Suppose that X is a Banach space and C is an
injective operator in B(X), the space of all bounded linear
operators on X. In this note, a two-parameter C-semigroup
(regularized semigroup) of operators is introduced, and some of its
properties are discussed. As an application we show that the
existence and uniqueness of solution of the 2-abstract Cauchy
problem (∂/(∂ti))u(t1,t2)=Hiu(t1,t2),i=1,2, ti>0, u(0,0)=x, x∈C(D(H1)∩D(H2)) is closely related to the two-parameter C-semigroups of
operators.

1. Introduction and Preliminaries

Suppose that X is a Banach space and A is a linear operator in X with domain D(A) and range R(A). For a given x∈D(A), the abstract Cauchy problem for A with the initial value x consists of finding a solution u(t) to the initial value problem
ACP(A;x){du(t)dt=Au(t),t∈ℝ+,u(0)=x,
where by a solution we mean a function u:ℝ+→X, which is continuous for t≥0, continuously differentiable for t>0, u(t)∈D(A) for t∈ℝ+, and ACP(A;x) is satisfied.

If C∈B(X), the space of all bounded linear operators on X, is injective, then a one-parameter C-semigroup (regularized semigroup) of operators is a family {T(t)}t∈ℝ+⊂B(X) for which T(0)=C, T(s+t)C=T(s)T(t), and for each x∈X, the mapping t↦T(t)x is continuous. An operator A:D(A)→X with
D(A)={x∈X:limt→0T(t)x-Cxtexists in the range ofC},
and where, for x∈D(A), Ax:=C-1limt→0((T(t)x-Cx)/t) is called the infinitesimal generator of T(t).

Regularized semigroups and their connection with the ACP(A;x) have been studied in [1–6] and some other papers. Also the concept of local C-semigroups and their relation with the ACP(A;x) have been considered in [7–10].

In Section 2, we introduce the concept of two-parameter regularized semigroups of operators and their generator. Some basic properties of two-parameter regularized semigroups and their relation with the generators are studied in this section.

In Section 3, two-parameter abstract Cauchy problems are considered. It is proved that the existence and uniqueness of its solutions is closely related with two-parameter regularized semigroups of operators.

2. Two-Parameter Regularized Semigroups

In this section we introduce two-parameter regularized semigroup and its generator on Banach spaces. Then some properties of two-parameter regularized semigroups are studied.

Definition 2.1.

Suppose that X is a Banach space and C∈B(X) is an injective operator. A family {W(s,t)}s,t∈ℝ+⊂B(X) is called a two-parameter regularized semigroup (or two parameter C-semigroup) if

W(0,0)=C,

W(s+s′,t+t′)C=W(s,t)W(s′,t′), for all s,s′,t,t′∈ℝ+,

lim(s′,t′)→(s,t)W(s′,t′)x=W(s,t)x, for all x∈X.

It is called exponentially bounded if ∥W(s,t)∥≤Me(s+t)ω, for some M,ω>0.

Suppose that {W(s,t)}s,t∈ℝ+ is a two-parameter C-semigroup. Put u(s):=W(s,0) and v(t):=W(0,t), then it is easy to see that these families are two commuting one-parameter C-semigroups such that W(s,t)C=u(s)v(t). Also u(s) and v(t) commute with C. If H1 and H2 are their generators, respectively, then we will think of (H1,H2) as the generator of W(s,t).

From the one-parameter case (see [8]), one can prove that R(C)⊆D(H1)¯∩D(H2)¯, and C-1HiC=Hi, i=1,2.

Also if {U(s)}s∈ℝ+ and {V(t)}t∈ℝ+ are two commuting one-parameter C-semigroups, then one can see that W(s,t):=U(s)V(t) is a two-parameter C2-semigroup of operators.

The following is an example of a two-parameter C-semigroup which is not exponentially bounded.

Example 2.2.

Let X=L2(ℂ), and [W(s,t)f](z):=e-|z|2+(s+t)zf(z), (Cf)(z):=e-|z|2f(z), then W(s,t) is a two-parameter C-semigroup which is not exponentially bounded.

