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

**J**

**aroslaw**

** G**

**ôrnicki**

**(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, **

**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,**

*v g*

**C. In**

**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**

**{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,**

**(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,**

**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,**

**(B) Д Hm a„k = 0,**

*к n -» x*

### (Q A Z

*a nk*

### = *h*

*П k = 0*

*00*

**(D) lim £ Kfc + 1- e J = 0.**

" - o o fc= 0

**There is a lemma.**

**L**

**emma**

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

**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*

**(i) a sequence \x„)f=0 а X is (weak-)almost convergent to point x e X ,**

**(ii) for each strongly ergodic matrix A =**

oc

**(weak-) lim ** *ankxk — x.*

**(weak-) lim**

*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.**

**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**

**heorem**

**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*

**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**

**(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**

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

**k = 0**

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

**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**

**/ = 1, 2 , . . . , x e C .**

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\\Tkx - x 0\\ < £**

*k — 0 * *k — 0 * 1 ^

**||x —7x|| ** **. ** **. ** **_**

**^ - j — r^-sup {в*} ->0 ** **if n -*• 00.**

**if n -*• 00.**

**(1 ~ L ) к**

**(1 ~ L ) к**

oo

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

**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**

**orollary**

**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 **

**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**

*x e*

*C, the sequence *

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

**\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**

**xample**

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

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

**i— 1**X

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

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

**= { x e l 1: ||x|| < 1} and let**

**T: C -+ C be a mapping**

**= (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 **

**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. **

**= (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**

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

**Then**

**for n ^ k, ** **for n < k ,**

**for n ^ k,**

**for n < k ,**

**n, к = 0, 1, 2, . . .**

**n, к = 0, 1, 2, . . .**

**S„(e,)= I ** **= - J - r £ T*et**

**= - J - r £ T*et**

**k= 0 ** **W+ 1 fc= 0**

**k= 0**

**1 ** 1

**n+ 1 ’ ** **’ n + 1**

**’ n + 1**

(n+ l)-components

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

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

**Because, for each n = 0, **

**Because, for each n = 0,**

^{1}**,**

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

**1**

**,**

**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.**

**so S ,,^ ) f+Q as n - > oo; this**

**point Q.**

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

**weakly almost convergent to the fixed point Q.**

**weakly almost convergent to the fixed point Q.**

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

**D**

**efinition**

** 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**

**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. **

**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].**

**Lp [0, 2к], 1 < p < oo, p Ф 2, do not satisfy the Opial condition [22].**

**D****e 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 **

**tinuous convex functions y: [0, + oo) -> [0, + oo) with y(0) = 0. Let E о X be**

**T : E -* X is said to be of type**

**(y )**

**if **

**у**

**е Г and, for all x, у **

**е Г and, for all x, у**

*g*

**E and 0 < c < 1,**

**E and 0 < c < 1,**

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

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

**Obviously, every mapping of type **

**(y )**

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

**(y )**

**(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 **

**C cz X a closed convex weakly compact subset, and T: C -*■ C a non-**

**(y )**

**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.**

**for n — 1, 2, ... Then, for each**

**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**

**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.**

**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**

**emma**

** 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 *

**Let (X, || * ||) be a Banach space which satisfies the Opial**

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

**= |{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**

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

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

**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**

**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**

**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**

**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*

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

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

**{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+*

*2*

*~* 1

**m + 2 ’**

**(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**

**Since E x is metrizable, d(x\m), y0)**

**0 for m = 1, 2, ... Therefore from the**

**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**

**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.**

**lim ||xS{"mf+1i)) - > ’oll ^ *■>**

**m -» x ** **m -* -r**

**m -* -r**

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

**T**

**heorem**

** 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*

**Let (2f, ||*||) be a Banach space which satisfies the Opial**

*(T "x * *p*

**g**

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

**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**

**Proof. J.-P. Gossez and E. Lami Dozo in [9] proved that the set D**

**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**

**and T is a non-expansive mapping, we have**

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

**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 \\**

**^ 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**

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

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

**for m = 0, 1, 2, ... By Lemma 3.1, Tp = p. From T nx**

**p we get that**

**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.**

**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**

**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**

**<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,**

**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**

**Ty0 — Уо and**

**point. Let us assume that w,**

**and vv Ф y 0. Since cow{x) czFix(T),**

____ n-*oo

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

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

^{x}**— w|| > r for**

**— w|| > r for**

I - ► 0 0

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

**w and there**

p - ► 00

**exists the limit lim ||T"x — w||.**

n-^oo

**Collecting our results, we obtain**

**r < lim ||T"‘ x — **

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

**= 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-**

**This contradiction proves that cow(x) = {y0j, so T"x**

И —► CO

**pletes the proof of the theorem.**

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

**C**

**orollary**

** 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.**

**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.**

**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.**

**T****h e o r e m**

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

**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**

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

**П “*X**

**П “*X**

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

**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**

**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 .*

**S„{x) = - X T i+kx e W**

* 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).**

**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-**

**mapping T: let F : [0, 27i] — [0, 1] be the Cantor-Lebesgue function with**

**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**

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

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

**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:**

**E****x a m p l e**

**3.1 (DeMarr, [5]). The space **

**R 2****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**

**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)||) **

**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 =**

**cTx + (l — с) Ту— T(cx + (1 — c)y) — (1, 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**

**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) **

**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**

**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 .**

**< 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 **

**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.**

**{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.**

**C****o 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).**

**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.**

**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**

**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**

**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:**

**sequence { T fx } fL 0, 0 < A < 1, is referred before the result of Edelstein**

CO

**E****x a m p l e**

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

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

**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 **

**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 **

**triangle. This sequence is not convergent to the element Q, because ||T"/2^oli**

**= 1 for n — 1, 2, . . .**

**= 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**

**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