus,|Φ_𝐴(𝑡,𝑡₀)|=max_{1≤𝑗≤𝑛}|𝑥^𝑗(𝑡,𝑡₀)|<𝛾,𝑡≥𝑡₀, (3.17)where 𝛾=2𝜀/𝛿. We see that |Φ_𝐴(𝑡,𝑡₀)|≤𝛾 for all 𝑡,𝑡₀∈𝕋 with 𝑡≥𝑡₀.
Conversely, suppose that there exists a 𝛾>0 such that |Φ_𝐴(𝑡,𝑡₀)|≤𝛾 for all 𝑡,𝑡₀∈𝕋 with 𝑡≥𝑡₀. For any 𝑡₀ and 𝑥(𝑡₀)=𝑥₀, the solution of (3.2) satisfies|𝑥(𝑡)|=|Φ_𝐴(𝑡,𝑡₀)𝑥₀|≤|Φ_𝐴(𝑡,𝑡₀)||𝑥₀|≤𝛾|𝑥₀|,𝑡≥𝑡₀. (3.18)Thus, uniform stability of (3.2) is established.

Pinto [4] gave the characterization of ℎ-system. We obtain the time scale version in what follows.

Theorem 3.13. Equation (3.2) is an ℎ-system if and only if there exist a positive function ℎ defined on 𝕋 and a constant 𝑐≥1 such that|Φ_𝐴(𝑡,𝑡₀)|≤𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹,𝑡≥𝑡₀∈𝕋, (3.19)where Φ_𝐴(𝑡,𝑡₀) is a transition matrix of (3.2).

Proof. Suppose that (3.2) is an ℎ-system. Let 𝑥₀ be a vector (𝛿/2)𝑒^𝑗 for 𝑗=1,…,𝑛. Then, we have|𝑥(𝑡,𝑡₀,𝑥₀)|=|Φ_𝐴(𝑡,𝑡₀)𝑥₀|=|𝑥^𝑗(𝑡,𝑡₀𝛿)|2≤|𝑥₀|𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹,𝑡≥𝑡₀,𝑗=1,2,…,𝑛, (3.20)where 𝑥^𝑗(𝑡,𝑡₀) is the 𝑗th column of Φ_𝐴(𝑡,𝑡₀).
Thus, we obtain|Φ_𝐴(𝑡,𝑡₀2)|≤𝛿|𝑥₀|𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹=𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹,𝑡≥𝑡₀. (3.21) Conversely, we have|𝑥(𝑡,𝑡₀,𝑥₀)|=|Φ_𝐴(𝑡,𝑡₀)𝑥₀|≤|Φ_𝐴(𝑡,𝑡₀)||𝑥₀|≤𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹|𝑥₀|,𝑡≥𝑡₀. (3.22)Hence, (3.2) is an ℎ-system.

Remark 3.14. Theorems 3.11, 3.12, and 3.13 also hold for the linear dynamic systems with regressivity.

Now, we study the ℎ-stability of the nonlinear perturbed dynamic system without the regressivity condition on the right-hand side of the equation via Gronwall's inequality and Bihari's inequality on time scales.

We consider the perturbed systems of (3.2):𝑥^Δ=𝐴(𝑡)𝑥+𝐹(𝑡)𝑥,𝑥(𝑡₀)=𝑥₀, (3.23)where 𝐴,𝐹∈𝐶_rd(𝕋,𝑀_𝑛(ℝ)).

We can obtain the following variation-of-constants formula [11].

Lemma 3.15 (See [11]). The solution 𝑥(𝑡,𝑡₀,𝑥₀) of system (3.24) 𝑥^Δ(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝑔(𝑡,𝑥),𝑡∈𝕋, (3.24)where 𝐴∈𝐶_rd(𝕋,𝑀_𝑛(ℝ)) and 𝑔∶𝕋×ℝ^𝑛→ℝ^𝑛 is rd-continuous in the first argument with 𝑔(𝑡,0)=0, with the initial value 𝑥(𝑡₀)=𝑥₀ is given by𝑥(𝑡,𝑡₀,𝑥₀)=Φ_𝐴(𝑡,𝑡₀)𝑥₀+^𝑡_𝑡0Φ_𝐴(𝑡,𝜎(𝑠))𝑔(𝑠,𝑥(𝑠))Δ𝑠,𝑡≥𝑡₀, (3.25)where Φ_𝐴(𝑡,𝑡₀) is a transition matrix of (3.2).

