𝜓-Exponential Stability of Nonlinear Impulsive Dynamic Equations on Time Scales

Abstract

The purpose of this paper is to present the sufficient 𝜓-exponential, uniform exponential, and global exponential stability conditions for nonlinear impulsive dynamic systems on time scales.

1. Introduction

In recent years, a significant progress has been made in the stability theory of impulsive systems [1, 2], and in [3] authors studied the 𝜓-exponential stability for nonlinear impulsive differential equations. There are various types of stability of dynamic systems on time scales such as asymptotic stability [4, 5], exponential and uniform exponential stability [6–8], and ℎ-stability [9]. In the past decade, many authors studied impulsive dynamic systems on time scales [10–14]. There are some papers on the theory of the stability of impulsive dynamic systems on time scales. In [15], stability criteria for impulsive systems are given and in [16], authors studied 𝜓-uniform stability of linear impulsive dynamic systems.

In this paper, we consider the 𝜓-exponential stability of the zero solution of the first-order nonlinear impulsive dynamic system 𝑥^Δ(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑡∈𝕋⁺_𝑡0,𝑡≠𝑡_𝑘,𝑥𝑡⁺_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,𝑡=𝑡_𝑘𝑥𝑡,𝑘=1,2,…,𝑛,₊₀=𝑥₀, (1) where 𝕋 is a time scale which has at least finitely many right-dense points of impulsive 𝑡_𝑘, 𝑓∶[0,∞)×ℝ^𝑛→ℝ^𝑛 is a nonlinear function and rd continuous in (𝑡_𝑘−1,𝑡_𝑘]×ℝ^𝑛, 𝐼_𝑘∈𝐶_rd[ℝ^𝑛,ℝ^𝑛], and 0≤𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑛<𝑡 are fixed moments of impulsive effect. Let 𝜓_𝑖∶𝕋→(0,∞), 𝑖=1,2,…,𝑛, be rd continuous functions and let 𝜓=diag[𝜓₁,𝜓₂,…,𝜓_𝑛]. Throughout the paper, we assume that 𝑓(𝑡,0)=0, for all 𝑡 in the time scale interval [0,∞), and call the zero function the trivial solution of (1) and we consider 𝕋⁺_𝑡0={𝑡∈𝕋∶𝑡≥𝑡₀}. Existence and uniqueness of solutions of (1) have been studied in [10].

In the following part we present some basic concepts about time scale calculus and we refer the reader to resource [17] for more detailed information on dynamic equations on time scales.

2. Preliminaries

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. For 𝑡∈𝕋 we define the forward jump operator 𝜎∶𝕋→𝕋 by 𝜎(𝑡)∶=inf{𝑠∈𝕋∶𝑠>𝑡} (2) while the backward jump operator 𝜌∶𝕋→𝕋 is defined by 𝜌(𝑡)∶=sup{𝑠∈𝕋∶𝑠<𝑡}. (3)

If 𝜎(𝑡)>𝑡, we say that 𝑡 is right scattered, while if 𝜌(𝑡)<𝑡, we say that 𝑡 is left scattered. Also, if 𝜎(𝑡)=𝑡, then 𝑡 is called right dense, and if 𝜌(𝑡)=𝑡, then 𝑡 is called left dense. The graininess function 𝜇∶𝕋→[0,∞) is defined by 𝜇(𝑡)∶=𝜎(𝑡)−𝑡. (4)

We introduce the set 𝕋^𝜅 which is derived from the time scale 𝕋 as follows. If 𝕋 has a left-scattered maximum 𝑚, then 𝕋^𝜅=𝕋−{𝑚}; otherwise 𝕋^𝜅=𝕋.

A function 𝑓 on 𝕋 is said to be delta differentiable at some point 𝑡∈𝕋 if there is a number 𝑓^Δ(𝑡) such that for every 𝜀>0 there is a neighborhood 𝑈⊂𝕋 of 𝑡 such that 𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)≤𝜀|𝜎(𝑡)−𝑠|,𝑠∈𝑈. (5)

The function 𝑝∶𝕋→ℝ is said to be regressive provided 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋^𝜅. The set of all regressive rd-continuous functions 𝑓∶𝕋→ℝ is denoted by ℜ.

