_ The concept is a generalization of convergence in its usual meaning. Of

ROCZNIICI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVIII (1988)

J

G

(Rzeszôw)

Some remarks on almost convergence of the Picard iterates for non-expansive mappings in Banach spaces

which satisfy the Opial condition

Abstract. In this paper, a theorem on almost convergent contraction in Banach space has been prooved. There is discussed possibility of generalization of this theorem for the case of non- expansive mappings. In particular, it is proved a non-linear ergodic theorem for non-expansive mappings in Banach spaces satisfying the Opial condition. There is also investigated the iterative process of M. A. Krasnoselskii in the spaces of this type. The paper is illustrated by some examples in ll space.

I. Introduction. Let (A", || • ||) be a Banach space, С с: X a non-empty subset. A mapping T : C -> C is said to be contraction if there exists a real number L, 0 < L < 1, such that, for all x, y e C , ||7x — Ty\\ < L||x — x||. A mapping T is said to be non-expansive if ||7x—7v|| ^ ||x — y|| for x,

C. In

[20], G. G. Lorentz introduced the idea of almost convergence: a sequence {x„]*=0 c: X is said to be (weak-)almost convergent to a point x e X iff

1 я- l

(weak-) lim - £ x k+i = x,

n-+ oo П k = 0

uniformly in / = 0, 1, 2, _

The concept is a generalization of convergence in its usual meaning. Of course, each convergent sequence is almost convergent but not on the contrary [20]. In order to describe this idea in a Banach space, the concept of a strongly ergodic matrix has been usually used.

An infinite real matrix A = [fl„*]o«n.*<r is called strongly ergodic if (A) Д а * > 0,

n,k

(B) Д Hm a„k = 0,

к n -» x

(Q A Z

= h

П k = 0

00

(D) lim £ Kfc + 1- e J = 0.

" - o o fc= 0

There is a lemma.

L

1.1 ([2], [20]). I f (X, ||*||) is a Banach space, then the following conditions are equivalent:

(i) a sequence \x„)f=0 а X is (weak-)almost convergent to point x e X , (ii) for each strongly ergodic matrix A = we have

oc

(weak-) lim ankxk — x.

n -r k=o

N o t a t i o n . ww(x) is defined to be the set of weak subsequential limits of \T nx}%L0. The weak convergence of a sequence will be denoted by x„ -^x , and strong convergence by x„ — x. The set of fixed points of a mapping T will be denoted by Fix(T); the closed convex hull of a set S by convS.

II. Almost convergence contractions.

T

2.1. Let (X, ||-||) be a Banach space, С с X a non-empty convex closed subset, T : C -+ C a contraction, and A = [fl«*]o <#»,*<<» an infite real matrix such that

(a) Д > 0,

n,k

(b) lim (sup {a*}) = 0,

я-*оо к

00

(c) A £ a nk = !•

n k= 0

ao

Then the sequence S„(x) = £ ank Tkx, n = 0, 1, 2 , . . . , x e C , converges

to a unique f x e d point Fix(T) = {x0}.

Proof. From the Banach contraction principle we get

||T' x —x0|| < Y ^ -Цх-ГхЦ. / = 1, 2 , . . . , x e C . Then

CO

l | S . ( * ) - * o l l 4 I I ^ ^ x - X o | | = ||Z e« k (7 * x -x 0)||

l±

k = 0 k = 0

oo I к

< £ <*»k\\Tkx - x 0\\ < £ ||x -T x ||

k — 0 k — 0 1 ^

||x —7x|| . . _

^ - j — r^-sup {в*} ->0 if n -*• 00.

(1 ~ L ) к

oo

Hence S„(x) - 1 * . 7? -► x0 if n * со, x бС. ■

k = 0

Remark. We can notice that the elements of each strongly ergodic matrix comply with requirements (a)-(c) of Theorem 2.1.

This observation and Lemma 1.1 led us to make the following con­

clusion.

