Pacific Journal ofMathematics

COMMON FIXED POINTS AND ITERATION OF COMMUTINGNONEXPANSIVE MAPPINGS

SHIRO ISHIKAWA

Vol. 80, No. 2 October 1979

PACIFIC JOURNAL OF MATHEMATICSVol. 80, No. 2, 1979

COMMON FIXED POINTS AND ITERATION OFCOMMUTING NONEXPANSIVE MAPPINGS

SHIRO ISHIKAWA

The following result is shown. Let Tt(i = 1, 2, , v) becommuting nonexpansive self-mappings on a compact convexsubset D of a Banach space and let x be any point in ZλThen the sequence

IΓ π Γ&, l ί T Γs, π \s2 itIL v-i-iL ^ - 2 = i L L w i = 1 L % o = 1

converges to a common fixed point of {Γ}ϊ=i, where S< =(1 — αjJ + (XiTi, 0 < at < 1, I is the identity mapping.

In [2], DeMarr proved that if Tt(i eJ,J is an index set) arecommuting nonexpansive self-mappings on a compact convex subsetD of a Banach space (i.e., || Tx — Ty\\ ̂ ||a? — y\\ for all α?, ?/ in D,and T ^ = TjT, for all i, i e J ) , then TlίeJ) have a common fixedpoint in Ώ.

The problem we shall consider in this paper is that of constructinga sequence of points {xn}n=ι in D that converges to the common fixedpoint of Ti (i e J, J is a finite index set).

If a Banach space is strictly convex (i.e., \\ax + (1 — a)y\\ <max {|| a? 11, \\y\\) for a? Φ y, 0 < α: < 1), the problem was solved in [5].

Throughout this paper, we denote an identity mapping by I andthe set of fixed points of T by F[T]. And we define UtS Tt =Tn+1(Ut=i Ti) for any positive integer n and Π U Tt = 2V

We have the following main theorem.

THEOREM. Let Tt{i = 1, 2, v) be commuting nonexpansivemappings from a compact convex subset D of a Banach space intoitself, and let x be any point in D.

Then Cΐί=ιF[Ti] is nonempty and the sequence {x^J converges toa point in Πϊ^-PΊTiL where xn^ is defined for each positive integernt by

Γ π Γ-s, ϊ ϊ 1 Γsu • . Γs3 π \s2 π sj\... J\]x

where S< = (1 - α4)/ + α^T*, 0 < α* < l(i = 1, 2, , v).

Before proving the theorem, we first prove the following lemmason which the proof of theorem is based.

LEMMA 1. Let T and P be nonexpansive mappings from a

493

494 SHIRO ISHIKAWA

bounded convex subset D of a Banach space into itself that satisfythe conditions

( 1 ) P{D) = F[P] and T(P(D)) c P(D) .

Let x0 be any point in D and let a be any number such that 0 < a < 1.Then the sequences {xn — Txn}%=Q and {xn — Pxn}n=o respectively convergeto zero, where xn is defined for each positive integer n by

( 2 ) xn = (1 - a)yn + aTyn , yn = P

that, is xn = (SP)nx0, where S = (1 - ά)I + aT.

Proof. We see from (1) that for all n ^ 1

( 3 ) yn = Pyn and Tyn

Since T and P are nonexpansive mappings, we have, from (2) and(3), for all n ^ 0

\\yn+1 - Tyn+1\\ = \\Pxn - PTyn+1\\ ^ \\xn - Tyn+1\\

and, from (2) and (3), for all n ^ 1

II*• - Γ^n + 1 | | ^ | | ^ - Tyn\\ + \\Tyn - Γ».+ 1 | |

^ (1 — α) | | l/ — Tyn\\ + \\yn-yn+1\\

^ (1 - a)\\yn - Tyn\\ + | \Py% - P x J (

- Γy.H + a\\yn - Γ y J | = \\yn - ΓyJI

from which, we obtain

\\yn+1 - Γ ^ + 1 | | ^ | |xw - Γy n + 1 | | ^ \\yn - Γy.H for all n ^ l .

Hence the sequence {\\yn — 2fyJ|}ίU, which is nonincreasing andbounded below, has a limit.

Suppose t h a t lim \\yn — Tyn\\ = r > 0, t h a t is, for any ε > 0,there is an integer m such t h a t

( 4 ) r^\\yn-Tyn\\<,(l + ε)r for all n^m.

Also, from the boundedness of D, we can choose M such t h a t

( 5 ) L <; (If — m)r < 2L , where L is a diameter of D .

