Strongly continuous operator semigroups. Alternative approach Текст научной статьи по специальности «Математика»

MSC 60H30

STRONGLY CONTINUOUS OPERATOR SEMIGROUPS. ALTERNATIVE APPROACH

A.A. Zamyshlyaeva, South Ural State University, Chelyabinsk, Russian Federation, [email protected]

Inheriting and continuing the tradition, dating back to the Hill-Iosida-Feller-Phillips-Miyadera theorem, the new way of construction of the approximations for strongly continuous operator semigroups with kernels is suggested in this paper in the framework of the Sobolev type equations theory, which experiences an epoch of blossoming. We introduce the concept of relatively radial operator, containing condition in the form of estimates for the derivatives of the relative resolvent, the existence of Co-semigroup on some subspace of the original space is shown, the sufficient conditions of its coincidence with the whole space are given. The results are very useful in numerical study of different nonclassical mathematical models considered in the framework of the theory of the first order Sobolev type equations, and also to spread the ideas and methods to the higher order Sobolev type equations.

Keywords: Sobolev type equation, strongly continuous semigroups of operators with

kernals, approximations of semigroups.

Introduction

Let U and F be Banach spaces, operator L € L(U;F), operator M € Cl(U;F), function f (•) : R -^-F. Consider the Cauchy problem

u(0) = uo (1)

for the operator-differential equation

L U = Mu + f. (2)

L

of equivalent equations

U = Su + h, g = Tg + f. (3)

Here the operators S = L-1M € Cl(U), dom S = dom M, T = ML-1 € Cl(F), dom T =

L[dom M], the function h = L-1f : R -^-U. It is convenient to consider the equation (3) in the

frame of the equation

v = Av + 2 (4)

on the Banach space V. Here A : dom A ^ V, dom A = V, z(^) : R ^V.

The Cauchy problem

v(0) = v0, v0 € dom A (5)

for the homogeneous equation

v = Av (6)

is completely studied with the help of the semigroups theory. The main result of the classical semigroups theory [lj is a theorem of Hill-Iosida-Feller-Phillips-Miyadera (the HIFPM theorem), establishing a bijection between the resolving semigroup of the homogeneous equation (6) and

AA

being the infinitesimal generator of a semigroup (or generating the semigroup) are some conditions on the resolvent R^(A) = (pi — A)-1 of the operator A. Depending on these conditions, operator A generates the analytical group, analytical semigroup or strongly continuous (Co) semigroup.

The theory of degenerate operator semigroups developed by G.A.Sviridyuk and his disciples generalizes these results to the case of the Sobolev type equations [2-6]. It also consists of three parts: analytical groups, analytical semigroups and, finally, strongly continuous semigroups with kernels. We suggest the alternative (in comparison to [7]) method of construction of C0-semigroup for the equation (2). To our opinion, these results are very useful for the numerical modelling of different processes based on the first order Sobolev type equations and to spread methods to the higher order Sobolev type equations [Sj.

1. Relatively radial operators

Following [2, 7], introduce the L-resolvent set pL(M) = {p € C : (pL — M)-1 € L(F; U)} and the L-spectrum aL(M) = C \ pL(M) of operator M. The operator functions (pL — M)-1, RL(M) = (pL — M)-1L, LL(M) = L(pL — M)-1 are called L-resolvent, right LM

Definition 1. The operator M is called radial with respect to operator L (shortly, L-radial), if

(i) 3a € R Vp>a p € pL(M)

(ii) 3K > 0 Vp > a Vn € N

a=0

Remark 2. If there exists the operator L-1 € L(F; U), then opera tor M is L-radial exactly, when the operator L-1M € Cl(U) (or, equivalently, the operator ML-1 €Cl(F)) is radial.

Set U0 = ker L F° = ker LL(M)^y L0 (M0) denote restriction of the operator L (M)

to lineal U0 (dom M0 = U0 Pi dom M).

