DOI: 10.24193/subbmath.2020.4.08
Perturbations of local C -cosine functions
Chung-Cheng Kuo
Abstract. We show thatA+Bis a closed subgenerator of a localC-cosine function T(·) on a complex Banach spaceX defined by
T(t)x=
∞
X
n=0
Bn Z t
0
jn−1(s)jn(t−s)C(|t−2s|)xds
for all x∈X and 0≤t < T0, ifA is a closed subgenerator of a localC-cosine functionC(·) onX and one of the following cases holds: (i)C(·) is exponentially bounded, andBis a bounded linear operator onD(A) so thatBC=CBonD(A) and BA ⊂AB; (ii) B is a bounded linear operator on D(A) which commutes with C(·) onD(A) and BA ⊂AB; (iii) B is a bounded linear operator on X which commutes withC(·) onX. Herejn(t) =tn!n for allt∈R, and
Z t
0
j−1(s)j0(t−s)C(|t−2s|)xds=C(t)x for allx∈X and 0≤t < T0.
Mathematics Subject Classification (2010):47D60, 47D62.
Keywords: Local C-cosine function, subgenerator, generator, abstract Cauchy problem.
1. Introduction
Let X be a complex Banach space with norm k · k, and let L(X) denote the set of all bounded linear operators on X. For each 0 < T0 ≤ ∞ and each injection C ∈L(X), a family C(·) (={C(t)|0 ≤t < T0}) in L(X) is called a localC-cosine function onX if it is strongly continuous,C(0) =C onX and satisfies
2C(t)C(s) =C(t+s)C+C(|t−s|)C (1.1) onX for all 0≤t, s, t+s < T0(see [5], [7], [14], [15], [21], [23], [25]). In this case, the generator ofC(·) is a closed linear operatorA inX defined by
D(A) ={x∈X| lim
h→0+2(C(h)x−Cx)/h2∈R(C)}
andAx=C−1 lim
h→0+2(C(h)x−Cx)/h2 forx∈D(A). Moreover, we say thatC(·) is locally Lipschitz continuous, if for each 0< t0< T0 there exists aKt0 >0 such that
kC(t+h)−C(t)k ≤Kt0h (1.2) for all 0≤t, h, t+h≤t0; exponentially bounded, ifT0=∞and there existK, ω≥0 such that
kC(t)k ≤Keωt (1.3)
for allt≥0; exponentially Lipschitz continuous, ifT0=∞and there existK, ω≥0 such that
kC(t+h)−C(t)k ≤Kheω(t+h) (1.4) for allt, h≥0. In general, a localC-cosine function is also called aC-cosine function if T0 = ∞ (see [2], [12], [14], [16]) or a cosine function if C = I (identity operator onX) (see [1], [4], [6]), and aC-cosine function may not be exponentially bounded (see [16]). Moreover, a local C-cosine function is not necessarily extendable to the half line [0,∞) (see [21]) except forC=I(see [1], [4], [6]) and the generator of aC- cosine function may not be densely defined (see [2]). Perturbations of localC-cosine functions have been extensively studied by many authors appearing in [1], [2], [4], [9], [11], [17], [18], [19]. Some interesting applications of this topic are also illustrated there. In particular, a classical perturbation result of cosine functions shows that ifA is the generator of aC-cosine functionC(·) onX, andB a bounded linear operator onX, thenA+B is the generator of aC-cosine function onX whenC=I, but the conclusion may not be true whenC is arbitrary, and is still unknown until now even though B and C(·) are commutable, which can be completely solved in this paper and several new additive perturbation theorems concerning local C-cosine functions are also established as results in [20] for the case of C-semigroup and in [8], [13] for the case of local C-semigroup. A new representation of the perturbation of a local C-cosine function is given in (1.5) below. We show that if C(·) is an exponentially boundedC-cosine function onX with closed subgeneratorAandB a bounded linear operator on D(A) such that BC = CB on D(A) and BA ⊂AB, then A+B is a closed subgenerator of an exponentially boundedC-cosine functionT(·) onX defined by
T(t)x=
∞
X
n=0
Bn Z t
0
jn−1(s)jn(t−s)C(|t−2s|)xds (1.5) for allx∈X and 0≤t < T0 (see Theorem 2.6 below). Herejn(t) = tn!n for allt∈R, and
Z t
0
j−1(s)j0(t−s)C(|t−2s|)xds=C(t)x
for allx∈Xand 0≤t < T0. Moreover,T(·) is also exponentially Lipschitz continuous or norm continuous ifC(·) is. We then show that the exponential boundedness ofT(·) can be deleted andC-cosine functions can be extended to the context of localC-cosine functions when the assumption of BC(·) = C(·)B on D(A) is added (see Theorem 2.7 below). Moreover,T(·) is locally Lipschitz continuous or norm continuous ifC(·) is. We also show that A+B is a closed subgenerator of a local C-cosine function T(·) on X ifA is a closed subgenerator of a local C-cosine function C(·) on X and
B a bounded linear operator onX such thatBC(·) =C(·)B onX (see Theorem 2.8 below). A simple illustrative example of these results is presented in the final part of this paper.
