Practical Stability in terms of Two Measures for Set Differential Equations on Time Scales

Abstract

We present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle.

1. Introduction

Stability theory in the sense of Lyapunov is now well known. Its basic theory and applications can be found in the monographs of Lasalle and Lefschetz [1] and Rouche et al. [2]. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been proved to be very powerful (see the monograph [3] and the papers [4–6]).

The practical stability is a very important problem in the field of application, which deals with the question of whether the system state evolves within certain subsets of the state-space. It is very useful in estimating the worst-case transient and steady-state responses and in verifying pointwise in time constraints imposed on the state trajectories. Thus practical stability is concerned with quantitative analysis as opposed to Lyapunov analysis which is qualitative in nature. There are some relative results for practical stability of various dynamic systems. We can refer to the monograph of Lakshmikantham et al. [7] and the papers of Zhang and Sun [8], Wang and Liu [9], Wang et al. [10], Sun et al. [11], and Hristova and Georgieva [12] and the references cited therein.

Recently, the study of set differential equations in a semilinear metric space has gained much attention due to its applicability to multivalued differential inclusions and fuzzy differential equations and its inclusion of ordinary differential systems as a special case [13], and some basic results of interest are obtained in [14–21]. However, we notice that there are very few results for set valued differential equation on time scales. For example, Girton [22] gave the results of the existence and uniqueness of the solution of an initial value problem that involve set valued differential equations on time scales. Ahmad and Sivasundaram [23] and Hong [24, 25] discussed some basic problems of set valued differential equation on time scales and obtained some stability criteria, respectively. In this paper, we present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle. Consequently, this paper is organized as follows. In Section 2, we introduce the concepts of the time scales and the set valued differential equations, the propositions of the set valued differential equations on time scales. In Section 3, the relatively new comparison principle and conditions of stability are given.

2. Preliminaries

Let 𝕋 be a time scale with 𝑡₀≥0 as minimal element and no maximal element. Firstly, we give some relative definitions, which can be found in [1–3].

Definition 1. The mappings 𝜎,𝜌: 𝕋→𝕋 defined as 𝜎(𝑡)=inf(𝑠∈𝕋;𝑠>𝑡),𝜌(𝑡)=sup(𝑠∈𝕋;𝑠<𝑡) (1) are called jump operators.

Definition 2. A nonmaximal element 𝑡∈𝕋 is said to be right-scattered (rs) if 𝜎(𝑡)>𝑡 and right-dense (rd) if 𝜎(𝑡)=𝑡. A nonminimal element 𝑡∈𝕋 is called left-scattered (ls) if 𝜌(𝑡)<𝑡 and left-dense (ld) if 𝜌(𝑡)=𝑡.

Definition 3. Let 𝕋^𝑘=𝕋⧵𝜌sup𝕋,sup𝕋ifsup𝕋<∞,𝕋ifsup𝕋=∞. (2)

Definition 4. The mapping 𝑔 is called regulated, if in each left-dense 𝑡∈𝕋 the left sided and in each right-dense 𝑡∈𝕋 the right sided limit exist.

Definition 5. The mapping 𝑔∶𝕋→ℝ^𝑛 is called rd continuous if (i)it is continuous at each right-dense 𝑡∈𝕋;(ii)at each left-dense point the left-sided limit 𝑔(𝑡⁻) exists.
Let 𝐶_rd[𝕋,ℝ^𝑛] denote the set of rd-continuous mappings from 𝕋 to ℝ^𝑛.

Definition 6. The mapping 𝑓∶𝕋×ℝ^𝑛→ℝ^𝑛 is said to be right-dense (rd) continuous and is denoted by 𝑓∈𝐶_rd[𝕋×ℝ^𝑛,ℝ^𝑛] if(i)it is continuous at each (𝑡,𝑥) with right-dense or maximal 𝑡;(ii)the limits 𝑓(𝑡⁻,𝑥)=lim_{(𝑠,𝑦)→(𝑡−,𝑥)}𝑓(𝑠,𝑦) and lim_𝑦→𝑥𝑓(𝑡,𝑦) exist at each (𝑡,𝑥) with left-dense 𝑡.
Let 𝐾(ℝ^𝑛) denote the collection of all nonempty, compact, and convex subsets of ℝ^𝑛. Define the Hausdorff metric 𝐷[𝐴,𝐵]=maxsup_𝑥∈𝐵𝑑(𝑥,𝐴),sup_𝑦∈𝐴𝑑𝑦,𝐵, (3) where 𝑑[𝑥,𝐴]=inf[𝑑(𝑥,𝑦)∶𝑦∈𝐴], 𝐴,𝐵 are bounded sets in ℝ^𝑛. We note that 𝐾(ℝ^𝑛) with this metric is a complete metric space.
It is known that if the space 𝐾(ℝ^𝑛) is equipped with the natural algebraic operations of addition and nonnegative scalar multiplication, then 𝐾(ℝ^𝑛) becomes a semilinear metric space which can be embedded as a complete cone into a corresponding Banach space.
The Hausdorff metric (3) satisfies the following properties: 𝐷[][][][],𝐷[][],𝐷[][][],𝐴+𝐶,𝐵+𝐶=𝐷𝐴,𝐵,𝐷𝐴,𝐵=𝐷𝐵,𝐴𝜆𝐴,𝜆𝐵=𝜆𝐷𝐴,𝐵𝐴,𝐵≤𝐷𝐴,𝐶+𝐷𝐶,𝐵 (4) for all 𝐴,𝐵,𝐶∈𝐾(ℝ^𝑛) and 𝜆∈ℝ₊.

