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Fixed point theorems for Lipschitzian mappings in Banach spaces
NonlinearAnalysis, Theory, Methods & Applications, Vol. 26, No. 12, pp. 1905-1911, 1996 Copyright © 1996ElsevierScience Ltd Printed in Great Britain. All rights reserved 0362-546X/96 $15.00+ 0.00
Pergamon
0362-546X(95)00050-X FIXED
POINT
THEOREMS
FOR
LIPSCHITZIAN
IN BANACH
MAPPINGS
SPACES
TAE H W A KIMt~: and W. A. KIRK§ ~tDepartment of Applied Mathematics, National Fisheries University of Pusan, P u s a n 608-737, Korea; and § Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A. (Received 11 January 1995; received f o r publication 21 March 1995) Key words and phrases: Fixed point theorems, Lipschitzian mappings, nonexpansive mappings.
1. I N T R O D U C T I O N
Let X be a Banach space and D _ X. A mapping T: D ~ X is said to be n o n e x p a n s i v e if for each x, y • D, IIT ( x ) - T(y)ll - IIx - yll. A rich fixed point theory for mappings of this class exists (e.g. see [1]). On the other hand, if T is merely assumed to be k-Lipschitzian, that is, for some fixed k >__ 0, liT(x) - T(y)ll - kllx - yl[ for each x , y • D , then no comparable theory exists if the Lipschitz constant k > 1. Indeed, it is known (cf. [2]) that for any k > 1 there exists a k-Lipschitzian self-mapping of the unit ball of the infinite dimensional Hilbert space/2 which has no fixed point. It has been known for some time, however, that such mappings T will always have fixed points if k is sufficiently near 1 and if appropriate constraints are placed on the iterates of T. One of the first results of this type is due to Goebel [3] who proved that if K is a weakly compact convex subset of a strictly c o n v e x Banach space, and if K has normal structure (see below), then a mapping T: K ~ K will always have a fixed point if T 2 is nonexpansive and if T is k-Lipschitzian for k < 2. In Section 2 we substantially generalize this result, showing in particular that it remains true without the strict convexity assumption. In Section 3 we show that analogs of the results of Section 2 hold for asymptotically nonexpansive mappings as well. A mapping T: D ~ D is said to be a s y m p t o t i c a l l y n o n e x p a n s i v e if there exists an integer N such that for n _ N, liT(x) - T(y)[I -<
k.llx
- yll
( x , y • O),
where limn ~ ~ kn = 1. This is a class of mappings introduced by Goebel and Kirk in [4], where it is shown that if K is a n o n e m p t y bounded closed and convex subset o f a uniformly convex Banach space and if T: K --, K is asymptotically nonexpansive, then T has a fixed point and, moreover, the set Fix(T) of fixed points of T is closed and convex. For concrete examples of spaces to which our results apply in full generality we need some definitions. Let X be a Banach space and B 1 the unit ball in X. The m o d u l u s o f c o n v e x i t y of X is the function fi: [0, 2] ~ [0, 1] defined by ~(e)=infll
[[x + yl~[ : x' y e B t ' llx - yll >- e
t This research was carried out while the first author was visiting the University of Iowa, and supported by the Y o n a m Foundation, 1994. 1905
1906
TAE HWA KIM and W. A. KIRK
The characteristic o f convexity of X is the number eo(X) = suple : ~(e) = 01. It is easy to see [3] that X is uniformly convex iff eo(X) = 0; uniformly nonsquare iff eo(X) < 2; and strictly convex iff J(2) = 1. Moreover, if eo(X) < 1, then X has normal structure, that is, each bounded convex subset H o f X which contains more than one point contains a point x0 such that sup[llxo - xll : x ~ H I < diam(H). The fact pertinent to the results we obtained below is this. If eo(X) < 1, then every nonempty bounded closed and convex subset o f X has the fixed point property for asymptotically nonexpansive (hence nonexpansive) mappings [5]. 2. NONEXPANSIVE MAPPINGS In [2] it is shown that if K is a weakly compact convex subset of a Banach space X, and if K has normal structure, then every nonexpansive T: K ~ K has a fixed point. In [3] Goebel extended this result by proving that if, in addition, X is strictly convex, then it suffices to assume only that T 2 is nonexpansive provided T satisfies the Lipschitz condition
IIT(x) - T(y)II -<
kllx
-
Yll