Theorem 3.16. Suppose that (3.2) is ℎ-stable. Then, (3.23) is ℎ-stable if there exists a positive constant 𝛽 such that for all 𝑡₀∈𝕋,^∞_𝑡0ℎ(𝑠)ℎ(𝜎(𝑠))|𝐹(𝑠)|Δ𝑠≤𝛽. (3.26)

Proof. Since (3.2) is ℎ-stable, there exist a constant 𝑐≥1 and a positive bounded function ℎ(𝑡) such that|Φ_𝐴(𝑡,𝑡₀)|≤𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹ (3.27)for all 𝑡≥𝑡₀∈𝕋. By Lemma 3.15, the solution 𝑥(𝑡) of (3.23) satisfies𝑥(𝑡)=Φ_𝐴(𝑡,𝑡₀)𝑥₀+^𝑡_𝑡0Φ_𝐴(𝑡,𝜎(𝑠))𝐹(𝑠)𝑥(𝑠)Δ𝑠,𝑡≥𝑡₀, (3.28)where Φ_𝐴(𝑡,𝑡₀) is a transition matrix of (3.2).
By taking the norms of both sides of (3.28), we have|𝑥(𝑡)|≤𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹|𝑥₀|+𝑐^𝑡_𝑡0ℎ(𝑡)ℎ(𝜎(𝑠))⁻¹|𝐹(𝑠)||𝑥(𝑠)|Δ𝑠,𝑡≥𝑡₀. (3.29)Dividing by ℎ(𝑡)>0 on both sides,|𝑥(𝑡)|ℎ(𝑡)≤𝑐|𝑥₀|ℎ(𝑡₀)+𝑐^𝑡_𝑡0ℎ(𝑠)ℎ(𝜎(𝑠))|𝐹(𝑠)||𝑥(𝑠)|ℎ(𝑠)Δ𝑠,𝑡≥𝑡₀. (3.30)In view of Gronwall's inequality on time scales in [9], we obtain|𝑥(𝑡)|ℎ(𝑡)≤𝑐|𝑥₀|ℎ(𝑡₀)𝑒_{𝑐(ℎ(𝑠)/ℎ(𝜎(𝑠)))|𝐹(𝑠)|}(𝑡,𝑡₀)=𝑐|𝑥₀|ℎ(𝑡₀)exp(^𝑡_𝑡0log(1+𝜇(𝑠)𝑐(ℎ(𝑠)/ℎ(𝜎(𝑠)))|𝐹(𝑠)|)𝜇(𝑠)Δ𝑠)if𝑐𝜇≠0,|𝑥₀|ℎ(𝑡₀)exp(^𝑡_𝑡0𝑐ℎ(𝑠)ℎ(𝜎(𝑠))|𝐹(𝑠)|Δ𝑠)if𝜇=0≤𝑐|𝑥₀|ℎ(𝑡₀)exp(^𝑡_𝑡0𝑐ℎ(𝑠)ℎ(𝜎(𝑠))|𝐹(𝑠)|Δ𝑠)≤𝑐|𝑥₀|ℎ(𝑡₀)exp(^∞_𝑡0𝑐ℎ(𝑠)ℎ(𝜎(𝑠))|𝐹(𝑠)|Δ𝑠)≤𝑐|𝑥₀|ℎ(𝑡₀)𝑒^𝑐𝛽 (3.31)for all 𝑡≥𝑡₀∈𝕋. Thus,|𝑥(𝑡)|≤𝑑|𝑥₀|ℎ(𝑡)ℎ(𝑡₀)⁻¹,𝑡≥𝑡₀, (3.32)where 𝑑=𝑐𝑒^𝑐𝛽≥1. Hence, (3.23) is ℎ-stable.