LM

n=1

ML

(i) any vector ^ € ker L \ {0} does not have M-adjoint vectors;

(ii) ker RL(M) P im RL(M) = {0}, ker LL(M) n im LL(M) = {0}.

(in) there exists the operator M-1 € C(F°; U°).

By U1 (F1) the closure of the lineal im RL(M) ( im LL(M)) by norm of the space

U (F).

ML

(i) lim pRL(M)u = u Vu €U1;

(ii) liin pLlL(M)f = f Vf €f1.

By U (F) the closure of the lineal U0 -+ im R^Lu p) (M) (F0 + im L^u p) (M)) by norm

of the space U (F). Obviously, U1 (F1) is the subspace in U (F).

Lemma 2. Let the operator M be weakly L-radial. Then U = U0 ®U1, JF = F0 ®F1.

2. The resolving operator semigroups

Consider two equivalent forms of the equation (2)

RT(M)U = (aL — M)-1Mu, LT(M)f = M(aL — M)-1f

as concrete interpretations of the equation

Av = Bv,

defined on a Banach space V, where the operators A,B € L(V)

(7)

(8)

(9)

Definition 3. The vector-function v € C(R+; V), differentiable on R+ and satisfying (9) is called a solution of the equation (9).

A little away from the standard [lj, following [TJ define

Definition 4. The mapping V' € C(R+; L(V)) is called a semigroup of the resolving operators fa resolving semigroup) of the equation (9), if

(i) VsVtv = Vs+tv for all s,t > 0 and any v from the space V;

(ii) v(t) = Vtv is a solution of the equation (9) for any v from a dense in V set.

The semigroup is called uniformly bounded, if

3C> 0 Vt > 0 HVi^v) < C.

ML

strogly continuous resolving semigroup of the equation (7) ((8)), treated on the subspaceU (F), presented in the form:

U = s — lim

(_1)

k-1

(F = s _ lim

k^+ж (k — 1)! у t (_1)k-1 / k

dk

1

zr RTT(M)

k^+те (k — 1)! V t

Proof. Denote the following families of operators:

d^-1 » dk-1

di

k-1 lTt(m )

*= hi

)•

uk = ( 1)

k

(k - 1)! t

k

di

k- RT(M)

Note that

Vu € U0 Uku = 0. (10)

M L K

Definition 1:

U Wou) < K Vt € R+ Vk € N. (11)

u dom M

d = d (_1)k- 1 (k

dtUku dt I (k — 1)! Vt

k1

di

—rTt(m )

u=

k

k

k

k

k

k

( 1)fc 1 kk (—ктгттRL(M) - RUM)

»= t

(k -1)! V tk+l d^_l »v J tk+2 d^k »

_ (k )k+l (-dц_1RL(M > + t$_- (Ri(M » RL(M ))

u =

u =

»=t

(-1)k- l (kT'd- RL(M))(kRL(M) - 0

(k - 1)! V t / d^k~l v » ' у t

u=

»= t

= tL - M^ Mu

Thus,

dUku = Ulk(L - jM\ Mu Уи € dom M. (12)

dt k

dt

Now let u € imR^M), i.e. u = R^(M)v in some /3 > 0 h v €U. Proof, then

tlirri Ulu = u. (13)

Make the change n — k

l

( 1)k— l dk_ l

lim Utu = lim —-------— nk—-RL,(M)u =

t^o+ k (k - 1)!^ d^k~l »v 7

k~ l / ( 1)m dm ( 1)m_ l dm~l \

lim V ( ------:— jim+1-— RL(M) - --------— ц'm--- R»L(M)) u + lim jj,RL(M)u =

y m! ^ d^m J (m - 1)Г d^m~l 1J ~ »v 7

k l (_1)m- l / ц dm dm- l \

lim -----------— nm(------- — R»L(M)----- --iRL(M)) u + lim fxRL(M)u =

(m - 1)!p V mdam J dam_l »K J » 1

m= l

k l (-1)m_ l m dm_l

= E (m-1)!dkm-1 (Rl(M))(^rL(m) “1 )u + ^rL(m)u (14)

m= 1

Due to W (—-i)!i1™d™-1 (Rl(M))Wl(u) < K to any n> 0 under L-radiality of the operator

