On Regularized Quasi-Semigroups and Evolution Equations

Abstract

We introduce the notion of regularized quasi-semigroup of bounded linear operators on Banach spaces and its infinitesimal generator, as a generalization of regularized semigroups of operators. After some examples of such quasi-semigroups, the properties of this family of operators will be studied. Also some applications of regularized quasi-semigroups in the abstract evolution equations will be considered. Next some elementary perturbation results on regularized quasi-semigroups will be discussed.

1. Introduction and Preliminaries

The theory of quasi-semigroups of bounded linear operators, as a generalization of strongly continuous semigroups of operators, was introduced in 1991 [1], in a preprint of Barcenas and Leiva. This notion, its elementary properties, exponentially stability, and some of its applications in abstract evolution equations are studied in [2–5]. The dual quasi-semigroups and the controllability of evolution equations are also discussed in [6].

Given a Banach space 𝑋, we denote by 𝐵(𝑋) the space of all bounded linear operators on 𝑋. A biparametric commutative family {𝑅(𝑠,𝑡)}_{𝑠,𝑡≥0}⊆𝐵(𝑋) is called a quasi-semigroup of operators if for every 𝑠,𝑡,𝑟≥0 and 𝑥∈𝑋, it satisfies(1)𝑅(𝑡,0)=𝐼, the identity operator on 𝑋, (2)𝑅(𝑟,𝑠+𝑡)=𝑅(𝑟+𝑡,𝑠)𝑅(𝑟,𝑡), (3)lim_{(𝑠,𝑡)→(𝑠0,𝑡0)}‖𝑅(𝑠,𝑡)𝑥−𝑅(𝑠₀,𝑡₀)𝑥‖=0,𝑥∈𝑋,(4)‖𝑅(𝑠,𝑡)‖≤𝑀(𝑠+𝑡), for some continuous increasing mapping 𝑀∶[0,∞)→[0,∞).

Also regularized semigroups and their connection with abstract Cauchy problems are introduced in [7] and have been studied in [8–12] and many other papers.

We mention that if 𝐶∈𝐵(𝑋) is an injective operator, then a one-parameter family {𝑇(𝑡)}_≥0⊆𝐵(𝑋) is called a 𝐶-semigroup if for any 𝑠,𝑡≥0 it satisfies 𝑇(𝑠+𝑡)𝐶=𝑇(𝑠)𝑇(𝑡) and 𝑇(0)=𝐶.

In this paper we are going to introduce regularized quasi-semigroups of operators.

In Section 2, some useful examples are discussed and elementary properties of regularized quasi-semigroups are studied.

In Section 3 regularized quasi-semigroups are applied to find solutions of the abstract evolution equations. Also perturbations of the generator of regularized quasi-semigroups are also considered in this section. Our results are mainly based on the work of Barcenas and Leiva [1].

2. Regularized Quasi-Semigroups

Suppose 𝑋 is a Banach space and {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} is a two-parameter family of operators in 𝐵(𝑋). This family is called commutative if for any 𝑟,𝑠,𝑡,𝑢≥0, 𝐾(𝑟,𝑡)𝐾(𝑠,𝑢)=𝐾(𝑠,𝑢)𝐾(𝑟,𝑡). (2.1)

Definition 2.1. Suppose 𝐶 is an injective bounded linear operator on Banach space 𝑋. A commutative two-parameter family {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} in 𝐵(𝑋) is called a regularized quasi-semigroups (or 𝐶-quasi-semigroups) if (1)𝐾(𝑡,0)=𝐶, for any 𝑡≥0; (2)𝐶𝐾(𝑟,𝑡+𝑠)=𝐾(𝑟+𝑡,𝑠)𝐾(𝑟,𝑡), 𝑟,𝑡,𝑠≥0; (3){𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} is strongly continuous, that is, lim_{(𝑠,𝑡)→(𝑠0,𝑡0)}‖𝐾(𝑠,𝑡)𝑥−𝐾(𝑠₀,𝑡₀)𝑥‖=0,𝑥∈𝑋; (2.2)(4)there exists a continuous and increasing function 𝑀∶[0,∞)→[0,∞), such that for any 𝑠,𝑡>0, ‖𝐾(𝑠,𝑡)‖≤𝑀(𝑠+𝑡). For a 𝐶-quasi-semigroups {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} on Banach space 𝑋, let 𝐷 be the set of all 𝑥∈𝑋 for which the following limits exist in the range of 𝐶: lim_𝑡→0+𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡=lim_𝑡→0+𝐾(𝑠−𝑡,𝑡)𝑥−𝐶𝑥𝑡,𝑠>0lim_𝑡→0+𝐾(0,𝑡)𝑥−𝐶𝑥𝑡. (2.3) Now for 𝑥∈𝐷 and 𝑠≥0, define 𝐴(𝑠)𝑥=𝐶⁻¹lim_𝑡→0+𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡. (2.4){𝐴(𝑠)}_𝑠≥0 is called the infinitesimal generator of the regularized quasi-semigroup {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0}. Somewhere we briefly apply generator instead of infinitesimal generator.