In the following theorem we can see some elementary properties of a two-parameter C-semigroup.

Theorem 2.3.

Suppose that W(s,t) is a two-parameter C-semigroup with the infinitesimal generator (H1,H2). Then, one has the following.

For each x∈X and for every s,t≥0, ∫0t∫0sW(μ,ν)xdμdν, is in D(H1)∩D(H2). Also
lim(h,k)→(0,0)1hk∫tt+h∫ss+kW(μ,ν)xdμdν=W(s,t)x.

For each x∈X, and for every s,t∈ℝ+, ∫0sW(μ,t)xdμ∈D(H1) and ∫0tW(s,ν)xdν∈D(H2); furthermore
H1∫0sW(μ,t)xdμ=W(s,t)x-W(0,t)x,H2∫0tW(s,ν)xdν=W(s,t)x-W(s,0)x.

R(C)¯⊆D(H1)∩D(H2)¯ and H1 and H2 are closed.

For any x∈D(H1)∩D(H2), and each s,t>0, u(s)x and v(t)x are in D(H1)∩D(H2). Also for this x, and i=1,2,
∂∂tiW(t1,t2)x=HiW(t1,t2)x=W(t1,t2)Hix.

For any a,b>0, T(t):=W(ta,tb) is a one-parameter C-semigroup whose generator is an extension of aH1+bH2.

Proof.

To prove (i), suppose x∈X. First we note that for any ν≥0,
limh→01h∫tt+hW(μ,ν)Cxdμ=W(0,ν)limh→01h∫tt+hW(μ,0)xdμ=W(0,ν)W(t,0)x=W(t,ν)Cx.
Thus
1h(W(h,0)∫0s∫0tW(μ,ν)xdμdν-C∫0s∫0tW(μ,ν)xdμdν)=1hC(∫0s∫ht+hW(μ,ν)xdμdν-∫0s∫0tW(μ,ν)xdμdν)=∫0s(1h[∫tt+hW(μ,ν)Cxdμ-∫0hW(μ,ν)Cxdμ])dν,
which tends to C∫0s(W(t,ν)-W(0,ν))xdν as h→0. This implies that ∫0s∫0tW(μ,ν)xdμdν is in D(H1) and
H1∫0s∫0tW(μ,ν)xdμdν=∫0s(W(t,ν)-W(0,ν))xdν.
A similar argument implies that it is in D(H2) and
H2∫0s∫0tW(μ,ν)xdμdν=∫0t(W(μ,s)-W(μ,0))xdν.
For the second part, from the continuity of C we have
Clim(h,k)→(0,0)1hk∫tt+h∫ss+kW(μ,ν)xdμdν=lim(h,k)→(0,0)1hk∫tt+h∫ss+kW(μ,ν)Cxdμdν=lim(h,k)→(0,0)1h∫tt+hW(0,ν)1k∫ss+kW(μ,0)xdμdν=limh→01h∫tt+hW(0,ν)(limk→01k∫ss+kW(μ,0)xdμ)dν=W(0,t)W(s,0)x=W(s,t)Cx.
Now the fact that C is injective completes the proof of this part.

The proof of (ii) has a process similar to the first part of (i).

To prove (iii), we first note that H1 and H2 are closed as a trivial consequence of the one-parameter case (see [2]). For any x∈X we saw that
1h∫0h∫0hW(μ,ν)xdμdν∈D(H1)∩D(H2),
which tends to W(0,0)x=Cx∈R(C), as h→0. This implies that R(C)¯⊆D(H1)∩D(H2)¯.

To prove (iv), we let x∈D(H1)∩D(H2). If u(s)=W(s,0) and v(t)=W(s,t), there is y∈X such that
lims→0u(s)x-Cxs=Cy.
Hence
lims→0u(s)v(t)x-Cv(t)xs=v(t)Cy=Cv(t)y,
which is in the R(C), and this implies that v(t)x is in D(H1), similarly it is in D(H2).