2. Perturbation theorems
In this section, we first note some basic properties of a local C-cosine function with its subgenerator and generator.
Definition 2.1. (see [10], [14]) Let C(·) be a strongly continuous family in L(X). A linear operatorAinX is called a subgenerator ofC(·) if
C(t)x−Cx= Z t
0
Z s
0
C(r)Axdrds for allx∈D(A) and 0≤t < T0, and
Z t
0
Z s
0
C(r)xdrds∈D(A) andA Z t
0
Z s
0
C(r)xdrds=C(t)x−Cx
for all x ∈ X and 0 ≤ t < T0. A subgenerator A of C(·) is called the maximal subgenerator ofC(·) if it is an extension of each subgenerator ofC(·) to D(A).
Proposition 2.2. (see [4], [5], [10], [14], [21])LetAbe the generator of a localC-cosine function C(·)onX. Then
C(t)x∈D(A) andC(t)Ax=AC(t)x (2.1) for allx∈D(A)and0≤t < T0;
C−1AC=A andR(C(t))⊂D(A) (2.2) for all0≤t < T0;
x∈D(A) andAx=yxif and only if C(t)x−Cx= Z t
0
Z s
0
C(r)yxdrds (2.3) for all0≤t < T0;
A0 is closable andC−1A0C=A (2.4) for each subgeneratorA0 of C(·);
Ais the maximal subgenerator of C(·). (2.5) From now on, we always assume thatA:D(A)⊂X →X is a closed linear operator so thatCA⊂AC.
Theorem 2.3. (see [10], [16]) A strongly continuous family C(·) in L(X) satisfying (1.3) is a C-cosine function onX with subgeneratorAif and only if CC(·) =C(·)C, λ2∈ρC(A), andλ(λ2−A)−1C=Lλ onX for allλ > ω. Here
Lλx= Z ∞
0
e−λtC(t)xdtforx∈X.
Lemma 2.4. (see [1])LetC(·)(={C(t)|0≤t < T0})be a strongly continuous family in L(X). We setC(−t) =C(t)for0≤t < T0. ThenC(·)is a localC-cosine function on X if and only if 2C(t)C(s)=C(t+s)C+C(t−s)ConX for all|t|,|s|,|t−s|,|t+s|< T0. In this case,
S(−t) =−S(t) (2.6)
for all0≤t < T0;
S(t+s)C=S(t)C(s) +C(t)S(s)on X (2.7) for all|t|,|s|,|t+s|< T0.Here S(t) =j0∗C(t)for all|t|< T0.
By slightly modifying the proof of [3, Lemma 2], the next lemma is also attained.
Lemma 2.5. Let C(·)(={C(t)|0≤t < T0}) be a local C-cosine function onX, and C(−t) =C(t)for0≤t < T0. Assume thatS∗n+1denotes the(n+ 1)-fold convolution of S forn∈N∪ {0}, that is
S∗2(t)x= Z t
0
S(t−s)S(s)xds
and
S∗n+1(t)x= Z t
0
S∗n(t−s)S(s)xds.
Then
S∗n+1(t) = Z t
0
jn−1(s)jn(t−s)S(t−2s)Cnds= Z t
0
jn(s)jn(t−s)C(t−2s)Cnds
onX for all |t|< T0. HereS(t) =j0∗C(t)and Z t
0
j−1(s)j0(t−s)S(t−2s)C0ds=S(t) =S∗1(t)
for all|t|< T0.