x, y ~ K,
where k satisfies the condition
T h r o u g h o u t we shall refer to the above Lipschitz condition as Goebel's Lipschitz condition. Note especially that this condition always holds if k < 2, so that all the ensuing results are new for arbitrary Banach spaces. On the other hand, Goebel's condition holds for some k _ 2 in any space X for which eo(X ) < 1. For example, the condition is satisfied for k < x/5 in any Hilbert space. Thus, in particular spaces, even sharper results hold. (Of course since a Hilbert space has strictly convex norm, our results in this case are subsumed by Goebel's original results.) Our approach is based on a fundamental idea of Khamsi and Bruck. We begin with a concept introduced by Bruck in 1973 [6]. A closed and convex subset K of a Banach space X is said to have the conditional fixed point property (CFPP) if every nonexpansive mapping F: K ~ K satisfies: Either Fix(T) ;~ O , or T has a fixed point in every closed convex T-invariant subset of K. As noted in [1], if a Banach space X has the property that each nonempty closed and convex subset of its unit sphere has the fixed point property for nonexpansive mappings, then every closed and convex subset K of X which is locally weakly compact has the C F P P . Bruck proved in [7] that if a closed convex subset K of a Banach space satisfies both the fixed point property for nonexpansive mappings and the C F P P , and if K is either weakly c o m p a c t or bounded and separable, then the c o m m o n fixed point set of any commutative family of nonexpansive self mappings of K is a n o n e m p t y nonexpansive retract of K. (Subsequently it has been noted that Bruck's p r o o f works for certain topologies weaker than the weak topology; in particular for the weak* topology (see [8-10]).) There is a more immediate fact, sufficient for most of our purposes, which requires another definition. If F c_ K ¢_ X, then F is said to be a l-local retract o f K if every family [Bi, i • I} of closed balls centered at points of F has the property: ( n i ~ t B i ) O K ~ ~ = ( n i ~ t B i ) O F ~ Q . This concept is due to Khamsi [11, 12], who used it to prove the existence of c o m m o n
Fixed point theorems
1907
fixed points for commuting families o f nonexpansive mappings in a more general context. It is easy to see that a l-local retract of a convex set is metrically convex, and a l-local retract of a closed set must itself be closed. It is easy to check that nonexpansive retracts are always l-local retracts. Our first observation is quite simple and generally known. LEMMA 1. Suppose K is a nonempty, closed, and convex subset of a Banach space, and suppose K has both the fixed point property for nonexpansive mappings and the C F P P . Then if T: K --, K is nonexpansive, Fix(T) is a n o n e m p t y l-local retract of K.
Proof. By assumption Fix(T) # Q . Let x ~ K, and let C = O z ~ r i x t r ) B ( z : l l z - xll). Then since liT(y) - zll -- liT(y) - T(z)ll --- IlY - zll -< IIz - xll for a l l y e C, T: C ~ C. S i n c e K has the C F P P , C O Fix(T) ;~ Q . Our principal observation is the following variant of a result o f [3]. LEMMA 2. Let X be a Banach space, let H be a n o n e m p t y subset o f X, and suppose H is a l-local retract of c--0-ffT(H). Suppose T: H ~ H satisfies Goebel's Lipschitz condition, and suppose T 2 = I. Then T has a fixed point.
Proof. Define a mapping G: H ~ H as follows. Let x e H and let m x = ½(x + T(x)). Since mx ~ ('lz~nB(z; [Iz - mxll) and since H is a l-local retract o f conv(H), there exists a point G(x) ~ H such that G(x) ~ H n ( z g n B ( z ; llz - rnxI')) • Note in particular that G(x) e B(x; IIx - rnrll) n B(T(x); IIT(x) - mrll), so
IIx - G(x)ll -- IIG(x) - Z(x)ll = ½1Ix - T(x)ll.
Also, since G ( x )
~ HA
(nz~nB(z; IIz - mxll) c_ liT*
a(x)
-
a(x)l[-<
G(x); IITo
B(To
liT*
a(x)
G(x) -
m~ll), we have
m~ll.
-
On the other hand,
IITo G(x) - xll -- IITo
G(x)
<- k l l G ( x )
k IIx 2
-
-
T2(x)ll
T(x)ll
r(x)ll
and
liT°
G(x) -
Thus,
Ilro
G(x) -
m~ll ~
r(x)ll ~
((D) 1 - 6
k[IG(x) -
k xll = ~ IIx - r(x)ll.
~ IIx - r(x)ll =
1 - 6
kllx - G(x)ll.
1908
TAE HWA KIM and W. A. KIRK
Note also that IIC(x) -
aZ(x)ll
= IlGZCx) - T o C(x)lJ = ½ l i T * C ( x ) - G ( x ) l l .
