On Weak Exponential Expansiveness of Evolution Families in Banach Spaces

Abstract

The aim of this paper is to give several characterizations for the property of weak exponential expansiveness for evolution families in Banach spaces. Variants for weak exponential expansiveness of some well-known results in stability theory (Datko (1973), Rolewicz (1986), Ichikawa (1984), and Megan et al. (2003)) are obtained.

1. Introduction

In recent years, the exponential stability theory of one parameter semigroups of operators and evolution families has witnessed significant development. A number of long-standing open problems have been solved, and the theory seems to have obtained a certain degree of maturity. One of the most important results of the stability theory is due to Datko, who proved in 1970 in [1] that a strongly continuous semigroup of operators {𝑇(𝑡)}_𝑡≥0 is uniformly exponentially stable if and only if for each vector 𝑥 from the Banach space 𝑋 the function 𝑡→‖𝑇(𝑡)𝑥‖ lies in 𝐿²(ℝ₊). Later, Pazy generalizes the result in [2] for 𝐿^𝑝(ℝ₊), 𝑝≥1. In 1973, Dakto [3] generalized the results above and proved that an evolutionary process 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} with uniform exponential growth is uniformly exponentially stable if and only if there exists an exponent 𝑝≥1 such that sup_𝑠≥0∫^∞_𝑠‖𝑈(𝑡,𝑠)𝑥‖^𝑝𝑑𝑡<∞, for each 𝑥∈𝑋. This result was improved by Rolewicz in 1986 (see [4]). In [5, 6], the authors generalized the results above in the case of 𝐶₀-semigroups and evolutionary process, respectively, and presented a unified treatment in terms of Banach function spaces.

In the last few years, new concepts of exponential expansiveness and in particular, of exponential instability, have been introduced and characterized (see[7–14]). The cases of uniform exponential instability have been considered in [8] for evolution families and in [10] for linear skew-product flows.

In the present paper, we introduce the concept of weak exponential expansiveness for evolution families which is an extension of classical concept of exponential expansiveness. Our main objective is to give some characterizations for weak exponential expansiveness properties of evolution families in Banach spaces, and variants for weak exponential expansiveness of some well-known results in stability theory (Datko [3], Rolewicz [4], Ichikawa [15], and Megan et al. [8]) are obtained.

2. Preliminaries

Let 𝑋 be a real or complex Banach space. The norm on 𝑋 and on the space 𝐵(𝑋) of all bounded linear operators on 𝑋 will be denoted by ‖⋅‖.

Definition 1. A family 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} of bounded linear operators is called an evolution family if the following conditions are satisfied:(i)𝑈(𝑡,𝑡)=𝐼, the identity operator on 𝑋, for all 𝑡≥0;(ii)𝑈(𝑡,𝑟)𝑈(𝑟,𝑠)=𝑈(𝑡,𝑠) for all 𝑡≥𝑟≥𝑠≥0;(iii)there exist 𝑀≥1 and 𝜔>0 such that ‖𝑈(𝑡,𝑠)𝑥₀‖≤𝑀𝑒^{𝜔(𝑡−𝑠)}‖𝑥₀‖ for all 𝑡≥𝑠≥0 and 𝑥₀∈𝑋;(iv)for every 𝑥₀∈𝑋 and every 𝑡₀≥0, the mapping 𝑟↦‖𝑈(𝑟,𝑡₀)𝑥₀‖ is continuous on [𝑡₀,∞);(v)for every 𝑡≥𝑠≥0, the operator 𝑈(𝑡,𝑠) is injective.

Definition 2. An evolution family 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} is called uniformly expansive if there exists a constant 𝑁>0 such that 𝑈𝑡,𝑡₀𝑥₀≥𝑁𝑈𝑟,𝑡₀𝑥₀, (1) for all 𝑡≥𝑟≥𝑡₀≥0 and 𝑥₀∈𝑋.

