Nonlinear Analysis 42 (2000) 175–183www.elsevier.nl/locate/na

Asymptotic behavior for commutative semigroupsof asymptotically nonexpansive-type mappings(

Gang Li ∗

Department of Mathematics, Yangzhou University, Yangzhou 225002, People’s Republic of China

Received 5 May 1997; accepted 10 August 1998

Keywords: Asymptotic behavior; Semitopological semigroup; Asymptotically nonexpansive-typemappings

1. Introduction

Let C be a nonempty subset of a Banach space X . A mapping T :C 7→C is said tobe of asymptotically nonexpansive-type if TN is continuous for some N ≥ 1 and if

lim supn→∞

supy∈C(‖T nx − T ny‖−‖x − y‖)≤ 0 for every x∈C:

Let S = {T (t)x: t≥ 0} be a family of mappings from C into itself. S is said to bean asymptotically nonexpansive-type semigroup on C if T (t + s)=T (t)T (s) for everyt; s≥ 0, and if T (t) is continuous for some t≥ 0 and if

lim supt→∞

supy∈C(‖T (t)x − T (t)y‖ − ‖x − y‖)≤ 0 for all x∈C:

A weak convergence theorem for nonexpansive semigroups was �rst established byMiyadera and Kobayasi [10] in a uniformly convex Banach space, and it was gen-eralized to that for commutative semigroup of asymptotically nonexpansive mappingsby Oka [11]. After works of Lau and Takahashi [5], we proved the theorems for gen-eral reversible semigroups of asymptotically nonexpansive-type mappings [6,7]. On the

( Research supported in part by the NSF of China.∗ Correspondence address: Department of Mathematics, Kyungnam University, Masan, Kyungnam, 631-701,

South Korea.E-mail address: ligang@hanma.kyungnam.ac.kr (G. Li)

0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(98)00338 -1

176 G. Li / Nonlinear Analysis 42 (2000) 175–183

other hand, Lin et al. [8] proved the weak convergence theorem for asymptotically non-expansive mapping in a Banach space which has the uniformly Opial property, and Lin[9] show the similar result for asymptotically nonexpansive-type mappings in a Banachspace which has the Opial property and the norm is Uniform Kadec-Klee (UKK).The purpose of this paper is to provide the weak convergence theorem for almost-

orbits of commutative semigroups of asymptotically nonexpansive-type mappings in aBanach space X which has the Opial property and the norm of X is UKK. Using ourtheorem, we can simultaneously handle weak convergence theorems for asymptoticallynonexpansive-type mapping and semigroups. Moreover, our theorem seems to be new,even for uniformly convex Banach spaces.

2. Preliminaries

Throughout this paper X denotes a Banach space, C a nonempty closed convexsubset of X , and G a commutative topological semigroup with the identity. The normof X is said to be UKK if give �¿0 there exist �¿0 such that for �-separate {xn}(i.e. sep(xn) := inf{‖xn − xm‖: n 6=m}≥ �) in B1, where B1 = {x∈X : ‖x‖≤ 1}, {xn}converges to x weakly implies ‖x‖≤ 1 − �. X is said to be nearly uniformly convex(UNC) if it is re exive and has a UKK norm. It is known [3] that every uniformlyconvex Banach space is UNC.X is said to have the Opial property if {xn} converges to x weakly implieslim sup

n‖xn − x‖¡ lim sup

n‖xn − y‖

for all y 6= x.Let S = {T (t): t ∈G} be a family of mappings from C into itself. S is said to be

a commutative semigroup of asymptotically nonexpansive-type on C if the followingconditions are satis�ed:(a) T (s+ t)x=T (s)T (t)x for all s; t ∈G and x∈C;(b) T (t) is continous for all t ∈G;(c) for each t ∈G and x∈C, there exist r(t; x)¿0 such that

‖T (t)x − T (t)y‖≤ r(t; x) + ‖x − y‖ for all y∈C;with

limt∈Gr(t; x)= 0; (1)

where limt∈G r(t; x) denote the limit of a net r(·; x) on the directed system (G; ≤ )and the binary relation ≤ on G is de�ned by a≤ b if and only if there is c∈Gwith a+ c= b (In this case, we write c= b− a). A function u(·) :G 7→C is saidto be an almost-orbit of S = {T (t): t ∈G} if

lims∈G

[supt∈G

‖u(t + s)− T (t)u(s)‖]=0: (2)

Denote by F(S) the set of common �xed points of {T (t): t ∈G}.