We have from (3), (2) and (4) t h a t for any n^ m and k ^ 0

\\yn - yn+k+1\\ ^ \\y» - 2/»+ill + ll2/»+i - 2/ + 2 | | + ••• + ll3/»+* - 2

\yn+k - xn+k\

ITERATION OF COMMUTING NONEXPANSIVE MAPPINGS 495

( 6 ) ^ a(k + 1)(1 + β)r .

Now we shall prove by induction that

( 7 ) (1 + ak){l + e)r - (1 - a)~hr S II TyM - yM-k\\

for any k such that 0 ^ k <̂ M — m.When & = 0, the result is trivial. Now we assume that (7) is

true for some k such that 0 ^ k <L M — m — 1. We see, from (3)and (2), that

\\TyM - yM-k\\ = \\PTyM - P ^ _ ( f c + 1 ) | | ^ \\TyM - ^_ ( f c + 1 ) | |

= | | (1 - a){TyM - yM-{k+1)) + a(TyM - TyM_{k+1))\\

^ ( 1 - ά)\\TyM - VM-U+DW +OL\\yM- yM-ik+i)\\

from which and (6), it follows that

II TyM - yM-k\\ ^ (1 - α ) | | TyM - yM-{k+1)\\ + a\k + 1)(1 + e)r .

From this and the assumption by induction, we have

(1 + ak)(l + e)r - (1 - ά)-kεr

^ (1 - α ) | | TyM - yM-{h+ι)\\ + ^2(& + 1)(1 + ε)r

and it is clear that this inequality is equal to (7) with k + 1 for k.Hence, by induction, it follows that (7) is true for any k such that0 ^ k ^ M - m.

Since log(l + ί) ^ £ for all £e( — 1, <*>), we have from (5) that

(1 - a)-'M~m) = expΓ(M - m) log (l + a )Ί

exp Γ(M - m ) — £ - ] ^ exp ( 2L

L 1 αJ] ^ exp (

Thus it follows from (7) with M — m for k that

2 LTyM - yu\\ ^ (1 + α(Jlf - m))(l + e)r - er exp

— a)r

Since ε is any positive number, this inequality is imcompatible withthe definition of L. Hence we obtain that r = 0, that is,

(8) lim\\yn-Tyn\\ - 0 .ίl.->oo

Now since T and P are nonexpansive mappings, we have from(2) and (3) that, for all n ^ 1,

4% SHIRO ISHIKAWA

\\xn - Tx%\\ = | |(1 + a)yn + aTy, - Γ((l - a)yn + aTy.)\\

= ||(1 - α)y. - (1 - a)Tyn + 2ty. - Γ((l - a)yn + aTyn)\\

^{l-a)\\yn-Tyn\\+a\\yn-Tyn\\

= \\V.- 2V.II

and

||». - PxJI = ||(1 - a)yn + aTyn - P((l - a)yn + aTyn

- ||(1 - a)[Py, - P((l - α)y. + aTy.)]

+ a[PTyn - P((l - a)yn + aTyn)]\\

tZ2a(l-a)\\v.-Ty.\\.

Therefore we obtain that from (8) that

K - Txn\\ = lim \\xn - Px,\\ = 0 .

LEMMA 2. Let T and P be nonexpansive mappings from acompact convex subset D of a Banach space into itself such that

( 9 ) P(D) = F[P] and T(P(D)) c P(D) .

Let x0 be any point in D. Define xn = Pnx0 for each positive integern, where Pn = (SP)n, S = (1 - a)I + aT, 0 < a < 1. Then it followsthat

(10) for any x0 in D, lim^^ (SP)nx0 — Px0 exists, which is, denotedby Px0,

(11) P{D) = F[P] =F[T]Π F[P]

and

(12) {PJn=i converges uniformly to P .

Proof. Since D is compact, there exists a subsequence {xn.}Z=iof {xn} that converges to a point u in D. From the boundedness ofD, Lemma 1 is applicable, so we have,

| |u - Tu\\ ̂ \im{\\u - xnt\\ + \\xnt - Tx%ί\\ + \\Txni - Tu\\}i—*oo

^ lim {21|α^ -n\\ + \\xni - TxH\\} = 0 ,

and similarly \\u — Pu\\ = 0.From this, it follows that

(13) ueF[T]ΓιF[P] .

Since (9) implies (3), we see from (13) and (3) that for all n ^ 0,

ITERATION OF COMMUTING NONEXPANSIVE MAPPINGS 497

n - xn+1\\ = \\u - ((1 - a)yn+ί + aTyn+ί)\\

^ (1 - α)| |t* - yn+1\\ + α | | 2V - Tyn+ι\\

S \\u - ^ + 1 | | = \\Pu - Pa . 11 ̂ Hit - xn\\ .