Let 𝑝∈ℜ and 𝜇(𝑡)≠0 for all 𝑡∈𝕋. The exponential function on 𝕋, defined by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠1𝜇(𝑧)log1+𝜇(𝑧)𝑝(𝑧)Δ𝑧, (6) is the solution to the initial value problem 𝑦^Δ=𝑝(𝑡)𝑦, 𝑦(𝑠)=1. Properties of the exponential function on 𝕋 are given in [6].

In [6] authors defined the Lyapunov function on time scales, type I Lyapunov function 𝑉 as, 𝑉(𝑥)=^𝑛_𝑖=1𝑉_𝑖𝑥_𝑖=𝑉₁𝑥₁+⋯+𝑉_𝑛𝑥_𝑛, (7) and Δ derivative of type I Lyapunov function as follows: [𝑉(𝑥(𝑡))]^Δ=^𝑛_𝑖∑₌₁𝑉_𝑖𝑥_𝑖+𝜇(𝑡)𝑓_𝑖(𝑡,𝑥)−𝑉_𝑖𝑥_𝑖𝜇(𝑡)for𝜇(𝑡)(𝑥)()≠0,∇𝑉⋅𝑓𝑡,𝑥for𝜇(𝑡)=0. (8)

We start introducing notations that will be used in the following sections. In the Euclidean 𝑛-space, norm of a vector 𝑥={𝑥₁,𝑥₂,…,𝑥_𝑛}^𝑇 is given by ‖𝑥‖=max{|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|}. The induced norm of an 𝑛×𝑛 matrix 𝐴 is defined to be ‖𝐴‖=sup_{‖𝑥‖≤1}‖𝐴𝑥‖.

Now, we give definition of 𝜓-exponential, 𝜓-uniform exponential, 𝜓-global exponential stability, and stability conditions for the solution of nonlinear impulsive dynamic system (1).

3. 𝝍-Exponential Stability

Definition 1. The trivial solution to (1) is 𝜓 exponentially stable on [0,∞) if any solution 𝑥(𝑡,𝑡₀,𝑥₀) of the system (1) satisfies for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, 𝜓(𝑡)𝑥𝑡,𝑡₀,𝑥₀≤𝐶𝑥₀,𝑡₀𝑒_⊖𝑀𝑡,𝑡₀^𝑑, (9) where 𝑑 is a positive constant and 𝐶(ℎ,𝑡)∈ℝ⁺×𝕋⁺_𝑡0→ℝ⁺ is a nonnegative increasing function, 𝑀>0. If the function 𝐶 is independent of 𝑡₀, then the trivial solution to system (1) is said to be 𝜓 uniformly exponentially stable on [0,∞).

Definition 2. The trivial solution to (1) is 𝜓 globally exponentially stable on [0,∞) if there exist some constants 𝛿>0 and 𝑀≥1 such that any solution 𝑥(𝑡,𝑡₀,𝑥₀) of (1), for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, we have 𝜓(𝑡)𝑥𝑡,𝑡₀,𝑥₀≤𝑀𝑒_⊖𝛿𝑡,𝑡₀. (10)

Now, we shall present sufficient conditions for the 𝜓-exponential stability, 𝜓 uniformly exponential stability, and 𝜓 globally exponentially stability of(1).

Theorem 3. Assume that 𝐷⊂ℝ^𝑛 contains the origin and there exists a type I Lyapunov function 𝑉∶𝕋⁺_𝑡0×𝐷→[0,∞) such that, for all (𝑡,𝑥)∈𝕋⁺_𝑡0×𝐷 and 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, 𝜆₁(𝑡)𝜓(𝑡)𝑥(𝑡)^𝑝≤𝑉(𝑡,𝑥)≤𝜆₂(𝑡)𝜓(𝑡)𝑥(𝑡)^𝑞,𝑉^Δ(𝑡,𝑥)≤−𝜆₃(𝑡)𝜓(𝑡)𝑥(𝑡)^𝑟−𝐿(𝑀⊖𝛿)𝑒_⊖𝛿𝑡,𝑡₀(𝑡),1+𝑀𝜇𝑉(𝑡,𝑥)−𝑉^𝑟╱𝑞(𝑡,𝑥)≤𝛾𝑒_⊖𝛿𝑡,𝑡₀, (11)(12)(13) where 𝜆₁(𝑡), 𝜆₂(𝑡), and 𝜆₃(𝑡) are positive functions, where 𝜆₁(𝑡) is nondecreasing; 𝑝,𝑞,𝑟, and 𝛾 are positive constants; 𝐿 is a nonnegative constant, and 𝛿>𝑀∶=inf_𝑡≥0𝜆₃(𝑡)╱[𝜆₂(𝑡)]^𝑟╱𝑞>0. Then the trivial solution to (1) is 𝜓 exponentially stable on [0,∞).