Now from [2, Theorem 2.4(b)], for x∈D(H1)∩D(H2), from the fact that v(t)x is in D(H1),
∂∂sW(s,t)Cx=dds(u(s)(v(t)x))=H1u(s)(v(t)x)=H1W(s,t)Cx=CH1W(s,t)x.
On the other hand from the part (ii) and closedness of H1,
∫0sH1W(μ,t)xdμ=H1∫0sW(μ,t)xdμ=W(s,t)x-W(0,t)x,
which implies that (∂/∂s)W(s,t)x exists. Hence from the continuity of CC∂∂sW(s,t)x=∂∂sW(s,t)Cx=CH1W(s,t)x.
But C is injective so
∂∂sW(s,t)x=H1W(s,t)x=W(s,t)H1x.
The second one is similar.

To prove (v), first we note that T(t) is a one-parameter C-semigroup. Now if x∈D(aH1+bH2)=D(H1)∩D(H2),
Climt→0+T(t)x-Cxt=limt→0+W(ta,0)W(0,tb)x-W(ta,0)Cx+W(ta,0)Cx-C2xt=blimt→0+W(ta,0)W(0,tb)x-Cxbt+alimt→0+W(at,0)Cx-C2xt=bC2H2x+aH1C2x.
Now the fact that C is injective implies that
C-1limt→0+T(t)x-Cxt=aH1x+bH2x.

For an exponentially bounded one-parameter C-semigroup T(t) with the generator A, from [1] the existence of Lλ(A)x=∫0∞e-λtT(t)xdt is guaranteed for sufficiently large λ∈ℝ. Now we have the following lemma for one-parameter C-semigroups of operators which is similar to the Yosida-approximation theorem for strongly continuous semigroups. This will be applied in our study of two-parameter regularized semigroups.

Lemma 2.4.

Let {T(t)}t∈ℝ+ be a one-parameter C-semigroup satisfying the condition ∥T(t)∥≤Meωt, for some ω>0 and M>0, with the generator A. If for λ>ω, Aλ:=λALλ(A), then one has the following.

For any x∈X, ∥Lλ(A)x∥≤(M/(λ-ω))∥x∥, Aλ=λ2Lλ(A)-λC, and so Aλ is bounded. Also S(t):=CetAλ is a one-parameter C-semigroup which is exponentially bounded.

For any x∈D(A)¯, limλ→∞λLλ(A)x=Cx and for all x∈D(A), limλ→∞Aλx=CAx. Also if R(C) is dense in X, then the first equality holds on X.

For any x∈D(A)¯, T(t)x=limλ→∞CetAλx.

Proof.

The first inequality of (i) is trivial. From [2, Lemma 2.8], we know that for any x∈X, (λ-A)Lλ(A)x=Cx; thus,
-λ(λ-A)Lλ(A)x=-λCx.
This implies our desired equality.

For the second part, first we show that CAλ=AλC. For this we note that
CLλ(A)=C∫0∞e-λtT(t)xdx=∫0∞Ce-λtT(t)xdx=∫0∞e-λtT(t)Cxdx=Lλ(A)Cx.
This and the first part imply that CAλ=AλC. Now we prove the C-semigroup properties of S(t). Trivially S(0)=C. Also from the last equality,
S(s+t)C=Ce(s+t)AλC=CesAλCetAλ=S(s)S(t).
The fact that Aλ, λ>ω, is a bounded operator trivially implies that S(·) is exponentially bounded. Now the continuity of the mapping t↦S(t)x at zero implies the strongly continuity of S(t).

To prove (ii), for x∈D(A), from (i) and the fact that A is closed, we have
∥λLλ(A)x-Cx∥=∥ALλ(A)x∥=∥Lλ(A)Ax∥≤∥Lλ(A)∥∥Ax∥≤M(λ-ω)∥Ax∥→0asλ→∞.
The continuity of C and Lλ(A) implies that for any x∈D(A)¯, limλ→∞λLλ(A)x=Cx.

Now for x∈D(A),
limλ→∞Aλx=limλ→∞λLλ(A)Ax=CAx=ACx.
For the last part of (ii), if C has a dense range, then by [8, Lemma 1.1.3], R(C)⊆D(A)¯, and so X=R(C)¯⊆D(A)¯⊆X, which means that D(A)¯=X.