Definition 3. An evolution family 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} is said to be uniformly exponentially expansive if there are 𝑁,𝑣>0 such that 𝑈𝑡,𝑡₀𝑥₀≥𝑁𝑒^{𝑣(𝑡−𝑟)}𝑈𝑟,𝑡₀𝑥₀, (2) for all 𝑡≥𝑟≥𝑡₀≥0 and 𝑥₀∈𝑋.

Remark 4. It is obvious that an evolution family 𝒰 is uniformly exponentially expansive if and only if there are 𝑁,𝑣>0 such that 𝑈𝑡,t₀𝑥₀≥𝑁𝑒^{𝑣(𝑡−𝑡0)}𝑥₀, (3) for all 𝑡≥𝑡₀≥0 and 𝑥₀∈𝑋.

Remark 5. If the evolution family 𝒰 is uniformly exponentially expansive, then it is uniformly expansive. The converse is not necessarily valid. To show this, we consider the following example.

Example 6. Let 𝑓∶ℝ₊→ℝ₊ be a monotone increasing and bounded continuous function, 𝑋=ℝ₊. The evolution family 𝒰 defined by 𝑈𝑡,𝑡₀𝑥₀=𝑒^{𝑓(𝑡)−𝑓(𝑡0)}𝑥₀, (4) for all 𝑡≥𝑡₀≥0 and 𝑥₀∈𝑋.

Proof. As a first step, we prove that 𝒰 is uniformly expansive. The evolution family 𝒰 satisfies the inequality 𝑈𝑡,𝑡₀𝑥₀=𝑒^{𝑓(𝑡)−𝑓(𝑡0)}𝑥₀≥𝑒^{𝑓(𝑟)−𝑓(𝑡0)}𝑥₀=𝑁𝑈𝑟,𝑡₀𝑥₀, (5) for all 𝑡≥𝑟≥𝑡₀≥0 and 𝑥₀∈𝑋, where 𝑁=1. Hence, 𝒰 is uniformly expansive.
As a second step, we prove that 𝒰 is not uniformly exponentially expansive. If we suppose that 𝒰 is uniformly exponentially expansive, then, by Remark 4, there exist some constants 𝑁,𝑣>0 such that 𝑒^{𝑓(𝑡)−𝑓(𝑡0)}≥𝑁𝑒^{𝑣(𝑡−𝑡0)}, (6) for all 𝑡≥𝑡₀≥0.
In particular, for 𝑡₀=0, we obtain 𝑒^{𝑓(𝑡)−𝑓(0)}≥𝑁𝑒^𝑣𝑡, which is absurd for 𝑡→∞. Hence, 𝒰 is not uniformly exponentially expansive.

Definition 7. An evolution family 𝒰={𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0} is called weakly exponentially expansive if there are 𝑁,𝑣>0 such that for all 𝑥₀∈𝑋 there exists 𝑡₀≥0 with 𝑈𝑡,𝑡₀𝑥₀≥𝑁𝑒^{𝑣(𝑡−𝑟)}𝑈𝑟,𝑡₀𝑥₀, (7) for all 𝑡≥𝑟≥𝑡₀.

Remark 8. If the evolution family 𝒰 is uniformly exponentially expansive, then it is weakly exponentially expansive.

The following example shows that the converse is not valid.

Example 9. Let 𝑋=ℝ² with the Euclidean norm. Consider the evolution family generated by the matrix 𝑈(𝑡,𝑡₀)=𝑃(𝑡,𝑡₀)𝑄(𝑡₀), where 𝑃𝑡,𝑡₀=𝑒^𝑡−𝑡0sin𝑡𝑒^{−(𝑡−𝑡0)}cos𝑡−𝑒^𝑡−𝑡0cos𝑡𝑒^{−(𝑡−𝑡0)},𝑄𝑡sin𝑡₀=cos𝑡₀sin𝑡₀sin𝑡₀−cos𝑡₀. (8)

