[0, co) with limsup^oo/b(0 < 1 such that \\T(t)x - T(t)y\\ < b(t)\\x-y\\ for x, y in C and t > 0;. (c) asymptotically nonexpansive [6] if for each x in C, lim sup J sup[|| F(r)x ...
PROCEEDINGSof the AMERICANMATHEMATICAL SOCIETY Volume 117, Number 2, February 1993
ASYMPTOTIC BEHAVIOR OF ALMOST-ORBITS OF NONLINEAR SEMIGROUPS OF NON-LIPSCHITZIAN MAPPINGS IN HILBERT SPACES KOK-KEONGTAN AND HONG-KUN XU (Communicated by Palle E. T. Jorgensen)
Abstract. Let C be a nonempty closed convex subset of a Hilbert space H, &~ = {T(t): t > 0} be a continuous nonlinear asymptotically nonexpansive semigroup acting on C with a nonempty fixed point set F{£?~), and u: [0, oo) —» C be an almost-orbit of &~. Then {«(/)} almost converges weakly to a fixed point of 3~, i.e., there exists an element y in F(f?~) such that
1 /•' weak-lim - / u(r + h)dr = y uniformly for h > 0.
' Jo
This implies that {u(t)} converges weakly to a fixed point of y if and only if {u(t + h) - u(t)} converges weakly to zero as / tends to infinity for each
h >0.
1. Introduction Let C be a nonempty closed convex subset of a Hilbert space H and let y
= {T(t): t > 0} be a (one-parameter)
semigroup acting on C, i.e., each
T(t) is a self-mapping of C and for each x in C and t, s > 0 we have
(i) T(0)x = x and (ii) T(s + t)x = T(s)T(t)x. &~ is said to be continuous if for each x in C, the mapping t —>T(t)x is continuous. Recall that a semigroup & on C is said to be (a) nonexpansive if \\T(t)x - T(t)y\\ < \\x - y\\ for x, y in C and J > 0; (b) uniformly asymptotically nonexpansive [6] if there is a function b: [0, oo) -►[0, co) with limsup^oo/b(0 < 1 such that \\T(t)x - T(t)y\\ < b(t)\\x-y\\
for x, y in C and t > 0; (c) asymptotically nonexpansive [6] if for each x in C,
limsupJ sup[||F(r)x - F(r)y|| - ||x - y ||]1 < 0. f-»oo
\yec
J
Received by the editors October 20, 1990 and, in revised form, June 4, 1991.
1991 Mathematics Subject Classification. Primary 47H10, 47H20. Key words and phrases. Almost-orbit, asymptotically nonexpansive semigroup, weakly asymptotically regular, nonexpansive retraction, fixed point, metric projection, asymptotic center. ©1993 American Mathematical Society
0002-9939/93 $1.00 + $.25 per page
385
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386
K.-K. TAN AND H.-K. XU
It is easily seen that (a) => (b) => (c) and that both the inclusions are proper
(cf. [6, p. 112]). Recently, Miyadera and Kobayasi [9] introduced the notion of almost-orbits of nonexpansive semigroups on C and proved the weak almost convergence of such an almost-orbit in a uniformly convex Banach space with a Frechet differentiable norm. Takahashi and Zhang [10] extended this notion to uniformly asymptotically nonexpansive semigroups in Hilbert spaces. In this paper, we introduce the concept of almost-orbits for asymptotically nonexpansive semigroups. We shall prove that if u: [0, co) —► C is an almostorbit of an asymptotically nonexpansive semigroup & = {T(t): t > 0} such that each T(t) is continuous and the fixed point set F(SF) of !F is nonempty, then it almost converges weakly to some y in F(f?~), i.e.,
1 f'
weak-lim - / u(r + h)dr = y t^oo
t Jo
uniformly for h > 0. This extends Theorem 4.1 of Miyadera and Kobayasi [9] to semigroups of non-Lipschitzian mappings in Hilbert spaces and implies that {u(t)} converges weakly to a member of F(&~) if and only if u is weakly asymptotically regular, that is, u(t + h) - u(t) —>0 weakly as t —► co for each h > 0. We also prove the existence of a nonexpansive retraction from the set of almost-orbits of the semigroup SF onto the fixed point set F(f?) of &. Related results to our paper can be found in Brack [2], Kirk [5], Lau [7], Takahashi and Zhang [10], and You and Xu [11]. 2. Preliminaries Throughout this section, let C be a nonempty closed convex subset of a Hilbert space H and SF = {T(t) : t > 0} be an asymptotically nonexpansive semigroup acting on C. We shall first introduce the notion of almost-orbit &~ and prove some lemmas that will be used later.