Definition 7. Let 𝐴,𝐵∈𝐾(ℝ^𝑛). The set 𝐶∈𝐾(ℝ^𝑛) satisfying 𝐴=𝐵+𝐶 is known as the Hukuhara difference of the sets 𝐴 and 𝐵 and is denoted by the symbol 𝐴−𝐵.

Definition 8. The forward derivative of 𝐹: 𝕋→𝐾(ℝ^𝑛), denoted by Δ𝐹, is defined by (𝑡)=𝐹(𝜎(𝑡(𝑡)Δ𝐹))−𝐹𝜎(𝑡)(𝑡)−𝑡,if𝑡isright-scattered,Δ𝐹=lim_𝑠→𝑡+𝐹(𝑠)−𝐹(𝑡)𝑠−𝑡,if𝑡isright-dense. (5) Analogously, the backward derivative ∇𝐹 is defined as (𝑡)=𝐹(𝑡)𝜌(𝑡)∇𝐹−𝐹(𝑡)(𝑡)𝑡−𝜌,if𝑡isleft-scattered,∇𝐹=lim_{𝑠→𝑡−}𝐹(𝑡)−𝐹(𝑠)𝑡−𝑠,if𝑡isleft-dense. (6)

Remark 9. The usual scalar subtraction occurs in the scalar coefficient at the beginning of the above expressions and the Hukuhara set difference occurs inside the brackets. Clearly, the existence of Hukuhara difference ensures the existence of the derivative in Definition 8; in the forthcoming analysis, we will be considering the forward derivative as the result for backward derivative followed immediately with suitable changes. The backward derivative has the following properties:(i)let 𝐹,𝐺∶𝕋→𝐾(ℝ^𝑛) be differentiable at 𝑡; then Δ(𝐹+𝐺)(𝑡)=Δ𝐹(𝑡)+Δ𝐺(𝑡);(ii)Δ(𝛼𝐹)(𝑡)=𝛼(Δ𝐹(𝑡)), 𝛼∈ℝ₊.

Remark 10. If the sets in 𝐾(ℝ^𝑛) are singletons only, then there is only one selector possible, namely, 𝐹 itself. In this case, the integral reduces to the generalized integral from time scales into (ℝ^𝑛). The assumption 𝑛=1 generalizes into the integral for time scales. When 𝕋 is everywhere right-dense, then we have Δ𝑡=𝑑𝑡, which results in the conventional formulation for integration of set valued functions. Moreover, for 𝑎,𝑏,𝑐∈𝕋, we have (i)∫^𝑐_𝑎𝐹(𝑡)Δ𝑡=∫^𝑏_𝑎𝐹(𝑡)Δ𝑡+∫^𝑐_𝑏𝐹(𝑡)Δ𝑡;(ii)∫^𝑏_𝑎𝐹(𝑡)Δ𝑡 is closed and convex but need not be necessarily compact.

Definition 11. Let 𝐹,𝐺: 𝕋→𝐾(ℝ^𝑛). Then the Hausdorff distance between 𝐹 and 𝐺, 𝐷[𝐹(⋅),𝐺(⋅)]: 𝕋→ℝ₊, is Δ-integrable and 𝐷^𝑡_𝑡0𝐹(𝑠)Δ𝑠,^𝑡_𝑡0𝐺(𝑠)Δ𝑠≤^𝑡_𝑡0𝐷[𝐹(𝑠),𝐺(𝑠)]Δ𝑠. (7)