C

2.2. Let (X, ||-||) be a Banach space, С с X a non-empty convex closed subset, T : C -* C a contraction. Then for all

C, the sequence

\T " x )f=0 is almost convergent to a fixed point Fix(T) = |x 0].

Our conclusion agrees with our expectation. It appears the problem if it is possible to get an analogical result for a non-expansive mapping. Without an additional assumption on a space and a mapping it is impossible to do this. This is shown by Example 2.1. We write about positive solutions of this problem in the third part.

00

E

2.1. Let us take a space Z1 = \x = (xl5 x2, ...): £ |x,-| < oo,

X

x,e/?}. This space with a norm ||x|| = £ |х,| is a Banach space. Let C

= { x e l 1: ||x|| < 1} and let T: C -+ C be a mapping T(x1, x 2, ...)

= (0, x l5 x 2, ...). It is a non-expansive mapping having only one fixed point Q = (0, 0, 0,...). We can point in the set C a point x0 such that the sequence {Tnx 0}%L0 is not almost convergent to the point Q. It is the point x0 = ex

= (1, 0, 0, ...). It is noticeable that Tex = e2 = (0, 1, 0, .. .), T 2ex = e3, etc.

Let A = [a„k] 0!S„'k<00 be a real strongly ergodic matrix where

Then

for n ^ k, for n < k ,

n, к = 0, 1, 2, . . .

S„(e,)= I = - J - r £ T*et

k= 0 W+ 1 fc= 0

1 1

n+ 1 ’ ’ n + 1

(n+ l)-components

, 0, 0, . . . , n = 0, 1, 2, ...

Because, for each n = 0,

2, . . . , ||Sn(e1)|| =

so S ,,^ ) f+Q as n - > oo; this means that the sequence (T’"e1|£L 0 is not almost convergent to the fixed point Q.

Remark. On the ground of I. Schur’s theorem [25], the sequence is not

weakly almost convergent to the fixed point Q.

III. Non-linear ergodic theorem for the Banach spaces which satisfy the Opial condition.

D

3.1. Let (X , || ||) be a normed linear space.- We say that a space X satisfies the Opial condition of for each sequence jx„} *=0 с X weakly convergent to a point x 0 e X and for all у Ф x0, y e X

(1) lim Цх„-у|| > lim Цх„-х0||.

It is known that (1) is equivalent to the analogous condition obtained by replacing lim by lim (see [19]). The example of a Banach space which satisfies the Opial condition is every Hilbert space, the spaces lp, 1 ^ p < oo.

But not all uniformly convex Banach spaces satisfy this condition. The spaces Lp [0, 2к], 1 < p < oo, p Ф 2, do not satisfy the Opial condition [22].

De f in it io n

3.2. We denote by Г the set of all strictly increasing con­

tinuous convex functions y: [0, + oo) -> [0, + oo) with y(0) = 0. Let E о X be a non-empty convex subset of a normed linear space (X, || ||). A mapping T : E -* X is said to be of type

if

е Г and, for all x, у

E and 0 < c < 1,

у (||c73c + (1 - с) Ту - T(cx + (1 - c) y)||) < ||x - y\\ - 1| Tx - Ty\\.

Obviously, every mapping of type

is non-expansive but not every non-expansive mapping is of type

(see Example 3.1). For mappings of type

(y ),

Bruck [3] proved the mean ergodic theorem for non-expansive mappings:

Let (X, || • II) be a Banach space with the Fréchet differentiable norm, C cz X a closed convex weakly compact subset, and T: C -*■ C a non- expansive mapping such that T" is of type

for n — 1, 2, ... Then, for each x in C, {Tnxj ® о is weakly almost convergent to the unique point of Fix(T) nconv a>w(x). In the case of the uniformly convex Banach spaces which satisfy the Opial condition, Hirano [11] has got the following.