Therefore, we have
IIG(x) - aZ(x)ll = ½ l l r o < ½11To
G(x)
-
O(x)[I
G(x)
-
mxll
(
_<1-6
2 1-~
x
~(x)ll.
Since (k/2)(1 - ~(2/k)) < 1 we conclude that the sequence [G°(x)} is Cauchy, and since H is closed, hence complete, there exists y e H such that C°(x) ~ y as n ~ oo. F r o m this, we obtain
½11G°(x) -
To G"(x)I[
= IIG°(x) -
Gn+l(x)ll
However, since both [1.11 and T are continuous, T(y) = y.
~
0
as
n ~
oo.
] ] G " ( x ) - To G"(x)]l--" I I Y - T(x)[], so
THEOREM 1. Let K be a nonempty, closed, and convex subset of a Banach space, and suppose K has both the fixed point property for nonexpansive mappings and the C F P P . Suppose T: K ~ K is a mapping for which T 2 is nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. Proof. By assumption H :-- Fix(T 2) ;~ ~3, and by l e m m a 1 H is a 1-10cal retract of K. Since an obvious calculation shows that T: H ~ H, all the assumptions of lemma 2 are fulfilled. Immediately we have the following generalization of Goebel's result. COROLLARY 1. Let K be a weakly compact convex subset of a Banach space, and suppose K has normal structure. Suppose T: K ~ K is a mapping for which T 2 is nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. Proof. By the theorem of [2], K has both the fixed point property for nonexpansive mappings and the C F P P . Finally, we note that in theorem 1 one only need know that the fixed point set of T z is a l-local retract of K or, more particularly, a nonexpansive retract of K. This can sometimes be assured by conditions on the mapping rather than on its domain. A mapping T: K ~ K is said to be (a, n)-rotative for a < n if for each x ~ K,
IIx - T"(x)ll -<
allx -
T(x)ll,
T is said to be n-rotative if it is (a, n)-rotative for some a < n, and rotative if it is n-rotative for some n ~ N. Goebel and Koter [13] (also, see [1, p. 176]) have shown that if K is a nonempty, closed and convex subset of a Banach space, then any nonexpansive rotative mapping
Fixed point theorems
1909
T: K --, K has a nonempty fixed point set which is a nonexpansive retract of K. Note in particular that this result does not require any compactness assumption on K nor does it require special geometric conditions on the underlying Banach space X. Thus, we have the following theorem. THEOREM 2. Suppose K is a nonempty, closed and convex subset of a Banach space, and suppose T: K --, K is a mapping which has the property that T 2 is nonexpansive and rotative. Then if T satisfies Goebel's Lipschitz condition, T has a fixed point. 3. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS As noted in the previous section, knowledge about the structure of the fixed point set of nonexpansive mappings can have rather striking implications about the existence of fixed points of wider classes of mappings. In this section we observe that this is true for asymptotically nonexpansive mappings as well. Goebel and Kirk [4] initially proved that if K is a n o n e m p t y bounded closed and convex subset of a uniformly convex Banach space and if T: K --, K is asymptotically nonexpansive, then Fix(T) is n o n e m p t y closed and convex. Subsequently it was shown [5] that the existence part of the above is true if eo(X) < 1. (The asymptotic assumption was weakened in [14] as well.) However, if eo(X) < 1, it is no longer possible to conclude that Fix(T) is convex, but it is the case that Fix(T) is a l-local retract of K. THEOREM 3. Suppose K is a closed and convex subset of a Banach space, and suppose each closed and convex subset H of K has the fixed point property for asymptotically nonexpansive mappings. Then the fixed point set of any asymptotically nonexpansive T: K ~ K is a (nonempty) l-local retract of K. The above fact combined with lemma 2 immediately yields the following theorem. THEOREM 4. Suppose K is a closed and convex subset of a Banach space, and suppose each closed and convex subset H of K has the fixed point property for asymptotically nonexpansive mappings. Suppose T: K --, K is a mapping for which T 2 is asymptotically nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. Proof. Fix(T z) is a nonempty l-local retract of its closed convex hull. COROLLARY 2. Suppose K is a bounded closed and convex subset of a Banach space X, and suppose e o ( X ) < 1. Suppose T: K ~ K is a mapping for which T z is asymptotically nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. P r o o f o f theorem 3. By assumption Fix(T) # Q3. Suppose xi ~ Fix(T) and ri >- 0 for i e / , and suppose S o := ( ( ' l i ~ i B ( x i ; r~)) (q K # Q~. For each x e K and i e I, let r(lT"(x)l; xi) = lim sup [[T"(x) - xi [[, n --* c o
and let S 1 = {x ~ K : r([T"(x)], xi) <- ri].