Here are some useful examples of regularized quasi-semigroups.

Example 2.2. Let {𝑇_𝑡}_𝑡≥0 be an exponentially bounded strongly continuous 𝐶-semigroup on Banach space 𝑋, with the generator 𝐴. Then 𝐾(𝑠,𝑡)∶=𝑇_𝑡,𝑠,𝑡≥0, (2.5) defines a 𝐶-quasi-semigroup with the generator 𝐴(𝑠)=𝐴, 𝑠≥0, and so 𝐷=𝐷(𝐴).

Example 2.3. Let 𝑋=BUC(ℝ), the space of all bounded uniformly continuous functions on ℝ with the supremum-norm. Define 𝐶,𝐾(𝑠,𝑡)∈𝐵(𝑋), by 𝐶𝑓(𝑥)=𝑒^−𝑥2𝑓(𝑥),𝐾(𝑠,𝑡)𝑓(𝑥)=𝑒^−𝑥2𝑓(𝑡²+2𝑠𝑡+𝑥),𝑠,𝑡≥0. (2.6) One can see that {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} is a regularized 𝐶-quasi-semigroup of operators on 𝑋, with the infinitesimal generator 𝐴(𝑠)𝑓=2𝑠𝑓̇ on 𝐷, where 𝐷={𝑓∈𝑋∶𝑓̇∈𝑋}.

Example 2.4. Let {𝑇_𝑡}_𝑡≥0 be a strongly continuous semigroup of operators on Banach space 𝑋, with the generator 𝐴. If 𝐶∈𝐵(𝑋) is injective and commutes with 𝑇_𝑡, 𝑡≥0, then 𝐾(𝑠,𝑡)∶=𝐶𝑒^{𝑇𝑠+𝑡−𝑇𝑠},𝑠,𝑡≥0, (2.7) is a 𝐶-quasi-semigroup with the generator 𝐴(𝑠)=𝐴𝑇_𝑠. Thus 𝐷=𝐷(𝐴). In fact, for 𝑥∈𝐷, boundedness of 𝐶 implies that 𝐶𝐴(𝑠)𝑥=lim_𝑡→0+𝐶𝑒^{𝑇𝑠+𝑡−𝑇𝑠}𝑥−𝐶𝑥𝑡=𝐶lim_𝑡→0+𝑒^{𝑇𝑠+𝑡−𝑇𝑠}𝑥−𝑥𝑡𝑑=𝐶∣𝑑𝑠^𝑡=0(𝑇_𝑠+𝑡−𝑇_𝑠)𝑥=𝐶𝐴𝑇_𝑠𝑥. (2.8) Now injectivity of 𝐶 implies that 𝐴(𝑠)𝑥=𝐴𝑇_𝑠𝑥, and so 𝐷=𝐷(𝐴).

Example 2.5. Let {𝑇_𝑡}_𝑡≥0 be a strongly continuous exponentially bounded 𝐶-semigroup of operators on Banach space 𝑋, with the generator 𝐴. For 𝑠,𝑡≥0, define 𝐾(𝑠,𝑡)=𝑇(𝑔(𝑠+𝑡)−𝑔(𝑠)),𝑠,𝑡≥0, (2.9) where 𝑔(𝑡)=∫^𝑡₀𝑎(𝑠)𝑑𝑠, and 𝑎∈𝐶[0,∞), with 𝑎(𝑡)>0. We have 𝐾(𝑠,0)=𝑇(0)=𝐶 and the 𝐶-semigroup properties of {𝑇(𝑡)}_𝑡≥0 imply that 𝐶𝐾(𝑟,𝑠+𝑡)=𝐶𝑇(𝑔(𝑟+𝑡+𝑠)−𝑔(𝑟))=𝐶𝑇(𝑔(𝑟+𝑡+𝑠)−𝑔(𝑡+𝑟)+𝑔(𝑡+𝑟)−𝑔(𝑟))=𝑇(𝑔(𝑟+𝑡+𝑠)−𝑔(𝑡+𝑟))𝑇(𝑔(𝑡+𝑟)−𝑔(𝑟))=𝐾(𝑟+𝑡,𝑠)𝐾(𝑟,𝑡). (2.10) So {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} is a 𝐶-quasi-semigroup (the other properties can be also verified easily). Also 𝐷=𝐷(𝐴) and for 𝑥∈𝐷, 𝐴(𝑠)𝑥=𝑎(𝑠)𝐴𝑥.

Some elementary properties of regularized quasi-semigroups can be seen in the following theorem.