Definition 12. Assume 𝐹: 𝕋→𝐾(ℝ^𝑛) is set valued function and 𝑡∈𝕋^𝑘. Let Δ_𝐻𝐹(𝑡) be an element of 𝐾(ℝ^𝑛) (provided it exists) with the property that given any 𝜖>0, there exists a neighborhood 𝑈_𝑡 of 𝑡 (i.e., 𝑈_𝑡=(𝑡−𝛿,𝑡+𝛿)⋂𝕋 for some 𝛿>0) such that 𝐷𝐹(𝑡+ℎ)−𝑓(𝜎(𝑡)),Δ_𝐻𝐹(𝑡)(𝑡)(𝑡),𝐷𝑓(𝜎(𝑡()ℎ−𝜇≤𝜖ℎ−𝜇))−𝐹𝑡−ℎ,Δ_𝐻𝐹(𝑡)𝜇(𝑡)𝜇(𝑡)+ℎ≤𝜖+ℎ (8) for all (𝑡−ℎ,𝑡+ℎ)∈𝑈_𝑡 with 0≤ℎ<𝛿, where 𝜇(𝑡)=𝜎(𝑡)−𝑡. We call Δ_𝐻𝐹(𝑡) the Δ_𝐻-derivative of 𝐹 at 𝑡. We say that 𝐹 is Δ_𝐻-differentiable at 𝑡 if its Δ_𝐻-derivative exists at 𝑡. Moreover, we say 𝐹 is Δ_𝐻-differentiable on 𝕋^𝑘 if its Δ_𝐻-derivative exists at each 𝑡∈𝕋^𝑘. The multivalued function Δ_𝐻𝐹(𝑡): 𝕋^𝑘→𝐾(ℝ^𝑛) is then called the Δ_𝐻-derivative of 𝐹 on 𝕋^𝑘.

Proposition 13. Let??𝐹: 𝕋→𝐾(ℝ^𝑛), Δ_𝐻𝐹(𝑡)∈𝐾(ℝ^𝑛)??(provided it exists). Then some easy and useful relationships concerning the Δ_𝐻-derivative hold.(i)If the??Δ_𝐻-derivative of??𝐹??at??𝑡??exists, then it is unique. Hence, the??Δ_𝐻-derivative is well defined.(ii)Assume??𝐹:??𝕋→𝐾(ℝ^𝑛) ??is a multivalued function and let??𝑡∈𝕋^𝑘. Then one has the following.(1)If??𝐹??is??Δ_𝐻-differentiable at??𝑡, then??𝐹??is continuous at??𝑡.(2)If??𝐹??is continuous at??𝑡??and??𝑡??is right-scattered, then??𝐹??is??Δ_𝐻-differentiable at??𝑡??withΔ_𝐻𝐹(𝑡)=𝐹(𝜎(𝑡))−𝐹(𝑡)𝜇(𝑡). (9)(3)If???𝑡??is right-dense, ??then??𝐹??is??Δ_𝐻-differentiable at??𝑡??if the limitslim_ℎ→0+𝐹(𝑡+ℎ)−𝐹(𝑡)ℎ,lim_ℎ→0+𝐹(𝑡)−𝐹(𝑡−ℎ)ℎ (10)?exist and satisfy the equations lim_ℎ→0+𝐹(𝑡+ℎ)−𝐹(𝑡)ℎ=lim_ℎ→0+𝐹(𝑡)−𝐹(𝑡−ℎ)ℎ=Δ_𝐻𝐹(𝑡). (11)(4)If??𝐹??is differentiable at??𝑡, ??then𝐹(𝜎(𝑡))=𝐹(𝑡)+𝜇(𝑡)Δ_𝐻𝐹(𝑡). (12)

Remark 14. Let 𝐹: 𝕋→𝐾(ℝ^𝑛), Δ_𝐻𝐹(𝑡)∈𝐾(ℝ^𝑛) (provided it exists). We consider the two cases 𝕋=ℝ and 𝕋=ℤ, where ℤ stands for the set consisting of all integers. (1)If 𝕋=ℝ, then 𝐹: ℝ→𝐾(ℝ^𝑛) is Δ_𝐻-differentiable at 𝑡∈ℝ if and only if 𝐷_𝐻𝐹(𝑡)=lim_ℎ→0+𝐹(𝑡+ℎ)−𝐹(𝑡)ℎ=lim_ℎ→0+𝐹(𝑡)−𝐹(𝑡−ℎ)ℎ (13)exists, that is, if and only if 𝐹 is differentiable in the Hukuhara sense at 𝑡. In this case, we have 𝐷_𝐻𝐹(𝑡)=Δ_𝐻𝐹(𝑡). (14)(2)If 𝕋=ℤ, then 𝐹: ℤ→𝐾(ℝ^𝑛) is Δ_𝐻-differentiable at 𝑡∈ℤ with Δ_𝐻𝐹(𝑡)=𝐹(𝜎(𝑡))−𝐹(𝑡)𝜇(𝑡)=𝐹(𝑡+1)−𝐹(𝑡)=Δ𝐹(𝑡), (15)where Δ is the usual forward multivalued difference operator.