Proof. We divide the proof into two steps.
As a first step, we prove that 𝒰 is weakly exponentially expansive. For every 𝑥₀∈ℝ², there exist 𝜌≥0 and 𝑡₀∈[0,2𝜋) such that 𝑥₀=(𝜌cos𝑡₀,𝜌sin𝑡₀)^𝑇.
It is easy to see that 𝑈𝑡,𝑡₀𝑥₀=𝑃𝑡,𝑡₀𝑄𝑡₀𝑥₀=𝑃𝑡,𝑡₀𝜌,0^𝑇=𝜌𝑒^𝑡−𝑡0sin𝑡,−𝜌𝑒^𝑡−𝑡0cos𝑡^𝑇, (9) and hence 𝑈𝑡,𝑡₀𝑥₀=𝜌𝑒^𝑡−𝑡0≥𝑁𝑒^𝑡−𝑟𝑈𝑟,𝑡₀𝑥₀, (10) for all 𝑡≥𝑟≥𝑡₀ with 𝑁=1, which shows that 𝒰 is weakly exponentially expansive.
As a second step, we prove that 𝒰 is not uniformly exponentially expansive. If we assume that 𝒰 is uniformly exponentially expansive, then there exist some constants 𝑁,𝑣>0 such that 𝑈𝑡,𝑡₀𝑦₀≥𝑁𝑒^{𝑣(𝑡−𝑟)}𝑈𝑟,𝑡₀𝑦₀, (11) for all 𝑡≥𝑟≥𝑡₀≥0 and 𝑦₀∈𝑋.
In particular, for 𝑦₀=(sin𝑡₀,−cos𝑡₀)^𝑇, we obtain 𝑈𝑡,𝑡₀𝑦₀=𝑃𝑡,𝑡₀(0,1)^𝑇=𝑒^{−(𝑡−𝑡0)}cos𝑡,𝑒^{−(𝑡−𝑡0)}sin𝑡^𝑇, (12) and hence 𝑈𝑡,𝑡₀𝑦₀=𝑒^{−(𝑡−𝑡0)}=𝑒^{−(𝑡−𝑟)}𝑈𝑟,𝑡₀𝑦₀, (13) which shows that 𝒰 is not uniformly exponentially expansive.

3. The Main Results

Theorem 10. The following assertions are equivalent:(i)𝒰 is weakly exponentially expansive;(ii)there are 𝛿>0 and 𝑐>1 such that for every 𝑥₀∈𝑋 there exist 𝑡₀≥0 and ℎ₀∈(0,𝛿] with 𝑈ℎ₀+𝑟,t₀𝑥₀≥𝑐𝑈𝑟,𝑡₀𝑥₀, (14) for all 𝑟≥𝑡₀;(iii) there exist 𝛿>0 and 𝑐>1 such that for each 𝑥₀∈𝑋 there exists 𝑡₀≥0 with the property that for every 𝑟≥𝑡₀ there is ℎ₀∈(0,𝛿] with 𝑈ℎ₀+𝑟,𝑡₀𝑥₀≥𝑐𝑈𝑟,𝑡₀𝑥₀. (15)

