AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation78542810.1155/2010/785428785428Research ArticleOn Regularized Quasi-Semigroups and Evolution EquationsJanfadaM.RuessWolfgangDepartment of MathematicsSabzevar Tarbiat Moallem UniversityP.O. Box 397, SabzevarIransttu.ac.ir201023052010201026112009160420102010Copyright © 2010This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 , 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 . The dual quasi-semigroups and the controllability of evolution equations are also discussed in .

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,t0B(X) is called a quasi-semigroup of operators if for every s,t,r0 and xX, 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,xX,

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  and have been studied in  and many other papers.

We mention that if CB(X) is an injective operator, then a one-parameter family {T(t)}0B(X) is called a C-semigroup if for any s,t0 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 .

2. Regularized Quasi-Semigroups

Suppose X is a Banach space and {K(s,t)}s,t0 is a two-parameter family of operators in B(X). This family is called commutative if for any r,s,t,u0, 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,t0 in B(X) is called a regularized quasi-semigroups (or C-quasi-semigroups) if

K(t,0)=C, for any t0;

CK(r,t+s)=K(r+t,s)K(r,t), r,t,s0;

{K(s,t)}s,t0 is strongly continuous, that is, lim(s,t)(s0,t0)K(s,t)x-K(s0,t0)x=0,xX;

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,t0 on Banach space X, let D be the set of all xX for which the following limits exist in the range of C: limt0+K(s,t)x-Cxt=limt0+K(s-t,t)x-Cxt,s>0limt0+K(0,t)x-Cxt. Now for xD and s0, define A(s)x=C-1limt0+K(s,t)x-Cxt.{A(s)}s0 is called the infinitesimal generator of the regularized quasi-semigroup {K(s,t)}s,t0. Somewhere we briefly apply generator instead of infinitesimal generator.

Here are some useful examples of regularized quasi-semigroups.

Example 2.2.

Let {Tt}t0 be an exponentially bounded strongly continuous C-semigroup on Banach space X, with the generator A. Then K(s,t):=Tt,s,t0, defines a C-quasi-semigroup with the generator A(s)=A, s0, and so D=D(A).

Example 2.3.

Let X=BUC(), the space of all bounded uniformly continuous functions on 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,t0. One can see that {K(s,t)}s,t0 is a regularized C-quasi-semigroup of operators on X, with the infinitesimal generator A(s)f=2sḟ on D, where D={fX:ḟX}.

Example 2.4.

Let {Tt}t0 be a strongly continuous semigroup of operators on Banach space X, with the generator A. If CB(X) is injective and commutes with Tt, t0, then K(s,t):=CeTs+t-Ts,s,t0, is a C-quasi-semigroup with the generator A(s)=ATs. Thus D=D(A). In fact, for xD, boundedness of C implies that CA(s)x=limt0+CeTs+t-Tsx-Cxt=Climt0+eTs+t-Tsx-xt=Cddst=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}t0 be a strongly continuous exponentially bounded C-semigroup of operators on Banach space X, with the generator A. For s,t0, define K(s,t)=T(g(s+t)-g(s)),s,t0, where g(t)=0ta(s)ds, and aC[0,), with a(t)>0. We have K(s,0)=T(0)=C and the C-semigroup properties of {T(t)}t0 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,t0 is a C-quasi-semigroup (the other properties can be also verified easily). Also D=D(A) and for xD, 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,t0 is a C-quasi-semigroup with the generator {A(s)}s0 on Banach space X. Then

for any xD and s0,t00, K(s0,t0)xD and K(s0,t0)A(s)x=A(s)K(s0,t0)x;

for each x0D,   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 x0D and r0, K(r,t)x0=Cx0+0tA(r+s)K(r,s)x0ds,t0;

let f:[0,)X be a continuous function; then for every t[0,), limh01htt+hK(s,u)f(u)du=K(s,t)f(t);

Let C'B(X) be injective and for any s,t0, 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)}s0,

Suppose {R(s,t)}s,t0 is a quasi-semigroup of operators on Banach space X with the generator {A(s)}s0, and CB(X) commutes with every R(s,t), s,t0. Then K(s,t):=CR(s,t) is a C-quasi-semigroup of operators on X with the generator {A(s)}s0.

Proof.