From this, we obtain that limn_+oo\\u — xn\\=\imn.^oo\\u — xn.\\=0. Hencewe have proved that (10) is true, that is, for any xQ in D, P(x0) =lim^(SP)*x0 is well-defined. From (13), we see that P(x0) eF[T]Γ\ F[P]for all x0 in D, that is,

(14) P(D) (Z F(T) Π F[P] .

And we have that, for any v in F[T] Π -P[P],

= Km (SP) t; = Pv ,

so we see that

(15) F[T]nF[P]aF[P].

Also, clearly w = PweP(D) for all w in F[P]. From this, (14) and(15), we get (11).

Finally we shall prove (12). Let ε be any positive number.Since D is compact, there are finite points {x\, x\, •••, x%) such that,for any x in D,

(16) min{ | | x - α?S||i 1 ^ i ^ k} < ± .3

F r o m (10), we can choose N such t h a t

(17) | | (SP) a ? ί - P α ? ί | | < — for all n ^ N and l ^ i ^ k .3

Let #o be any point in D. From (16), we can take x{ such that

(18) Hso-ao'lK-f-.o

Since >SP is nonexpansive, clearly P is also nonexpansive. Hence weobtain from (17) and (18) that, for all n^N,

\\(spyχo - PxQ\\

^ \\(spyχo - (spyxiw + \\(SP)*χi - Pχt\\ + np&ί - Pa?0||

^ 2||aj0 - x{\\ + \\(SPyxi - Pxi\\ ^ ε

which implies (12).

LEMMA 3. Let T and Pn(n — 1, 2, •) δe nonexpansive mappingsfrom a compact convex subset D of a Banach space into itself. Assume

498 SHIRO ISHIKAWA

that the following conditions are satisfied:

(19) for any x in D, lim% ̂ Pnx — Px exists ,

( 2 0 ) P{D) = F[P] c i ^ [ P J / o r α ί ϊ n ^ l

( 2 1 ) PTO converges uniformly to P

and

(22) T(P(D)) c P(D) .

Then it follows that

(23) for any x in D, l inv^ Pnx — Px exists, where Pn = Π?=i (SP<),S = (1 - α)I + αΓ, 0 < a < 1,

( 2 4 ) P C D ) - F [ P ] - F [ T ] ΓΊ F [ P ] c F [ P J for all n ^ l

and

(25) Pn converges uniformly to P .

Proof. Let ε be any positive number. Since P satisfies theconditions of P in Lemma 2, from (12), we can choose N such that

(26) \\(SP)Ny -py\\<± for all y i n ' f l ,Δ

where P is defined as in Lemma 2.Prom (21), there exists M such that

\\SPx - SPnx\\ ^ \\Px - Pnx\\ ^ ε

2N

for all n Ξ£ M and all x in Z>.This implies that, for all n such that n ^ M

W

From this and (26), we have that, for all n such that n >̂ max {AT, M},

\\Pnx- P{P^Nx)\\ £ \\Pnx - ^

< ε .

ITERATION OF COMMUTING NONEXPANSIVE MAPPINGS 499

Since Lemma 2 says that P(D) = J^[JΓ] Π F[P], this implies that thereexists a subsequence {Pn.x}ΐ=1 that converges to a point u in F[T] ΠF[P]. Also we see, from (20), for all n ^ 1,

llΛ+i* - u\\ = \\SPn+1Pnx

Hence we get that lim^^W Pnx — u\\ = lim^JIP^.a; — w|| = 0 , that is,Pnx converges to a point in F[T] Π ̂ [ P ] for any x in D. This implies(23), and

(27)

If v e F[T] Π i*TP], then v = P%v = l i m ^ Pnv = Pv, so we see

(28) JP[Γ]Π F[P]dF[P].

Since clearly F [ P ] c P(D) and F[T] n F[P] c F [ P J for all w ^ 1, (24)follows from (27) and (28).

Now we shall prove (25). Let ε be any positive number. Asin the proof of Lemma 2, we can choose finite points {x\, xl, , x%)from D satisfying (16). From (23), we can choose N' such that

(29) WPX-PxlW^— for all n ^ N' and l ^ ί ^ k .3

Let x0 be any point in D. By (16), we can take x( that satisfies (18).Since P is nonexpansive, we obtain from (18) and (29) that, for

all n ^ N',

\\Pnx0~-Px0\\^ \\Pnx0 - Pnxi\\ + \\Pnxi - Pxi\\ + \\Pxi - Pxo\\

^ 2 | | α ? o - » o i l l + \\Pnxί - Px{\\ ̂ e .