of
Definition. A continuous function u: [0, co) —► C is said to be an almost-orbit
of & if lim (sup \\u(s + t)- T(s)u(t)\\) '^°°
\s>0
= 0. )
Clearly, for each x in C, the orbit {T(t)x : t > 0} of & at x is an almostorbit of & provided &~ is continuous.
Denote by F(9r) the fixed point set of &, i.e., F(9r) = {x e C: T(t)x = x
for all t > 0}. Lemma 2.1. If each T(t) is continuous then F(!F) is closed and convex. Proof. It needs only to prove the convexity of F(9~). Suppose f, g £ F(Sr) and let p = \(f + g). For an arbitrary e > 0, since & is asymptotically nonexpansive, there exists a to > 0 such that for t > to ,
sup[\\q - T(t)y\\ - \\q - y\\] to,
\\T(t)p -p\\2 = {\\T(t)p - f\\2 + \\\T{t)p - g\\2 - \\\f-g\\2
0
and
y(t) = sup\\v(s + t)-T(s)v(t)\\. s>0
Then lim,^,*, 0, one can find a 5n (depending upon t) such that for s>s0 sup(\\T(s)u(t)-T(s)y\\-\\u(t)-y\\) So,
\\u(t + s)- v(t + s)\\< \\u(t + s)- T(s)u(t)\\ + \\T(s)u(t) - T(s)v(t)\\ + \\v(t + s) - T(s)v(t)\\ 0, limsup||«(s)
- v(s)\\ = limsup||«(/ + s) - v(t + s)\\
S—'OO
S—+OG
0 and a fixed t > 0, we can choose an 5n so large that for s > So
sup(\\Pu(t) - T(s)y\\ - \\Pu(t) - y\\) < e. yec It then follows that for s > So, \\Pu(t + s)- u(t + s)\\ < \\Pu(t) - u(t + s)\\ 0.
Proof. Since F(5F) is nonempty, {u(t)} is bounded by Theorem 3.1 and Lemma 2.2 and {Pu(t)} converges strongly to some z in F(9r) by Lemma 2.3. It remains only to prove that if {t„} | co , {/?„} c [0, co), and
1 ['" — \ u(r + h„)dr->y ln Jo
weakly, then y = z. To show this, let A7 = sup(>0 \\Pu(t) - u(t)\\. every r > 0,
Re(u(r) - Pu(r), Pu(r) -f)>0
Since for
for all / in F(&),
we have Re(/ - z, u(r) - Pu(r)) < Re(Pu(r) - z, u(r) - Pu(r))
< M\\Pu(r) - z\\ for all / £ F(9r). It follows that
Relf-z, \
1 fn(u(r + h„)-Pu(r + hn))dr) < ¥- f " \\Pu(r + hn) - z\\dr. tn Jo
I
Since lim^oo \\Pu(t) - z\\ =0, above inequality yields
In Jo
taking the limit as n goes to infinity in the
Re(/ - z, y - z) < 0 for all / e F(9r); since y € F(SF), we then have ||y - z||2 = Re(y - z, y - z) < 0 and thus y = z . This completes the proof. □ Theorem 3.3. Under the same assumptions of Theorem 3.2, for each almost-orbit {u(t)} of &~, the following statements are equivalent: (i) {u(f)} converges weakly to a fixed point of £F; (ii) {u(t)} is weakly asymptotically regular, that is, {u(t + h) - u(t)} converges weakly to zero as t tends to infinity for each h > 0; (iii) {u(t + h) - u(t)} converges weakly to zero as t tends to infinity uniformly
for h>0. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
ALMOST-ORBITSOF NONLINEAR SEMIGROUPS
391
Proof. By Lemma 2.2 and Theorem 3.1, we need only to prove (iii) =>■(i). By Theorem 3.2, {u(t)} almost converges weakly to z = limr_>00.Pw(J). So for each w £ H and e > 0, we can choose a tx for which
(w,—
l u(r + h)dr-z\
0.