G. Li / Nonlinear Analysis 42 (2000) 175–183 177

3. Lemmas

Lemma 1. Let {v1(t): t ∈G} and {v2(t): t ∈G} be almost-orbits of S. Then,limt∈G ‖v1(t)− v2(t)‖ exists.

Proof. Put

�1 = supt∈G

‖v1(t + s)− T (t)v1(s)‖

and

�2 = supt∈G

‖v2(t + s)− T (t)v2(s)‖

for s∈G. Then lims∈G �1(s)= lims∈G �2(s)= 0. Since

‖v1(t + s)− v2(t + s)‖ ≤ ‖v1(t + s)− T (t)v1(s)‖+ ‖T (t)v1(s)− T (t)v2(s)‖

+ ‖T (t)v2(s)− v2(t + s)‖

≤�1(s) + �2(s) + r(t; v1(s)) + ‖v1(s)− v2(s)‖for every t; s∈G, we have

inft∈Gsupt≤�

‖v1(�)− v2(�)‖≤�1(s) + �2(s) + ‖v1(s)− v2(s)‖

and then

inft∈Gsupt≤�

‖v1(�)− v2(�)‖≤ supt∈G

inft≤�

‖v1(�)− v2(�)‖:

Thus, limt∈G ‖v1(t)− v2(t)‖ exists.

Lemma 2. Let C be a weakly compact subset of a Banach space which has theOpial property, S = {T (t): t ∈G} be an asymptotically nonexpansive-type semigroupon C, and let u(·) be an almost-orbit of S. Suppose that every weak limit point of{u(t): t ∈G} is a �xed point of S, then {u(t)} converges weakly.

Proof. Let W be the set of all weak limit points of subset of the set {u(t): t ∈G}.Clearly, W is nonvoid since C is a weakly compact subset. Let vi ∈W; i=1; 2, andv1 =w− lim�∈A u(t�), v2 =w− lim�∈B u(t�), where {t�: �∈A} and {t�: �∈B} are twosubnets of G. Suppose that v1 6= v2. Then, by Lemma 1 and Opial’s property

limt∈G

‖u(t)− v1‖ = lim�∈A

‖u(t�)− v1‖

¡ lim�∈A

‖u(t�)− v2‖

= limt∈G

‖u(t)− v2‖:

178 G. Li / Nonlinear Analysis 42 (2000) 175–183

In the same way we have limt∈G ‖u(t)−v2‖¡ limt∈G ‖u(t)−v1‖. This is a contradiction.Consequently, v1 = v2 and hence W is a singleton.

Lemma 3. Suppose that X has Opial’s property and its norm is UKK. Let C bea nonempty weakly compact convex subset of X, and let S = {T (t): t ∈G} be acommutative semigroup of asymptotically nonexpansive-type mappings on C. If x∈Cand if {T (t)x: t ∈G} converges to x weakly, then x is a �xed point of S.

Proof. We �rst prove that there exists limt∈G ‖T (t)x− x‖. In fact, the condition T (t)xconverges to x weakly implies that

‖T (t)x − x‖ ≤ lim infs∈G

‖T (t)x − T (s+ t)x‖

≤ r(t; x) + lim infs∈G

‖T (s)x − x‖

for all t ∈G. This implies that lim supt∈G ‖T (t)x − x‖≤ lim inf s∈G ‖T (s)x − x‖, andhence limt∈G ‖T (t)x − x‖ exists. Let r= limt∈G ‖T (t)x − x‖. For any n∈N , it thenfollows from Eq. (1) that we can choose sn ∈G such that for all t≥ sn,

r − 1n≤‖T (t)x − x‖≤ r + 1

n

and

r(t; x)¡1n

Let

�= {{tn}: n∈N; t1≥ s1; tn+1≥ tn + sn+1}:It is clear that � is nonempty and for any {t n}∈�,

r − 1n≤‖T (t n)x − x‖≤ r + 1n

and

r(t n; x)¡1n:

Moreover, {t + t n}∈� for any t ∈G.Now we claim that for any {t n}∈�, {T (t n)x: n∈N} is relative compact. If it is not

true, then r¿0 and {T (t n)x} has a subsequence {T (tn k )x} such that sep(T (tn k )x)=�¿0.Since the norm of X is UKK, there is �¿0 such that ‖x − z‖¡r − � wheneverlim supk→∞ ‖T (tnk )x − z‖≤ r + �. For m∈N , if nk¿m, then tn k ≥ tm + sm, let hmk =tn k − tm (i.e. hmk ∈G such that tn k = tm + hmk ), then hmk ≥ sm, and hence

‖T (tnk )x − T (tm)x‖≤ r(tm; x) + ‖T (hmk )x − x‖≤ r +2m:

G. Li / Nonlinear Analysis 42 (2000) 175–183 179

This implies that if 2=m¡�, then

lim supk→∞

‖T (tnk )x − T (tm)x‖≤ r + �;

‖T (tm)x − x‖¡r − �:This is a contradiction, and {T (t n)x} must be relatively compact. Next we claim that{T (t n)x} converges to x as n → ∞. In fact, let z be a limit point of {T (t n)x}, thenfor any k ∈N there exists nk ≥ k such that

‖z − T (tnk )x‖¡1k:

This implies that

‖T (t + tnk )x − z‖ ≤ ‖T (t + tnk )x − T (tnk )x‖+1k

≤ r(tnk ; x) + ‖T (t)x − x‖+ 1k

≤ ‖T (t)x − x‖+ 2k:

This implies that lim supt∈G ‖T (t)x − z‖≤ r + 2=k, since k ∈N is arbitrary, we havelim supt∈G ‖T (t)x−z‖≤ r, and then the Opial’s property implies that z= x. This impliesT (t n)x must converge to x. On the other hand, since {t + t n}∈� for every t ∈G and{t n}∈�, we also have T (t + t n)x converges to x. By the continuity of T (t), we getT (t)x= x. This completes the proof.

4. Main results

Theorem 1. Suppose that X has the Opial’s property and the norm of X is UKK.Let C be a weakly compact convex subset of X, S = {T (t): t ∈G} be a commutativesemigroup of asymptotically nonexpansive-type on C, and let u(·) be an almost-orbitof S. Then {u(t): t ∈G} is weakly convergent (to a �xed point) if and only if it isweakly asymptotically regular (i.e. (u(t + h)− u(t)) converges to 0 weakly for everyh∈G).

Proof. Suppose that u(t+h)−u(t) converges weakly to 0 for every h∈G. By Lemma 2and 3, it is enough to show that for every weak limit y of {u(t): t ∈G}, {T (t)y: t ∈G}converges to y weakly.Let {t�: �∈A} be a subnet of G and w − lim�∈A u(t�)=y. Then for any h∈Gw − lim

�∈Au(t� + h)=y:

Let I1 be the set of all �nite nonempty subset of X ∗, and let N be the positive integerset, let I = I1×N = {(B; n): B∈ I1; n∈N}, and for any �=(B; n)∈ I , we write P1�=B

180 G. Li / Nonlinear Analysis 42 (2000) 175–183

and P2�= n. In this case, (I; ≤ ) is a directed system and the binary ≤ on I is de�nedby �1≤ �2 if and only if P1�1⊆P1�2 and P2�1≤P2�2.Let � be the weak topology on C, and for �∈ I , let

O�={x∈C: |f(x)− f(y)|¡ 1

P2�; ∀f∈P1�

}:

It is easily seen that {O�: �∈ I} be a open base at y and O�1 ⊇O�2 if �1≤ �2.Let ’(t)= suph∈G ‖u(t + h) − T (h)u(t)‖. Since limt∈G ’(t)= limt∈G r(t; y)= 0, so

for any �∈ I , there exists t0� ∈G such that if t≥ t0� then

’(t)¡1P2�

and

r(t; y)¡1P2�

:

Let b(t)= lim sup�∈A ‖u(t� + t)− y‖, then for t; s∈G

b(t + s) = lim sup�∈A

‖u(t� + t + s)− y‖

≤ lim sup�∈A

‖u(t� + t + s)− T (t)y‖ (Opial’s property)

≤ lim sup�∈A

(‖u(t� + t + s)− T (t)u(t� + s)‖+ ‖T (t)u(t� + s)− T (t)y‖)

≤ r(t; y) + lim sup�∈A

‖u(t� + s)− y‖

= r(t; y) + b(s):

Since lim supt∈G r(t; y)= 0, we have

limt∈Gb(t)= b := inf{b(s): s∈G}:

For �∈ I , we can choose t1� ≥ t0� such that

b(t)= lim sup�∈A

‖u(t� + t)− y‖≤ b+ 1P2�

(3)

for all t≥ t1�.Suppose that z is a weak limit point of {T (t)y: t ∈G}. Then for any �∈ I; z ∈ co

{T (t)y: t≥ t1�}, it follows that there exist an integer p(¿2) and nonnegative numbersa1�; a

2�; : : : ; a

p� with

∑pi=1 a

i�=1 and s

1�; s

2�; : : : ; s

p� ∈{t ∈ G: t≥ t1�} such that∥∥∥∥∥z −

p∑i=1

ai�T (si�)y

∥∥∥∥∥¡ 1P2�

: (4)

Let l�=∑p

i=1 si� and l

i�= l� − si�, 1≤ i≤p.

G. Li / Nonlinear Analysis 42 (2000) 175–183 181

Note: for �∈ I ,

w − lim�∈A

u(t� + l�)=y:

Select �1� ∈A such that if �≥ �1�, thenu(t� + l�)∈O�: (5)

Since li�≥ t1� for all 1≤ i≤p, it then follows from Eq. (3) that for all 1≤ i≤p,

b(li�)= lim sup�∈A

‖u(t� + li�)− y‖≤ b+1P2�

:

Therefore, there exists �2� ∈A such that

‖u(t� + li�)− y‖≤ b+2P2�

(6)

for all 1≤ i≤p, and �≥ �2�. Now since b≤ b(l�)= lim sup�∈A ‖u(t� + l�)− y‖, thereexists �� ∈A such that ��≥ �1�; �2� and

‖u(t�� + l�)− y‖≥ b−1P2�

: (7)

Since ��≥ �1�; �2�, it then follows from Eqs. (5) and (6) that

u(t�� + l�)∈O�; (8)

‖u(t�� + li�)− y‖≤ b+2P2�

for all 1≤ i≤p: (9)

Eq. (8) implies that u(t�� + l�) is convergent to y weakly, and Eqs. (4), (7) and (9)imply that

‖u(t�� + l�)− z‖ ≤∥∥∥∥∥u(t�� + l�)−

p∑i=1

ai�T (si�)y

∥∥∥∥∥+∥∥∥∥∥p∑i=1

ai�T (si�)y − z

∥∥∥∥∥≤∥∥∥∥∥u(t�� + l�)−

p∑i=1

ai�T (si�)u(t�� + l

i�)

∥∥∥∥∥+

p∑i=1

ai�‖T (s i�)u(t�� + l i�)− T (s i�)y‖+1P2�

≤p∑i=1

ai�’(t�� + li�) +

p∑i=1

ai�(r(si�; y)

+ ‖u(t�� + li�)− y‖) +1P2�

182 G. Li / Nonlinear Analysis 42 (2000) 175–183

≤ b+ 5P2�

≤ ‖u(t�� + l�)− y‖) +6P2�

:

This implies that lim sup�∈I ‖u(t�� + l�) − z‖≤ lim sup�∈I ‖u(t�� + l�) − y‖. By theOpial’s property, we have z=y. This completes the proof.