This implies (25).

LEMMA 4. Let Γ^i = 1, 2, •••,&) be a commuting family ofmappings. Then it follows that

Proof. Let x be any point in f\ϊ=i F[Tt\. We see that Tkx =ΓfcTίίc = T̂ΓfcCC for all i such that 1 ^ ί ^ fc — 1, which implies thatTkx belongs to F[T%] for all 1 ^ i ^ & - 1.

Proof of theorem. For all i such that 1 ^ i ^ v, put

Γ Π Γs* IΪ 1 ΪS^ Γs2 Π Sx~\ ΊΊΊ* = Pi«aj .

500 SHIRO ISHIKAWA

We shall prove the theorem by induction. Let us assume thatthe following conditions are true for some integer j such that 1 <Ξ3 ^ v - 1:

(30) for any x in D, lim^ ̂ P{J]x = PU)x exists,

(31) PU)(D) = F[Pij)] = h F f Γ J c f f P ί ] for all integers ns S 1 ,ί = l

(32) {-Pn]}̂ -=i converges uniformly to P ( i )

and

(33)

Since P ^ x = [Πϊjΐϊ 0S;+ 1P^)Jk we can apply Lemma 3 byregarding T^lf S, +1, P

u\ P%, Ptf™, Pi3'+1) and conditions (30)-(33) asT, S, P, PΛ, PΛ, P and conditions (19)-(22). Hence we have,

(34) for any x in D, lim% i + 1_0 OP^+1^ = Pu+1)x exists ,

Γ(35) Pu+ί)(D) = F[P{j+ί)] - Γ\F[Tt] c ίΊP&ϊ"] for all wyί = l

and

(36) { P ^ J ^ + ^ i converges uniformly to P o + 1 ) .

Moreover, if j + 2 ^v, Lemma 4 shows from (35) that

(37) Tj+1(P{j+1))(D) c P{j+1\D) .

When i = 1, conditions (30)-(32) immediately follow by regarding Pin Lemma 2 as an identity mapping. Also from (31) and Lemma 4,we get (33).

Therefore, by induction, it follows that lim^oo P^x — PMx ePM(D) - Γ\Ui F[Tt]. This completes the proof of the theorem.

From the finite intersection property, we have the followingresult due to DeMarr [2]. And note that we do not assume Zorn'slemma in our proof.

COROLLARY 1. Let Tt(i e J, J is an index set) he commutingnonexpansίve mapping from a compact convex subset of a Banachspace into itself. Then there exists a point u in D such that T{w — ufor all ie J.

When v = 1 and at = 1/2, we have the following corollary, whichis essentially equal to the result we have obtained as a Corollary 2in [3].

ITERATION OF OCMMUTING NONEXPANSIVE MAPPINGS 501

COROLLARY 2. Let T be a nonexpansive mapping from a compactconvex subset D of a Banach space into itself. Then {((I + T)/2)wx}~=1

converges to a fixed point of T.

The author would like to thank the referee for letting me knowabout reference [5].

REFERENCES

1. L. P. Belluce and W. A. Kink, Fixed point theorems for families of contractionmappings, Pacific J. Math., 18 (1966), 213-217.2. R. E. DeMarr, Common fixed-points for commuting contraction mappings, PacificJ. Math., 13 (1963), 1139-1141.3. S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banachspace, Proc. Amer. Math. Soc, 59 (1976), 65-71.4. T. C. Lim, A fixed point theorem for families of nonexpansive mappings, PacificJ. Math., 53 (1974), 487-493.5. J. Linhart, Beitrdge zur Fixpunkttheorie nichtexpandierender Operatoren, Mona-tshefte fur Mathematik, 76 (1972), 239-249.

Received May 30, 1978.

FACULTY OF ENGINEERING KEIO UNIVERSITY

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Pacific Journal of MathematicsVol. 80, No. 2 October, 1979

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nonexpansive mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493Philip G. Laird, On characterizations of exponential polynomials . . . . . . . . . . 503Y. C. Lee, A Witt’s theorem for unimodular lattices . . . . . . . . . . . . . . . . . . . . . . . 509Teck Cheong Lim, On common fixed point sets of commutative

mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517R. S. Pathak, On the Meijer transform of generalized functions . . . . . . . . . . . . 523T. S. Ravisankar and U. S. Shukla, Structure of 0-rings . . . . . . . . . . . . . . . . . . . 537Olaf von Grudzinski, Examples of solvable and nonsolvable convolution

equations in K′p, p ≥ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561Roy Westwick, Irreducible lengths of trivectors of rank seven and eight . . . . 575

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1979Vol.80,N

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