Then we choose a t2 such that
|(k; , u(t + r) - u(t))\ < xe
for all t > t2 and 0 < r < tx.
Hence for t > max{tx, t2} , we have
\(w,u(t)-z)\= (w,j J\(t)dr-z\ < lw,— l u(r + t)dr- z\
+ (w, j-J\u(r + t)-u(t))drj 1
0} be an asymptotically nonexpansive semigroup acting on C. Suppose u is an almost-orbit of &~. Then for each h > 0, the function v: [0, oo) —»C defined by v(t) = T(h)u(t)
is also an almost-orbit of !fF.
Proof. As before, we set tp(t) = supi>0 \\u(s + t) - T(s)u(t)\\.
Since
IK* + t) - T(s)v(t)\\ = \\T(h)u(s + t)- T(s)T(h)u(t)\\ < \\T(h)u(s + t)-u(h + s + OH+ \\u(h + s + t) - T(s + h)u(t)\\ < tp(s + t) + tp(t),
the result follows. □ Lemma 4.2. Under the same assumptions of Lemma 4.1, the set
f)co{u(t):t>s}nF($r) s>0
is either empty or the singleton set {z} where z = v/eak-lim,^00(l/t)
f0u(r)dr
= lim^oo Pu(t).
Proof. If w is in f|.s>o^>{wW: t > •*}nf(/) then F(ZF) is nonempty, and hence by Theorem 3.2 and Lemma 2.4, z := weak-lim,_00(l/7)/0 u(r)dr = lim^oo Pu(t) is the asymptotic center of {u(t)} in F(cF). We conclude the
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392
K.-K. TAN AND H.-K. XU
proof by showing w = z.
By Lemma 2.2, r(w) := lim,-,,*, \\u(t) - w\\ and
r :—lim^oo \\u(t) - z\\ exist and r(w) > r. Now, since
||z - w\\2 = \\u(s) - w\\2 - \\u(s) - z||2 - 2 • Re(z - w, u(s) - z), we get
||z - w\\2 + 2 • lim Re(z - w , u(s) - z) s—oo
= lim \\u(s) - w\\2 - lim \\u(s) - z||2 = r(w)2 - r2 > 0. s—+oo
s—*oo
For any e > 0, there is a So so large that for s >so, 2 • Re(z - w , u(s) - z) > -\\z - w\\2 - e.
Since w £ co{u(t): t > Sq) , it follows that 2 • Re(z - w, w - z) > -\\z - w\\2 - e, that is, ||z - io ||2 < e; since e > 0 is arbitrary, we must have z = w . □
Theorem 4.1. Let C be a nonempty closed convex subset of a Hilbert space H and let SF = {T(t): t > 0} be a continuous asymptotically nonexpansive semigroup acting on C such that each T(t) is continuous and F(!F) is nonempty.
Then the limit
1 f' Q(u) := weak- lim - / u(r) dr '-*00 t Jo
is the unique retraction from the set AO(^") of all almost-orbits of & onto F(^) such that (i) Q is nonexpansive in the sense that \\Qu-Qv\\
< Hm-uHoo :=sup||w(f)
-v(t)\\
foru,v
£ AO(y);
l>0
(ii) QT(h)u = T(h)Qu = Qu for u £ AO(^) and h>0; and (iii) Q(u) £ f\>0co{"(r): t>s} for u£ AO(^). Proof, (i) By the weak lower semicontinuity
of the norm of H, we derive that
\\Qu-Qv\\< liminf \ f \\u(r) - v(r)\\dr < \\u- wlU ; ,-oo t Jo that is, Q is nonexpansive and (i) is proven.
(ii) Let u £ AO(^") and h > 0. First we observe that QT(h)u is well defined by Lemma 4.1. Since Qu £ F(3r), we have T(h)Qu = Qu. Thus it remains to prove QT(h)u = Qu. In fact, we have
QT(h)u = weak-lim - I T(h)u(r)dr «-°° t Jo 1 f' —weak- lim - / u(h + r) dr + weak- lim-
i rl / (T(h)u(r) - u(h + r))dr
'->°° t Jo
1 f
= weak- lim - / u(r + h) dr = Q(u) t->oo t Jo
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ALMOST-ORBITSOF NONLINEARSEMIGROUPS
393
since
- / \\T(h)u(r)-u(h
t Jo
+ r)\\dr