To prove (iii), for any x∈D(A), we have
∥CetAλx-CetAμx∥=∥∫01dds(CetsAλet(1-sAμ)x)∥≤∫01t∥CetsAλet(1-sAμ)(Aλx-Aμx)∥ds≤t∥C∥∥Aλx-Aμx∥≤t∥C∥(∥Aλx-ACx∥+∥ACx-Aμx∥).
This and the previous part prove the existence of limλ→∞CetAλx.

Using this theorem we may find the following approximation theorem for two-parameter regularized semigroups.

Corollary 2.5.

Suppose that (H,K) is the infinitesimal generator of an exponentially bounded two-parameter C-semigroup W(s,t), then for each x∈D(H)∩D(K),
W(s,t)x=Climλ→∞esHλ+tKλx.

For exponentially bounded C-semigroup W(s,t) satisfying ∥W(s,t)∥≤Me(s+t)ω, with the infinitesimal generator (H,K), define Lλ1(H)x:=∫0∞e-λ1sW(s,0)xds and Lλ2(K)x:=∫0∞e-λ2tW(0,t)xdt, where Re(λi)>ω. From the previous Lemma Lλ1(H) and Lλ2(K) are bounded operators.

Theorem 2.6.

(i) Let (H,K) be the generator of an exponentially bounded two-parameter C-semigroup, then for large enough λ1, λ2Lλ1(H)Lλ2(K)=Lλ2(K)Lλ1(H).

Let (H,K) be the generator of an exponentially bounded two-parameter C-semigroup, then D(H)∩D(HK)⊆D(KH), and for x∈D(H)∩D(HK),
HKx=KHx.

Suppose that H and K are the generators of two exponentially bounded one-parameter C-semigroups {u(s)}s∈ℝ+ and {v(t)}t∈ℝ+, respectively. If their resolvents commute and R(C) is dense in X, then W(s,t):=u(s)v(t) is a two-parameter C2-semigroup.

Proof.

The proof of (i) follows trivially from the properties of two-parameter C-semigroups.

To prove (ii), we let x∈D(H)∩D(HK); from the strongly continuity of W(s,t) and the fact that K is closed, we have
C2HKx=Clims→0W(s,0)Kx-CKxs=lims→01s(W(s,0)(limt→0W(0,t)x-Cxt)-limt→0W(0,t)x-Cxt)=lims→0limt→01st(W(s,0)W(0,t)x-W(s,0)Cx-W(0,t)x+Cx)=lims→0limt→01st(W(0,t)W(s,0)x-W(s,0)Cx-W(0,t)x+Cx)=lims→0limt→01t(W(0,t)(W(s,0)x-Cxs)-W(s,0)x-Cxs)=Clims→0K(W(s,0)x-Cxs)=C2KHx.
However, C is injective, and this completes the proof of (i).

To prove (iii), from our hypothesis, for sufficiently large λ, λ′, we know that
Lλ(H)Lλ′(K)=Lλ′(K)Lλ(H).
By Lemma 2.4, Hλ=λ2Lλ(H)-λC and Kλ′=λ′2Lλ′(H)-λ′C, thus HλKλ′=Kλ′Hλ. From (iii) of Lemma 2.4, for each x∈D(H)∩D(K),
u(s)x=limλ→∞CesHλx,v(t)=limλ′→∞CetKλ′x.
So
u(s)v(t)x=Climλ→∞esHλv(t)x=C2limλ→∞esHλ(limλ′→∞etKλ′x),(esHλis continuous)=C2limλ→∞limλ′→∞esHλetKλ′x=C2limλ→∞limλ′→∞etKλ′esHλx=Climλ→∞v(t)esHλx=v(t)u(s)x.
Now the continuity of u(s) and v(t) and the fact that D(H)∩D(K)¯=R(C)¯=X imply that for each x∈X, u(s)v(t)x=v(t)u(s)x. Thus
W(s,t)W(s′,t′)=u(s)v(t)u(s′)v(t′)=u(s)u(s′)v(t)v(t′)=Cu(s+s′)Cv(t+t′)=W(s+s′,t+t′)C2.
On the other hand W(0,0)=C2, which completes the proof.