Corollary 3.17. Suppose that (3.2) is ℎ-stable with bounded ℎ(𝑡)/ℎ(𝜎(𝑡)) for each 𝑡∈𝕋. Then, (3.23) is ℎ-stable if there exists a positive constant 𝛽 such that for all 𝑡₀∈𝕋,^∞_𝑡0|𝐹(𝑠)|Δ𝑠≤𝛽. (3.33)

Corollary 3.18. When 𝕋=ℝ, (3.23) is ℎ-stable if there exists a positive constant 𝛽 such that for all 𝑡₀∈ℝ,^∞_𝑡0|𝐹(𝑠)|𝑑𝑠≤𝛽. (3.34)

Corollary 3.19. When 𝕋=𝛿ℤ with a positive constant 𝛿, (3.23) is ℎ-stable if there exists a positive constant 𝛽 such that for all 𝑡₀∈𝛿ℤ,^∞_𝑠=𝑡0ℎ(𝑠)ℎ(𝑠+𝛿)|𝐹(𝑠)|≤𝛽. (3.35)

We consider the nonlinear perturbed dynamic system𝑧^Δ(𝑡)=𝐴(𝑡)𝑧(𝑡)+𝐺(𝑡,𝑧(𝑡)), (3.36)where 𝐴∈𝐶_rd(𝕋,𝑀_𝑛(ℝ)) and 𝐺 is rd-continuous on 𝕋×ℝ^𝑛.

We recall the notion of the class 𝐻.

Definition 3.20. A function 𝑤∶ℝ⁺→ℝ⁺ belongs to the class 𝐻 if
(H₁)𝑤(𝑢) is nondecreasing and continuous for 𝑢≥0 and positive for 𝑢>0,(H₂) there exists a continuous function 𝜙 on ℝ⁺ with 𝑤(𝛼𝑢)≤𝜙(𝛼)𝑤(𝑢) for 𝛼>0,𝑢≥0,(H₃)lim_𝑢→0⁺(𝑤(𝑢)/𝑢) exists.

Theorem 3.21. Suppose that (3.2) is ℎ-stable and|𝐺(𝑡,𝑥)|≤𝐹(𝑡)𝑤(|𝑥|),𝑡≥𝑡₀, (3.37)where 𝐹 is positive and rd-continuous, and 𝑤∈𝐻 with corresponding multiplier function 𝜙 and𝜆(𝑡)=ℎ(𝑡₀)𝐹(𝑡)ℎ(𝜎(𝑡))|𝑧₀|𝜙(|𝑧₀|ℎ(𝑡)ℎ(𝑡₀)),𝑑=𝑊⁻¹[𝑊(𝑐)+𝑐^∞_𝑡0𝜆(𝑠)Δ𝑠] (3.38)in [13, Theorem 5.8]. Then, (3.36) is ℎ-stable.

Proof. It follows from Lemma 3.15 that the solution 𝑧(𝑡) of (3.36) is given by𝑧(𝑡)=Φ_𝐴(𝑡,𝑡₀)𝑧₀+^𝑡_𝑡0Φ_𝐴(𝑡,𝜎(𝑠))𝐺(𝑠,𝑧(𝑠))Δ𝑠,𝑡≥𝑡₀, (3.39)where Φ_𝐴(𝑡,𝑡₀) is a transition matrix of (3.2).
By ℎ-stability of (3.2), there exists a positive bounded function ℎ(𝑡) such that|Φ_𝐴(𝑡,𝑡₀)|≤𝑐ℎ(𝑡)ℎ(𝑡₀)⁻¹ (3.40)for all 𝑡≥𝑡₀∈𝕋.
The rest of the proof is the same as that of [3, Theorem 2.10].

Now, we obtain some results about the ℎ-stability of system (3.36) in the following corollaries.

Corollary 3.22. Suppose that (3.2) is ℎ-stable with a nondecreasing function ℎ(𝑡) and|𝐺(𝑡,𝑥)|≤𝐹(𝑡)𝑤(|𝑥|),𝑡≥𝑡₀, (3.41)where 𝑤∈𝐻 with corresponding multiplier function 𝜙(𝛼)=𝛼 for all 𝛼>0 and𝑑=𝑊⁻¹[𝑊(𝑐)+𝑐^∞_𝑡0𝐹(𝑠)Δ𝑠]. (3.42)Then, (3.36) is ℎ-stable.

