fixed point theorems for asymptotically nonexpansive ...

Nonlinear Anolysls. Theory, Methods & Applications,

Vol. 22, No. II, pp. 1345-1355. 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X194 $7.00+ .OQ

Pergamon

FIXED POINT THEOREMS FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS TECK-CHEONG

LIME

and HONG-KUN

Xu$

TDepartment of Mathematics, George Mason University, Fairfax, VA 22030, U.S.A.; and $.Institute of Applied Mathematics, East China University of Science and Technology, Shanghai 200237, People’s Republic of China (Received

3 1 May 1992; received for publication

4 June 1993)

Key words and phrases: Fixed points, asymptotically nonexpansive mappings, uniformly structure, uniformly smooth Banach space, Opial’s condition, duality map, Maluta’s constant.

normal

1. INTRODUCTION

LET X be a Banach to be a Lipschitzian

space and C be a nonempty subset of X. Then a mapping mapping if, for each integer n 1 1, there exists a constant

T: C -+ C is said k, > 0 such that

for all x, y in C. A Lipschitzian mapping T is called uniformly k-Lipschitzian if k, = k for all n I 1, nonexpansive if k,, = 1 for all n L 1, and asymptotically nonexpansive if lim k, = 1, ” respectively. In 1965, Kirk [l] proved that if C is a weakly compact convex subset of a Banach space with normal structure, then every nonexpansive self-mapping T of C has a fixed point. (A nonempty convex subset C of a normed linear space is said to have normal structure if each bounded convex subset K of C consisting of more than one point contains a nondiametral point, i.e. an x E Ksuch that supl[lx - yII:y E K] < sup(Ilu - u 11. . u, u E K) = diam K.) Seven years later, in 1972, Goebel and Kirk [2] proved that if the space Xis further assumed to be uniformly convex, then every asymptotically nonexpansive self-mapping T of C has a fixed point. This result has now been generalized to a 2-uniformly rotund Banach space by Yu and Dai [3] and generally to a k-uniformly rotund (k-UR) Banach space for any integer k 2 1 independently by Martinez Yanez [4] and Xu [5], and more generally to a nearly uniformly convex (NUC) Banach space by the latter author [6]. (We refer the reader to [7,8] for definitions of k-UR and NUC Banach spaces.) However, it remains an open question whether normal structure implies the existence of fixed points of asymptotically nonexpansive mappings. In this paper we present some results on fixed points of asymptotically nonexpansive mappings. Precisely, we prove in Section 2 the strong convergence (under certain assumptions) of an approximating fixed point for an asymptotically nonexpansive mapping in a uniformly smooth Banach space. This partially extends a celebrated convergence theorem due to Reich [9] for nonexpansive mappings. In Section 3 we verify the existence and weak convergence of fixed points of asymptotically nonexpansive mappings in a space with a weakly continuous duality map. Finally, in Section 4, we provide two fixed point theorems for asymptotically nonexpansive mappings which connect with Maluta’s constant for a Banach space. 1345

1346

T.-C. LIM and H.-K. Xu 2. UNIFORMLY

Let X be a Banach space. sphere of X, the limit

SMOOTH

BANACH

SPACES

Recall that X is said to be smooth lim Ilx + tY II t+0

if, for each x E S,,

the unit

llxll (2.1)

t

exists for all y E S, . If the limit (2.1) exists and is attained uniformly in x, y E S,, then X is said to be uniformly smooth. In this case, the (normalized) duality map J: X + X*, the dual space of X, defined by J(x) = (x*

E

x*:

(x,x*>

=

llxl12= llx*l12),

xex

is single-valued and uniformly continuous on bounded subsets of X when both X and X* are endowed with their norm topologies. Now suppose X is a uniformly smooth Banach space, C is a bounded closed convex subset of X, and T: C + C is a nonexpansive mapping. Then to a fixed u E C and each integer n 2 1, by the Banach Contraction Principle, we have a unique x, E C such that