3. Main Results

Consider the space 𝐾ℝ^𝑛𝑁=𝐾ℝ^𝑛×𝐾ℝ^𝑛×𝐾ℝ^𝑛,…,𝐾ℝ^𝑛,𝑁times,𝑁∈𝑍, (16) together with either of the following metrics on the space 𝐾(ℝ^𝑛)^𝑁: 𝐷[𝑋,𝑌]=𝐷𝑋₁,𝑌₁,𝐷𝑋₂,𝑌₂,…,𝐷𝑋_𝑁,𝑌_𝑁, (17) where 𝑋,𝑌∈𝐾_𝑐(𝑅^𝑛)^𝑁,𝑋_𝑖,𝑌_𝑖∈𝐾_𝑐(𝑅^𝑛), 𝑖=1,2,…,𝑁.

Consider the initial value problem Δ_𝐻𝑋=𝐹(𝑡,𝑋),𝑋𝑡₀=𝑋₀, (18) where 𝐹∈𝐶_rd[𝕋×𝐾(ℝ^𝑛)^𝑁,𝐾(ℝ^𝑛)^𝑁], 𝑋∈𝐾(ℝ^𝑛)^𝑁, 𝑋=(𝑋₁,𝑋₂,…,𝑋_𝑁) such that for each 𝑖, 𝑖=1,2,…,𝑁, 𝑋_𝑖∈𝐾(ℝ^𝑛).

First of all, we define the following classes of function: 𝒦=𝛼∶ℝ₊→ℝ₊,𝛼isstrictlyincreasingand𝛼(0)=0,𝒞𝒦=𝜙∶𝕋×ℝ₊→ℝ₊(),ℝ,𝜙𝑡,𝑠∈𝒦foreach𝑡Γ=ℎ∶𝕋×𝐾^𝑛𝑁→ℝ₊,inf_{𝑋∈𝐾(ℝ𝑛)𝑁}ℎ(𝑡,𝑋)=0,Σ=𝑄∶ℝ₊→ℝ₊(0),𝑆=)ℝ,𝑄=0and𝑄isincreasingℎ,𝛾(𝑡,𝑋∈𝕋×𝐾^𝑛𝑁,ℎ∈Γandℎ(𝑡,𝑋)<𝛾,𝛾>0. (19)

In order to discuss the stability of the solution of set valued differential systems (18), we state some notions and definitions.

On the vector Lyapunov function on time scales, we defined the Dini derivative of the function 𝑉∈𝐶_rd[𝕋×𝐾(ℝ^𝑛)^𝑁,ℝ^𝑁] along with the solutions of (18) by 𝐷⁻_Δ𝑉(𝑡,𝑋)=liminf_{𝜇(𝑡)→0−}𝑉(𝑡,𝑋)−𝑉𝑡−𝜇(𝑡),𝑋(𝑡)−𝜇(𝑡)𝐹(𝑡,𝑋)𝜇(𝑡),𝐷⁺_Δ𝑉(𝑡,𝑋)=limsup_{𝜇(𝑡)→0+}𝑉𝑡+𝜇(𝑡),𝑋(𝑡)+𝜇(𝑡)𝐹(𝑡,𝑋)−𝑉(𝑡,𝑋)𝜇(𝑡). (20)

Definition 15. A function 𝐺: 𝕋×ℝ^𝑁₊→ℝ^𝑁(𝑁≥1) is said to be upper quasi-monotone nondecreasing in 𝑈, if 𝑈,𝑊∈ℝ^𝑁₊ and ‖𝑈‖_𝑀≤‖𝑊‖_𝑀 imply ‖𝐺(𝑡,𝑈)‖_𝑀≤‖𝐺(𝑡,𝑊)‖_𝑀, where ‖⋅‖_𝑀=max_{1≤𝑖≤𝑁}𝑈_𝑖.

In the following, we will prove the comparison result in terms of vector Lyapunov functions relative to the set differential system on time scales.