Proof. (i)⇒(ii) If 𝒰 is weakly exponentially expansive, then by Definition 7, there exist 𝑁,𝑣>0 such that for all 𝑥₀∈𝑋 there exists 𝑡₀≥0 with the property 𝑈𝑡,𝑡₀𝑥₀≥𝑁𝑒^{𝑣(𝑡−𝑟)}𝑈𝑟,𝑡₀𝑥₀, (16) for all 𝑡≥𝑟≥𝑡₀. Let 𝛿>0 satisfy that 𝑁𝑒^𝑣𝛿>1. Then, for ℎ₀=𝛿, we have 𝑈𝑟+ℎ₀,𝑡₀𝑥₀≥𝑁𝑒^𝑣𝛿𝑈𝑟,𝑡₀𝑥₀=𝑐𝑈𝑟,𝑡₀𝑥₀, (17) for all 𝑟≥𝑡₀.
(ii)⇒(iii) It is obvious.
(iii)⇒(i) We define 𝑁=1/𝑀𝑒^𝜔𝛿 and 𝑣=ln𝑐/𝛿, where 𝛿>0 and 𝑐>1 are given by (iii).
From (iii), it results that for each 𝑥₀∈𝑋, there exists 𝑡₀≥0 with the property that for every 𝑟≥𝑡₀ there is ℎ₀∈(0,𝛿] such that 𝑈ℎ₀+𝑟,𝑡₀𝑥₀≥𝑐𝑈𝑟,𝑡₀𝑥₀. (18)
Let 𝑟≥𝑡₀, and we have that there is ℎ₁∈(0,𝛿] with 𝑈ℎ₁+ℎ₀+𝑟,𝑡₀𝑥₀≥𝑐𝑈ℎ₀+𝑟,𝑡₀𝑥₀≥𝑐²𝑈𝑟,𝑡₀𝑥₀. (19) By induction, we have that 𝑈𝑟_𝑛+𝑟,𝑡₀𝑥₀≥𝑐^𝑛𝑈𝑟,𝑡₀𝑥₀,∀𝑛∈ℕ, (20) where 𝑟_𝑛=0,𝑛=0,_{𝑛−1𝑖=0}ℎ_𝑖,𝑛∈ℕ^∗,ℎ_𝑖∈(0,𝛿]. (21)
It is easy to see that (𝑟_𝑛) is unbounded. In fact, if (𝑟_𝑛) is bounded, then there exists 𝑟^∗∈ℝ with 𝑟_𝑛→𝑟^∗ (𝑛→∞). From the relation (20) and 𝑐>1, it follows that 𝑈𝑟+𝑟^∗,𝑡₀𝑥₀≥lim_𝑛→∞𝑐^𝑛𝑈𝑟,𝑡₀𝑥₀⟶∞, (22) which is a contradiction because {𝑈(𝑡,𝑠)}_{𝑡≥𝑠≥0}⊆𝐵(𝑋).
So, (𝑟_𝑛) is unbounded, and then for 𝑡≥𝑟, there is 𝑛∈ℕ such that 𝑟_𝑛≤𝑡−𝑟≤𝑟_𝑛+1≤(𝑛+1)𝛿. (23)
Then, 𝑈𝑟_𝑛+1+𝑟,𝑡₀𝑥₀≤𝑀𝑒^{𝜔(𝑟+𝑟𝑛+1−𝑡)}𝑈𝑡,𝑡₀𝑥₀≤𝑀𝑒^{𝜔(𝑟𝑛+1−𝑟𝑛)}𝑈𝑡,𝑡₀𝑥₀≤𝑀𝑒^𝜔𝛿𝑈𝑡,𝑡₀𝑥₀, (24) and hence 𝑈𝑡,𝑡₀𝑥₀≥1𝑀𝑒^𝜔𝛿𝑈𝑟+𝑟_𝑛+1,𝑡₀𝑥₀≥1𝑀𝑒^𝜔𝛿𝑐^𝑛+1𝑈𝑟,𝑡₀𝑥₀=1𝑀𝑒^𝜔𝛿𝑒^{𝑣(𝑛+1)𝛿}𝑈𝑟,𝑡₀𝑥₀≥𝑁𝑒^{𝑣(𝑡−𝑟)}𝑈𝑟,𝑡₀𝑥₀. (25)

Remark 11. Theorem 10 can be considered a generalization of some results from uniform exponential instability proved in [8].

An important set in what follows is ℱ₁, the set of all nondecreasing functions 𝐹∶ℝ₊→ℝ₊ with the properties:(f1)𝐹(𝑡𝑟)≤𝐹(𝑡)𝐹(𝑟), for all (𝑡,𝑟)∈ℝ²₊;(f2)𝐹(𝑡)>0, for every 𝑡>0.