If H and K are two closed operators on X, then X1:=D(H)∩D(K) with ∥x∥1=∥x∥+∥Hx∥+∥Kx∥, x∈X1, is a Banach space.

Proposition 2.7.

Suppose that C∈B(X) is injective and {W(s,t)} is a two-parameter C-semigroup with the generator (H,K). Then W1(s,t):=W(s,t)|X1 defines a two-parameter C1-semigroup, with the generator (H1,K1), where C1=C|X1, and H1, K1 are the part of H and K on X1, respectively.

Proof.

The C1-semigroup properties of W1(s,t) are obvious. Let (A,B) be the generator of W1(s,t); we show that A=H1 and B=H2. First we note that
D(H1)={x∈X1:Hx∈X1}={x∈D(H)∩D(K):x∈D(H2)∩D(KH)}=D(K)∩D(H2)∩D(KH).
Let x∈D(H1). So we have
W1(s,0)x-C1xt=W(s,0)x-Cxt→CHx=C1H1x,HW1(s,0)x-C1xt=W(s,0)Hx-CHxt→CH2x=HC1H1x,KW1(s,0)x-C1xt=W(s,0)Kx-CKxt→CHKx=KCHx=KC1H1x.
These show that (W1(s,0)x-C1x)/t→C1H1x in ∥·∥1, that is, x∈D(A) and Ax=H1x. Hence H1⊆A. Conversely, if x∈D(A)⊆X1, then
∥·∥1-limt→0W(s,0)x-Cxt=∥·∥1-limt→0W1(s,0)x-C1xt=C1Ax=CAx,
so Hx=Ax∈X1. Hence x∈D(K)∩D(H2)∩D(KH)=D(H1) and H1x=Hx=Ax.

A similar argument shows that K1=B, which completes the proof.

3. Two-Parameter Abstract Cauchy Problems

Suppose that Hi:D(Hi)⊆X→X, i=1,2, is linear operator. Consider the following two-parameter Cauchy problem:
2-ACP(H1,H2;x){∂∂tiu(t1,t2)=Hiu(t1,t2),ti>0,i=1,2,u(0,0)=x,x∈C(D(H1)∩D(H2)).
We mean by a solution a continuous Banach-valued function u(·,·):[0,∞)×[0,∞)→X which has continuous partial derivative and satisfies 2-ACP(H1,H2;x).

In this section first we prove that if (H1,H2) is the infinitesimal generator of a two-parameter C-semigroup of operators, then 2-ACP(H1,H2;x) has a unique solution for any x∈C(D(H1)∩D(H2)). Next it is proved that under some condition on C, existence and uniqueness of solutions of 2-ACP(H1,H2;Cx), for every x∈D(H1)∩D(H2), imply that this unique solution is induced by a two-parameter regularized semigroup.

Theorem 3.1.

Suppose that an extension of (H1,H2) is the generator of a two-parameter C-semigroup W(s,t), then 2-ACP(H1,H2;x) has the unique solution u(s,t;x):=W(s,t)C-1x, for all x∈C(D(H1)∩D(H2)).

Proof.