1910
TAE HWA KIM and W. A. KIRK
It is easy to see that So - $1 ; indeed, if x e So then
r(l Tn(x)}; xi) = lim sup II Tn(x) - xi II t t ~
=
lim
sup IIT~(x) - Tn(xi)l[ tl --~ oo
_< (lim sup kn)llx - x, ll n~oo
-- IIx - xill <_ r i .
Thus, $1 ;~ Q). We now show that S 1 is closed a n d convex.
Convexity. Suppose x, y ~ S1, let z = a x + fly, where a _> 0, fl _> 0, and a + fi = 1, and let i e I. Then (as in the preceding argument)
r([Tn(Z)];Xi)
<
IIz - x~ll -< ~llx - xill + filly - x, ll -< q .
Closededness. Suppose [uml c_ $1 with Um --~ X (m --~ 0o). N o t e that for each i r([Tn(um)l; x i) <~ ri. Thus, for each m, lim sup IlTn(x) - x, II -< lim sup n ---~ oo
IITn(x)
-
Tn(Um)ll
n~at~
+ lim sup
[ITn(Um)
-
xill
n~oo
_< (lim sup ko)llx - umll + r(lT~(um)l;xi) t/~oo
-- IIx-
umll + r(IT~(Um)]; xi)
-< IIx - umll + r,. Since Um ~ x as m ~ oo the conclusion follows. Finally, since T:S1 ~ S~, it must be the case that $1 N F i x ( T ) ; ~ . However, S~ n Fix(T) = So n Fix(T). To see this, note that So n Fix(T) _ S 1 n Fix(T) since So c_ S1. Conversely, suppose x e S~ n Fix(T). Then for each i e I,
r([T"(x)];xi)
=
r([x];xi)
= IIx - x,l[ ~ r~,
R e m a r k . It is perhaps interesting to note that if K is a b o u n d e d closed and convex subset o f a Banach space X for which eo(X) < l, and if T: K ~ K, then Fix(T) is a nonexpansive retract o f K if T is nonexpansive, and a l-local retract o f K if T is asymptotically nonexpansive. W e do not k n o w whether Fix(T) is actually a nonexpansive retract in the latter case. REFERENCES 1. GOEBEL K. & KIRK W. A., Topics in M e t r i c F i x e d P o i n t Theory. Cambridge University Press, Cambridge (1990). 2. KIRK W. A., A fixed point theorem for mappings which do not increase distances, A m . m a t h . M o n t h l y 72, 1004-1006 (1965).
Fixed point theorems
1911
3. GOEBEL K., Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compos. Math. 22, 269-274 (1970). 4. GOEBEL K. & KIRK W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. math. Soc. 35, 171-174 (1972). 5. GOEBEL K., KIRK W. A. & THELE R. L., Uniformly Lipschitzian families of transformations in Banach spaces, Can. J. Math. 26, 1245-1256 (1974). 6, BRUCK R. E., Properties of fixed point sets of nonexpansive mappings in Banach spaces, Trans. Am. math. Soe. 179, 251-262 (1973). 7. BRUCK R. E., A common fixed point theorem for a commuting family of nonexpansive mappings, Pacif. J. Math. 53, 59-71 (1974). 8. KUCZUMOW T., Fixed point theorems in product spaces, Proc. Am. math. Soc. 108, 727-729 (1990). 9. KUCZUMOW T. & STACHURA A., Bruck's retraction method, in Fixed Point Theory and its Applications (Edited by J. B. BAILLON and M. THI~RA), Pitman Research Notes in Mathematics, Vol. 252, pp. 285-292. Pitman, New York (1991). 10. KIRK W. A., Nonexpansive mappings and nonexpansive retracts in Banach spaces, in Fixed Point Theory and Applications (Edited by K. K. TAN), pp. 137-155. World Scientific, Singapore (1992). 11. KHAMSI M. A., Etude de la propri6t6 du point fixe dans les espaces de Banach et les espaces de Banach et les espaces m&riques, Th+se de Doctorat de L'Universit6 Paris V1 (1987). 12. KHAMSI M. A., One-local retract and common fixed point for commuting mappings in metric spaces, preprint. 13. GOEBEL K. & KOTER M., A remark on nonexpansive mappings, Can. math. Bull. 24, 113-115 (1981). 14. KIRK W. A., A fixed point theorem for mappings of asymptotically nonexpansive type, Israel J. Math. 17, 339-346 (1974).