Theorem 2.6. Suppose {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} is a 𝐶-quasi-semigroup with the generator {𝐴(𝑠)}_𝑠≥0 on Banach space 𝑋. Then(i)for any 𝑥∈𝐷 and 𝑠₀,𝑡₀≥0, 𝐾(𝑠₀,𝑡₀)𝑥∈𝐷 and 𝐾(𝑠₀,𝑡₀)𝐴(𝑠)𝑥=𝐴(𝑠)𝐾(𝑠₀,𝑡₀)𝑥; (2.11)(ii)for each 𝑥₀∈𝐷, 𝜕𝜕𝑡𝐾(𝑟,𝑡)𝐶𝑥₀=𝐴(𝑟+𝑡)𝐾(𝑟,𝑡)𝐶𝑥₀=𝐾(𝑟,𝑡)𝐴(𝑟+𝑡)𝐶𝑥₀; (2.12)(iii)if 𝐴(𝑠) is locally integrable, then for each 𝑥₀∈𝐷 and 𝑟≥0, 𝐾(𝑟,𝑡)𝑥₀=𝐶𝑥₀+^𝑡₀𝐴(𝑟+𝑠)𝐾(𝑟,𝑠)𝑥₀𝑑𝑠,𝑡≥0; (2.13)(iv)let 𝑓∶[0,∞)→𝑋 be a continuous function; then for every 𝑡∈[0,∞), lim_ℎ→01ℎ_𝑡^𝑡+ℎ𝐾(𝑠,𝑢)𝑓(𝑢)𝑑𝑢=𝐾(𝑠,𝑡)𝑓(𝑡); (2.14)(v)Let 𝐶^′∈𝐵(𝑋) be injective and for any 𝑠,𝑡≥0, 𝐶^′𝐾(𝑠,𝑡)=𝐾(𝑠,𝑡)𝐶^′. Then 𝑅(𝑠,𝑡)∶=𝐶^′𝐾(𝑠,𝑡) is a 𝐶𝐶^′-quasi-semigroup with the generator {𝐴(𝑠)}_𝑠≥0,(vi)Suppose {𝑅(𝑠,𝑡)}_{𝑠,𝑡≥0} is a quasi-semigroup of operators on Banach space 𝑋 with the generator {𝐴(𝑠)}_𝑠≥0, and 𝐶∈𝐵(𝑋) commutes with every 𝑅(𝑠,𝑡), 𝑠,𝑡≥0. Then 𝐾(𝑠,𝑡)∶=𝐶𝑅(𝑠,𝑡) is a 𝐶-quasi-semigroup of operators on 𝑋 with the generator {𝐴(𝑠)}_𝑠≥0.

Proof. First we note that from the commutativity of {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0}; 𝐶𝐾(𝑠,𝑡)=𝐾(𝑠,𝑡)𝐶𝑠,𝑡≥0. (2.15) Also 𝑥∈𝐷 implies that lim_𝑡→0+𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡=𝐶𝐴(𝑠)𝑥𝑠≥0. (2.16) Thus from continuity of 𝐾(𝑠₀,𝑡₀), we have lim_𝑡→0+𝐾(𝑠,𝑡)𝐾(𝑠₀,𝑡₀)𝑥−𝐶𝐾(𝑠₀,𝑡₀)𝑥𝑡=𝐾(𝑠₀,𝑡₀)lim_𝑡→0+𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡=𝐾(𝑠₀,𝑡₀)𝐶𝐴(𝑠)𝑥=𝐶𝐾(𝑠₀,𝑡₀)𝐴(𝑠)𝑥. (2.17) Thus 𝐾(𝑠₀,𝑡₀)𝑥∈𝐷 and 𝐴(𝑠)𝐾(𝑠₀,𝑡₀)=𝐾(𝑠₀,𝑡₀)𝐴(𝑠)𝑥.