Let C be a closed convex subset of an uniformly convex Banach space which satisfies the Opial condition, let T : C -> C be a non-expansive map­

ping with a fixed point, x eC . Then the sequence {T "x}fL 0 is weakly almost convergent to a fixed point of T.

We are going to give a result for the case of a Banach space which satisfies the Opial condition. We can use results of T. Kuczumov.

L

3.1. Let (X, || * ||) be a Banach space which satisfies the Opial condition, and let D a X be a convex weakly compact subset. I f M

= |{x|m)}i= 1, m = 1, 2, ... I is a family o f sequences which are contained in D

and each o f them is weakly convergent to a point y 0 ED and z e D , and

^ lim ||x|m+1)- z || > rm+l

for all m = 1, 2, then z = y 0, where rm:= inf {lim ||xjm) — x||).

xeD . I - » x

Proof. Let E x = conv (x|m), z). c D is a separable set. It assures

m = 1 ,2 ,...

f= 1,2....

that the set E x, as weakly compact, is metrizable [23]; it means that a metric d marked in £ , induces the weak topology in this set. From the inequality in the lemma, the sequence \гт\%=1 is non-increasing and bounded, so it is convergent and there exists lim rm = r. By the Opial condition, rm

m *ol>

— lim ||xjm) — y0||, m = 1, 2, ..., and this limit is realized by a subsequence i~* oo

{ip}p=i, i-e., rm= lim ||х И - у 0||. Therefore we have

(

2

Л V Л ll*!” * 1’

m j 0 i ^ j о

~Уо\\ > r m+

~ 1

(3) Л V Л +

m in i ^ i n JW

Since E x is metrizable, d(x\m), y0) 0 for m = 1, 2, ... Therefore from the I “+QO

family of sequences M we can choose a sequence {x|Ü }*=1 convergent in the metric d to y 0, i.e., y0 as tn-*co. So from condition (2) we have

lim ||xS{"mf+1i)) - > ’oll ^ *■> and from condition (3) we have lim ||x^Vi) — z|| ^ r.

m -» x m -* -r

Therefore, the Opial condition assures that y0 = z. ■

T

3.1. Let (2f, ||*||) be a Banach space which satisfies the Opial condition, D cz X a convex weakly compact subset, and T: D -> D a non- expansive mapping. Then for all x e D

(T "x p

Fix( T)) <=>(weak- lim (T"+1 x — Г"х) = 0).

Proof. J.-P. Gossez and E. Lami Dozo in [9] proved that the set D determined in this way has a normal structure, therefore, from the Kirk theorem [16], Fix(7] Ф 0 . Let us assume that T"x p. We can show that

И -+ 0 0

peFix(T). From assumption, each subsequence T”‘ p. Then for all m

i ->oo

= 0, 1, 2, ... we have also T(",+m)x p. Let us pay attention to the fact

i - * GO

that if a sequence {T"x}£L0 is weakly convergent, then a sequence of the

norms {||Г"х||}£о is bounded. Since the space satisfies the Opial condition

and T is a non-expansive mapping, we have

inf(lim ||7^"i+’">X —17||: v<=D) = lim ||Т0ч + м)х - р ||

^ lim \\Т(щ+т+1)х~Тр\\ > lim \\T{”i+m+1)x - p \\

i - * QO l - ► 00

= inf(lim ||T<"i+m+1)x —y||: veD ) i — ► со

for m = 0, 1, 2, ... By Lemma 3.1, Tp = p. From T nx p we get that rt cc

weak- lim (Tn+1 x - T"x) = 0.

П -► QO

Now we are going to show the implication in the other way. Let cow(x) determine the set of weakly limits of subsequences of a sequence {T nx}%L0.