M

k l (-1)m_l m dm_l

£ ji-ljiIr i^m-l(RL(M>^RL(M) - Du

<

U

< (k - 1)K||^L(M)u - u)||U ^ 0, ц ^ +rc>.

u

u

l

Uku - Ulu = У dds (Ul_sUsku) ds =

Ul_sU^L - SM^J _M- Ul_sU^L - ^j^M^ M J uds =

l

k

l

L -- M k

-l

ML

ts

-l

M M(Rl(M)) vds =

= i ut-sut{ k -

s t - s\k~L/M\ l VL T\1\-1*1\2

-RL(M)--------RLL_(M)((3L - M)-1M) vds.

l / s s t — s t-s

Taking into account (11), we get

t

\\Utu - Ufu\\u < K2 |

<

[ s

J k

0

2 t K

kl

K2

s(t - s)

kl

\\((3L - M)-1M)2v\\uds <

s(t-s)

1 + 1) W((3L - M)-lM)2v\\u■

(15)

From (15), (10), (11) and density of im RL(M) in U1 it follows the existence of the limit

U = s - lim Uk, U €C(U), \\Utbnl\ < K Vt> 0.

k—m L(u)

Inequality (15) shows that Uku uniformly with respect to t € (0,T] converges to Utu. Thus, the family {Ut : t > 0} is strongly continuous with respect to t, because due to continuity of right L-resolvents of the operator it follows a strong continuity of the family {Uk : t > 0} to any k € N. In order to extend the strong continuity of {U1 : t > 0} up to zero, we define an element of the family of operators U0 as a strong limit:

U0 = s - lim Ut.

t——0+

Due to (10)

Vu € U° U°u = lim Utu = lim lim Uku = 0

t—— 0+ t—— 0+ k——^o

In addition, using the above-mentioned uniform convergence, it can be shown that

Vu € U1 U°u = lim Uu = lim lim Uku = lim lim Uku = u.

t——0+ t——0+ k—k—t——0+

So, we get that U7° = P. Note that

rTt = Mr-1 / kl\ Ukl = (kl - 1)! ^ j)

kl

dkl

1

ZT RL(M)

di

(-1)

kl 2

kl

(kl - 2)! V t

kl

dkl-2

¿¡¡k— RL(M)) RL(M)

(-1)

l1

kl

(l - 1)! V t

kl

¿i—1

^ RL(M)) (RL(M))(k-1)l

di

-L\1 (-1)1-1 dl-1 [rl (m )]N t/k) (l - 1)! di1-1 R (M)]

= ( Uk

(16)

t

t

2

k

k

lim Ul

i—+my l

U k . Indeed, at и € U

t_

Uk

и — ( U и

и

k-1, k-m-1

£ (Uk

m=0 '

~ t \ m Uk

Uk — Uk

и

<

U

k

k

< kKk-1

Tending in identity (16) l ^ +rc>, we obtain

и

Ut = U k

(17)

Let us show that hence and from the strong continuity it follows that {Ut : t > 0} is a semigroup. Taking rational s = k/l h t = m/n using (17) twice, we get

- s - t - kn - Im / - 1 \ kn / ~ 1 \ / - 1 \ kn+lm - kn + lm - s + t

UsUt = U m U m = (u [U = (U = U^^ = Us+t.

For arbitrary real numbers s,t > 0 there exist sequences of rational numbers {sn : n € N}, {tn :

im tn = t. Then,

— m

Vn € N UsnUtn = Usn+tn. (18)

n € N} such that lim sn = s, lim tn = t. Then

П^Ж П^Ж

Since,

||USnUtnи — USUtu\\u < \\USn\\ф)

\\Utnи — Uu\\u + ||USnU и — USU u\\u <

< K\\Utnи — U u\\u + ||USnUtu — USU u\\u,

(19)

tending n ^ to in (18), (19) and using the strong continuity with respect to t of the operators family {Ut : t > 0}, we obtain the desired.