To prove (ii), consider the quotient 𝐾(𝑟,𝑡+𝑠)𝐶𝑥₀−𝐾(𝑟,𝑡)𝐶𝑥₀𝑠=𝐾(𝑟+𝑡,𝑠)𝐾(𝑟,𝑡)𝑥₀−𝐾(𝑟,𝑡)𝐶𝑥₀𝑠=𝐾(𝑟,𝑡)𝐾(𝑟+𝑡,𝑠)𝑥₀−𝐶𝑥₀𝑠, (2.18) which tends to 𝐾(𝑟,𝑡)𝐶𝐴(𝑟+𝑡)𝑥₀ as 𝑠→0⁺.
Also for 𝑠<0, 𝐾(𝑟,𝑡+𝑠)𝐶𝑥₀−𝐾(𝑟,𝑡)𝐶𝑥₀𝑠=𝐾(𝑟,𝑡)𝐶𝑥₀−𝐾(𝑟,𝑡+𝑠)𝐶𝑥₀=−𝑠𝐾(𝑟+𝑡+𝑠,−𝑠)𝐾(𝑟,𝑡+𝑠)𝑥₀−𝐾(𝑟,𝑡+𝑠)𝐶𝑥₀−𝑠=𝐾(𝑟,𝑡+𝑠)𝐾(𝑟+𝑡+𝑠,−𝑠)𝑥₀−𝐶𝑥₀1−𝑠=𝐾(𝑟,𝑡+𝑠)−𝑠(𝐾(𝑟+𝑡+𝑠,−𝑠)𝑥₀−𝐾(𝑟+𝑡,−𝑠)𝑥₀+𝐾(𝑟+𝑡,−𝑠)𝑥₀−𝐶𝑥₀). (2.19) Now the strongly continuity of {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} implies that lim_𝑠→0−𝐾(𝑟+𝑡+𝑠,−𝑠)𝑥₀−𝐾(𝑟+𝑡,−𝑠)𝑥₀=0. (2.20) Thus lim_𝑠→0−𝐾(𝑟+𝑡+𝑠,−𝑠)𝑥₀−𝐶𝑥₀−𝑠=𝐶𝐴(𝑟+𝑡)𝑥₀. (2.21) Hence by the strongly continuity of 𝐾(𝑠,𝑡), lim_𝑠→0−𝐾(𝑟,𝑡+𝑠)𝐶𝑥₀−𝐾(𝑟,𝑡)𝐶𝑥₀𝑠=𝐾(𝑟,𝑡)𝐶𝐴(𝑟+𝑡)𝑥₀. (2.22) Thus (𝜕/𝜕𝑡)𝐾(𝑟,𝑡)𝐶𝑥₀=𝐾(𝑟,𝑡)𝐶𝐴(𝑟+𝑡)𝑥₀. The second equality holds by (i).
Now integrating of this equation, we have 𝐾(𝑟,𝑡)𝐶𝑥₀−𝐶𝑥₀=𝐶^𝑡₀𝐾(𝑟,𝑠)𝐴(𝑟+𝑠)𝑥₀𝑑𝑠. (2.23) Hence injectivity of 𝐶 implies (iii).(iv)is trivial from continuity of 𝑓 and strongly continuity of {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0}. In (v), obviously {𝑅(𝑠,𝑡)}_{𝑠,𝑡≥0} is a 𝐶^′𝐶-quasi-semigroup. For 𝑥∈𝐷, we have 𝑅(𝑠,𝑡)𝑥−𝐶𝐶^′𝑥𝑡=C'𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡, (2.24) which tends to 𝐶^′𝐶𝐴(𝑠), as 𝑡→0⁺. This proves (v).(vi)can be seen easily.