Proof. Let 𝑥 be a solution to (1) that stays in 𝐷 for all 𝑡≥𝑡₀. As 𝑀∶=inf_𝑡≥0𝜆₃(𝑡)╱[𝜆₂(𝑡)]^𝑟╱𝑞>0,  𝑒_𝑀(𝑡,𝑡₀) is well defined and positive. Thus 𝜆₃(𝑡)╱[𝜆₂(𝑡)]^𝑟╱𝑞≥𝑀. Consider 𝑉(𝑡,𝑥(𝑡))𝑒_𝑀𝑡,𝑡₀^Δ=𝑉^Δ(𝑡,𝑥(𝑡))𝑒^𝜎_𝑀𝑡,𝑡₀+𝑉(𝑡,𝑥(𝑡))𝑒^Δ_𝑀𝑡,𝑡₀,≤−𝜆₃(𝑡)𝜓(𝑡)𝑥(𝑡)^𝑟−𝐿(𝑀⊖𝛿)𝑒_⊝𝛿𝑡,𝑡₀𝑒_𝑀𝑡,𝑡₀+𝑀𝑉(𝑡,𝑥(𝑡))𝑒_𝑀𝑡,𝑡₀=−𝜆₃(𝑡)𝜓(𝑡)𝑥(𝑡)^𝑟+𝑀𝑉(𝑡,𝑥(𝑡))−𝐿(𝑀⊖𝛿)𝑒_⊖𝛿𝑡,𝑡₀×𝑒_𝑀𝑡,𝑡₀≤−𝜆₃(𝑡)𝜆₂(𝑡)^𝑟╱𝑞𝑉^𝑟╱𝑞((𝑡((𝑡()𝑒𝑡,𝑥))+𝑀𝑉𝑡,𝑥))−𝐿𝑀⊖𝛿_⊖𝛿𝑡,𝑡₀𝑒_𝑀𝑡,𝑡₀≤𝑀𝑉(𝑡,𝑥(𝑡))−𝑉^𝑟╱𝑞(𝑡,𝑥(𝑡))−𝐿(𝑀⊖𝛿)𝑒_⊖𝛿𝑡,𝑡₀×𝑒_𝑀𝑡,𝑡₀≤𝑀𝛾−𝐿(𝑀⊖𝛿)𝑒_𝑀⊖𝛿𝑡,𝑡₀. (14) Integrating both sides of above inequality from 𝑡₀ to 𝑡 with 𝑥₀=𝑥(𝑡₀), we obtain, for 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑉(𝑡,𝑥)𝑒_𝑀𝑡,𝑡₀≤𝑉𝑡₀,𝑥₀+^𝑡_𝑡0𝑀𝛾−𝐿(𝑀⊖𝛿)𝑒_𝑀⊖𝛿𝜏,𝑡₀𝑡Δ𝜏=𝑉₀,𝑥₀+𝑀𝛾𝑒𝑀⊖𝛿−𝐿_𝑀⊖𝛿𝑡,𝑡₀+𝑀𝛾𝑡𝛿⊖𝑀+𝐿≤𝑉₀,𝑥₀+𝑀𝛾𝛿⊖𝑀+𝐿. (15) From condition 𝑉(𝑡₀,𝑥₀)≤𝜆₂(𝑡₀)‖𝜓(𝑡₀)𝑥₀‖^𝑞𝑉(𝑡,𝑥)𝑒_𝑀𝑡,𝑡₀≤𝜆₂𝑡₀𝜓𝑡₀𝑥₀^𝑞+𝑀𝛾𝛿⊖𝑀+𝐿. (16) Letting 𝜆₂𝑡₀𝜓𝑡₀𝑥₀^𝑞+𝑀𝛾𝑥𝛿⊖𝑀+𝐿=𝐶₀,𝑡₀>0 (17) we get, 𝑉(𝑡,𝑥)𝑒_𝑀𝑡,𝑡₀≤𝐶𝑥₀,𝑡₀. (18) By condition (11), we have 𝜓(𝑡)𝑥(𝑡)≤𝜆₁^−1╱𝑝(𝑡)(𝑉(𝑡,𝑥))^1╱𝑝 (19) And by the fact that 𝜆₁(𝑡)≥𝜆₁(𝑡₀), we obtain 𝜓(𝑡)𝑥(𝑡)≤𝜆₁^−1╱𝑝𝑡₀(𝑉(𝑡,𝑥))^1╱𝑝. (20) From (18) and (20) we obtain the result for all, 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, 𝜓(𝑡)𝑥(𝑡)≤𝜆₁^−1╱𝑝𝑡₀𝐶𝑥₀,𝑡₀^1╱𝑝𝑒_⊖𝑀𝑡,𝑡₀^1╱𝑝. (21) By Definition 1 system (1) is 𝜓 exponentially stable.