Further, let u1 = (RL(M))2v for some 3 > 0 and v €U. Then

\\Ulul - JJtu1WU = lim \\Ulu1 - Ukul\\u <

4+2

K 4t

< lim ----------

k^x> 2

4+2

K 4t

1 + k ) \\((вЬ — M)~M)2v\\u = — \№L — M)-Mf v\\u

due to (15).

= u1 + KL (M )RL(M )(0L — M )-1Mv =

= u1 + t(L — tM )-1(L — tM + tM )KjL(M )(pL — M )-1Mv =

= tKp(M )(0L — M )-1Mv + tKL (M )K%(M )((0L — M )-1M )2v

Therefore

Ut - I

u1 — KL (M )(PL — M )-1Mv

<

и

k

<

u{ 11 - rL (

t

1Mv Ut - u\ u

+

u t

<

u

K 4t < tK\\((3L - M)-1M)2v\\u + — WRl(M)((3L - M)-1M)2v\\u.

Tending t ^ 0+ we obtain, that

Ut - I

lim --- u1 = RL(M)(3L - M)-1 Mv. (20)

t——0+ t

Act on (20) by the operator Us and get the differentiability on the right of the semigroup at this element u1 = (RL(M))2v at point s > 0. In order to show the differentiability on the left at this point, one can consider the expression

Us-t - Us Us-t(Ut -1)

----------u =----------------u , s > t > 0,

-t t

proceed to the limit when t ^ 0+, using the uniform boundedness of the semigroup. So, by virtue of (20)

d

—^(RLM))2v = UtRL(M)(3L - M)-1Mv.

Act on last identity by the operator RL(M).

By construction Ut commutes with the operators RL(M) and (aL - M)-1M for the u1

d

RL(M) Jt^u1 = (aL - M)-1MUtu1. (21)

Obviously, for u0 € U0 Us(u0 + u1) = UsuK Then one can change u1 by u = u0 + u1 € U0-+ im(RL(M))2 in identity (21). Thus, the function u(t) = Utu is the solution of the equation (7) for arbitrary u from the dense in ¿/lineal U0+ im(RL(M ))2.

(For the semigroup Fl = s - lim which is constructed by means of the left L-resolvent,

k—m

the proof is identical).

□

The semigroup Ut (i^) at first is defined not on the whole space U (F), but on some subspace U (F). Introduce the sufficient condition of their coincidence: U = U (F = F).

Theorem 3. [2j Let the space U (F) be reflexive, the operator M be weakly L-radial. Then

U = U0 ®Ul (F = F0 ®F1').

M L L

and

\\RL(M)(AL - M)-1Mu||u < con^(u^ Vu € dom M

O

FF

\-1r / const(f) O

\\M(XL - Mr1 L^(M)f \\f Vf €F)

for any X, i > 0.

M L U

U0 ®U (F = F0 &F1).

Remark 3. Obviously, that under the condition of strong L-radiality of operator M on the

right (011 the left) the resolving semigroup of the equation (7) ((8)) is defined 011 the whole space

U (F), and the projector P lim цЕ^(М) (Q = s — lim ¡iL^M)) is it’s unit.

References

1. Hille E., Phillips R.S. Functional Analysis and Semi-Groups. American Mathematical Society, Providence, Rhode Island, 1957.

2. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Köln, Tokyo, VSP, 2003.

3. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems Not Solvable with Respect to the Highest Order Derivative. N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 2003.

4. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. N.Y., Basel, Hong Kong, Marcel Dekker, Inc., 1999.

5. Sidorov N., Loginov B., Sinithyn A., Falaleev M. Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications. Dordrecht, Boston, London, Kluwer Academic Publishers, 2002.

6. Arshin A. B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Series in Nonlinear Analisys and Applications, 15, De Gruyter, 2011.