Since D is weakly compact, by the Eberlein-Smulian Theorem [23] each sequence concluded in D contains a weakly convergent subsequence, i.e., for a sequence \T nx}%L0 cz D, cow(x) Ф 0 . Now we are going to show that

<uw(x) cz Fix(T). From the assumption weak- lim (Tn+1 x — T nx) — 0 we have

,„. + m)

" _>0°

T 1 x y 0 for m = 0, 1, 2, ... And, as the weakly convergent sequences,

i —► QO

they are bounded. From Lemma 3.1, like in the first part of the proof, we get Ty0 — Уо and c=Fix(T). We can now show that <uw(x) has a unique point. Let us assume that w, and vv Ф y 0. Since cow{x) czFix(T), the sequence of the norms {||T"x — y0ll}^=o is convergent and lim ||T"x —y0|l

____ n-*oo

= r. On the other hand, from the Opial condition, lim ||T"1

— w|| > r for

I - ► 0 0

w # y 0- Since vveci)w(x), there exists a subsequence T"px w and there

p - ► 00

exists the limit lim ||T"x — w||.

n-^oo

Collecting our results, we obtain

r < lim ||T"‘ x —

= lim ||T"px — wj| < lim ||T”px — y0ll = r.

i -+ oo p -► oo p->oo

This contradiction proves that cow(x) = {y0j, so T"x y0 and this com-

И —► CO

pletes the proof of the theorem.

From the proof of Theorem 3.1 we have got a corollary immediately:

C

3.1. Let (X , || -||) be a Banach space which satisfies the Opial condition, C cz X a convex weakly compact subset, and T : С -* C a non- expansive mapping such that for each x e C weak- lim (T”+1 x — T nx) = 0.

Then the set Fix(T) nconv <uw(x) has a unique point.

These results led us to give a non-linear ergodic theorem for a Banach space which satisfies the Opial condition.

Th e o r e m

3.2. Let (X , ||-||) be a Banach space which satisfies the Opial

condition, C cz X a convex weakly compact subset, and T

C -> C a mapping of type (y) such that, for all x e C , weak- lim (T"+1 x — T nx) = 0. Then the

П “*X

sequence \T nx}fL 0, xeC , is weakly almost convergent to the unique point Fix ( T) n conv cow (x).

Proof. Since Г is a mapping of type (y), so it is a non-expansive mapping, {T nx}«L0 c C for any x e C and lim ||T"+1 x — T"x|| = 0. Therefore

п-+ 0 0

the assumptions of Theorem 1.1 of [3] are satisfied. It means that for each weak neighbourhood W of the set Fix (7) Ф 0 there exists a natural number N such that

1 " - 1

S„{x) = - X T i+kx e W for n ^ N , k ^ O .

n i =0

Now, it is sufficient to show that the sequence !S„(x)|^=1 is weakly conver­

gent. We have learned that all limits of subsequences of the sequence (x)] ! are included in Fix(T). On the other hand, they belong to the set

00

____ ____

n conv ITkx: k ^ i } which is equal to convmw(x) (see [2], Lemma 1.2).

;=o

Therefore all weak limits of subsequences of the sequence |5„(х)},^ x belong to the set Fix (T) n conv cow(x), which, by Corollary 3.1, has a unique

point, ш

Remark. Bruck in paper [2] gave an example of a Hilbert space and a mapping T: let F : [0, 27i] — [0, 1] be the Cantor-Lebesgue function with dissection ratio Let \EX) be the resolution of the identity on the complex space L2(0, 1) defined by Exf = f mC[0tFW], where Cs denotes the character-

2n

istic function of the set S. Put T = f ea dEx. Then T is unitary, in particular ô

non-expansive, and the sequence [Tn\) fL 0 is not weakly convergent to 0.

Not every non-expansive mapping of a compact convex subset of a Banach space which satisfies the Opial condition into itself is of type (y). The following example shows it:

Ex a m p l e

3.1 (DeMarr, [5]). The space

with the norm ||(n, b)||

= max {|a|, |b|] is a Banach space which satisfies the Opial condition. Let B(0, 1) = { x e R 2: ||x|| ^ 1] and T: B (0, 1) ->£((), 1) described by the for­

mula T((a, b)) = (|b|, b) is a non-expansive mapping. Let us assume that the mapping T is of type (y). Consider x = (1, 1), у = (1, - 1 ) , c = Then cTx + (l — с) Ту— T(cx + (1 — c)y) — (1, 0); this means that y(||(l, 0)||)

<11(1, 1 )-(1 , —1)|| — ||(1, 1)—(1, -1 )11= 0, so y(l) = 0. Since у(0) = 0, we get the result which is contradictory to the fact that у is a strictly increasing function.