If we consider 𝜓 as scaler function independent of 𝑡, then we get a sufficient condition for 𝜓 uniformly exponential stability as stated below.

Theorem 4. In Theorem 3 if 𝜓 is a constant function independent of 𝑡 and 𝜆_𝑖(𝑡)=𝜆_𝑖, 𝑖=1,2,3, are positive constants, then the trivial solution to system (1) is 𝜓 uniformly exponentially stable on [0,∞).

Proof. The proof is similar to proof of Theorem 3 by taking 𝛿>𝜆₃╱[𝜆₂]^𝑟╱𝑞 and 𝑀=𝜆₃╱[𝜆₂]^𝑟╱𝑞, hence omitted.

Theorem 5. Assume that 𝐷⊂ℝ^𝑛 contains the origin and there exists a type I Lyapunov function 𝑉∶𝕋⁺_𝑡0×𝐷→[0,∞) such that, for all (𝑡,𝑥)∈𝕋⁺_𝑡0×𝐷 and 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, 𝜆₁𝜓𝑥(𝑡)^𝑝≤𝑉(𝑥),𝑉^Δ(𝑡,𝑥)≤−𝜆₂𝑉(𝑥)−𝐿(𝑀⊖𝛿)𝑒_⊖𝛿(𝑡,0)(𝑡),1+𝑀𝜇 (22)(23) where 𝜓 is a constant function independent of 𝑡. 𝜆₁,𝜆₂,𝑝,𝛿>0,  𝐿≥0 are constants and 0<𝑀<min{𝜆₂,𝛿}. Then the trivial solution to (1) is 𝜓 uniformly exponentially stable on [0,∞).

Proof. Let 𝑥 be a solution to (1) that stays in 𝐷 for all 𝑡≥𝑡₀. Since 𝑀∈ℜ⁺, 𝑒_𝑀(𝑡,0) is well defined and positive. Now consider 𝑉(𝑥(𝑡))𝑒_𝑀(𝑡,0)^Δ=𝑉^Δ(𝑡,𝑥(𝑡))𝑒^𝜎_𝑀(𝑡,0)+𝑀𝑉(𝑥(𝑡))𝑒_𝑀(),≤𝑡,0−𝜆₂𝑉(𝑥(𝑡))−𝐿(𝑀⊖𝛿)𝑒_⊖𝛿(𝑡,0)𝑒_𝑀()(𝑥(𝑡𝑒𝑡,0+𝑀𝑉))_𝑀()=𝑡,0−𝜆₂𝑉(𝑥(𝑡))+𝑀𝑉(𝑥(𝑡))−𝐿(𝑀⊖𝛿)𝑒_⊖𝛿(𝑡,0)𝑒_𝑀()≤𝑡,0𝑀−𝜆₂𝑉(𝑥(𝑡))−𝐿(𝑀⊖𝛿)𝑒_⊖𝛿(𝑡,0)𝑒_𝑀()()𝑒𝑡,0≤−𝐿𝑀⊖𝛿_⊖𝛿(𝑡,0)𝑒_𝑀()()𝑒𝑡,0=−𝐿𝑀⊖𝛿_𝑀⊖𝛿(𝑡,0). (24) Integrating both sides of the above inequality from 𝑡₀ to 𝑡, we obtain, for 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑉(𝑥(𝑡))𝑒_𝑀(𝑡,0)≤𝑉𝑥₀𝑒_𝑀𝑡₀,0−𝐿𝑒_𝑀⊖𝛿(𝑡,0)+𝐿𝑒_𝑀⊖𝛿𝑡₀𝑥,0≤𝑉₀𝑒_𝑀𝑡₀,0+𝐿𝑒_𝑀⊖𝛿𝑡₀≤𝑉𝑥,0₀+𝐿𝑒_𝑀𝑡₀,0. (25) This implies that 𝑉(𝑥(𝑡))≤𝑉𝑥₀+𝐿𝑒_𝑀𝑡₀,0𝑒_⊖𝑀()=𝑉𝑥𝑡,0₀+𝐿𝑒_⊖𝑀𝑡,𝑡₀. (26) From (26) and by invoking condition (22) we obtain, for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, 𝜓𝑥(𝑡)≤𝜆₁^−1╱𝑝𝑉𝑥₀+𝐿𝑒_⊖𝑀𝑡,𝑡₀^1╱𝑝. (27) By Definition 1 system (1) is 𝜓 uniformly exponentially stable.