Theorem 12. An evolution family 𝒰 is weakly exponentially expansive if and only if there are 𝐹∈ℱ₁ and 𝐾>0 such that for every 𝑥₀∈𝑋⧵{0} there is 𝑡₀≥0 with ^∞_𝑟𝐹1𝑈𝜏,𝑡₀𝑥₀1𝑑𝜏≤𝐾𝐹𝑈𝑟,𝑡₀𝑥₀, (26) for all 𝑟≥𝑡₀.

Proof. Necessity. If 𝒰 is weakly exponentially expansive, then by Definition 7, there are 𝑁,𝑣>0 such that for all 𝑥₀∈𝑋⧵{0} there exists 𝑡₀≥0 with ^∞_𝑟1𝑈𝜏,𝑡₀𝑥₀𝑑𝜏≤^∞_𝑟1𝑁𝑒^{𝑣(𝜏−𝑟)}𝑈𝑟,𝑡₀𝑥₀=1𝑑𝜏𝑁𝑣𝑈𝑟,𝑡₀𝑥₀, (27) for all 𝑟≥𝑡₀.
Thus, the inequality (26) is satisfied for 𝐹(𝑡)=𝑡 and 𝐾=1/𝑁𝑣.
Sufficiency. We assume for a contradiction that for all 𝛿>0 and 𝑐>1 there exists 𝑥₀∈𝑋 such that for every 𝑡₀≥0 there is 𝑟≥𝑡₀ with 𝑈𝑡+𝑟,𝑡₀𝑥₀<𝑐𝑈𝑟,𝑡₀𝑥₀, (28) for all 𝑡∈(0,𝛿].
In particular, for 𝛿=𝐾𝐹(3) and 𝑐=3, the inequality (28) implies 𝐹(3)^∞_𝑟𝐹1𝑈𝑡,𝑡₀𝑥₀≥𝑑𝑡_∞𝑟𝐹3𝑈𝑡,𝑡₀𝑥₀=𝑑𝑡_∞0𝐹3𝑈𝜏+𝑟,𝑡₀𝑥₀≥𝑑𝜏_𝛿0𝐹3𝑈𝜏+𝑟,𝑡₀𝑥₀>𝑑𝜏_𝛿0𝐹1𝑈𝑟,𝑡₀𝑥₀1𝑑𝜏=𝛿𝐹𝑈𝑟,𝑡₀𝑥₀, (29) which contradicts the inequality (26). This contradiction proves that 𝒰 is weakly exponentially expansive.

It makes sense to consider also the set ℱ₂ all non-decreasing functions 𝐹∶ℝ₊→ℝ₊ with the properties:(g1)𝐹(𝑡𝑟)≥𝐹(𝑡)𝐹(𝑟), for all (𝑡,𝑟)∈ℝ²₊;(g2)𝐹(𝑡)>0, for every 𝑡>0.

Theorem 13. An evolution family 𝒰 is weakly exponentially expansive if and only if there are 𝐹∈ℱ₂ and 𝐾>0 such that for every 𝑥₀∈𝑋⧵{0} there is 𝑡₀≥0 with the relation (26).

Proof. Necessity. This is a simple verification for 𝐹(𝑡)=𝑡.
Sufficiency. It is similar to the proof of Theorem 12. Indeed, from (28) for 𝑐∈(1,+∞) and 𝛿=2𝐾/𝐹(1/𝑐), we have ^∞_𝑟𝐹1𝑈𝑡,𝑡₀𝑥₀𝑑𝑡=^∞₀𝐹1𝑈𝜏+𝑟,𝑡₀𝑥₀≥𝑑𝜏_𝛿0𝐹1𝑈𝜏+𝑟,𝑡₀𝑥₀≥𝑑𝜏_𝛿0𝐹1𝑐𝑈𝑟,𝑡₀𝑥₀1𝑑𝜏≥𝛿𝐹𝑐𝐹1𝑈𝑟,𝑡₀𝑥₀1>𝐾𝐹𝑈𝑟,𝑡₀𝑥₀, (30) which contradicts the inequality (26).