Corollary 3.23. Suppose that (3.2) is ℎ-stable and|𝐺(𝑡,𝑥)|≤𝐹(𝑡)𝑤(|𝑥|),𝑡≥𝑡₀∈𝕋=ℝ, (3.43)where 𝑤∈𝐻 with corresponding multiplier function 𝜙 and𝜆(𝑡)=ℎ(𝑡₀)𝐹(𝑡)ℎ(𝑡)|𝑧₀|𝜙(|𝑧₀|ℎ(𝑡)ℎ(𝑡₀)),𝑑=𝑊⁻¹[𝑊(𝑐)+𝑐^∞_𝑡0𝜆(𝑠)𝑑𝑠]. (3.44)Then, (3.36) is ℎ-stable.

Corollary 3.24. Suppose that (3.2) is ℎ-stable and|𝐺(𝑡,𝑥)|≤𝐹(𝑡)𝑤(|𝑥|),𝑡≥𝑡₀∈𝕋=𝛿ℤ, (3.45)where 𝑤∈𝐻 with corresponding multiplier function 𝜙 and𝜆(𝑡)=ℎ(𝑡₀)𝐹(𝑡)ℎ(𝑡+𝛿)|𝑧₀|𝜙(|𝑧₀|ℎ(𝑡)ℎ(𝑡₀)),𝑑=𝑊⁻¹[𝑊(𝑐)+𝑐𝛿_{𝑠∈𝛿ℤ𝑡0}𝜆(𝑠)]. (3.46)Then, (3.36) is ℎ-stable.

We examine the property of ℎ-stability for the perturbed dynamic system (3.24) on time scale 𝕋.

Lemma 3.25. Suppose that 𝑘∈𝐶_rd(𝕋×ℝ₊,ℝ) is nondecreasing in the second argument 𝑥 for each fixed 𝑡≥𝑡₀ with the property
𝑥(𝑡)−^𝑡_𝑡0𝑘(𝑠,𝑥(𝑠))Δ𝑠≤𝑦(𝑡)−^𝑡_𝑡0𝑘(𝑠,𝑦(𝑠))Δ𝑠,𝑡≥𝑡₀∈𝕋, (3.47)for 𝑥,𝑦∈𝐶_rd(𝕋,ℝ₊). If 𝑥(𝑡₀)<𝑦(𝑡₀), then 𝑥(𝑡)<𝑦(𝑡) for all 𝑡≥𝑡₀∈𝕋.

Proof. Suppose that there exists a 𝜏∈𝕋 with 𝑡₀<𝜏 such that 𝑥(𝑠)<𝑦(𝑠) for 𝑡₀≤𝑠<𝜏 and 𝑥(𝜏)≥𝑦(𝜏). Also, we obtain𝑥(𝜏)−^𝜏_𝑡0𝑘(𝑠,𝑥(𝑠))Δ𝑠>𝑦(𝜏)−^𝜏_𝑡0𝑘(𝑠,𝑦(𝑠))Δ𝑠,𝜏≥𝑡₀∈𝕋. (3.48)This is a contradiction.

If 𝕋=ℝ and 𝕋=𝛿ℤ with a positive constant 𝛿, then we can obtain the following results as corollaries of Lemma 3.25.

Corollary 3.26 (See [14, Lemma 2.1]). Suppose that 𝑘∈𝐶(ℝ×ℝ₊,ℝ) is nondecreasing in the second argument 𝑥 for 𝑡≥𝑡₀ with the property𝑥(𝑡)−^𝑡_𝑡0𝑘(𝑠,𝑥(𝑠))𝑑𝑠≤𝑦(𝑡)−^𝑡_𝑡0𝑘(𝑠,𝑦(𝑠))𝑑𝑠,𝑡≥𝑡₀∈ℝ, (3.49)for 𝑥,𝑦∈𝐶(𝕋,ℝ₊). If 𝑥(𝑡₀)<𝑦(𝑡₀), then 𝑥(𝑡)<𝑦(𝑡) for all 𝑡≥𝑡₀∈ℝ.