Theorem 16. Assume that (𝐻₁)𝑉∈𝐶_rd[𝕋×𝐾(ℝ^𝑛)^𝑁,ℝ^𝑁], 𝑉 is locally Lipschitzian in 𝑋; that is, for 𝑋,𝑌∈𝐾(ℝ^𝑛)^𝑁, one has |𝑉(𝑡,𝑋)−𝑉(𝑡,𝑌)|≤𝐴𝐷[𝑋,𝑌], where 𝐴 is an 𝑁×𝑁 matrix of nonnegative elements, |𝑉(𝑡,𝑋)−𝑉(𝑡,𝑌)|=(|𝑉₁(𝑡,𝑋)−𝑉₁(𝑡,𝑌)|,|𝑉₂(𝑡,𝑋)−𝑉₂(𝑡,𝑌)|,…,|𝑉_𝑁(𝑡,𝑋)−𝑉_𝑁(𝑡,𝑌)|). Here, by |𝑉(𝑡,𝑋)| one means the vector (|𝑉₁(𝑡,𝑋)|,|𝑉₂(𝑡,𝑋)|,…,|𝑉_𝑁(𝑡,𝑋)|), where 𝑉_𝑖 are the components of 𝑉, 𝑖=1,2,…,𝑁;(𝐻₂)𝑉 satisfies 𝐷⁺_Δ𝑉(𝑡,𝑋)≤𝐺(𝑡,𝑉(𝑡,𝑋)), (21)where 𝐺∈𝐶_rd[𝕋×ℝ^𝑁,ℝ^𝑁] is upper quasi-monotone nondecreasing in 𝑈 for each 𝑡∈𝕋; (𝐻₃)𝑈∈ℝ^𝑁, 𝑈(𝑡)=𝑈(𝑡,𝑡₀,𝑈₀) is the solution of 𝑈^Δ=𝐺(𝑡,𝑈),𝑈𝑡₀=𝑈₀. (22)Then for any solution 𝑋(𝑡) of (18), one has ‖𝑉(𝑡,𝑋)‖_𝑀≤‖𝑈(𝑡)‖_𝑀,𝑡∈𝕋, (23) provided ‖𝑉(𝑡₀,𝑋₀)‖_𝑀≤‖𝑈₀‖_𝑀.

Proof. Let 𝑈(𝑡) be the solution of (22) for 𝑡≥𝑡₀, 𝑡∈𝕋. Set 𝑚(𝑡)=𝑉(𝑡,𝑋(𝑡)) as an application of the properties of Hausdorff metric; we obtain the estimation 𝑚(𝜎(𝑡(𝑡)(𝜎(𝑡)(𝜎(𝑡((𝑡(𝜎(𝑡)(𝜎(𝑡(𝜎(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡(𝜎(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡((𝑡𝐷[𝑋(𝜎(𝑡(𝑡)+(𝜎(𝑡))𝐹((𝑡(𝜎(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡((𝑡.))−𝑚=𝑉,𝑋)))−𝑉𝑡,𝑋))=𝑉,𝑋)))−𝑉,𝑋−𝑡𝑡,𝑋)))+𝑉,𝑋−𝑡𝑡,𝑋)))−𝑉𝑡,𝑋))≤𝐴)),𝑋−𝑡𝑡,𝑋))]+𝑉,𝑋−𝑡𝑡,𝑋)))−𝑉𝑡,𝑋)) (24) Letting 𝑋(𝜎(𝑡))=𝑋(𝑡)+𝑌(𝑡), where 𝑌(𝑡) is the Hukuhara difference of 𝑋(𝜎(𝑡)) and 𝑋(𝑡) for small ℎ>0 and is assumed to exist, hence, 𝑚(𝜎(𝑡(𝑡)𝐷[𝑋(𝑡)(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡(𝜎(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡((𝑡𝐷[𝑌(𝑡),(𝜎(𝑡))𝐹((𝑡(𝜎(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡((𝑡𝐷[𝑋(𝜎(𝑡(𝑡),(𝜎(𝑡))𝐹((𝑡(𝜎(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡((𝑡.))−𝑚≤𝐴+𝑌,𝑋−𝑡𝑡,𝑋))]+𝑉,𝑋−𝑡𝑡,𝑋)))−𝑉𝑡,𝑋))=𝐴−𝑡𝑡,𝑋))]+𝑉,𝑋−𝑡𝑡,𝑋)))−𝑉𝑡,𝑋))=𝐴))−𝑋−𝑡𝑡,𝑋))]+𝑉,𝑋−𝑡𝑡,𝑋)))−𝑉𝑡,𝑋)) (25) Consequently, we find that 𝐷⁺_Δ𝑚(𝑡)=limsup_{𝜎(𝑡)−𝑡→0+}𝑚(𝜎(𝑡))−𝑚(𝑡)𝜎(𝑡)−𝑡=limsup_{𝜎(𝑡)−𝑡→0+}1𝜎(𝑡)𝐴𝐷[𝑋(𝜎(𝑡(𝑡),(𝜎(𝑡))𝐹((𝑡+𝑉(𝜎(𝑡)(𝑡)+(𝜎(𝑡))𝐹((𝑡((𝑡−𝑡))−𝑋−𝑡𝑡,𝑋))],𝑋−𝑡𝑡,𝑋)))−𝑉𝑡,𝑋))𝜎(𝑡)𝐷𝑋(𝜎(𝑡(𝑡)−𝑡≤𝐴))−𝑋𝜎(𝑡)((𝑡−𝑡,𝐹𝑡,𝑋))+𝐷⁺_Δ𝑉()𝐷Δ𝑡,𝑋=𝐴_𝐻𝑋(𝑡),𝐹(𝑡,𝑋(𝑡))+𝐷⁺_Δ𝑉(𝑡,𝑋)=𝐷⁺_Δ𝑉(𝑡,𝑋), (26) where 𝐷⁺_Δ𝑚(𝑡) is the right-derivative of 𝑚(𝑡). Hence, we have 𝐷⁺_Δ𝑚(𝑡)≤𝐺(𝑡,𝑚(𝑡)). (27) It is said that 𝐷⁺_Δ𝑚(𝑡)_𝑀≤‖𝐺(𝑡,𝑚(𝑡))‖_𝑀. (28) Because ‖𝑉(𝑡₀,𝑋₀)‖_𝑀≤‖𝑈₀‖_𝑀, we can obtain that ‖𝑉(𝑡,𝑋)‖_𝑀≤‖𝑈(𝑡)‖_𝑀,𝑡∈𝕋. (29) The proof is complete.