Pergamon
0362-546X(95)00050-X FIXED
POINT
THEOREMS
FOR
LIPSCHITZIAN
IN BANACH
MAPPINGS
SPACES
TAE H W A KIMt~: and W. A. KIRK§ ~tDepartment of Applied Mathematics, National Fisheries University of Pusan, P u s a n 608-737, Korea; and § Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A. (Received 11 January 1995; received f o r publication 21 March 1995) Key words and phrases: Fixed point theorems, Lipschitzian mappings, nonexpansive mappings.
1. I N T R O D U C T I O N
Let X be a Banach space and D _ X. A mapping T: D ~ X is said to be n o n e x p a n s i v e if for each x, y • D, IIT ( x ) - T(y)ll - IIx - yll. A rich fixed point theory for mappings of this class exists (e.g. see [1]). On the other hand, if T is merely assumed to be k-Lipschitzian, that is, for some fixed k >__ 0, liT(x) - T(y)ll - kllx - yl[ for each x , y • D , then no comparable theory exists if the Lipschitz constant k > 1. Indeed, it is known (cf. [2]) that for any k > 1 there exists a k-Lipschitzian self-mapping of the unit ball of the infinite dimensional Hilbert space/2 which has no fixed point. It has been known for some time, however, that such mappings T will always have fixed points if k is sufficiently near 1 and if appropriate constraints are placed on the iterates of T. One of the first results of this type is due to Goebel [3] who proved that if K is a weakly compact convex subset of a strictly c o n v e x Banach space, and if K has normal structure (see below), then a mapping T: K ~ K will always have a fixed point if T 2 is nonexpansive and if T is k-Lipschitzian for k < 2. In Section 2 we substantially generalize this result, showing in particular that it remains true without the strict convexity assumption. In Section 3 we show that analogs of the results of Section 2 hold for asymptotically nonexpansive mappings as well. A mapping T: D ~ D is said to be a s y m p t o t i c a l l y n o n e x p a n s i v e if there exists an integer N such that for n _ N, liT(x) - T(y)[I -<
k.llx
- yll
( x , y • O),
where limn ~ ~ kn = 1. This is a class of mappings introduced by Goebel and Kirk in [4], where it is shown that if K is a n o n e m p t y bounded closed and convex subset o f a uniformly convex Banach space and if T: K --, K is asymptotically nonexpansive, then T has a fixed point and, moreover, the set Fix(T) of fixed points of T is closed and convex. For concrete examples of spaces to which our results apply in full generality we need some definitions. Let X be a Banach space and B 1 the unit ball in X. The m o d u l u s o f c o n v e x i t y of X is the function fi: [0, 2] ~ [0, 1] defined by ~(e)=infll
[[x + yl~[ : x' y e B t ' llx - yll >- e
t This research was carried out while the first author was visiting the University of Iowa, and supported by the Y o n a m Foundation, 1994. 1905
1906
TAE HWA KIM and W. A. KIRK
The characteristic o f convexity of X is the number eo(X) = suple : ~(e) = 01. It is easy to see [3] that X is uniformly convex iff eo(X) = 0; uniformly nonsquare iff eo(X) < 2; and strictly convex iff J(2) = 1. Moreover, if eo(X) < 1, then X has normal structure, that is, each bounded convex subset H o f X which contains more than one point contains a point x0 such that sup[llxo - xll : x ~ H I < diam(H). The fact pertinent to the results we obtained below is this. If eo(X) < 1, then every nonempty bounded closed and convex subset o f X has the fixed point property for asymptotically nonexpansive (hence nonexpansive) mappings [5]. 2. NONEXPANSIVE MAPPINGS In [2] it is shown that if K is a weakly compact convex subset of a Banach space X, and if K has normal structure, then every nonexpansive T: K ~ K has a fixed point. In [3] Goebel extended this result by proving that if, in addition, X is strictly convex, then it suffices to assume only that T 2 is nonexpansive provided T satisfies the Lipschitz condition
IIT(x) - T(y)II -<
kllx
-
Yll
x, y ~ K,
where k satisfies the condition