(Such a sequence [x,) is said to be an approximating fixed point of T since it possesses the property that 1imII TX, - x,II = 0.) The celebrated convergence theorem of Reich [9] now states that lx,) does c”onverge strongly to a fixed point of T. In this section we shall partially extend this result to asymptotically nonexpansive mappings. But first we begin with some preliminaries. Let E be a nonempty bounded closed convex subset of a Banach space X and let d(E) = sup(Ilx - y 11: x, y E E) be the diameter of E. For each x E E, let T(X, E) = sup] IIx - y 11:y E EJ and let r(E) = inf(r(x, E): x E E ), the Chebyshev radius of E relative to itself. The normal structure coefficient of X is defined [lo] as the number N(X)

= inf(d(E)/r(E):

E bounded

closed convex

subset of X with d(E) > 0).

A space X such that N(X) > 1 is said to have uniformly normal structure. It is known that a space with uniformly normal structure is reflexive and that all uniformly convex or uniformly smooth Banach spaces have uniformly normal structure (cf. for example [l 11). Now we prove the following result that is a slight generalization of a theorem of Casini and Maluta [12]. THEOREM 1. Suppose X is a Banach space with uniformly normal structure, C is a nonempty bounded subset of X, and T: C + C is a uniformly k-Lipschitzian mapping with k < N(X)1’2. Suppose also there exists a nonempty bounded closed convex subset E of C with the following property (P): xeE

(P) where o,(x)

is the weak o-limit

implies

o,(x)

c

E,

set of Tat x, i.e. the set

(y E X: y = weak-lim T”jx for some nj t co). _i Then

T has a fixed point

in E.

Asymptotically

nonexpansive

1347

mappings

Proof. Take any x0 in E and consider, for each integer n 2 1, the sequence According to the definition of N(X), we have a y, E Co{T';r,: j z n) (F6 denotes convex hull) such that llmll T’% - ~~11 5 &X)A(lT’x~ljzJ, i where &r(X) = N(X)-’ number

and A@,,) denotes

the asymptotic

liIIl(SLlp(IlZi Zjll:

i,

?I

diameter

of the sequence

( T'X~t, ,,. the closed (2.2) (z,), i.e. the

j > n)).

Since X is reflexive, 1~~) admits a subsequence (yn,J converging weakly to some x1 E X. From (2.2) and the W-1.s.c. of the functional hmllT”x, - ~11, it follows that n hm(l T”x, - xl/l 5 #(X)A((T”x,)). (2.3) n It is also easily seen that x1 belongs

to the set

n;=, cO(Tjx,

llz - xi11 5 limllz - T”x,II n

: j 2 n) and that

for all z E X.

(2.4)

Observing property (P) and the fact that n;= 1E6[ T’x,: j 1 n) = Co q,,(x) which is easy to prove by using the Separation Theorem (cf. [13]), we get that x1 actually lies in E. So we can repeat the above process and obtain a sequence lx,Jf= 1 in E with the properties: for all integers m 2 1, limIIT”x,,_,

- x,1/ I &X)A((T”x,,_,]),

n

l/z - x,11 I limllz - T”x,,_111

for all z E X.

n

Now proceeding with the same argument as in the proof of Casini and Maluta (x,} converges strongly to a fixed point of T. n A direct consequence theorem 2 below.

of theorem

1 is the following

(2.5) (2.6) [ 121, we see that

result that will be used in the proof

of

COROLLLARY 1. Suppose X is a uniformly smooth Banach space, C is a nonempty bounded subset of X, and T: C + C is an asymptotically nonexpansive mapping. Suppose also there exists a nonempty closed bounded convex subset E of C with the property (P) above. Then T has a fixed point in E.

Suppose now C is a bounded closed convex subset of a Banach space X and T: C + C is an asymptotically nonexpansive mapping (we may always assume k, 2 1 for all n 2 1). Fix a u in C and define for each integer n 2 1 the contraction S,: C --) C by S,(x)

=

1 - ;

(

n>

u + $ T”x,

n

Principle where (t,) c [0, 1) is any sequence such that t, + 1. Then the Banach Contraction yields a unique point x, fixed by S,, . Now the question gives rise to whether the sequence (x,,) converges strongly to a fixed point of T. The following is a partial answer.