Corollary 3.27 (See [5, Lemma 9]). Suppose that 𝑘∶𝛿ℤ×ℝ₊→ℝ is nondecreasing in the second argument 𝑥 for 𝑡≥𝑡₀ with the property𝑥(𝑡)−^𝑡−𝛿_𝑠=𝑡0𝑘(𝑠,𝑥(𝑠))𝛿≤𝑦(𝑡)−^𝑡−𝛿_𝑠=𝑡0𝑘(𝑠,𝑦(𝑠))𝛿,𝑡≥𝑡₀∈𝛿ℤ, (3.50)for 𝑥,𝑦∶𝛿ℤ→ℝ₊. If 𝑥(𝑡₀)<𝑦(𝑡₀), then 𝑥(𝑡)<𝑦(𝑡) for all 𝑡≥𝑡₀∈𝛿ℤ.

Theorem 3.28. Assume that 𝑥=0 of (3.2) is uniformly stable. Suppose that|𝑔(𝑡,𝑥)|≤𝜄(𝑡,|𝑥|), (3.51)where 𝜄∈𝐶_rd(𝕋×ℝ₊,ℝ₊) is strictly increasing in 𝑢 for each fixed 𝑡∈𝕋_𝑡0 with 𝜄(𝑡,0)=0. Consider the scalar dynamic equation𝑢^Δ=𝑐𝜄(𝑡,𝑢),𝑢(𝑡₀)=𝑢₀,𝑡∈𝕋_𝑡0, (3.52)where 𝑐≥1 is a constant. If 𝑢=0 of (3.52) is ℎ-stable, then (3.24) is h-stable whenever 𝑢₀=𝑐|𝑥₀|.

Proof. It follows from Lemma 3.15 that the solution 𝑥(𝑡) of (3.24) is given by𝑥(𝑡)=Φ_𝐴(𝑡,𝑡₀)𝑥₀+^𝑡_𝑡0Φ_𝐴(𝑡,𝜎(𝑠))𝑔(𝑠,𝑥(𝑠))Δ𝑠,𝑡∈𝕋_𝑡0. (3.53)Since 𝑥=0 of (3.2) is uniformly stable, we obtain|𝑥(𝑡)|≤|Φ_𝐴(𝑡,𝑡₀)||𝑥₀|+^𝑡_𝑡0|Φ_𝐴(𝑡,𝜎(𝑠))||𝑔(𝑠,𝑥(𝑠))|Δ𝑠≤𝑐|𝑥₀|+𝑐^𝑡_𝑡0𝜄(𝑠,|𝑥(𝑠)|)Δ𝑠. (3.54)Thus, we have|𝑥(𝑡)|−𝑐^𝑡_𝑡0𝜄(𝑠,|𝑥(𝑠)|)Δ𝑠≤𝑐|𝑥₀|≤𝑢₀=𝑢(𝑡)−𝑐^𝑡_𝑡0𝜄(𝑠,𝑢(𝑠))Δ𝑠,𝑡∈𝕋_𝑡0. (3.55)By Lemma 3.25, we have |𝑥(𝑡)|<𝑢(𝑡) for all 𝑡∈𝕋_𝑡0. Since 𝑢=0 of (3.52) is ℎ-stable, we obtain|𝑥(𝑡)|≤𝑐₁𝑢₀ℎ(𝑡)ℎ(𝑡₀)⁻¹≤𝑐₁𝑐|𝑥₀|ℎ(𝑡)ℎ(𝑡₀)⁻¹=𝑀|𝑥₀|ℎ(𝑡)ℎ(𝑡₀)⁻¹,𝑡∈𝕋_𝑡0, (3.56)where 𝑀=𝑐₁𝑐≥1 is a constant. This completes the proof.

Acknowledgment

This work was supported by the Korea Research Foundation Grant founded by Korea Government (MOEHRD, KRF-2005-070-C00015).