Now, let us consider the mapping Tx : = X1 + (1—X)T, 0 < X < 1, where

~ Roczniki PTM — Prace Matematyczne XXVIII

T: C -> C is a non-expansive mapping of a closed convex subset C of a linear normed space (then Tx is a non-expansive mapping and Fix(T)

= Fix(7})). Browder and Petryshyn [1] proved the following theorem.

I f ( X , ||*||) is a uniformly convex Banach space and T : C —> C is a non- expansive self-mapping on a closed bounded convex subset C, then Tx, 0 < Â

< 1, is asymptotically regular, i.e., lim \\TX+X x — Tfx\\ = 0, x e C .

«-► 00

Edelstein and O’Brien in [ 6] observed that, in a Banach space which satisfies the Opial condition, if T is a non-expansive mapping of a weakly compact convex subset C e l into itself, then for any x e C the sequence

{T”x}fL о is weakly convergent to a fixed point of T.

This result in connection with Theorems 3.1 and 3.2 gives the following.

Co r o l l a r y

3.2. Let (X , || *||) be a Banach space which satisfies the Opial condition, C c X a weakly compact convex subset and T: C —> C a non- expansive mapping o f type (y). Then the sequence {T"x}fL0, x e C , 0 < A < 1, is weakly almost convergent to the point Fix (7) r> conv tow (x).

IV. Some remarks on the Krasnoselskii iterative process. In [18] Kras- noselskii proved that if X is a uniformly convex Banach space, C c X is a compact subset, and T: C -* C is a non-expansive mapping, then for any x e C the sequence of iterates |((/+ T)/2)n{x)\f=0 converges to a fixed point of Schaefer in [24] observed that the same is true for the sequence \T fx \? =0 with 0 < A < 1. Next, Ishikawa proved in [12], among other results that if D a X is a closed subset in a Banach space and T: D -> T(D) c D is a non- expansive mapping such that T(D) is a compact subset of D, then T has a fixed point in D and the sequence {((/ + T)/2)n{x))f=0 converges to a fixed point of T.

Genel and Lindenstrauss proved in [7] that it is impossible to improve the results obtained. They proved that in the Hilbert space l2 there is a non- expansive mapping T determined on a closed convex bounded subset for which the sequence {((I+T)/2)"(*)}®=0 does not contain a convergent subse­

quence. A generalization of the presented results on weak convergence of the sequence { T fx } fL 0, 0 < A < 1, is referred before the result of Edelstein and O’Brien. But it is still impossible to generalize this result as well. The following example shows it:

CO

Ex a m p l e

4.1. Let X = l1, ||x|| = £ W This is a Banach space with the i=i

Opial condition. Let C = { x e l1: ||x|| < 1}. On the ground of the Alaoglu theorem [23], it is a weak-(*)-compact set. Let T : C -> C be a non-expansive mapping such that T ( x t , x 2, ...) = (0, xt , x 2, ...). This mapping has a fixed point Q = (0, 0, 0, ...). Let us consider the sequence [T"/2x0} ^ 0, where x0

= (1 ,0 , 0, ...). It is easy to notice that

where in the numerator we have successive terms of nth line of the Pascal triangle. This sequence is not convergent to the element Q, because ||T"/2^oli

= 1 for n — 1, 2, . . .

V. Open problems. 1. Let (X , ||*||) be a Banach space which satisfies the Opial condition, С <= X a convex weakly compact subset, and Г: C -> C a non-expansive mapping. Is the sequence {T nx}%L0, x e C , under these condi­

tions, weakly almost convergent to a fixed point of the mapping T?