Remark 14. The preceding theorems are variants for the case of weak exponential expansiveness property of a well-known theorem due to Rolewicz [4].

Corollary 15. An evolution family 𝒰 is weakly exponentially expansive if and only if there are 𝑝>0 and 𝐾>0 such that for all 𝑥₀∈𝑋⧵{0} there exists 𝑡₀≥0 with ^∞_𝑟1𝑈(𝑡,𝑡₀)𝑥₀^𝑝1𝑑𝑡≤𝐾𝑈(𝑟,𝑡₀)𝑥₀^𝑝, (31) for all 𝑟≥𝑡₀.

Proof. It is immediate from Theorem 12 for 𝐹(𝑡)=𝑡^𝑝.

Remark 16. Corollary 15 is the version of a well-known theorem due to Datko [3], for the case of weak exponential expansiveness of evolution families.

In the following corollary, we give a discrete version of Theorems 12 and 13.

Corollary 17. An evolution family 𝒰 is weakly exponentially expansive if and only if there are 𝐹∈ℱ₁∪ℱ₂ and 𝐾>0 such that for all 𝑥₀∈𝑋⧵{0} there exists 𝑡₀≥0 with ^∞_𝑛=0𝐹1𝑈𝑟+𝑛,𝑡₀𝑥₀1≤𝐾𝐹𝑈𝑟,𝑡₀𝑥₀, (32) for all 𝑟≥𝑡₀.

Proof. Necessity. This is a simple verification for 𝐹(𝑡)=𝑡.
Sufficiency. By Definition 1, we know that ‖𝑈(𝑟+𝜏,𝑟+𝑛)‖≤𝑀𝑒^𝜔, 𝜏∈[𝑛,𝑛+1] and 𝜏↦‖𝑈(𝑟+𝜏,𝑟+𝑛)‖ is continuous on [𝑛,𝑛+1] for all 𝑛∈ℕ, so there exist 𝐿_𝑛>0 such that 𝐿_𝑛=min_{𝜏∈[𝑛,𝑛+1]}‖𝑈(𝑟+𝜏,𝑟+𝑛)‖,𝑛∈ℕ. (33)
Let 𝐿=inf_𝑛𝐿_𝑛. We suppose that 𝐹∈ℱ₁, and from (32), it results that ^∞_𝑟𝐹1𝑈𝑡,𝑡₀𝑥₀=𝑑𝑡_{∞𝑛=0𝑛𝑛+1}𝐹1𝑈(𝑟+𝜏,𝑟+𝑛)𝑈𝑟+𝑛,𝑡₀𝑥₀≤𝑑𝜏_{∞𝑛=0𝑛𝑛+1}𝐹1𝐿𝑈𝑟+𝑛,𝑡₀𝑥₀1𝑑𝜏≤𝐹𝐿^∞_𝑛=0𝐹1𝑈𝑟+𝑛,𝑡₀𝑥₀1≤𝐾𝐹𝐿𝐹1𝑈𝑟,𝑡₀𝑥₀, (34) for all 𝑟≥𝑡₀.
If 𝐹∈ℱ₂, in a similar way we have 𝐹(𝐿)^∞_𝑟𝐹1𝑈𝑡,𝑡₀𝑥₀≤𝑑𝑡_∞0𝐹𝐿𝑈𝑟+𝜏,𝑡₀𝑥₀=𝑑𝜏_{∞𝑛=0𝑛𝑛+1}𝐹𝐿𝑈(𝑟+𝜏,𝑟+𝑛)𝑈𝑟+𝑛,𝑡₀𝑥₀≤𝑑𝜏_{∞𝑛=0𝑛𝑛+1}𝐹1𝑈𝑟+𝑛,𝑡₀𝑥₀1𝑑𝜏≤𝐾𝐹𝑈𝑟,𝑡₀𝑥₀, (35) for all 𝑟≥𝑡₀.
Applying Theorems 12 and 13, we conclude that 𝒰 is weakly exponentially expansive.