The fact that u(s,t;x):=W(s,t)C-1x is a solution of 2-ACP(H1,H2;x) is obvious from Theorem 2.3. It is enough to show that 2-ACP(H1,H2;x) has the unique solution u(s,t)=0, for the initial value x=0. From one-parameter case (see [2]), we know that the systems
du(t)dt=H1u(t),t∈ℝ+,u(0)=0,dv(t)dt=H2v(t),t∈ℝ+,v(0)=0
have the unique solution zero. Now if u(s,t;0) is a solution of 2-ACP(H1,H2;0), then
u1(s):=W(s,0)C-1u(0,t;0),u2(s):=u(s,t;0)
are two solutions of (3.2), for the initial value u(0,t;0), since
ddsu1(s)=ddsW(s,0)C-1u(0,t;0)=H1W(s,0)C-1u(0,t;0)=H1u1(s),ddsu2(s)=∂∂su(s,t;0)=H1u(s,t;0)=H1u2(s).
The uniqueness of solution in one-parameter case implies that u1(s)=u2(s). So
W(s,0)C-1u(0,t;0)=u(s,t;0).
Also v1(t):=W(0,t)C-1u(s,0;0) and v2(t):=u(s,t;0) are two solutions of (3.3) for the initial value u(s,0;0). From the uniqueness of solution in (3.3), W(0,t)C-1u(s,0;0)=u(s,t;0), for all s,t≥0. Thus
u(s,t;0)=W(s,0)C-1u(0,t;0)=W(s,0)C-1W(0,t)u(0,0;0)=0.

The uniqueness of solution 2-ACP(H,K;Cx), for all x∈D(H)∩D(K), also leads us to a two-parameter C-semigroup. This will be shown in the following theorem.

In this theorem X1 and C1 have their meaning in Proposition 2.7.

Theorem 3.2.

Suppose that C∈B(X) is injective and H,K are two closed operators satisfying
Cx∈X1,KCx=CKx,HCx=CHx,∀x∈X1.
If, for each x∈X1, the Cauchy problem 2-ACP(H,K;Cx) has a unique solution u(·,·;Cx), then there exists a two-parameter C1-semigroup W1(·,·) on X1 such that u(·,·;Cx)=W1(·,·)x. Moreover, the infinitesimal generator of W1(·,·) is a restriction of (H1,K1), where H1 and K1 are the part of H and K on X1, respectively.

Proof.

Suppose that, for any x∈X1, 2-ACP(H,K;Cx) has a unique solution u(·,·;Cx)∈C1([0,∞)×[0,∞),X). For x∈X1 and 0<s,t<∞, define W1(s,t)x:=u(s,t;Cx).

From the uniqueness of solution W1(s,t) is a well-defined and linear operator on X1 and
W1(0,0)x=u(0,0;x)=Cx.
By uniqueness of solutions one can see that
W1(s+s′,t+t′)C1=W1(s,t)W1(s′,t′).

We are going to show that W1(s,t) is a bounded operator on (X1,∥·∥1). Let 0<s,t<∞. Define the mapping ϕs,t:X1→C([0,s]×[0,t],X1) by ϕs,tx=W1(·,·)x=u(·,·;Cx). Obviously ϕs,t is linear. We claim that this mapping is closed. Suppose that xn∈X1, xn→x and u(·,·;Cxn)=ϕs,t(xn)→y in C([0,s]×[0,t],X1) with its usual supremum norm. From the Cauchy problem we know that
u(μ,ν;Cxn)=Cxn+∫0μHu(η,ν;Cxn)dη,u(μ,ν;Cxn)=Cxn+∫0νKu(μ,η;Cxn)dη.
Letting n→∞, we obtain
y(μ,ν)=Cx+∫0μHy(η,ν)dη,y(μ,ν)=Cx+∫0νKy(μ,η)dη
for any (μ,ν)∈[0,s]×[0,t]. Now define ỹ on [0,∞)×[0,∞) by
ỹ(μ,ν)={Cy(μ,ν),0≤μ≤s,0≤ν≤t,W1(0,ν-t)y(μ,t),0≤μ≤s,t<ν<∞,W1(μ-s,0)y(s,ν),s<μ<∞,0≤ν≤t,W1(μ-s,ν-t)y(s,t),s<μ<∞,t<ν<∞.
One can see that ỹ is a solution of 2-ACP(H,K;C2x). Indeed from (3.12)
ỹ(0,0)=Cy(0,0)=C2x.
Also (3.12) and the fact that C commutes with H and K imply that
∂∂μỹ(μ,ν)={Hy(μ,ν),0≤μ≤s,0≤ν≤t,HW1(0,ν-t)y(μ,t),0≤μ≤s,t<ν<∞,HW1(μ-s,0)y(s,ν),0<μ<∞,0≤ν≤t,HW1(μ-s,ν-t)y(s,t),0<μ<∞,0<ν<∞,=Hỹ(μ,ν).
Similarly
∂∂νỹ(μ,ν)=Kỹ(μ,ν).
Uniqueness of the solution implies that
ỹ(·,·)=u(·,·;Cx2)=W1(·,·)Cx=CW1(·,·)x.
In particular for 0≤μ≤s and 0≤ν≤s,
Cy(μ,ν)=ỹ(μ,ν)=CW1(μ,ν)x=Cϕs,t(x)(μ,ν).
The fact that C is injective implies that y=ϕs,t(x), which shows that ϕs,t is closed operator.