T h r o u g h o u t we shall refer to the above Lipschitz condition as Goebel's Lipschitz condition. Note especially that this condition always holds if k < 2, so that all the ensuing results are new for arbitrary Banach spaces. On the other hand, Goebel's condition holds for some k _ 2 in any space X for which eo(X ) < 1. For example, the condition is satisfied for k < x/5 in any Hilbert space. Thus, in particular spaces, even sharper results hold. (Of course since a Hilbert space has strictly convex norm, our results in this case are subsumed by Goebel's original results.) Our approach is based on a fundamental idea of Khamsi and Bruck. We begin with a concept introduced by Bruck in 1973 [6]. A closed and convex subset K of a Banach space X is said to have the conditional fixed point property (CFPP) if every nonexpansive mapping F: K ~ K satisfies: Either Fix(T) ;~ O , or T has a fixed point in every closed convex T-invariant subset of K. As noted in [1], if a Banach space X has the property that each nonempty closed and convex subset of its unit sphere has the fixed point property for nonexpansive mappings, then every closed and convex subset K of X which is locally weakly compact has the C F P P . Bruck proved in [7] that if a closed convex subset K of a Banach space satisfies both the fixed point property for nonexpansive mappings and the C F P P , and if K is either weakly c o m p a c t or bounded and separable, then the c o m m o n fixed point set of any commutative family of nonexpansive self mappings of K is a n o n e m p t y nonexpansive retract of K. (Subsequently it has been noted that Bruck's p r o o f works for certain topologies weaker than the weak topology; in particular for the weak* topology (see [8-10]).) There is a more immediate fact, sufficient for most of our purposes, which requires another definition. If F c_ K ¢_ X, then F is said to be a l-local retract o f K if every family [Bi, i • I} of closed balls centered at points of F has the property: ( n i ~ t B i ) O K ~ ~ = ( n i ~ t B i ) O F ~ Q . This concept is due to Khamsi [11, 12], who used it to prove the existence of c o m m o n
Fixed point theorems
1907
fixed points for commuting families o f nonexpansive mappings in a more general context. It is easy to see that a l-local retract of a convex set is metrically convex, and a l-local retract of a closed set must itself be closed. It is easy to check that nonexpansive retracts are always l-local retracts. Our first observation is quite simple and generally known. LEMMA 1. Suppose K is a nonempty, closed, and convex subset of a Banach space, and suppose K has both the fixed point property for nonexpansive mappings and the C F P P . Then if T: K --, K is nonexpansive, Fix(T) is a n o n e m p t y l-local retract of K.
Proof. By assumption Fix(T) # Q . Let x ~ K, and let C = O z ~ r i x t r ) B ( z : l l z - xll). Then since liT(y) - zll -- liT(y) - T(z)ll --- IlY - zll -< IIz - xll for a l l y e C, T: C ~ C. S i n c e K has the C F P P , C O Fix(T) ;~ Q . Our principal observation is the following variant of a result o f [3]. LEMMA 2. Let X be a Banach space, let H be a n o n e m p t y subset o f X, and suppose H is a l-local retract of c--0-ffT(H). Suppose T: H ~ H satisfies Goebel's Lipschitz condition, and suppose T 2 = I. Then T has a fixed point.
Proof. Define a mapping G: H ~ H as follows. Let x e H and let m x = ½(x + T(x)). Since mx ~ ('lz~nB(z; [Iz - mxll) and since H is a l-local retract o f conv(H), there exists a point G(x) ~ H such that G(x) ~ H n ( z g n B ( z ; llz - rnxI')) • Note in particular that G(x) e B(x; IIx - rnrll) n B(T(x); IIT(x) - mrll), so
IIx - G(x)ll -- IIG(x) - Z(x)ll = ½1Ix - T(x)ll.
Also, since G ( x )
~ HA
(nz~nB(z; IIz - mxll) c_ liT*
a(x)
-
a(x)l[-<
G(x); IITo
B(To
liT*
a(x)
G(x) -
m~ll), we have
m~ll.
-
On the other hand,
IITo G(x) - xll -- IITo
G(x)
<- k l l G ( x )
k IIx 2
-
-
T2(x)ll
T(x)ll
r(x)ll
and
liT°
G(x) -
Thus,
Ilro
G(x) -
m~ll ~
r(x)ll ~
((D) 1 - 6
k[IG(x) -
k xll = ~ IIx - r(x)ll.
~ IIx - r(x)ll =
1 - 6
kllx - G(x)ll.
1908
TAE HWA KIM and W. A. KIRK
Note also that IIC(x) -
aZ(x)ll
= IlGZCx) - T o C(x)lJ = ½ l i T * C ( x ) - G ( x ) l l .
Therefore, we have
IIG(x) - aZ(x)ll = ½ l l r o < ½11To
G(x)
-
O(x)[I
G(x)
-
mxll
(
_<1-6
2 1-~
x
~(x)ll.