Proof. It is easy to see that
S∗n+1(t) = Z t
0
jn−1(s)jn(t−s)S(t−2s)Cnds
= Z t
0
jn(s)jn(t−s)C(t−2s)Cnds
onX forn= 0. By induction, we have
S∗n+1(t)x= Z t
0
S∗n(s)S(t−s)xds
= Z t
0
Z s
0
jn−1(r)jn−1(s−r)C(s−2r)Cn−1S(t−s)xdrds
=1 2
Z t
0
Z s
0
jn−1(r)jn−1(s−r)
S(t−2r) +S(t+ 2r−2s)
Cnxdrds
= Z t
0
Z s
0
jn−1(r)jn−1(s−r)S(t−2r)Cnxdrds
= Z t
0
Z t
r
jn−1(r)jn−1(s−r)S(t−2r)Cnxdsdr
= Z t
0
jn−1(r)jn(t−r)S(t−2r)Cnxdr
=1 2
Z t
0
jn−1(r)jn(t−r)−jn(r)jn−1(t−r)
S(t−2r)Cnxdr
=1 2
Z t
0
d
dr[jn(r)jn(t−r)]S(t−2r)Cnxdr
= Z t
0
jn(r)jn(t−r)C(t−2r)Cnxdr
for alln∈N,x∈X and|t|< T0.
Applying Theorem 2.3 we can obtain the next perturbation theorem concerning exponentially boundedC-cosine functions just as a corollary of [11, Corollary 2.6.6].
Theorem 2.6. LetA be a subgenerator of an exponentially boundedC-cosine function C(·) on X. Assume that B ∈ L(D(A)), BC = CB on D(A) and BA ⊂AB. Then A+B is a closed subgenerator of an exponentially bounded C-cosine function T(·) on X given as in (1.5). Moreover,T(·) is also exponentially Lipschitz continuous or norm continuous if C(·) is.
Proof. It is easy to see that
(λ2−A−B)−1C=
∞
X
n=0
Bn(λ2−A)−n−1C
for λ > ω, and the boundedness of {kC(t)k |0 ≤ t ≤ t0} for each t0 > 0 and the strong continuity of C(·) imply that the right-hand side of (1.5) converges uniformly on compact subsets of [0,∞). In particular, T(·) is a strongly continuous family in L(X). For simplicity, we may assume thatkC(t)k ≤Keωt for allt≥0 and for some
fixedK, ω≥0. ThenkT(t)k ≤Ke(ω+
√kBk)tfor allt≥0, and
(λ2−A−B)−1Cx=
∞
X
n=0
Bn Z ∞
0
e−λt Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
= Z ∞
0
∞
X
n=0
Bne−λt Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
= Z ∞
0
e−λtj0∗T(t)xdt forλ > ωandx∈X or equivalently,
λ(λ2−A−B)−1Cx= Z ∞
0
e−λtT(t)xdt forλ > ωandx∈X. Here
Z t
0
j−1(s)j0(t−s)S(t−2s)xds=S(t)xfort≥0.
Applying Theorem 2.3, we get thatT(·) is an exponentially boundedC-cosine function onX with closed subgeneratorA+B. Since
Z t
0
jn−1(r)jn(t−r)C(t−2r)xdr
− Z s
0
jn−1(r)jn(s−r)C(s−2r)xdr
= Z t
s
jn−1(r)jn(t−r)C(t−2r)xdr +
Z s
0
jn−1(r)[jn(t−r)C(t−2r)−jn(s−r)C(s−2r)]xdr
(2.8)
and
Z s
0
jn−1(r)[jn(t−r)C(t−2r)−jn(s−r)C(s−2r)]xdr
= Z s
0
jn−1(r)jn(s−r)[C(t−2r)−C(s−2r)]xdr +
Z s
0
jn−1(r)[jn(t−r)−jn(s−r)]C(t−2r)xdr
= Z s
0
jn−1(r)jn(s−r)[C(|t−2r|)−C(|s−2r|)]xdr +
Z s
0
jn−1(r)[jn(t−r)−jn(s−r)]C(|t−2r|)xdr
(2.9)
for alln∈N,x∈X andt≥s≥0, we observe from (1.5) thatT(·) is also exponen- tially Lipschitz continuous or norm continuous ifC(·) is.
Next we deduce a new perturbation theorem concerning localC-cosine functions.
In particular, the exponential boundedness ofT(·) in Theorem 2.6 can be deleted when the assumption ofBC(·) =C(·)B onD(A) is added.
Theorem 2.7. LetAbe a subgenerator of a localC-cosine functionC(·)onX. Assume that B is a bounded linear operator on D(A) such thatBC(·) =C(·)B onD(A)and BA ⊂AB. Then A+B is a closed subgenerator of a local C-cosine function T(·) on X given as in (1.5). Moreover, T(·)is also locally Lipschitz continuous or norm continuous if C(·) is.
Proof. Just as in the proof of Theorem 2.6, we observe from (2.8)-(2.9) and (1.5) that T(·) is also locally Lipschitz continuous or norm continuous ifC(·) is. Since
R(C(t))⊂D(A) andBC(·) =C(·)B onD(A), we have
CT(·) =T(·)ConX.