Now, we list some definitions about stability which will be used in the following discussion.

Definition 17. Let ℎ₀,ℎ∈Γ, 𝑡∈𝕋. Then one says that (i)ℎ₀ is finer than ℎ if there exists a 𝜌>0 and a function 𝜙∈𝒞𝒦 such that ℎ₀(𝑡,𝑥)<𝜌impliesℎ(𝑡,𝑥)≤𝜙𝑡,ℎ₀(𝑡,𝑥); (30)(ii)ℎ₀ is uniformly finer than ℎ if in (i) 𝜙 is independent of 𝑡.

Definition 18. Let 𝜆,𝐴,𝐵 be positive constants (𝜆<𝐴,𝐵<𝐴). The system (18) is said to be (i)practically stable if for any 0<𝜆<𝐴, the condition ‖𝑋₀‖_𝑀<𝜆 implies ‖𝑋(𝑡)‖_𝑀<𝐴, 𝑡≥𝑡₀ for 𝑡₀,𝑡∈𝕋, where 𝑋(𝑡)=𝑋(𝑡,𝑡₀,𝑋₀) is any solution of (18);(ii)practically quasi-stable if for any 𝜆,𝐵,𝑇>0 and some 𝑡₀∈𝕋 with 𝑡₀+𝑇∈𝕋, the condition ‖𝑋₀‖_𝑀<𝜆 implies ‖𝑋(𝑡)‖_𝑀<𝐵, 𝑡≥𝑡₀+𝑇, 𝑡∈𝕋;(iii)strongly practically stable if (i) and (ii) hold simultaneously;(iv)practically asymptotically stable if (i) holds and for any 𝜖>0 there exists 𝑇₀>0 such that 𝑡₀+𝑇₀∈𝕋 and ‖𝑋₀‖<𝜆 implies ‖𝑋(𝑡)‖<𝜖, 𝑡≥𝑡₀+𝑇₀.

Definition 19. Let 𝜆,𝐴,𝐵 be positive constants (𝜆<𝐴,𝐵<𝐴), ℎ,ℎ₀∈Γ. The system (18) is said to be (PS1)(ℎ₀,ℎ)-practically stable if for any 0<𝜆<𝐴, the condition ℎ₀(𝑡₀,𝑋₀)<𝜆 implies ℎ(𝑡,𝑋(𝑡))<𝐴, 𝑡≥𝑡₀, for some 𝑡₀,𝑡∈𝕋, where 𝑋(𝑡)=𝑋(𝑡,𝑡₀,𝑋₀) is any solution of (18);(PS2)(ℎ₀,ℎ)-practically quasi-stable if for any 𝜆,𝐵,𝑇>0 and some 𝑡₀∈𝕋 with 𝑡₀+𝑇∈𝕋, the condition ℎ₀(𝑡₀,𝑋₀)<𝜆 implies ℎ(𝑡,𝑋(𝑡))<𝐵, 𝑡≥𝑡₀+𝑇, 𝑡∈𝕋;(PS3)(ℎ₀,ℎ)-strongly practically stable if (PS1) and (PS2) hold simultaneously;(PS4)(ℎ₀,ℎ)-practically asymptotically stable if (PS1) holds and for any 𝜖>0 there exists 𝑇₀>0 such that 𝑡₀+𝑇₀∈𝕋 and ℎ₀(𝑡₀,𝑋₀)<𝜆 implies ℎ(𝑡,𝑋(𝑡))<𝜖, 𝑡≥𝑡₀+𝑇₀.
One can similarly define corresponding notion for the system (22).