Since (k/2)(1 - ~(2/k)) < 1 we conclude that the sequence [G°(x)} is Cauchy, and since H is closed, hence complete, there exists y e H such that C°(x) ~ y as n ~ oo. F r o m this, we obtain
½11G°(x) -
To G"(x)I[
= IIG°(x) -
Gn+l(x)ll
However, since both [1.11 and T are continuous, T(y) = y.
~
0
as
n ~
oo.
] ] G " ( x ) - To G"(x)]l--" I I Y - T(x)[], so
THEOREM 1. Let K be a nonempty, closed, and convex subset of a Banach space, and suppose K has both the fixed point property for nonexpansive mappings and the C F P P . Suppose T: K ~ K is a mapping for which T 2 is nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. Proof. By assumption H :-- Fix(T 2) ;~ ~3, and by l e m m a 1 H is a 1-10cal retract of K. Since an obvious calculation shows that T: H ~ H, all the assumptions of lemma 2 are fulfilled. Immediately we have the following generalization of Goebel's result. COROLLARY 1. Let K be a weakly compact convex subset of a Banach space, and suppose K has normal structure. Suppose T: K ~ K is a mapping for which T 2 is nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. Proof. By the theorem of [2], K has both the fixed point property for nonexpansive mappings and the C F P P . Finally, we note that in theorem 1 one only need know that the fixed point set of T z is a l-local retract of K or, more particularly, a nonexpansive retract of K. This can sometimes be assured by conditions on the mapping rather than on its domain. A mapping T: K ~ K is said to be (a, n)-rotative for a < n if for each x ~ K,
IIx - T"(x)ll -<
allx -
T(x)ll,
T is said to be n-rotative if it is (a, n)-rotative for some a < n, and rotative if it is n-rotative for some n ~ N. Goebel and Koter [13] (also, see [1, p. 176]) have shown that if K is a nonempty, closed and convex subset of a Banach space, then any nonexpansive rotative mapping
Fixed point theorems
1909
T: K --, K has a nonempty fixed point set which is a nonexpansive retract of K. Note in particular that this result does not require any compactness assumption on K nor does it require special geometric conditions on the underlying Banach space X. Thus, we have the following theorem. THEOREM 2. Suppose K is a nonempty, closed and convex subset of a Banach space, and suppose T: K --, K is a mapping which has the property that T 2 is nonexpansive and rotative. Then if T satisfies Goebel's Lipschitz condition, T has a fixed point. 3. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS As noted in the previous section, knowledge about the structure of the fixed point set of nonexpansive mappings can have rather striking implications about the existence of fixed points of wider classes of mappings. In this section we observe that this is true for asymptotically nonexpansive mappings as well. Goebel and Kirk [4] initially proved that if K is a n o n e m p t y bounded closed and convex subset of a uniformly convex Banach space and if T: K --, K is asymptotically nonexpansive, then Fix(T) is n o n e m p t y closed and convex. Subsequently it was shown [5] that the existence part of the above is true if eo(X) < 1. (The asymptotic assumption was weakened in [14] as well.) However, if eo(X) < 1, it is no longer possible to conclude that Fix(T) is convex, but it is the case that Fix(T) is a l-local retract of K. THEOREM 3. Suppose K is a closed and convex subset of a Banach space, and suppose each closed and convex subset H of K has the fixed point property for asymptotically nonexpansive mappings. Then the fixed point set of any asymptotically nonexpansive T: K ~ K is a (nonempty) l-local retract of K. The above fact combined with lemma 2 immediately yields the following theorem. THEOREM 4. Suppose K is a closed and convex subset of a Banach space, and suppose each closed and convex subset H of K has the fixed point property for asymptotically nonexpansive mappings. Suppose T: K --, K is a mapping for which T 2 is asymptotically nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. Proof. Fix(T z) is a nonempty l-local retract of its closed convex hull. COROLLARY 2. Suppose K is a bounded closed and convex subset of a Banach space X, and suppose e o ( X ) < 1. Suppose T: K ~ K is a mapping for which T z is asymptotically nonexpansive, and suppose T satisfies Goebel's Lipschitz condition. Then T has a fixed point. P r o o f o f theorem 3. By assumption Fix(T) # Q3. Suppose xi ~ Fix(T) and ri >- 0 for i e / , and suppose S o := ( ( ' l i ~ i B ( x i ; r~)) (q K # Q~. For each x e K and i e I, let r(lT"(x)l; xi) = lim sup [[T"(x) - xi [[, n --* c o
and let S 1 = {x ~ K : r([T"(x)], xi) <- ri].
