# On Two-Parameter Regularized Semigroups and the Cauchy Problem

Authors: Mohammad Janfada
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

Suppose  thatX 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.


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## Body

## 1. Introduction and Preliminaries

Suppose thatX 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
(1.1)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.IfC∈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
(1.2)D(A)={x∈X:limt→0T(t)x-Cxtexistsin 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 theACP(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 Section2, 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 Section3, 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 thatX 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(i)
W(0,0)=C,(ii)
W(s+s′,t+t′)C=W(s,t)W(s′,t′), for all s,s′,t,t′∈ℝ+,(iii)
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-parameterC-semigroup which is not exponentially bounded.Example 2.2.
LetX=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-parameterC-semigroup.Theorem 2.3.
Suppose thatW(s,t) is a two-parameter C-semigroup with the infinitesimal generator (H1,H2). Then, one has the following.(i)
For eachx∈X and for every s,t≥0, ∫0t∫0sW(μ,ν)xdμdν, is in D(H1)∩D(H2). Also
(2.1)lim(h,k)→(0,0)1hk∫tt+h∫ss+kW(μ,ν)xdμdν=W(s,t)x.(ii)
For eachx∈X, and for every s,t∈ℝ+, ∫0sW(μ,t)xdμ∈D(H1) and ∫0tW(s,ν)xdν∈D(H2);  furthermore
(2.2)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.(iii)
R(C)¯⊆D(H1)∩D(H2)¯ and H1 and H2 are closed.(iv)
For anyx∈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,
(2.3)∂∂tiW(t1,t2)x=HiW(t1,t2)x=W(t1,t2)Hix.(v)
For anya,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), supposex∈X. First we note that for any ν≥0,
(2.4)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
(2.5)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
(2.6)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
(2.7)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
(2.8)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 thatH1 and H2 are closed as a trivial consequence of the one-parameter case (see [2]). For any x∈X we saw that
(2.9)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 letx∈D(H1)∩D(H2). If u(s)=W(s,0) and v(t)=W(s,t), there is y∈X such that
(2.10)lims→0u(s)x-Cxs=Cy.
Hence
(2.11)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),
(2.12)∂∂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,
(2.13)∫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 C(2.14)C∂∂sW(s,t)x=∂∂sW(s,t)Cx=CH1W(s,t)x.
But C is injective so
(2.15)∂∂sW(s,t)x=H1W(s,t)x=W(s,t)H1x.
The second one is similar.
To prove (v), first we note thatT(t) is a one-parameter C-semigroup. Now if x∈D(aH1+bH2)=D(H1)∩D(H2),
(2.16)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
(2.17)C-1limt→0+T(t)x-Cxt=aH1x+bH2x.For an exponentially bounded one-parameterC-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.(i)
For anyx∈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.(ii)
For anyx∈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.(iii)
For anyx∈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,
(2.18)-λ(λ-A)Lλ(A)x=-λCx.
This implies our desired equality.
For the second part, first we show thatCAλ=AλC. For this we note that
(2.19)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,
(2.20)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), forx∈D(A), from (i)  and the fact that A is closed, we have
(2.21)∥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 forx∈D(A),
(2.22)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 anyx∈D(A), we have
(2.23)∥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),
(2.24)W(s,t)x=Climλ→∞esHλ+tKλx.For exponentially boundedC-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, λ2(2.25)Lλ1(H)Lλ2(K)=Lλ2(K)Lλ1(H).(ii)
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),
(2.26)HKx=KHx.(iii)
Suppose thatH 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-parameterC-semigroups.
To prove (ii), we letx∈D(H)∩D(HK); from the strongly continuity of W(s,t) and the fact that K is closed, we have
(2.27)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 largeλ, λ′, we know that
(2.28)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),
(2.29)u(s)x=limλ→∞CesHλx,v(t)=limλ′→∞CetKλ′x.
So
(2.30)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
(2.31)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.IfH 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 thatC∈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.
TheC1-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
(2.32)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
(2.33)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
(2.34)∥·∥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 thatK1=B, which completes the proof.


## 3. Two-Parameter Abstract Cauchy Problems

Suppose thatHi:D(Hi)⊆X→X, i=1,2, is linear operator. Consider the following two-parameter Cauchy problem:
(3.1)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 thatu(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
(3.2)du(t)dt=H1u(t),t∈ℝ+,u(0)=0,(3.3)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
(3.4)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
(3.5)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
(3.6)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
(3.7)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 theoremX1 and C1 have their meaning in  Proposition 2.7.Theorem 3.2.
Suppose thatC∈B(X) is injective and H,K are two closed operators satisfying
(3.8)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 anyx∈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 solutionW1(s,t) is a well-defined and linear operator on X1 and
(3.9)W1(0,0)x=u(0,0;x)=Cx.
By uniqueness of solutions one can see that
(3.10)W1(s+s′,t+t′)C1=W1(s,t)W1(s′,t′).
We are going to show thatW1(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
(3.11)u(μ,ν;Cxn)=Cxn+∫0μHu(η,ν;Cxn)dη,u(μ,ν;Cxn)=Cxn+∫0νKu(μ,η;Cxn)dη.
Letting n→∞, we obtain
(3.12)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
(3.13)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)
(3.14)ỹ(0,0)=Cy(0,0)=C2x.
Also (3.12) and the fact that C commutes with H and K imply that
(3.15)∂∂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
(3.16)∂∂νỹ(μ,ν)=Kỹ(μ,ν).
Uniqueness of the solution implies that
(3.17)ỹ(·,·)=u(·,·;Cx2)=W1(·,·)Cx=CW1(·,·)x.
In particular for 0≤μ≤s and 0≤ν≤s,
(3.18)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 Theoremϕs,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],
(3.19)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
(3.20)∥·∥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)(3.21)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:(3.22)∂∂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:
(3.23)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:
(3.24)∂∂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:
(3.25)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(3.26)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).



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*Source: 415847-2009-08-16.xml*