3. Evolution Equations and Regularized Quasi-Semigroups

Suppose 𝐶 is an injective bounded linear operator on Banach space 𝑋 and 𝑟>0. In this section, we study the solutions of the following abstract evolution equation using the regularized quasi-semigroups:𝑥̇(𝑡)=𝐴(𝑡+𝑟)𝑥(𝑡),𝑡>0,𝑥(0)=𝐶²𝑥₀,𝑥₀∈𝑋. (3.1) One can see [13, 14] for a comprehensive studying of abstract evolution equations.

Theorem 3.1. Let {𝐴(𝑠)}_𝑠≥0 be the infinitesimal generator of a 𝐶-quasi-semigroups {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0} on Banach space 𝑋, with domain 𝐷. Then for each 𝑥₀∈𝐷 and 𝑟≥0, the initial value problem (3.1) admits a unique solution.

Proof. Let 𝑥(𝑡)=𝐾(𝑟,𝑡)𝐶𝑥₀. By Theorem 2.6(ii), 𝑥(𝑡) is a solution of (3.1).
Now we show that this solution is unique. Suppose 𝑦(𝑠) is another solution of (3.1). Trivially 𝑦(𝑠)∈𝐷. Let 𝑡>0. For 𝑠∈[0,𝑡] and 𝑥∈𝑋, define 𝐹(𝑠)𝑥=𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑥,𝐺(𝑠)=𝐹(𝑠)𝐶𝑦(𝑠). (3.2) From 𝐶-quasi-semigroup properties, for small enough ℎ>0, we have 𝐾(𝑟+𝑠,𝑡−𝑠)𝐶=𝐾(𝑟+𝑠+𝑡−𝑠−(𝑡−𝑠−ℎ),𝑡−𝑠−ℎ)𝐾(𝑟+𝑠,𝑡−𝑠−(𝑡−𝑠−ℎ))=𝐾(𝑟+𝑠+ℎ,𝑡−𝑠−ℎ)𝐾(𝑟+𝑠,ℎ). (3.3) So 𝐹(𝑠+ℎ)𝑥−𝐹(𝑠)𝑥ℎ=𝐾(𝑟+𝑠+ℎ,𝑡−𝑠−ℎ)𝐶𝑥−𝐾(𝑟+𝑠+ℎ,𝑡−𝑠−ℎ)𝐾(𝑟+𝑠,ℎ)𝑥ℎ=−𝐾(𝑟+𝑠+ℎ,𝑡−𝑠−ℎ)[𝐾(𝑟+𝑠,ℎ)𝑥−𝐶𝑥ℎ]→−𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝐴(𝑟+𝑠)𝑥,asℎ→0. (3.4) This means that 𝐹̇(𝑠)𝑥=−𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝐴(𝑟+𝑠)𝑥. (3.5) Therefore, from this, the fact that 𝑦(𝑠) satisfies (3.1), and 𝐶𝐹(𝑠)=𝐹(𝑠)𝐶, we obtain that 𝐺̇(𝑠)=𝐹̇(𝑠)𝐶𝑦(𝑠)+𝐹(𝑠)𝐶𝑦(̇𝑠)=−𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝐴(𝑟+𝑠)𝐶𝑦(𝑠)+𝐾(𝑟+𝑠,𝑡−𝑠)𝐶²𝑦(̇𝑠)=−𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝐴(𝑟+𝑠)𝐶𝑦(𝑠)+𝐾(𝑟+𝑠,𝑡−𝑠)𝐶²𝐴(𝑟+𝑠)𝑦(𝑠)=0. (3.6) Hence for every 𝑠∈(0,𝑡), 𝐺̇(𝑠)=0. Consequently, 𝐺(𝑠) is a constant function on [0,𝑡]. In particular, 𝐺(0)=𝐺(𝑡). So from 𝑦(0)=𝐶𝑥₀, we have 𝐺(0)=𝐹(0)𝐶𝑦(0)=𝐾(𝑟,𝑡)𝐶²𝑥₀=𝐺(𝑡)=𝐹(𝑡)𝐶𝑦(𝑡)=𝐾(𝑟+𝑡,0)𝐶²𝑦(𝑡)=𝐶³𝑦(𝑡). (3.7) Hence 𝐶²𝐾(𝑟,𝑡)𝑥₀=𝐶³𝑦(𝑡). Now injectivity of 𝐶 implies that 𝑦(𝑡)=𝐾(𝑟,𝑡)𝐶𝑥₀, which proves the uniqueness of the solution.

Now with the above notation, we consider the inhomogeneous evolution equation𝑥̇(𝑡)=𝐴(𝑟+𝑡)𝑥(𝑡)+𝐶²𝑓(𝑡),0<𝑡≤𝑇,𝑥(0)=𝐶²𝑥₀,𝑥₀∈𝐷. (3.8) The following theorem guarantees the existence and uniqueness of solutions of (3.8) with some sufficient conditions on 𝑓.

Theorem 3.2. Let 𝐾(𝑠,𝑡) be a 𝐶-quasi-semigroup on Banach space 𝑋, with the generator {𝐴(𝑠)}_𝑠≥0 whose domain is 𝐷. If 𝑓∶[0,𝑇]→𝐷 is a continuous function, each operator 𝐴(𝑠) is closed, and 𝐶^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝑓(𝑠)𝑑𝑠∈𝐷,0<𝑡≤𝑇, (3.9) then the initial value equation (3.8) admits a unique solution 𝑥(𝑡)=𝐾(𝑟,𝑡)𝐶𝑥₀+^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠. (3.10)

Proof. For the existence of the solution, it is enough to show that 𝑥(𝑡) in (3.10) is continuously differentiable and satisfies (3.8).
Trivially 𝑥(0)=𝐶𝑥₀. We know that 𝑦(𝑡)=𝐾(𝑟,𝑡)𝐶𝑥₀ is a solution of (3.1) by Theorem 3.1. Define 𝑔(𝑡)=^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠, (3.11) which is in 𝐷 by our hypothesis. We have 𝑔(𝑡+ℎ)−𝑔(𝑡)ℎ=1ℎ[₀^𝑡+ℎ𝐾(𝑟+𝑠,𝑡+ℎ−𝑠)𝐶𝑓(𝑠)𝑑𝑠−^𝑡₀=1𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠]ℎ[^𝑡₀𝐾(𝑟+𝑠,𝑡+ℎ−𝑠)𝐶𝑓(𝑠)𝑑𝑠−^𝑡₀+𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠_𝑡𝑡+ℎ𝐾(𝑟+𝑠,𝑡+ℎ−𝑠)𝐶𝑓(𝑠)𝑑𝑠]. (3.12) On the other hand, the 𝐶-quasi-semigroup properties imply that 𝐾(𝑟+𝑠,𝑡+ℎ−𝑠)𝐶𝑓(𝑠)=𝐾(𝑟+𝑠+𝑡+ℎ−𝑠−ℎ,ℎ)𝐾(𝑟+𝑠,𝑡+ℎ−𝑠−ℎ)𝑓(𝑠)=𝐾(𝑟+𝑡,ℎ)𝐾(𝑟+𝑠,𝑡−𝑠)𝑓(𝑠). (3.13) So 𝑔(𝑡+ℎ)−𝑔(𝑡)ℎ=1ℎ[^𝑡₀𝐾(𝑟+𝑡,ℎ)𝐾(𝑟+𝑡,𝑡−𝑠)𝑓(𝑠)𝑑𝑠−^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠+_𝑡^𝑡+ℎ=𝐾(𝑟+𝑠,𝑡+ℎ−𝑠)𝐶𝑓(𝑠)𝑑𝑠]_𝑡0𝐾(𝑟+𝑡,𝑡−𝑠)(𝐾(𝑟+𝑡,ℎ)𝑓(𝑠)−𝐶𝑓(𝑠)ℎ+1)𝑑𝑠ℎ_𝑡^𝑡+ℎ𝐾(𝑟+𝑠,𝑡+ℎ−𝑠)𝐶𝑓(𝑠)𝑑𝑠. (3.14) Since the range of 𝑓 is in 𝐷, passing to the limit when ℎ→0, and using Theorem 2.6(v), we have 𝑔(̇𝑡)=^𝑡₀=𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝐴(𝑟+𝑡)𝑓(𝑠)𝑑𝑠+𝐾(𝑟+𝑡,𝑡−𝑡)𝐶𝑓(𝑡)_𝑡0𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝐴(𝑟+𝑡)𝑓(𝑠)𝑑𝑠+𝐶²𝑓(𝑡). (3.15) Therefore, 𝑔(̇𝑡) exists. Also by our hypothesis 𝐴(𝑟+𝑡) is closed, and ∫^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠∈𝐷, thus ^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝐴(𝑟+𝑡)𝑓(𝑠)𝑑𝑠=𝐴(𝑟+𝑡)^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠. (3.16) Consequently, 𝑔(̇𝑡)=𝐴(𝑟+𝑡)𝑔(𝑡)+𝐶²𝑓(𝑡),𝑡≥0. (3.17) Hence ̇𝜕𝑥(𝑡)=𝜕𝑡𝐾(𝑟,𝑡)𝐶𝑥₀+𝐴(𝑟+𝑡)^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠+𝐶²𝑓(𝑡)=𝐴(𝑟+𝑡)(𝐾(𝑟,𝑡)𝐶𝑥₀+^𝑡₀𝐾(𝑟+𝑠,𝑡−𝑠)𝐶𝑓(𝑠)𝑑𝑠)+𝐶²𝑓(𝑡)=𝐴(𝑟+𝑡)𝑥(𝑡)+𝐶²𝑓(𝑡). (3.18) This completes the proof.

We conclude this section with two simple perturbation theorems and some examples, as applications of our discussion.

Theorem 3.3. (a) Suppose 𝐵 is the infinitesimal generator of a strongly continuous semigroup {𝑇(𝑡)}_𝑡≥0 and {𝐴(𝑠)}_𝑠≥0 with domain 𝐷 is the generator of a regularized 𝐶-quasi-semigroup {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0}, which commutes with {𝑇(𝑡)}_𝑡≥0. Then {𝐴(𝑠)+𝐵}_𝑠≥0 with domain 𝐷∩𝐷(𝐵) is the infinitesimal generator of a regularized 𝐶-quasi-semigroup.
(b) Suppose 𝐵 is the infinitesimal generator of an exponentially bounded 𝐶-semigroup {𝑇(𝑡)}_𝑡≥0 and {𝐴(𝑠)}_𝑠≥0 with domain 𝐷 is the generator of a quasi-semigroup (resp., regularized 𝐶^′-quasi-semigroup) {𝐾(𝑠,𝑡)}_{𝑠,𝑡≥0}, which commutes with {𝑇(𝑡)}_𝑡≥0. Then {𝐴(𝑠)+𝐵}_𝑠≥0 with domain 𝐷∩𝐷(𝐵) is the infinitesimal generator of a 𝐶-regularized quasi-semigroup (resp., regularized 𝐶𝐶^′-quasi-semigroup).

Proof. In (a) and (b), define 𝑅(𝑠,𝑡)=𝑇(𝑡)𝐾(𝑠,𝑡). (3.19) One can see that {𝑅(𝑠,𝑡)}_s,𝑡≥0 is a 𝐶-regularized quasi-semigroup (in (b), resp., regularized 𝐶𝐶^′-quasi-semigroup). We just prove that {𝐴(𝑠)+𝐵}_𝑠≥0 is its generator. In (a), let {𝐵(𝑠)}_𝑠≥0 be the infinitesimal generator of {𝑅(s,𝑡)}_{𝑠,𝑡≥0} and 𝑥∈𝐷∩𝐷(𝐵). Hence lim_𝑡→0+𝑇(𝑡)𝑥−𝑥𝑡,lim_𝑡→0+𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡 (3.20) exist in 𝑋 and the range of 𝐶, respectively. Now the fact that 𝐶 commutes with 𝑇(𝑡) and strongly continuity of 𝑇(𝑡) implies that lim_𝑡→0+𝑇(𝑡)𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡 (3.21) exists in the range of 𝐶. So lim_𝑡→0+𝑅(𝑠,𝑡)𝑥−𝐶𝑥𝑡=lim_𝑡→0+𝑇(𝑡)𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡=lim_𝑡→0+𝑇(𝑡)𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡+𝐶lim_𝑡→0+𝑇(𝑡)𝑥−𝑥𝑡 (3.22) exists in the range of 𝐶 and 𝐶𝐵(𝑠)𝑥=lim_𝑡→0+𝑅(𝑠,𝑡)𝑥−𝐶𝑥𝑡=𝐶𝐴(𝑠)𝑥+𝐶𝐵𝑥. (3.23) By injectivity of 𝐶, 𝐵(𝑠)𝑥=𝐴(𝑠)𝑥+𝐵𝑥.
The proof the other parts is similar.