Corollary 18. An evolution family 𝒰 is weakly exponentially expansive if and only if there is 𝐾>0 such that for all 𝑥₀∈𝑋⧵{0} there exists 𝑡₀≥0 with ^∞_𝑛=01𝑈𝑟+𝑛,𝑡₀𝑥₀1≤𝐾𝑈𝑟,𝑡₀𝑥₀, (36) for all 𝑟≥𝑡₀.

Another characterization of the weak exponential expansiveness is given by the following.

Theorem 19. An evolution family 𝒰 is weakly exponentially expansive if and only if there are 𝐾,𝛼>0 such that for every 𝑥₀∈𝑋⧵{0} there is 𝑡₀≥0 with 1𝑡−𝑟^𝑡_𝑟𝑒^{𝛼(𝜏−𝑟)}1𝑈𝜏,𝑡₀𝑥₀1𝑑𝜏≤𝐾𝑈𝑟,𝑡₀𝑥₀, (37) for all 𝑡>𝑟≥𝑡₀.

Proof. Necessity. If 𝒰 is weakly exponentially expansive then by Definition 7, there are 𝑁,𝑣>0 such that for all 𝑥₀∈𝑋⧵{0} there is 𝑡₀>0 with the property that for 𝛼=𝑣/2 we have 1𝑡−𝑟^𝑡_𝑟𝑒^{𝛼(𝜏−𝑟)}1𝑈𝜏,𝑡₀𝑥₀≤1𝑑𝜏𝑁(𝑡−𝑟)^𝑡_𝑟𝑒^{−𝛼(𝜏−𝑟)}1𝑑𝜏𝑈𝑟,𝑡₀𝑥₀=1−𝑒^{−𝛼(𝑡−𝑟)}()1𝑁𝛼𝑡−𝑟𝑈𝑟,𝑡₀𝑥₀1≤𝐾𝑈𝑟,𝑡₀𝑥₀, (38) for all 𝑡>𝑟≥𝑡₀, where 𝐾=(1/𝑁)sup_𝜆>0((1−𝑒^−𝜆)/𝜆)<∞.
Sufficiency. Let 𝛿>0 be such that 𝑒^𝛼𝛿>1+3𝛼𝛿𝐾, where 𝐾 and 𝛼 are given by (37). We suppose that 𝒰 is not weakly exponentially expansive. Then, by Theorem 10, for 𝑐=3, there exists 𝑥₀∈𝑋 such that for all 𝑡₀≥0 and all 𝑢∈(0,𝛿] there is 𝑟≥𝑡₀ with 𝑈𝑢+𝑟,𝑡₀𝑥₀<3𝑈𝑟,𝑡₀𝑥₀. (39) Then, for 𝑡=𝑟+𝛿, we have 1𝑡−𝑟^𝑡_𝑟𝑒^{𝛼(𝜏−𝑟)}1𝑈𝜏,𝑡₀𝑥₀=1𝑑𝜏𝛿^𝛿₀𝑒^𝛼𝑢1𝑈𝑢+𝑟,𝑡₀𝑥₀≥1𝑑𝑢3𝛿^𝛿₀𝑒^𝛼𝑢1𝑑𝑢𝑈𝑟,𝑡₀𝑥₀=𝑒^𝛼𝛿−113𝛼𝛿𝑈𝑟,𝑡₀𝑥₀1>𝐾𝑈𝑟,𝑡₀𝑥₀, (40) which contradicts the inequality (37), the proof is completed.

Acknowledgments

The authors would like to thank the referee for helpful suggestions and comments. This work was supported by the “Fundamental Research Funds for the Central Universities” (no. 2012LWB53).