In the following, using Theorem 1, we provide weak convergence theorems forasymptotically nonexpansive-type mappings and semigroups. Let T be an asymptoti-cally nonexpansive-type mapping from C into itself and let {xn} be an almost-orbitof T , i.e.

limn→∞

[supm≥0

‖xn+m − T mxn‖]=0:

Let S = {T (t): t≥ 0} be an asymptotically nonexpansive-type semigroup on C andlet a function u(:) :R+ → C be an almost-orbit of S, i.e.

lims→

[supt≥0

‖u(t + s)− T (t)u(s)‖]=0:

Put G= {0; 1; 2 : : :}; S = {T i: i∈G} in Theorem 1, we get the following nonlinearweak convergence theorem for asymptotically nonexpansive mappings.

Theorem 2. Suppose that X has the Opial property and the norm of X is UKK, letC be a weakly compact convex subset of X, and let T :C 7→C be an asymptoticallynonexpansive-type mapping, and let {xn} be an almost-orbit of T. Then {xn} isweakly convergent (to a �xed point) if and only if it is weakly asymptotically regular(i.e. {xn+1 − xn} converges to 0 weakly).Put G=R+; S = {T (t): t ∈G} in Theorem 1. We get the nonlinear weak conver-

gence theorem for asymptotically nonexpansive-type semigroups.

Theorem 3. Suppose that X has the Opial property and the norm of X is UKK, let Cbe a weakly compact convex subset of X, and let S = {T (t): T ≥ 0} be an asymptot-ically nonexpansive-type semigroup, and let u(·) be an almost-orbit of S. Then u(t)converges weakly to some point of F(S) if and only if it is weakly asymptoticallyregular (i.e. u(t + h)− u(t) converges to 0 weakly as t→ 0 for all h≥ 0).Theorem 3 shows that the open question in [13, p. 550] has an a�rmative answer

for non Lipschitzian semigroup in a nonuniformly convex Banach space.

References

[1] R. Bruck, Convergence of iterates of asymptotically nonexpansive mappings in Banach space with theuniform Opial property, Colloq. Math. 65 (1993) 165–179.

G. Li / Nonlinear Analysis 42 (2000) 175–183 183

[2] N. Hirano, K. Kido, W. Takahashi, Asymptotic behavior of commutative semigroup of nonexpansivemappings in Banach spaces, Nonlinear Anal. 10 (1986) 229–249.

[3] R. Hu�, Banach spaces which are uniformly convex, Rocky Mountain J. Math. 10 (1980) 743–749.[4] W.A. Kirk, R. Torrejon, Asymptotically nonexpansive semigroup in Banach space, Nonlinear Anal. 1

(1979) 111–121.[5] A.T. Lau, W. Takahashi, Weak convergence and nonlinear ergodic theorems for reversible semigroups

of nonexpansive mapping, Paci�c J. Math. 12 (1987) 277–294.[6] G. Li, J.P. Ma, Nonlinear ergodic theorems for commutative semigroups of non-lipschitzian mappings

in Banach spaces, Acta Math. Sinica 39 (1996) 803–809.[7] G. Li, Weak convergence and nonlinear ergodic theorems for reversible topological semigroups of

non-lipschitzian mappings, J. Math. Anal. Appl. 206 (1997) 1411–1428.[8] P.K. Lin, K.K. Tan, H.K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically

nonexpansive mappings, Nonlinear Anal. 24 (1995) 929–946.[9] P.K. Lin, Asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. 26 (1996)

1137–1141.[10] I. Miyadera, K. Kobayasi, On the asymptotic behaviour of almost-orbits of nonlinear contraction

semigroups in Banach space, Nonlinear Anal. 6 (1982) 349–365.[11] H. Oka, Nonlinear ergodic theorems for commutative semigroups of asymptotically nonexpansive

mappings, Nonlinear Anal. 18 (1992) 619–635.[12] S. Prus, Banach spaces with the uniform Opial property, Nonlinear Anal. 18 (1992) 697–704.[13] S. Reich, A note on the mean ergodic theorem for nonlinear semigroups, J. Math. Anal. Appl. 91

(1983) 547–551.