Theorem 3.4. Let 𝐾(𝑠,𝑡) be a 𝐶-quasi-semigroup of operator on Banach space 𝑋 with the generator {𝐴(𝑠)} on domain 𝐷. If 𝐵∈𝐵(𝑋) commutes with 𝐾(𝑠,𝑡), 𝑠,𝑡≥0, and 𝐵²=𝐵, then {𝐵𝐴(𝑠)}_𝑠≥0 is the infinitesimal generator of 𝐶-regularized quasi-semigroup 𝑅(𝑠,𝑡)=𝐵(𝐾(𝑠,𝑡)−𝐶)+𝐶. (3.24)

Proof. The 𝐶-quasi-semigroup properties of {𝑅(𝑠,𝑡)}_{𝑠,𝑡≥0} can be easily verified. We just prove that its generator is {𝐵𝐴(𝑠)}_𝑠≥0. Let 𝑥∈𝐷; we have 𝑅(𝑠,𝑡)𝑥−𝐶𝑥𝑡=𝐵(𝐾(𝑠,𝑡)−𝐶)𝑥+𝐶𝑥−𝐶𝑥𝑡=𝐵𝐾(𝑠,𝑡)𝑥−𝐶𝑥𝑡 (3.25) which tends to 𝐵𝐴(𝑠)𝑥, as 𝑡→0. This completes the proof.

Example 3.5. Let 𝑟>0. Consider the following initial value problem: 𝜕𝜕𝜕𝑡𝑥(𝑡,𝜀)=2(𝑟+𝑡)𝜕𝜀𝑥(𝑡,𝜀)+𝜀𝑥(𝑡,𝜀),𝑥(0,𝜀)=𝑒^−4𝜀2𝑥₀(𝜀),𝜀,𝑡≥0. (3.26) Let 𝑋=BUC(ℝ), with the supremum-norm. Define 𝐶∈𝐵(𝑋) by 𝐶𝑥(𝜀)=𝑒^−𝜀2𝑥(𝜀), 𝑥(⋅)∈𝑋. Also define 𝐵∶𝐷(𝐵)→𝑋 by 𝐵𝑥(𝜀)=𝜀𝑥(𝜀), where 𝐷(𝐵)={𝑥∈𝑋∶𝐵𝑥∈𝑋}. It is well known that 𝐵 is the infinitesimal generator of 𝐶-regularized semigroup 𝑇(𝑡), defined by 𝑇(𝑡)𝑥(𝜀)=𝑒^{−𝜀2+𝜀𝑡}𝑥(𝜀). Now with 𝐷={𝑥∈𝑋∶𝑥̇∈𝑋}, if 𝐴(𝑠)∶𝐷→𝑋 is defined by 𝐴(𝑠)𝑥=2𝑠𝑥̇, then by Example 2.3, {𝐴(𝑠)}_𝑠≥0 is the infinitesimal generator of the regularized 𝐶²-quasi-semigroup 𝐾(𝑠,𝑡)𝑥(𝜀)=𝑒^−𝜀2𝑥(𝑡²+2𝑠𝑡+𝜀). Using Theorem 3.3 and the fact that 𝑇(𝑡)𝐾(𝑠,𝑟)=𝐾(𝑠,𝑟)𝑇(𝑡),  𝑠,𝑡,𝑡≥0, we obtain that {𝐴(𝑠)+𝐵} is the infinitesimal generator of regularized 𝐶²-quasi-semigroup 𝑅(𝑠,𝑡)=𝑇(𝑡)𝐾(𝑠,𝑡). Also using these operators, (3.26) can be written as 𝑥̇(𝑡)=(𝐴(𝑟+𝑡)+𝐵)𝑥(𝑡),𝑥(0)=𝐶⁴𝑥₀. (3.27) Thus by Theorem 3.1 for any 𝑥₀∈𝐷∩𝐷(𝐵), (3.26) has the unique solution 𝑥(𝑡,𝜀)=𝑅(𝑟,𝑡)𝐶²𝑥₀(𝜀)=𝑒^{−4𝜀2+𝜀𝑡}𝑥₀(𝑡²+2𝑟𝑡+𝜀). (3.28)

