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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)}

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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

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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.

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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 thatSn+1denotes the(n+ 1)-fold convolution of S forn∈N∪ {0}, that is

S2(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

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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

2−A−B)−1C=

X

n=0

Bn2−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

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fixedK, ω≥0. ThenkT(t)k ≤Ke(ω+

kBk)tfor allt≥0, and

2−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.

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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

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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)

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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)

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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

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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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References

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[10] Kostic, M.,Generalized Semigroups and Cosine Functions, Mathematical Institute Bel- grade, 2011.

[11] Kostic, M.,Abstract Volterra Integro-Differential Equations, Taylor and Francis Group, 2015.

[12] Kuo, C.-C.,On α-times integrated C-cosine functions and abstract Cauchy problem I, J. Math. Anal. Appl.,313(2006), 142-162.

[13] Kuo, C.-C.,On perturbation of α-times integratedC-semigroups, Taiwanese J. Math., 14(2010), 1979-1992.

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[15] Kuo, C.-C.,Local K-convolutedC-semigroups and complete second order abstract Cauchy problem, Filomat,32(2018), 6789-6797.

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[23] Takenaka, T., Piskarev, S.,Local C-cosine families and N-times integrated local cosine families, Taiwanese J. Math.,8(2004), 515-546.

[24] Travis, C.C., Webb, G.F.,Perturbation of strongly continuous cosine family generators, Colloq. Math.,45(1981), 277-285.

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Chung-Cheng Kuo

Fu Jen Catholic University, Department of Mathematics, New Taipei City, Taiwan 24205 e-mail:[email protected]

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