Letx∈X and 0≤t≤r < T0be fixed. Then Z t
0
jn−1(s)jn(t−s)S(t−2s)xds=1
2[j1(t)S(t)e − Z t
0
S(te −2s)xds]
forn= 1, and
Z t
0
jn−1(s)jn(t−s)S(t−2s)xds
= 1 2
Z t
0
[jn−2(s)jn(t−s)−jn−1(s)jn−1(t−s)]S(te −2s)xds for alln≥2. Here
S(·) =e j0∗S(·).
SinceBA⊂AB and
S(r)xe = Z r
0
Z t
0
C(s)xdsdt∈D(A), we have
AB Z r
0
j1(t)S(t)xe − Z t
0
S(te −2s)xds dt
=BA Z r
0
j1(t)S(t)xe − Z t
0
S(te −2s)xds dt
=B Z r
0
j1(t)[C(t)x−Cx]− Z t
0
[C(t−2s)x−Cx]ds dt
=B Z r
0
j1(t)C(t)xdt−B Z r
0
Z t
0
C(t−2s)xdsdt
=B Z r
0
j1(t)C(t)xdt−B Z r
0
S(t)xdt.
Since
Z r
0
j1(t)C(t)xdt=xj1(r)S(r)x−S(r)xe
and
j1(r)S(r)x= 2 Z r
0
j1(r−s)C(r−2s)xds,
we also have
AB Z r
0
j1(t)S(t)xe − Z t
0
S(te −2s)xds dt
= 2B Z r
0
j1(r−s)C(r−2s)xds−2B Z r
0
Z t
0
C(s)xdsdt.
(2.10)
Letn≥2 be fixed.
Using integration by parts, we have Z t
0
jn−1(s)jn(t−s)S(t−2s)xds
= 1 2
Z t
0
jn−2(s)jn(t−s)−jn−1(s)jn−1(t−s)
S(te −2s)xds.
Since Z r
0
Z t
0
jn−2(s)jn(t−s)Cxdsdt= Z r
0
Z t
0
jn−1(s)jn−1(t−s)Cxdsdt,
we have
A Z r
0
Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
= 1 2
Z r
0
Z t
0
jn−2(s)jn(t−s)AS(te −2s)xdsdt
− Z r
0
Z t
0
jn−1(s)jn−1(t−s)AS(te −2s)xdsdt
= 1 2
Z r
0
Z t
0
jn−2(s)jn(t−s)(C(t−2s)x−Cx)dsdt
− Z r
0
Z t
0
jn−1(s)jn−1(t−s)(C(t−2s)x−Cx)dsdt
= 1 2
Z r
0
Z t
0
jn−2(s)jn(t−s)C(t−2s)xdsdt
− 1 2
Z r
0
Z t
0
jn−1(s)jn−1(t−s)C(t−2s)xdsdt.
(2.11)
Since
Z r
0
Z t
0
jn−2(s)jn(t−s)C(t−2s)xdsdt
= Z r
0
Z r
s
jn−2(s)jn(t−s)C(t−2s)xdtds
= Z r
0
jn−2(s)
jn(r−s)S(r−2s)x
− Z r
s
jn−1(t−s)S(t−2s)xdt ds
= Z r
0
jn−2(s)jn(r−s)S(r−2s)xds
− Z r
0
jn−2(s) Z r
s
jn−1(t−s)S(t−2s)xdtds,
(2.12)
Z r
0
jn−2(s)jn(r−s)S(r−2s)xds
= Z r
0
jn−1(s)jn−1(r−s)S(r−2s)xds +2
Z r
0
jn−1(s)jn(r−s)C(r−2s)xds
=2 Z r
0
jn−1(s)jn(r−s)C(r−2s)xds
(2.13)
and
Z r
0
Z r
s
jn−2(s)jn−1(t−s)S(t−2s)xdtds
= Z r
0
Z t
0
jn−2(s)jn−1(t−s)S(t−2s)xdsdt, we have
Z r
0
Z t
0
jn−2(s)jn(t−s)C(t−2s)xdsdt
= 2 Z r
0
jn−1(s)jn(r−s)C(r−2s)xds
− Z r
0
Z t
0
jn−2(s)jn−1(t−s)S(t−2s)xdsdt.
(2.14)
By Lemma 2.5, we have Z r
0
Z t
0
jn(s)jn(t−s)C(t−2s)xdsdt
= Z r
0
Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt.