2. Is the non-linear means ergodic theorem true in uniformly convex spaces?

References

[1] F. E. B ro w d er, W. V. P e tr y s h y n , The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 571-575.

[2] R. E. В ruck, On the almost-convergence o f iterates a nonexpansive mappings in Hilbert space and the structure o f the weak co-limit set, Israel J. Math. 29 (1978), 1-16.

Г31 — , A simple proof o f the mean ergodic theorem for nonlinear contractions in Banach spaces, ibidem 32 (1979), 107-116.

[4] - , Asymptotic behavior o f nonexpansive mappings. Contemporary Math. 18 (1983), 1—47.

[5] R. D e M a r r , Common fixed points for commuting contraction mappings, Pacific J. Math. 13 (1963), 1139-1141.

[6] M. E d e is t e in , R. C. O’B rien , Nonexpansive mappings, asymptotic regularity and succes­

sive approximations, J. London Math. Soc. (2) 17 (1978), 547-557.

[7] A. G e n e l, J. L in d e n s t r a u s s , An example concerning fixed points, Israel J. Math. 22 (1975), 81-86.

[8] K. G o e b e l, S. R eich , Uniform convexity, hyperbolic geometry and nonexpansive mappings, Marcel Dekker, INC., New York-Basel 1984.

[9] J.-P. G o s sez, E. La mi D o z o , Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565-573.

[10] N. H ira no, A proof of the mean ergodic theorem for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 78 (1980), 361-365.

[H ] —, Nonlinear ergodic theorems and weak convergence theorems, J. Math. Soc. Japan 34 (1982), 35-46.

[12] S. Is h ik aw a, Fixed points and iteration o f a nonexpansive mapping in Banach spaces, Proc.

Amer. Math. Soc. 59 (1976), 65-71.

[13] L. V. К a r lo v it z , On nonexpansive mappings, ibidem 55 (1976), 321-325.

[14] W. A. K irk, A fixed point theorem for mappings with do not increase distances, Amer.

Math. Monthly 72 (1965), 1004-1006.

[15] - , On successive approximations for nonexpansive mappings in Banach spaces, Glasgow Math. J. 12 (1971), 6-9.

[16] — ; Fixed point theory for nonexpansive mappings, Fixed Point Theory, Proc. Workshop Univ. Sherbrooke 1980, Lecture Notes in Math. vol. 886, Springer-Verlag, Berlin 1981, 484-505.

[17] W. A. K irk , Fixed point theory for nonexpansive mappings II, Contemporary Math. 18 (1983), 121-140.

[18] M. A. K r a s n o s e ls k ii , Two observation about the method o f successive approximations (in Russian), Uspehi Mat. Nauk 10 (1955), 123-127.

[19] E. La mi D o z o , Multivalued nonexpansive mappings and O pials condition, Proc. Amer.

Math. Soc. 38 (1973), 286-292.

[20] G. G. L o r e n tz , A contribution to the theory o f divergent series, Acta Math. 80 (1948), 167-190.

[21] S. M a ssa , Some remarks on Opial spaces, Bolletino U.M.I. (6) 2A (1983), 65-69.

[22] Z. O p ia l, Weak convergence o f the sequence o f successive approximations fo r nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.

[23] S. R o le w ic z , M etric Linear Spaces, second enlarged edition, PWN and D. Reidel Publishing Company, Warszawa-Dordrecht-Boston-Lancaster 1984, 226-234.

[24] H. S c h a e fe r , Über die Methode sukzessiver Approximationen, Iber. Deutsch Math.

Verien. 59 (1957), 131-140.

[25] K. Y о si da, Functional Analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-Heidelberg-New York 1971.

INSTITUTE O F MATHEMATICS, PEDAGOGICAL UNIVERSITY RZFSZÔW, POLAND