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

R(t,0)=I, the identity operator on X,

R(r,s+t)=R(r+t,s)R(r,t),

lim(s,t)→(s0,t0)∥R(s,t)x-R(s0,t0)x∥=0,x∈X,

∥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).

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

K(t,0)=C, for any t≥0;

CK(r,t+s)=K(r+t,s)K(r,t), r,t,s≥0;

{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;

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:
limt→0+K(s,t)x-Cxt=limt→0+K(s-t,t)x-Cxt,s>0limt→0+K(0,t)x-Cxt.
Now for x∈D and s≥0, define
A(s)x=C-1limt→0+K(s,t)x-Cxt.{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 {Tt}t≥0 be an exponentially bounded strongly continuous C-semigroup on Banach space X, with the generator A. Then
K(s,t):=Tt,s,t≥0,
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.
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 {Tt}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 Tt, t≥0, then
K(s,t):=CeTs+t-Ts,s,t≥0,
is a C-quasi-semigroup with the generator A(s)=ATs. Thus D=D(A). In fact, for x∈D, boundedness of C implies that
CA(s)x=limt→0+CeTs+t-Tsx-Cxt=Climt→0+eTs+t-Tsx-xt=Cdds∣t=0(Ts+t-Ts)x=CATsx.
Now injectivity of C implies that A(s)x=ATsx, and so D=D(A).

Example 2.5.

Let {Tt}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,
where g(t)=∫0ta(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).
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

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;

for each x0∈D,
∂∂tK(r,t)Cx0=A(r+t)K(r,t)Cx0=K(r,t)A(r+t)Cx0;

if A(s) is locally integrable, then for each x0∈D and r≥0,
K(r,t)x0=Cx0+∫0tA(r+s)K(r,s)x0ds,t≥0;

let f:[0,∞)→X be a continuous function; then for every t∈[0,∞),
limh→01h∫tt+hK(s,u)f(u)du=K(s,t)f(t);

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,

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.
Also x∈D implies that
limt→0+K(s,t)x-Cxt=CA(s)xs≥0.
Thus from continuity of K(s0,t0), we have
limt→0+K(s,t)K(s0,t0)x-CK(s0,t0)xt=K(s0,t0)limt→0+K(s,t)x-Cxt=K(s0,t0)CA(s)x=CK(s0,t0)A(s)x.
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,
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-s=K(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-Cx0-s=K(r,t+s)1-s(K(r+t+s,-s)x0-K(r+t,-s)x0+K(r+t,-s)x0-Cx0).
Now the strongly continuity of {K(s,t)}s,t≥0 implies that
lims→0-K(r+t+s,-s)x0-K(r+t,-s)x0=0.
Thus
lims→0-K(r+t+s,-s)x0-Cx0-s=CA(r+t)x0.
Hence by the strongly continuity of K(s,t),
lims→0-K(r,t+s)Cx0-K(r,t)Cx0s=K(r,t)CA(r+t)x0.
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∫0tK(r,s)A(r+s)x0ds.
Hence injectivity of C implies (iii).

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,
which tends to C'CA(s), as t→0+. This proves (v).

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.
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).
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).
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.
This means that
Ḟ(s)x=-K(r+s,t-s)CA(r+s)x.
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)Cẏ(s)=-K(r+s,t-s)CA(r+s)Cy(s)+K(r+s,t-s)C2ẏ(s)=-K(r+s,t-s)CA(r+s)Cy(s)+K(r+s,t-s)C2A(r+s)y(s)=0.
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).
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.
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∫0tK(r+s,t-s)f(s)ds∈D,0<t≤T,
then the initial value equation (3.8) admits a unique solution
x(t)=K(r,t)Cx0+∫0tK(r+s,t-s)Cf(s)ds.

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)=∫0tK(r+s,t-s)Cf(s)ds,
which is in D by our hypothesis. We have
g(t+h)-g(t)h=1h[∫0t+hK(r+s,t+h-s)Cf(s)ds-∫0tK(r+s,t-s)Cf(s)ds]=1h[∫0tK(r+s,t+h-s)Cf(s)ds-∫0tK(r+s,t-s)Cf(s)ds+∫tt+hK(r+s,t+h-s)Cf(s)ds].
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).
So
g(t+h)-g(t)h=1h[∫0tK(r+t,h)K(r+t,t-s)f(s)ds-∫0tK(r+s,t-s)Cf(s)ds+∫tt+hK(r+s,t+h-s)Cf(s)ds]=∫0tK(r+t,t-s)(K(r+t,h)f(s)-Cf(s)h)ds+1h∫tt+hK(r+s,t+h-s)Cf(s)ds.
Since the range of f is in D, passing to the limit when h→0, and using Theorem 2.6(v), we have
ġ(t)=∫0tK(r+s,t-s)CA(r+t)f(s)ds+K(r+t,t-t)Cf(t)=∫0tK(r+s,t-s)CA(r+t)f(s)ds+C2f(t).
Therefore, ġ(t) exists. Also by our hypothesis A(r+t) is closed, and ∫0tK(r+s,t-s)Cf(s)ds∈D, thus
∫0tK(r+s,t-s)CA(r+t)f(s)ds=A(r+t)∫0tK(r+s,t-s)Cf(s)ds.
Consequently,
ġ(t)=A(r+t)g(t)+C2f(t),t≥0.
Hence
ẋ(t)=∂∂tK(r,t)Cx0+A(r+t)∫0tK(r+s,t-s)Cf(s)ds+C2f(t)=A(r+t)(K(r,t)Cx0+∫0tK(r+s,t-s)Cf(s)ds)+C2f(t)=A(r+t)x(t)+C2f(t).
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).
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
limt→0+T(t)x-xt,limt→0+K(s,t)x-Cxt
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
limt→0+T(t)K(s,t)x-Cxt
exists in the range of C. So
limt→0+R(s,t)x-Cxt=limt→0+T(t)K(s,t)x-Cxt=limt→0+T(t)K(s,t)x-Cxt+Climt→0+T(t)x-xt
exists in the range of C and
CB(s)x=limt→0+R(s,t)x-Cxt=CA(s)x+CBx.
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.