(2.15)
Combining (1.11) with (2.14) and (2.15), we have A
Z r
0
Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
= Z r
0
jn−1(s)jn(r−s)C(r−2s)xds
− Z r
0
Z t
0
jn−2(s)jn−1(t−s)S(t−2s)xdsdt.
(2.16)
It follows from (2.10)and (2.16) that we have A
Z r
0
∞
X
n=0
Bn Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
=A
∞
X
n=0
Bn Z r
0
Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
=
∞
X
n=0
ABn Z r
0
Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
=A Z r
0
Z t
0
C(s)xdsdt+AB Z r
0
Z t
0
j1(t−s)S(t−2s)xdsdt +
∞
X
n=2
BnA Z r
0
Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
=
C(r)x−Cx +B
Z r
0
j1(r−s)C(r−2s)xds− Z r
0
Z t
0
C(s)xdsdt
+
∞
X
n=2
Bn Z r
0
jn−1(s)jn(r−s)C(r−2s)xds
− Z r
0
Z t
0
jn−2(s)jn−1(t−s)S(t−2s)xdsdt
=
∞
X
n=0
Bn Z r
0
jn−1(s)jn(r−s)C(r−2s)xds−Cx− B Z r
0
Z t
0
C(s)xdsdt
− Z r
0
∞
X
n=1
Bn+1 Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
=
∞
X
n=0
Bn Z r
0
jn−1(s)jn(r−s)C(r−2s)xds−Cx
−B Z r
0
∞
X
n=0
Bn Z t
0
jn−1(s)jn(t−s)S(t−2s)xdsdt
(2.17)
for allx∈X and 0≤r < T0 or equivalently, (A+B)
Z r
0
Z t
0
T(s)xdsdt=T(r)x−Cx
for all x∈X and 0≤r < T0. SinceABn =BnAand BnC(t) =C(t)Bn onD(A), we have
Z r
0
Z t
0
T(s)(A+B)xdsdt= (A+B) Z r
0
Z t
0
T(s)xdsdt=T(r)x−Cx
for allx∈D(A) and 0≤r < T0. It follows from [14, Theorem 2.5] thatT(·) is a local C-cosine function onX with closed subgeneratorA+B, and is also locally Lipschitz
continuous or norm continuous ifC(·) is.
By slightly modifying the proof of Theorem 2.7 we also obtain the next perturbation theorem concerning localC-cosine functions which is still new even thoughT0=∞.
Theorem 2.8. LetAbe a subgenerator of a localC-cosine functionC(·)onX. Assume thatBis a bounded linear operator onX such thatBC(·) =C(·)B onX. ThenA+B is a closed subgenerator of a localC-cosine functionT(·)on X satisfying
T(t)x=
∞
X
n=0
Z t
0
jn−1(s)jn(t−s)C(|t−2s|)Bnxds (2.18) for all x∈X and 0≤t < T0. Moreover, T(·) is also locally Lipschitz continuous or norm continuous if C(·) is.
Proof. Suppose thatBis a bounded linear operator onX which commutes withC(·) onX. Then
T(t)x=
∞
X
n=0
Z t
0
jn−1(s)jn(t−s)C(|t−2s|)Bnxds
for allx∈Xand 0≤t < T0. Since the assumption ofBA⊂ABin the proof of Theo- rem 2.7 is only used to show that (2.10) and (2.17) hold, but both are automatically satisfied if BA ⊂ AB is replaced by assuming that B is a bounded linear operator onX which commutes with C(·) on X. Therefore, the conclusion of this theorem is
true.
We end this paper with a simple illustrative example.
Example 2.9. LetC(·) (={C(t)|0≤t <1}) be a family of bounded linear operators onc0 (family of all convergent sequences inCwith limit 0), defined by
C(t)x={xne−ncoshnt}∞n=1
for all x= {xn}∞n=1 ∈c0 and 0≤ t <1, then C(·) is a local C-cosine function on c0 with generator A defined by Ax = {n2xn}∞n=1 for all x = {xn}∞n=1 ∈ c0 with {n2xn}∞n=1 ∈c0. HereC =C(0). Let B be a bounded linear operator on c0 defined byBx={xne−ncoshn}∞n=1 for allx={xn}∞n=1∈D(A), thenC(·)B =BC(·) onc0. Applying Theorem 2.8, we get thatA+B generates a localC-cosine functionT(·) on c0satisfying (1.5).
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Chung-Cheng Kuo
Fu Jen Catholic University, Department of Mathematics, New Taipei City, Taiwan 24205 e-mail:[email protected]