7. Sviridyuk G.A. Linear Sobolev Type Equations and Strongly Continuous Semigroups of the Resolving Operators with Kernels [Lineynye uravneniya tipa Soboleva i sil’no nepreryvnye polugruppy razreshayushchikh operat.orov s yadramij. Doklady akadernii nauk, 1994, vol. 337, 110. 5, pp. 581-584.

8. Sviridyuk G.A., Zamyshlyaeva A.A. The Phase Spaces of a Class of Linear Higher-order Sobolev Type Equations. Differential Equations, 2006, vol. 42, 110. 2. pp. 269-278.

УДК 517.9 + 519.216.2

СИЛЬНО НЕПРЕРЫВНЫЕ ПОЛУГРУППЫ ОПЕРАТОРОВ. АЛЬТЕРНАТИВНЫЙ ПОДХОД

A.A. Замышляева

Наследуя и продолжая традицию, восходящую к теореме Хилле-Иосиды-Феллера-Филлипса-Миядеры, в данной работе в рамках теории уравнений соболевского типа, которая переживает эпоху своего расцвета, рассмотрен новый способ построения аппроксимаций сильно непрерывных полугрупп операторов с ядрами. Вводится понятие относительно радиального оператора, содержащее условие в виде оценки производной относительной резольвенты, показывается существование Со-полугруппы на некотором подпространстве исходного пространства, приводятся достаточные условия его совпадения со всем пространством. Результаты будут весьма полезными при численном исследовании многих неклассических математических моделей, рассматриваемых в рамках теории уравнений соболевского типа первого порядка, а также для распространения идей и методов на уравнения соболевского типа высокого порядка.

Ключевые слова: уравнение соболевского типа, сильно непрерывные полугруппы, операторов с ядрами, аппроксимации полугруппы,.

Литература

1. Хилле, Э. Функциональный анализ pi полугруппы / Э. Хилле, Р. Филлипс. - М.: ИЛ, 1962.

2. Sviridyuk, G.A. Linear Sobolev Type Equations and Degenerate Semigroups of Operators / G.A. Sviridyuk, V.E. Fedorov. - Utrecht; Boston; Köln; Tokyo: VSP, 2003.

3. Demidenko, G.V. Partial Differential Equations and Systems not Solvable with Respect to the Highest Order Derivative / G.V. Demidenko, S.V. Uspenskii. - N.Y.; Basel;Hong Kong: Marcel Dekker, Inc., 2003.

4. Favini, A. Degenerate Differential Equations in Banach Spaces / A. Favini, A. Yagi. - N.Y.; Basel; Hong Kong: Marcel Dekker, Inc., 1999.

5. Lyapunov-Shmidt Methods in Nonlinear Analysis and Applications / N. Sidorov, B. Loginov, A. Sinithyn, M. Falaleev. - Dordrecht; Boston; London: Kluwer Academic Publishers, 2002.

6. Arshin, A.B. Blow-up in Nonlinear Sobolev Type Equations /А.В. Arshin, M.O. Korpusov, A.G. Sveshnikov. - Series in nonlinear analisys and applications, 15, De Gruyter, 2011.

7. Свиридток, Г.А. Линейные уравнения типа Соболева pi срільтіо непрерывные полугруппы разрептатощріх операторов с ядрамрі / Г.А. Свріррідток // ДАН. - 1994. - Т. 337, №5. -С. 581-584.

8. Sviridyuk, G.A. The Phase Spaces of a Class of Linear Higher-order Sobolev Type Equations ./' G.A. Sviridyuk, A.A. Zamyshlyaeva /'/ Differential Equations. - 2006. - V. 42, №2. -P. 269-278.

Алена Александровна Замышляева, капдрідат фрізріко-математріческріх наук, доцент, кафедра «Уравнения математической физики>, Южно-Уральский государственный универ-срітєт (г. Челябрітіск, Россрійская Федератція), [email protected].

Поступила в редакцию 4 марта 2018 г.