Strong convergence of the resolvents and semigroups
associated with strongly accretive operators in general
Banach spaces
Citation for published version (APA):
Liu, G. Z. (1985). Strong convergence of the resolvents and semigroups associated with strongly accretive operators in general Banach spaces. (Eindhoven University of Technology : Dept of Mathematics :
memorandum; Vol. 8511). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1985
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EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of P{athematics and Computing Science
Memorandum 85 - 11 November 1985
STRONG CONVERGENCE OF THE RESOLVENTS AND SEMI GROUPS
ASSOCIATED WITH STRONGLY ACCRETIVE OPERATORS IN GENERAL BANACH SPACES
by
Liu Guizhong
University of Technology
Dept. of Mathematics
&
Computing Science Den Dolech 2, p.O. box 5135600 MB EINDHOVEN The Netherlands
STRONG CONVERGENCE OF THE RESOLVENTS AND SEMI GROUPS
ASSOCIATED WITH STRONGLY ACCRETIVE OPERATORS IN GENERAL BANACH SPACES
by
Liu Guizhong*)
Let E be a Banach Space with norm
II· II .
A subset AcE x E is said to be an w -accretive operator in E if1,2, Vt > 0, wt < 1
where w is some real number. Usually a O-accretive operator is just called an accretive operator. Any w -accretive operator A with w < 0
is known as a strongly accretive operator. Note that, a subset A of E x E is anw-accretive operator if and only if for each [xi'Yi]' i = 1,2, there exists an f E F(x
1 - x2) (F is the normalized (multi-valued) duality mapping) such that
w
I!
Xl - x2!I
2 + (y 1 - Y 2' f)~
O.*
Here (.,.) denotes the duality pairing of E and E •
The well known Crandall-LiglJettTheorem ([1]) asserts that, an accretive operator A in E satisfying the range condition
R(I + tA) :::> D(A) , Vt > 0
generates via the'exponential formula n
S(t)x
=
lim Jt/n x, x E D(A), t ~ 0
n+oo
*) Department of Mathematics, :ti'an Jiaotong University, Xi'an, Shaanxi Province, China.
(1)
(2)
(3)
2
-(J
t (I + tA) -1 is the resolvent) an oo-contractive semigroup S on D(A), that is
II S ( t ) x - S ( t ) y
Ii
;;; e oot !I x-y II, Vx,y E D(A), t ~ o.In a Hilbert Space H it is known that a strongly accretive operator A C H x H satisfying the range condition (3) has a unique zero pOint
*
- -
*
*
*
x and for any x E D(A), lim S(t)x
=
x , S(t)x=
x • "It ~ O.See Pazy ([3]). It seems that in General Banach Spaces there are no such results on asyn~totic properties, although there do exist
results on the limits of S(t)x/t and Jtx/t about accretive operators in special Banach Spaces
([2],[4]
and[5]).
In this short paper we present results on the strong limits of the resolvents and semi-groups associated with strongly accretive operators in general Banach Spaces.Theorem Assume that an oo-accretive (00 < 0) operator A in a Banach
Space E satisfies the range condition (3). Then there exists a
*
unique zero point x for A and
*
*
lim Jtx
=
lim S(t)x = x = S(t)xt+oo t + 00
Moreover, the following estimates hold true 11Jtx-x*
II
(l_oot)-1 !Ix-x* IIII
J t x - x * II ;;; I 00 I-It -1II
x - J t x IIIIS(t)x-x*
II :;;;
eoot Ilx-x* II"
II S(t)x *II ;;;
(2 -e~e:)(1
_ eWE)-1 max II S(s)x - xII
ewt, sE[O,s] (5) (7) (8) (9) Vt~O,E:>O. (10)Proof. Fix to > O. From the range condition (3) it is clear that J
3
-II
J t x - J yII
(1 - w to) -1II
x - yII,
Vx, y E D (A) .°
toSo, as w <
I,
according to the Banach contraction principle there*
exists a unique fixed point x of J
t in D(A) (actually in D(A».
°
Since the fixed points of J in D(A) are precisely the zero pOints
*
to
of A, x must be the unique zero point of A.
-1
*
For any x E D(A) j [Jtx, t (x - Jtx)] E A, [x ,0] E: A,
therefore we obtain from (2) that
w
II
J x - x*
II
:2 + {t -1 (x - J x). f)~
°
, t ' t*
where f E F (Jtx - x ). Substituting -1 -1*
-1*
(t (x-JtX),f) t (x-x , f ) - t (Jtx-x ,f) -1*
-1II
*
II
2 t (x - X J f) - t , J tX - x . in (12) yields (wt - 1)II
rl J x - x t*
II
II 2 + (x - x* ,
f)~
°
from which the estimate (7) is immediately obtained and therefore
*
lim JtX = x •
t+oo
-1
Noting that [Jet (x-Jtx)]
we have from (2) that
-1 E A, [J x,s (x-J x)] fA s s (t,s > 0)
II
11
2 -1 -1 w Jtx-Jsx, + (t (x-Jtx) -8 (x-Jsx,f),f) ~°
where f E F(J tX - J sX)' Substitution of -1 -1 (t (x-Jtx)-s (x-Jsx),f) -1 - 1 - 1 =(t (.Tsx-Jtx)+(t - s ) (x-Jsx,f) -1II
II
2 - 1 - 1=
t .. Jtx-Jsx + (t - s )(x-Jsx,p in (13) yields (wt - 1)II
Jtx - JsxIi
2 + (1- ts -1)(x - Jsx,f)~
°
and therefore (11) (12) (13)4
-from which the estimate ·(tS). follows by letting t + 00.
*
*
As x is the zero point of A, i t follows that in fact Jtx :::
Vt >
o .
So, from the exponential formula (4) we know that*
*
S(t)x = x
,
Vt ~ O. Estimate (9) is obtained by setting y ::: X*
(5) and therefore lim S(t)x::: x • t +oo
Let e > 0 be given. Set M = max
II
S(s)x - x!I
tE[O,e]
*
x
,
*
inExpress s ~ e as s = nE: + 0, where n is a positive integer, 0 ~ 0 < E:,
both uniquely determined oy s. Then
II
x - S(s)xII
~II
x - S(O)x" + II S(o)x - S(8) S(nE;)xII
from which we obtain
:ll AI + It x - S(ne)x II n-l
~M+
L
IIS(ke)x-S«k+l)e)xll k=O n-l M +I
ewek I! x - S(e)x!i
k=OIIS(t)x-S(t+s)
II~
ewt IIx-s(s)xII
~
(2 - eWE:) (1- eW£) -1 Mewt, Vt > O,s~
s.Now the estimate (10) follows from this inequality by letting s + 00.
· ..
- 5
-References.
[1J M.G. Crandall,T. Liggett, Generations of nonlinear transformations on general Banach Spaces, Amer. J. Math. 93 (1971), 108 - 132.
[2} I. Miyadera, On the infinitesimal generators and the asymptotic
behaviour of nonlinear contraction semigroups, Proc. Japan Acad., 58, Ser. A (1982), 1 - 4.
[3] A. Pazy, Semi groups of nonlinear contractions and their asymptotic behaviour, Nonlinear analysis and mechanics: Heriot - Watt symposium Volume III (1979), 36 - 134.
[4] A.T. Plant, The differentiability of nonlinear semigroups in uniformly convex spaces, Israel J. ].lath., 38 (1981), 257 - 268. [5] S. Reich, On the asymptotic behaviour of nonlinear semigroups and
the range of accretive operators II, J. ~~ath. Anal. Appl. 87 (1982), 134 - 146.