T.-C. LIMand H.-K. Xv

1348

THEOREM 2. Suppose

X is uniformly

smooth

lim(k, n

and (t,) is chosen

- l)/(k,

- t,) = 0.

(Such a sequence {r,) always exists, for example, Suppose in addition the following condition limllx, n holds.

Then the sequence

Proof. have

From corollary

lx,) converges

take

t, = mini1

- (k, - 1)1’2, 1 _ n-l).)

- Tx,II = 0

strongly

1, the fixed point

so that

(2.7)

to a fixed point

of T.

set F(T) of T is nonempty.

For each u E F(T),

we

(x, - T”x,,, J(x,, - u)) = (x,, - u, J(x,, - II)) + (u - T”x,, , J(x,, - II)) 1 11x, - d2

- 11~- T”x,,llIlx, - ~11

L -(k,

- 1)11x, - ul12

2 -(k,

- I)&,

where d = diam C. Since x, is a fixed point x,, -

of S,, it follows

T”x,, = k,

that

(u - x,) cl

and thus from the last inequality

above,

we get

(x,, - U, J(x,, - u)) I s,d2, where s, = t,,(k,, f: C + [0, 00) by

1)/(/c, -

t,) + 0 as n + 00. Now

f(z)

= LIM,tlxn

(2.8)

let LIM be a Banach

limit and define

- ~11~.

Since f is continuous and convex, f (z) + 00 as ll.zll -+ 00, and X is reflexive, f attains over C. Hence, the set E =

x E C: f(x)

its infimum

= y;‘gf(z)

l

1

is nonempty, closed and convex. Though E is not necessarily invariant under T, it does have the property (P). In fact, if x is in E and y = w-lim T”‘jx belongs to the weak o-limit set o,(x) of Tat x, then from the w-1.s.c. off f(y)

5 b

f(T”jx) j T= hm(LIM,llx,

5 i&LIM.Ilx.

m

= mEi; f(z).

and (2.7),‘we

have

I lim f(T”x) m T- Tmxnl12)= l~m(LIM,~~Tmx, m

- xii2 = LIM,llx,

- xii2

- Tmxlj2)

Asymptotically

nonexpansive

1349

mappings

This shows that y belongs to E and hence E satisfies the property (P). It follows from corollary 1 that T has a fixed z E E. Since z is also a minimizer off over C, it follows that lim

t-o+ for all x E C. This, together 46-471 for details)

f(z + ax - a) - f(z) > - () t

with the uniform LIM,(x

for all x E C; in particular,

of X, easily implies that (see [14, pp.

- z, J(x, - z)) I 0

(2.9)

- z, J(x, - z)) I 0.

(2.10)

we have LIM,(u

Combining

smoothness

(2.10) and (l.S),

we get

LIM,(x,

- z, J(x, - z)) = LIM,llx,

- ~(1’ I 0.

Therefore, there is a subsequence (x,,] of lx,) which converges strongly to U. To complete the proof, suppose there is another subsequence (xmj] of lx,) which converges strongly to (say) y. Then y is a fixed point of T by hypothesis (2.7). It then follows from (2.8) that (z - 24, J(z - Y)) 5 0, and (y - l4, J(y - z)> 5 0. Adding

these two inequalities

yields (z - Y, J(z - Y)) = llz - YV = 0

and thus z = y. This proves the strong

3. WEAKLY

convergence

CONTINUOUS

of lx,) to z.