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
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,ɛ)=2(r+t)∂∂ɛx(t,ɛ)+ɛx(t,ɛ),x(0,ɛ)=e-4ɛ2x0(ɛ),ɛ,t≥0.
Let X=BUC(ℝ), with the supremum-norm. Define C∈B(X) by Cx(ɛ)=e-ɛ2x(ɛ), x(·)∈X. Also define B:D(B)→X by Bx(ɛ)=ɛx(ɛ), 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(ɛ)=e-ɛ2+ɛtx(ɛ). 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(ɛ)=e-ɛ2x(t2+2st+ɛ). 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.
Thus by Theorem 3.1 for any x0∈D∩D(B), (3.26) has the unique solution
x(t,ɛ)=R(r,t)C2x0(ɛ)=e-4ɛ2+ɛtx0(t2+2rt+ɛ).

Example 3.6.

For a given sequence (pn)n∈ℕ of complex numbers with nonzero elements and (yn)n∈ℕ, consider the following equation:
ddtxn(t)=ein(t+1)xn(t)+pnxn(t),xn(0)=pn2yn,n∈ℕ.
Let X be the space c0, the set of all complex sequence with zero limit at infinity. For a bounded sequence p=(pn)n∈ℕ, define A:D(A):→X and Mp on X by
A(xn)n∈ℕ=(einxn)n∈ℕ,Mp(xn)n∈ℕ=(pnxn).
One can easily see that D(A)={(xn)n∈ℕ∈c0:(einxn)n∈ℕ∈c0} and Mp is a bounded linear operator which is injective. It is well known that A is the infinitesimal generator of strongly continuous semigroup
T(t)(xn)n∈ℕ=(einxn)n∈ℕ.
Thus by Example 2.4, {A(t)}t≥0, defined by
A(t)(xn)n∈ℕ:=AT(t)(xn)n∈ℕ=(ein(1+t)xn)n∈ℕ,
is the infinitesimal generator of the Mp-quasi-semigroup
K(s,t)=Mp(eT(s+t)-T(s)).
Using these operators, one can rewrite (3.29) as
ẋ(t)=(A(t)+MP)x(t),x(0)=Mp2y0,
where x0=(yn)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)+Mp}t≥0 is the infinitesimal generator of of Mp-quasi-semigroup
R(s,t)=T(t)K(s,t).
Also from Theorem 3.1, with r=0, for any y∈D(A), (3.34) has a unique solution
x(t)=R(0,t)Mpy=T(t)K(0,t)Mp2x0.
But from definition of K(s,t), for a given (xn)n∈ℕ∈c0,
K(0,t)(xn)n∈ℕ=eT(t)-I=e-1∑k=0∞Tk(t)k!(xn)n∈ℕ=e-1∑k=0∞(eikntxnk!)n∈ℕ=e-1(∑k=0∞eikntxnk!)n∈ℕ.
So the solution of (3.34) is
x(t)=R(0,t)Mpy=(∑k=0∞eikt(n+1)-1pn4ynk!)n∈ℕ,
or equivalently the solution of (3.29) is
xn(t)=∑k=0∞eikt(n+1)-1pn4ynk!,n∈ℕ.

BarcenasD.LeivaH.BárcenasD.LeivaH.Tineo MoyaA.The dual quasisemigroup and controllability of evolution equationsCucV.On the exponentially bounded C0-quasisemigroupsCucV.Non-uniform exponential stability of C0-quasisemigroups in Banch spaceCucV.A generalization of a theorem of Datko and PazyBárcenasD.LeivaH.Tineo MoyaA.Quasisemigroups and evolution equationsDaviesE. B.PangM. M. H.The Cauchy problem and a generalization of the Hille-Yosida theoremdeLaubenfelsR.C-semigroups and the Cauchy problemLiY.-C.ShawS.-Y.N-times integrated C-semigroups and the abstract Cauchy problemLiY.-C.ShawS.-Y.On characterization and perturbation of local C-semigroupsTanakaN.MiyaderaI.Exponentially bounded C-semigroups and integrated semigroupsTanakaN.MiyaderaI.C-semigroups and the abstract Cauchy problemPazyA.EngelK.-J.NagelR.