By the Closed Graph Theorem ϕs,t is a continuous operator from Banach space X1 into the Banach space C([0,s]×[0,t],X1). So if xn→x in X1, then ϕs,t(xn)→ϕs,t(x) in C([0,s]×[0,t],X1); thus for each (μ,ν)∈[0,s]×[0,t],
W1(s,t)xn=ϕs,t(xn)(μ,ν)→ϕs,t(x)(μ,ν)=W1(μ,ν)x.
But s and t were arbitrary; hence W1(μ,ν) is continuous for any μ,ν∈[0,∞). Also for every x∈X1, W1(·,·)x=ϕs,t(x) is continuous on [0,s]×[0,t]; that is, W1(·,·) is strongly continuous family of operators.

Now let (A,B) be its infinitesimal generator and x∈D(A), then
∥·∥1-lims→0W1(s,0)x-C1xs=C1Ax,
which implies that lims→0((W1(s,0)x-Cx)/s)=CAx, but D(A)⊆D(H)lims→0W1(s,0)x-Cxs=lims→0u(s,0;Cx)-Cxs=∂∂su(0,0;Cx)=HCx=CHx.
Hence CHx=CAx. The injectivity of C implies that Hx=Ax∈X1=D(H)∩D(K). Thus x∈D(K)∩D(H2)∩D(KH)=D(H1) and H1x=Ax. This shows that A is a restriction of H1. Similarly one can see that B is a restriction of K1, which completes the proof.

We conclude this section with a simple example as an application of our discussion. Consider the following sequence of initial value problems:

∂∂sun(s,t)=nun(s,t),∂∂tun(s,t)=n2un(s,t),n∈ℕ,un(0,0)=e-n2qn.
Suppose that X=c0, the space of all complex sequences in ℂ which vanish at infinity. Now define linear operators H and K in X and operator C on X as follows:
H(xn)n∈ℕ=(nxn)n∈ℕ,K(xn)n∈ℕ=(n2xn)n∈ℕ,C(xn)n∈ℕ=(e-n2xn)n∈ℕ.
Using these operators the initial value problem (3.22) can be rewrite as follows:
∂∂su(s,t)=Hu(s,t),∂∂tu(s,t)=Ku(s,t),u(0,0)=Cq,
where u(s,t)=(un(s,t))n∈ℕ and q=(qn)n∈ℕ. One can easily see that (H,K) is the generator of the following two-parameter C-semigroup:
W(s,t)(xn)n∈ℕ=(en2(t-1)+snxn)n∈ℕ
on X. Hence for every q=(qn)n∈ℕ∈D(H)∩D(K), by Theorem 3.1, the abstract Cauchy problem (3.24) has the unique solution

u(s,t)=W(s,t)q=(en2(t-1)+snqn)n∈ℕ.
This implies that for each n∈ℕ, un(s,t)=en2(t-1)+tnqn is a solution of (3.22).

Acknowledgment

The author is grateful to the referees for their very useful suggestions which helped him to improve the presentation considerably.

DaviesE. 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 problemTanakaN.OkazawaN.Local C-semigroups and local integrated semigroupsGaoM.Local C-semigroups and local C-cosine functionsShawS.-Y.KuoC.-C.Generation of local C-semigroups and solvability of the abstract Cauchy problemsWangS. W.GaoM. C.Automatic extensions of local regularized semigroups and local regularized cosine functions