1910
TAE HWA KIM and W. A. KIRK
It is easy to see that So - $1 ; indeed, if x e So then
r(l Tn(x)}; xi) = lim sup II Tn(x) - xi II t t ~
=
lim
sup IIT~(x) - Tn(xi)l[ tl --~ oo
_< (lim sup kn)llx - x, ll n~oo
-- IIx - xill <_ r i .
Thus, $1 ;~ Q). We now show that S 1 is closed a n d convex.
Convexity. Suppose x, y ~ S1, let z = a x + fly, where a _> 0, fl _> 0, and a + fi = 1, and let i e I. Then (as in the preceding argument)
r([Tn(Z)];Xi)
<
IIz - x~ll -< ~llx - xill + filly - x, ll -< q .
Closededness. Suppose [uml c_ $1 with Um --~ X (m --~ 0o). N o t e that for each i r([Tn(um)l; x i) <~ ri. Thus, for each m, lim sup IlTn(x) - x, II -< lim sup n ---~ oo
IITn(x)
-
Tn(Um)ll
n~at~
+ lim sup
[ITn(Um)
-
xill
n~oo
_< (lim sup ko)llx - umll + r(lT~(um)l;xi) t/~oo
-- IIx-
umll + r(IT~(Um)]; xi)
-< IIx - umll + r,. Since Um ~ x as m ~ oo the conclusion follows. Finally, since T:S1 ~ S~, it must be the case that $1 N F i x ( T ) ; ~ . However, S~ n Fix(T) = So n Fix(T). To see this, note that So n Fix(T) _ S 1 n Fix(T) since So c_ S1. Conversely, suppose x e S~ n Fix(T). Then for each i e I,
r([T"(x)];xi)
=
r([x];xi)
= IIx - x,l[ ~ r~,
R e m a r k . It is perhaps interesting to note that if K is a b o u n d e d closed and convex subset o f a Banach space X for which eo(X) < l, and if T: K ~ K, then Fix(T) is a nonexpansive retract o f K if T is nonexpansive, and a l-local retract o f K if T is asymptotically nonexpansive. W e do not k n o w whether Fix(T) is actually a nonexpansive retract in the latter case. REFERENCES 1. GOEBEL K. & KIRK W. A., Topics in M e t r i c F i x e d P o i n t Theory. Cambridge University Press, Cambridge (1990). 2. KIRK W. A., A fixed point theorem for mappings which do not increase distances, A m . m a t h . M o n t h l y 72, 1004-1006 (1965).
Fixed point theorems
1911
3. GOEBEL K., Convexity of balls and fixed point theorems for mappings with nonexpansive square, Compos. Math. 22, 269-274 (1970). 4. GOEBEL K. & KIRK W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Am. math. Soc. 35, 171-174 (1972). 5. GOEBEL K., KIRK W. A. & THELE R. L., Uniformly Lipschitzian families of transformations in Banach spaces, Can. J. Math. 26, 1245-1256 (1974). 6, BRUCK R. E., Properties of fixed point sets of nonexpansive mappings in Banach spaces, Trans. Am. math. Soe. 179, 251-262 (1973). 7. BRUCK R. E., A common fixed point theorem for a commuting family of nonexpansive mappings, Pacif. J. Math. 53, 59-71 (1974). 8. KUCZUMOW T., Fixed point theorems in product spaces, Proc. Am. math. Soc. 108, 727-729 (1990). 9. KUCZUMOW T. & STACHURA A., Bruck's retraction method, in Fixed Point Theory and its Applications (Edited by J. B. BAILLON and M. THI~RA), Pitman Research Notes in Mathematics, Vol. 252, pp. 285-292. Pitman, New York (1991). 10. KIRK W. A., Nonexpansive mappings and nonexpansive retracts in Banach spaces, in Fixed Point Theory and Applications (Edited by K. K. TAN), pp. 137-155. World Scientific, Singapore (1992). 11. KHAMSI M. A., Etude de la propri6t6 du point fixe dans les espaces de Banach et les espaces de Banach et les espaces m&riques, Th+se de Doctorat de L'Universit6 Paris V1 (1987). 12. KHAMSI M. A., One-local retract and common fixed point for commuting mappings in metric spaces, preprint. 13. GOEBEL K. & KOTER M., A remark on nonexpansive mappings, Can. math. Bull. 24, 113-115 (1981). 14. KIRK W. A., A fixed point theorem for mappings of asymptotically nonexpansive type, Israel J. Math. 17, 339-346 (1974).