Definition 20. Let 𝜆,𝐴,𝐵 be positive constants (𝜆<𝐴,𝐵<𝐴), 𝑄₀,𝑄∈Σ. Then we say that the system (22) is (𝑄₀,𝑄)-practically stable if for any 0<𝜆<𝐴, the condition 𝑄₀(‖𝑈₀‖_𝑀)<𝜆 implies 𝑄(‖𝑈(𝑡)‖_𝑀)<𝐴, 𝑡≥𝑡₀, 𝑡∈𝕋, where 𝑈(𝑡)=𝑈(𝑡,𝑡₀,𝑈₀) is any solution of (22).
Other practical stability notions can be defined similarly.

Theorem 21. Assume that (𝐴₁)𝜆,𝐴 are positive constants and 0<𝜆<𝐴;(𝐴₂)ℎ₀,ℎ∈Γ, 𝑡∈𝕋, ℎ₀ is nondecreasing in 𝑡, and ℎ₀ is uniformly finer than ℎ, ℎ(𝑡,𝑋)≤𝜙(ℎ₀(𝑡,𝑋)), 𝜙∈𝒦, where ℎ₀(𝑡,𝑋)<𝜆;(𝐴₃)there exists 𝑉∈𝐶_rd[𝕋×𝐾(ℝ^𝑛)^𝑁,ℝ^𝑁], such that 𝑉(𝑡,𝑋) is locally Lipschitzian in 𝑋 for each right-dense 𝑡∈𝕋 and for 𝑎,𝑏∈𝒦. It holds that 𝑏(ℎ(𝑡,𝑋))≤‖𝑉(𝑡,𝑋)‖_𝑀≤𝑎ℎ₀(𝑡,𝑋); (31)(𝐴₄)for (𝑡,𝑋)∈𝑆(ℎ,𝐴), 𝐷⁺_Δ𝑉(𝑡,𝑋)≤𝐺(𝑡,𝑉(𝑡,𝑋)), (32)where 𝐺∈𝐶_rd[𝕋×ℝ^𝑁,ℝ^𝑁] is upper quasi-monotone nondecreasing in 𝑈 for each 𝑡∈𝕋 and 𝑈(𝑡) is the solution of (22);?(𝐴₅)𝜙(𝜆)<𝐴,𝑎(𝜆)<𝑏(𝐴).

Then practical stability properties of system (22) imply the corresponding (ℎ₀,ℎ)-practical stability properties of (18).