First we note that from the commutativity of {K(s,t)}s,t0; CK(s,t)=K(s,t)Cs,t0. Also xD implies that limt0+K(s,t)x-Cxt=CA(s)xs0. Thus from continuity of K(s0,t0), we have limt0+K(s,t)K(s0,t0)x-CK(s0,t0)xt=K(s0,t0)limt0+K(s,t)x-Cxt=K(s0,t0)CA(s)x=CK(s0,t0)A(s)x. Thus K(s0,t0)xD 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 s0+.

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,t0 implies that lims0-K(r+t+s,-s)x0-K(r+t,-s)x0=0. Thus lims0-K(r+t+s,-s)x0-Cx0-s=CA(r+t)x0. Hence by the strongly continuity of K(s,t), lims0-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=C0tK(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,t0. In (v), obviously {R(s,t)}s,t0 is a C'C-quasi-semigroup. For xD, we have R(s,t)x-CC'xt=C'K(s,t)x-Cxt, which tends to C'CA(s), as t0+. 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,x0X. One can see [13, 14] for a comprehensive studying of abstract evolution equations.

Theorem 3.1.

Let {A(s)}s0 be the infinitesimal generator of a C-quasi-semigroups {K(s,t)}s,t0 on Banach space X, with domain D. Then for each x0D and r0, 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 xX, 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,ash0. 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<tT,x(0)=C2x0,x0D. 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)}s0 whose domain is D. If f:[0,T]D is a continuous function, each operator A(s) is closed, and C0tK(r+s,t-s)f(s)dsD,0<tT, 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        +1htt+hK(r+s,t+h-s)Cf(s)ds. Since the range of f is in D, passing to the limit when h0, 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)dsD, 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),t0. 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)}t0 and {A(s)}s0 with domain D is the generator of a regularized C-quasi-semigroup {K(s,t)}s,t0, which commutes with {T(t)}t0. Then {A(s)+B}s0 with domain DD(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)}t0 and {A(s)}s0 with domain D is the generator of a quasi-semigroup (resp., regularized C'-quasi-semigroup) {K(s,t)}s,t0, which commutes with {T(t)}t0. Then {A(s)+B}s0 with domain DD(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,t0 is a C-regularized quasi-semigroup (in (b), resp., regularized CC'-quasi-semigroup). We just prove that {A(s)+B}s0 is its generator. In (a), let {B(s)}s0 be the infinitesimal generator of {R(s,t)}s,t0 and xDD(B). Hence limt0+T(t)x-xt,limt0+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 limt0+T(t)K(s,t)x-Cxt exists in the range of C. So limt0+R(s,t)x-Cxt=limt0+T(t)K(s,t)x-Cxt=limt0+T(t)K(s,t)x-Cxt+Climt0+T(t)x-xt exists in the range of C and CB(s)x=limt0+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 BB(X) commutes with K(s,t), s,t0, and B2=B, then {BA(s)}s0 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,t0 can be easily verified. We just prove that its generator is {BA(s)}s0. Let xD; 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 t0. 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,t0. Let X=BUC(), with the supremum-norm. Define CB(X) by Cx(ɛ)=e-ɛ2x(ɛ), x(·)X. Also define B:D(B)X by Bx(ɛ)=ɛx(ɛ), where D(B)={xX:BxX}. 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={xX:ẋX}, if A(s):DX is defined by A(s)x=2sẋ, then by Example 2.3, {A(s)}s0 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,t0, 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 x0DD(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 of complex numbers with nonzero elements and (yn)n, consider the following equation: ddtxn(t)=ein(t+1)xn(t)+pnxn(t),xn(0)=pn2yn,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, define A:D(A):X and Mp on X by A(xn)n=(einxn)n,Mp(xn)n=(pnxn). One can easily see that D(A)={(xn)nc0:(einxn)nc0} 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=(einxn)n. Thus by Example 2.4, {A(t)}t0, defined by A(t)(xn)n:=AT(t)(xn)n=(ein(1+t)xn)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. Trivially T(t) commutes with K(r,s), for any r,s,t0. Now using Theorem 3.3 we obtain that {A(t)+Mp}t0 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 yD(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)nc0, K(0,t)(xn)n=eT(t)-I=e-1k=0Tk(t)k!(xn)n=e-1k=0(eikntxnk!)n=e-1(k=0eikntxnk!)n. So the solution of (3.34) is x(t)=R(0,t)Mpy=(k=0eikt(n+1)-1pn4ynk!)n, or equivalently the solution of (3.29) is xn(t)=k=0eikt(n+1)-1pn4ynk!,n.

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