Theorem 6. Assume that 𝐷⊂ℝ^𝑛 contains the origin and there exists a type I Lyapunov function 𝑉∶𝕋⁺_𝑡0×𝐷→[0,∞) such that, for all (𝑡,𝑥)∈𝕋⁺_𝑡0×𝐷 and 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, 𝜆₁𝜓(𝑡)𝑥(𝑡)^𝑝≤𝑉(𝑥)≤𝜆₂𝜓(𝑡)𝑥(𝑡)^𝑝,𝑉^Δ(𝑡,𝑥)≤−𝜆₃𝜓(𝑡)𝑥(𝑡)^𝑝−𝐿(𝐾⊖𝛿)𝑒_⊖𝛿(𝑡,0)(𝑡),1+𝐾𝜇 (28)(29) where 𝜆₁,𝜆₂,𝜆₃, and 𝑝 are positive constants, 𝐾=𝜆₃/𝜆₂, 𝐿≥𝜆₁ is a nonnegative constant, and 𝛿>𝜆₃/𝜆₂. Then the trivial solution to (1) is 𝜓 globally exponentially stable on [0,∞).

Proof. Let 𝑥 be a solution to (1) that stays in 𝐷 for all 𝑡≥𝑡₀. Since 𝐾=𝜆₃/𝜆₂, 𝑒_𝐾(𝑡,0) is well defined and positive. For all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, consider 𝑉(𝑥(𝑡))𝑒_𝐾(𝑡,0)^Δ=𝑉^Δ(𝑡,𝑥(𝑡))𝑒^𝜎_𝐾(𝑡,0)+𝑉(𝑥(𝑡))𝑒^Δ_𝐾(),≤𝑡,0−𝜆₃𝜓(𝑡)𝑥(𝑡)^𝑝−𝐿(𝐾⊖𝛿)𝑒_⊖𝛿(𝑡,0)𝑒_𝐾()(𝑥(𝑡𝑒𝑡,0+𝐾𝑉))_𝐾()=𝑡,0−𝜆₃𝜓(𝑡)𝑥(𝑡)^𝑝+𝐾𝑉(𝑥(𝑡))−𝐿(𝐾⊖𝛿)𝑒_⊖𝛿(𝑡,0)×𝑒_𝐾()≤𝑡,0−𝜆₃𝜆₂𝑉(𝑥(𝑡))+𝐾𝑉(𝑥(𝑡))−𝐿(𝐾⊖𝛿)𝑒_⊖𝛿(𝑡,0)𝑒_𝐾()=()𝑒𝑡,0−𝐿𝐾⊖𝛿_⊖𝛿(𝑡,0)𝑒_𝐾()()𝑒𝑡,0=−𝐿𝐾⊖𝛿_𝐾⊖𝛿(𝑡,0). (30) Integrating both sides of the above inequality from 𝑡₀ to 𝑡,𝑡≠𝑡_𝑘, with 𝑥₀=𝑥(𝑡₀), we obtain, 𝑉(𝑥(𝑡))𝑒_𝐾(𝑡,0)≤𝑉𝑥₀𝑒_𝐾𝑡₀𝑒,0+𝐿_𝐾⊖𝛿𝑡₀,0−𝑒_𝐾⊖𝛿()𝑥𝑡,0≤𝑉₀𝑒_𝐾𝑡₀,0+𝐿𝑒_𝐾⊖𝛿𝑡₀≤𝑉𝑥,0₀+𝐿𝑒_𝐾𝑡₀,0. (31) This implies that 𝑉(𝑥(𝑡))≤𝑉𝑥₀+𝐿𝑒_𝐾𝑡₀,0𝑒_⊖𝐾()=𝑉𝑥𝑡,0₀+𝐿𝑒_⊖𝐾𝑡,𝑡₀. (32) From (32), and by invoking condition (28), we obtain, for all 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…,𝑛, 𝜓(𝑡)𝑥(𝑡)≤𝜆₁^−1╱𝑝𝑉𝑥₀+𝐿𝑒_⊖𝐾𝑡,𝑡₀^1╱𝑝≤𝜆₁^−1╱𝑝𝑉𝑥₀+𝐿𝑒_⊖𝐾𝑡,𝑡₀^1╱𝑝. (33) If we set 𝑀∶=((𝑉(𝑥₀)+𝐿)/𝜆₁)^1╱𝑝, then (33) can be written as 𝜓(𝑡)𝑥(𝑡)≤𝑀𝑒_⊖𝐾𝑡,𝑡₀^1╱𝑝. (34) Since 𝑀≥1, by Definition 2 system (1) is 𝜓 globally exponentially stable.