Example 3.6. For a given sequence (𝑝_𝑛)_𝑛∈ℕ of complex numbers with nonzero elements and (𝑦_𝑛)_𝑛∈ℕ, consider the following equation: 𝑑𝑥𝑑𝑡^𝑛(𝑡)=𝑒^{𝑖𝑛(𝑡+1)}𝑥_𝑛(𝑡)+𝑝_𝑛𝑥_𝑛𝑥(𝑡),_𝑛(0)=𝑝²_𝑛𝑦_𝑛,𝑛∈ℕ. (3.29) Let 𝑋 be the space 𝑐₀, the set of all complex sequence with zero limit at infinity. For a bounded sequence 𝑝=(𝑝_𝑛)_𝑛∈ℕ, define 𝐴∶𝐷(𝐴)∶→𝑋 and 𝑀_𝑝 on 𝑋 by 𝐴(𝑥_𝑛)_𝑛∈ℕ=(𝑒^𝑖𝑛𝑥_𝑛)_𝑛∈ℕ,𝑀_𝑝(𝑥_𝑛)_𝑛∈ℕ=(𝑝_𝑛𝑥_𝑛). (3.30) One can easily see that 𝐷(𝐴)={(𝑥_𝑛)_𝑛∈ℕ∈𝑐₀∶(𝑒ⁱⁿ𝑥_𝑛)_𝑛∈ℕ∈𝑐₀} and 𝑀_𝑝 is a bounded linear operator which is injective. It is well known that 𝐴 is the infinitesimal generator of strongly continuous semigroup 𝑇(𝑡)(𝑥_𝑛)_𝑛∈ℕ=(𝑒^𝑖𝑛𝑥_𝑛)_𝑛∈ℕ. (3.31) Thus by Example 2.4, {𝐴(𝑡)}_𝑡≥0, defined by 𝐴(𝑡)(𝑥_𝑛)_𝑛∈ℕ∶=𝐴𝑇(𝑡)(𝑥_𝑛)_𝑛∈ℕ=(𝑒^{𝑖𝑛(1+𝑡)}𝑥_𝑛)_𝑛∈ℕ, (3.32) is the infinitesimal generator of the 𝑀_𝑝-quasi-semigroup 𝐾(𝑠,𝑡)=𝑀_𝑝(𝑒^{𝑇(𝑠+𝑡)−𝑇(𝑠)}). (3.33) Using these operators, one can rewrite (3.29) as 𝑥̇(𝑡)=(𝐴(𝑡)+𝑀_𝑃)𝑥(𝑡),𝑥(0)=𝑀²_𝑝𝑦₀, (3.34) where 𝑥₀=(𝑦_𝑛)_𝑛∈ℕ. Trivially 𝑇(𝑡) commutes with 𝐾(𝑟,𝑠), for any 𝑟,𝑠,𝑡≥0. Now using Theorem 3.3 we obtain that {𝐴(𝑡)+𝑀_𝑝}_𝑡≥0 is the infinitesimal generator of of 𝑀_𝑝-quasi-semigroup 𝑅(𝑠,𝑡)=𝑇(𝑡)𝐾(𝑠,𝑡). (3.35) Also from Theorem 3.1, with 𝑟=0, for any 𝑦∈𝐷(𝐴), (3.34) has a unique solution 𝑥(𝑡)=𝑅(0,𝑡)𝑀_𝑝𝑦=𝑇(𝑡)𝐾(0,𝑡)𝑀²_𝑝𝑥₀. (3.36) But from definition of 𝐾(𝑠,𝑡), for a given (𝑥_𝑛)_𝑛∈ℕ∈𝑐₀, 𝐾(0,𝑡)(𝑥_𝑛)_𝑛∈ℕ=𝑒^{𝑇(𝑡)−𝐼}=𝑒^∞−1_𝑘=0𝑇^𝑘(𝑡)𝑘!(𝑥_𝑛)_𝑛∈ℕ=𝑒^∞−1_𝑘=0(𝑒^{𝑖𝑘𝑛𝑡}𝑥_𝑛)𝑘!_𝑛∈ℕ=𝑒⁻¹(^∞_𝑘=0𝑒^{𝑖𝑘𝑛𝑡}𝑥_𝑛)𝑘!_𝑛∈ℕ. (3.37) So the solution of (3.34) is 𝑥(𝑡)=𝑅(0,𝑡)𝑀_𝑝𝑦=(^∞_𝑘=0𝑒^{𝑖𝑘𝑡(𝑛+1)−1}𝑝⁴_𝑛𝑦_𝑛)𝑘!_𝑛∈ℕ, (3.38) or equivalently the solution of (3.29) is 𝑥_𝑛(𝑡)=^∞_𝑘=0𝑒^{𝑖𝑘𝑡(𝑛+1)−1}𝑝⁴_𝑛𝑦_𝑛𝑘!,𝑛∈ℕ. (3.39)