DUALITY

n

MAP

Recall that a Banach space X is said to satisfy Opial’s condition [ 151 if, for any sequence in X, the condition that lx,) converges weakly to x E X implies that T-

-

lyllxn - XII< 14x, n

lx,)

Y II

for all y E X, y # x. It has been proved that Opial’s condition implies weakly normal structure and, hence, the fixed point property for nonexpansive mappings. However, it is not clear whether Opial’s condition implies the fixed point property for asymptotically nonexpansive mappings. Theorem 3 of this section will provide a partial answer to this question. By a gauge we mean a continuous strictly increasing function lp defined R+ := [0, 00) such that ~(0) = 0 and lim v(r) = w. We associate with a gauge ~1a (generally multivalued) duality T-Co map J, : X --t X* defined by

J,(x) = b* E X*: (x,x*) = llxll~dlxll)andIlx*ll = ul(llxll)).

1350

T.-C. LJM and H.-K. Xu

Clearly the (normalized) duality map J corresponds to the gauge p(t) = t. Browder [16] initiated the study of certain classes of nonlinear operators by means of a duality map J, . Set for t 1 0, I Q(t) = i0

v(r) dr.

Then it is known that .I, (x) is the subdifferential of the convex function Q( )I* 11)at x. Now recall that X is said to have a weakly continuous duality map if there exists a gauge a, such that the duality map J, is single-valued and continuous from X with the weak topology to X* with the weak* topology. A space with a weakly continuous duality map is easily seen to satisfy Opial’s condition (cf. [16]). Every lp (1 < p < 00) space has a weakly continuous duality map with the gauge p(t) = tP-‘. THEOREM 3. Suppose X is a Banach space with a weakly continuous duality weakly compact convex subset of X, and T: C + C is an asymptotically mapping. Then we have the following conclusions: (i) T has a fixed point in C; and (ii) if T is weakly asymptotically regular at some x E C, that is, w-lim(T”x n then (T”xJ converges weakly to a fixed point of T. Proof. (i) Let F be the family satisfy the following property

of subsets

XEK

(P)

K of C which are nonempty,

implies

map J,, C is a nonexpansive

- T”+‘x) = 0,

closed,

convex,

and

wW(x) c K.

(Here, as before, o,(x) is the weak w-limit set of Tat x.) F is then ordered by inclusion. weak compactness of C now allows one to use Zorn’s lemma to obtain a minimal element K in F. For each x in C, define the functional r, by 7 r,(_v) = ll$]Tflx - ~11.

The (say)

Then by lemma 1 of [6], when x lies in K, r, is a constant over y E K and this constant is independent of x E K, that is, Tfor all x, y E K. hm]lT”x - y]( = r (3.1) n Now fix x in K and let { T”‘x) be a subsequence of (T”xJ converging weakly to some y that is in K by property (P) and such that r’ := lim]]Tflix - yll exists. For any integers n, m L 1, noting i the identity 1

Wlx + VII) = Wxll) +

(Y, i 0

J,(x + 04) dt

for all x, y E X, we have @(/T”x

-

T”xll)

= @(Il(T”x - y) + (y - T”x)lj)

=

QdlT”x- ~11) +

(y - T”‘x, Jv(T”x

- y + t(y - Tmy))) dt.

Asymptotically

Substituting

ni for n and letting

nonexpansive

i go to infinity,

lim @((I Tnix - Tmxll) = Q(f)

mappings

1351

we get

+

(y - T”‘x, J,(t(y

i

- Tmx))) dt

0

; I/y - T?+(tlly

= C’(f) +

-

Tmxlj)dt

a’(+) + @(lly - Tmxll).

= It follows that

T”xll) > T"x(l)

I

-(

lim lim cD(k,,I(T"-"x m

> >

- x11)

n

= hm @(]IT”x - x/l) = Q(r), ” which implies that (P(T)) = 0. Hence, r’ = 0, i.e. (T”‘x] strongly converges to y. This proves that, for each x in K, the strong w-limit set o(x) := {y E X: y-strong-lim T”‘x for some ni t 00) of Tat x is nonempty. It is clearly closed. We further claim that w(x) is’norm-compact. In fact, given any sequence 1~~) in w(x). It is easy to construct a subsequence (Tmjx) of (T”x] such that f or all j 11. Repeating IIT”jx - Ujll