Proof. Assume that (22) is practically stable; then for given (𝑎(𝜆),𝑏(𝐴)), we have 𝑈_0𝑀<𝑎(𝜆)implying‖𝑈(𝑡)‖_𝑀<𝑏(𝐴),𝑡≥𝑡₀. (33) Then by (𝐴₂) and (𝐴₅), it follows that ℎ(𝑡₀,𝑋₀)≤𝜙(ℎ₀(𝑡₀,𝑋₀))<𝜙(𝜆)<𝐴. We claim that ℎ(𝑡,𝑋)<𝐴.
Indeed, if this were not true, there would exist a solution 𝑋(𝑡) of (18) with ℎ₀(𝑡₀,𝑋₀)<𝜆 and 𝑡₁>𝑡₀, 𝑡₁∈𝕋, such that ℎ(𝑡₁,𝑋(𝑡₁))≥𝐴, ℎ(𝑡,𝑋(𝑡))<𝐴, 𝑡₀≤𝑡<𝑡₁. As ℎ₀(𝑡₀,𝑋₀)<𝜆, Theorem 16 together with (𝐴₂), (𝐴₃) implies that ‖𝑉(𝑡,𝑋)‖_𝑀<‖𝑈(𝑡)‖_𝑀,𝑡₀≤𝑡<𝑡₁,𝑉𝑡₀,𝑋_0𝑀≤𝑎ℎ₀𝑡₀,𝑋₀(𝜆),𝑏(𝐴)ℎ𝑡<𝑎≤𝑏₁,𝑋𝑡₁,𝑋𝑡₁≤𝑉𝑡₁,𝑋_1𝑀≤𝑈𝑡_1𝑀≤𝑏(𝐴). (34) This contradiction proves that ℎ₀(𝑡₀,𝑋₀)<𝜆 implies ℎ(𝑡,𝑋)<𝐴, 𝑡≥𝑡₀.
Next we prove that system (18) is (ℎ₀,ℎ)-strongly practically stable. For given positive numbers 𝜆, 𝐴, 𝐵, and 𝑇, suppose that (22) is strongly practically stable for positive numbers 𝑎(𝜆), 𝑏(𝐴), 𝑏(𝐵), and 𝑇; this means we only need to prove (ℎ₀,ℎ)-practical quasi-stability of system (18). Practical quasi-stability of (22) means that ‖𝑈₀‖_𝑀<𝑎(𝜆) implies ‖𝑈(𝑡)‖_𝑀<𝑏(𝐵), 𝑡≥𝑡₀+𝑇 with 𝑡₀+𝑇∈𝕋.
From the foregoing argument, since ‖𝑉(𝑡,𝑋(𝑡))‖_𝑀<𝑎(𝜆), if ℎ₀(𝑡₀,𝑋₀)<𝜆, we have 𝑏(ℎ(𝑡,𝑈))≤‖𝑉(𝑡,𝑋)‖_𝑀≤‖𝑈(𝑡)‖_𝑀<𝑏(𝐵) for all 𝑡≥𝑡₀+𝑇 if ℎ₀(𝑡₀,𝑋₀)<𝜆; thus we have ℎ(𝑡,𝑋(𝑡))<𝐵, 𝑡≥𝑡₀+𝑇 provided ℎ₀(𝑡₀,𝑋₀)<𝜆. Hence system (18) is (ℎ₀,ℎ)-strongly practically stable.
Finally, we show that system (18) is (ℎ₀,ℎ)-practically asymptotically stable. Now, let us suppose that (22) is practically asymptotically stable. This implies we only need to prove that for any given 𝜖>0, there exists 𝑇₀>0 with 𝑡₀+𝑇₀∈𝕋 such that 𝑡₀+𝑇₀≥𝑇 and ℎ₀(𝑡₀,𝑋₀)<𝜆 implies ℎ(𝑡,𝑋(𝑡))<𝐵, 𝑡≥𝑡₀+𝑇₀ for system (18). Practical asymptotic stability of (22) means that 𝑈_0𝑀<𝑎(𝜆)implies𝑈𝑡,𝑡₀,𝑈_0𝑀<𝑏(𝐵),𝑡≥𝑡₀+𝑇₀. (35) From the argument above, since ‖𝑉(𝑡₀,𝑋₀)‖_𝑀<𝑎(𝜆) whenever ℎ₀(𝑡₀,𝑋₀)<𝜆, we obtain 𝑏(ℎ(𝑡,𝑋))≤‖𝑉(𝑡,𝑋)‖_𝑀≤‖𝑈(𝑡,𝑈)‖_𝑀<𝑏(𝐵) (36) for all 𝑡≥𝑡₀+𝑇₀, if ℎ₀(𝑡₀,𝑋₀)<𝜆. Thus we have ℎ(𝑡,𝑋)<𝐵, 𝑡≥𝑡₀+𝑇₀, provided ℎ₀(𝑡₀,𝑋₀)<𝜆. Hence system (18) is (ℎ₀,ℎ)-strongly practically stable.
The proof is complete.

Theorem 22. Suppose that the conditions of Theorem 21 are satisfied except that condition (A₃) is replaced by (𝐴₆)𝑄₀,𝑄∈Σ and for 𝑎,𝑏∈𝒦𝑄‖𝑉(𝑡,𝑋)‖_𝑀(ℎ(,𝑄≥𝑏𝑡,𝑋))₀‖𝑉(𝑡,𝑋)‖_𝑀≤𝑎ℎ₀(𝑡,𝑋). (37)Then (𝑄₀,𝑄)-practical stability properties of system (22) imply the corresponding (ℎ₀,ℎ)-practical stability properties of the system (18).

Proof. Assume that (22) is (𝑄₀,𝑄)-practically stable. Then we have (𝑎(𝜆), 𝑏(𝐴)), such that 𝑄₀𝑈_0𝑀<𝑎(𝜆)implies𝑄𝑈𝑡,𝑡₀,𝑈_0𝑀<𝑏(𝐴),𝑡≥𝑡₀. (38) Suppose that the (ℎ₀,ℎ)-practical stability of (18) does not hold; then, arguing as in Theorem 16, by (A₂), (A₆), we can show that 𝑄₀𝑉𝑡,𝑋𝑡,𝑡₀,𝑋_0𝑀≤𝑎ℎ₀𝑡,𝑋𝑡,𝑡₀,𝑋₀≤𝑎ℎ₀𝑡₁,𝑋𝑡₁,𝑡₀,𝑋₀≤𝑎(𝜆). (39) Then using (A₆), we have 𝑏(𝐴)≤𝑏ℎ𝑡₁,𝑋𝑡₁,𝑡₀,𝑋₀≤𝑄𝑉𝑡₁,𝑋𝑡₁,𝑡₀,𝑋_0𝑀≤𝑄𝑈𝑡₁,𝑈𝑡₁,𝑡₀,𝑈_0𝑀<𝑏(𝐴), (40) which is a contradiction. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).