4. Examples

Example 7. We consider Example (35) in [7] and extend the example by using impulse condition, 𝑥^Δ1=−𝑥+5𝑥^1/3𝑒_⊖𝛿(𝑡,0),𝑡≠𝑡_𝑘,𝑡∈𝕋,𝑥𝑡⁺_𝑘1=−3,𝑡=𝑘,𝑘=1,2,…,𝑛, (35)(36) where 𝛿>0 is a constant 𝑥₀∈ℝ. If there is a constant 0<𝑀<𝛿 such that 𝜇(𝑡)−11+𝑀𝜇(𝑡)≤−𝑀,231𝜇(𝑡25)^3/2+(2/5)−(2/5)𝜇(𝑡)³3(𝑡(𝑡),1+𝑀𝜇)≤−𝐿𝑀⊖𝛿)( (37)(38) for some constant 𝐿≥0 and all 𝑡≠𝑘, (35) is 𝜓 uniformly exponentially stable.
Under above assumptions, we will show that the conditions of Theorem 4 are satisfied. Let 𝜓(𝑡)=1/2, choose 𝐷=ℝ and 𝑉(𝑥)=𝑥², 𝑡≠𝑘, then (11) holds with 𝑝=𝑞=2, 𝜆₁=𝜆₂=4. If we calculate 𝑉^Δ, for all 𝑡≠𝑘, 𝑉^Δ1=2𝑥−𝑥+5𝑥^1/3𝑒_⊖𝛿((𝑡)1𝑡,0)+𝜇−𝑥+5𝑥^1/3𝑒_⊖𝛿(𝑡,0)², (39) we have the following comparison: 𝑉^Δ1=2𝑥−𝑥+5𝑥^1/3𝑒_⊖𝛿((𝑡)1𝑡,0)+𝜇−𝑥+5𝑥^1/3𝑒_⊖𝛿(𝑡,0)²≤𝜇(𝑡)−1𝑥²+231𝜇(𝑡25)^3/2+(2/5)−(2/5)𝜇(𝑡)³3𝑒_⊖𝛿(𝑡,0). (40) Dividing and multiplying the right-hand side by (1+𝑀𝜇(𝑡)), we see that (12) holds under the above assumptions with 𝑟=2 and 𝜆₃=4𝑀. Also, since 𝑝=𝑞=2, we have 𝑉(𝑥)−𝑉^𝑟╱𝑞(𝑥)=𝑥²−𝑥^22/2=0≤𝛾𝑒_⊖𝛿𝑡,𝑡₀, (41) for all 𝑡≠𝑘. Therefore (13) is satisfied. Hence, all hypotheses of Theorem 4 are satisfied and we conclude that the trivial solution to (35) is 𝜓 uniformly exponentially stable. We consider following two special cases of (35).

Case 1. If 𝕋=ℝ, then 𝜇(𝑡)=0. It is easy to see that (37) holds for 𝑀=1. Also for 𝐿=8/[375(𝛿−𝑀)], condition (38) is satisfied. Hence, we conclude that if 𝛿>1, then the trivial solution to (35) is 𝜓 uniformly exponentially stable.

Case 2. If 𝕋=(1/2)ℤ, then 𝜇(𝑡)=1/2. In this case rewriting (37) we have −12𝑀1+2≤−𝑀, (42) then (37) holds for 2/3>𝑀>0. Also for 𝐿=((6+√2)/2250(𝛿−𝑀))(1−(𝑀/2))(1−(𝛿/2)), condition (38) is satisfied. Therefore for 𝛿>2/3, then the trivial solution to (35) is 𝜓 uniformly exponentially stable.