Convergence of Krasnoselskii-Mann iterations of nonexpansive operators

MATHEMATICAL COMPUTER MODELLING PERGAMON

Mathematical

and Computer

Modelling 32 (2000)

1423-1431 www.elsevier.nl/locate/mcm

Convergence of Krasnoselskii-Mann Iterations of Nonexpansive Operators S. Department

REICH

AND

of Mathematics,

A.

J. ZASLAVSKI

The Technion-Israel 32000

Haifa,

Institute

of Technology

Israel

Abstract-In

this paper, we show that a generic nonexpansive operator on a closed and convex, but not necessarily bounded, subset of a hyperbolic space has a unique fixed point which attracts the Krasnoselskii-Mann iterations of this operator. @ 2000 Elsevier Science Ltd. All rights reserved.

Keywords-Complete sive operator.

metric space, Generic property, Hyperbolic space, Iterations,Nonexpan-

1.

INTRODUCTION

Iterations of nonexpansive operators find application in many areas of mathematics (see, for example, [l-7] and the references mentioned there). In this paper, we study the convergence of Krasnoselskii-Mann iterations of nonexpansive operators on a closed and convex, but not necessarily bounded, subset of a hyperbolic space. More precisely, we show that in an appropriate complete metric space of nonexpansive operators, there exists a subset which is a countable intersection of open everywhere dense sets such that each operator belonging to this subset has a (necessarily) unique fixed point and the Krssnoselskii-Mann iterations of the operator converge to it. Results of this kind for powers of a nonexpansive operator on a closed bounded convex subset of a Banach space had already been established by DeBlasi and Myjak [8]. The approach used by them and in the present paper is common in global analysis and the theory of dynamical systems [9,10]. Recently, it has also been used in the study of the structure of extremals of variational and optimal control problems [11,12]. Thus, instead of considering a certain convergence property for a single operator, we investigate it for a space of all such operators equipped with some natural metric, and show that this property holds for most of these operators. This allows us to establish convergence without restrictive assumptions on the space and on the operators themselves. Let (X, p) be a metric space and let R1 denote the real line. We say that a mapping c : R’ -+ X is a metric embedding of R’ into X if p(c( s), c(t)) = 1s - tJ for all real s and t. The image of R1 under a metric embedding will be called a metric line. The image of a real interval [a, b] = (t E R1 : a < t < b) under such a mapping will be called a metric segment. Assume that (X,p) contains a family M of metric lines such that for each pair of distinct points x and y in X there is a unique metric line in M which passes through x and y. This

08957177/09/% - see front matter PII: SO895-7177(90)00214-4

@ 2000 Elsevier Science Ltd.

All rights reserved.

Typeset

by &S-m

1424

S. REICH AND

metric line determines

A. J.ZASLAVSKI

a unique metric segment joining x and Y. We denote this segment by

[z, y]. For each 0 5 t I 1, there is a unique point z in [z, y] such that and

P(G z) = @(x7 Y)

P(G Y) = (1 - GJ(?

Y).

This point will be denoted by (1 - t)x @ ty. We will say that X, or more precisely (X, p, M), is a hyperbolic

space if p

+3

fY+I+

I $(y,z),

(

>

for all x, y, and z in X. An equivalent requirement is that

for all z, Y, z, and w in X. A set K C X is called p-convex if [z, y] c K for all 2 and y in K. It is clear that all normed linear spaces are hyperbolic. A discussion of more examples of hyperbolic spaces and, in particular, of the Hilbert ball can be found, for example, in (13,141. Let (X,p,M) b e a complete hyperbolic space and let K be a closed p-convex subset of X. Denote by A the set of all operators A : K -+ K such that P(k

for all 2, y E K.

AY) I P(S, Y),

(1.1)

Fix some t9 E K, and for each s > 0 set

B(s) =

{x E

K : p(x,B) _
(1.2)

For the set A we consider the uniformity determined by the following base: E(n)

= {(A, B) E A x A : p(Az, Bz) < n-l,

for all 2, y E B(n)}

,

(1.3)

where n is a natural number. Clearly, the uniform space A is metrizable and complete [15]. A mapping A : K + K is called regular if there exists a necessarily unique zA E K such that lim Anx = XA, n--*m A mapping A : K -+ K is called super-regular that for each s > 0,

A”x -+

XA,

for all x E K. if there exists a necessarily unique zA, E K such

as n +

03

uniformly on B(s).

Denote by I the identity operator. For each pair of operators A, B : K -+ K and each T E [0,11, define as operator rA ~3 (1 - r)B by

(Y-A@ (1 - r)B)(z)

= TAX CD(1 - r)Bz,

x E K.

In this paper, we establish the following three results. THEOREM 1.1. Let A : K -+ K be super-regular and let E, s be positive numbers. Then, there exist a neighborhood U of A in A and an integer no > 2 such that for each B E U, eacl! x E B(s), and each integer rz 2 no, the following inequality holds: p(zA, Bnx) 5 E. THEOREM 1.2. There exists a set 30 c A which is a countable intersection dense sets in A such that each A E 30 is super-regular. Let {fn}F=r

be a sequence of positive numbers from the interval (0,l) lim Fn = 0 n+oo

and

c n=l

Fn = co.

of open everywhere

such that

Krasnoselskii-Mann Iterations THEOREM

1425

There exists a set 3 c A which is a countable intersection of open everywhere

1.3.

dense sets in A such that each A E 3 is super-regular and the following

holds.

assertion

Let XA E K be the unique fixed point of A E 3 and let 8, s > 0. Then, there exist a neighborhood U of A in A and an integer no 2 1 such that for each sequence of positive numbers {T~}F&

T, E [Fn, 11, n = 1,2,. . . , and each B E U, the following

satisfying

relations

(r,B@(l-r,)l)x...~(r~B@(l-r$)y) (i) p((r,BW --rn)l) x. ..x(rlB@(l-rl)l)z, for each integer n 2 no and each x, y E B(s);

hold: I b,

(ii) if B E U is regular, then

p((r,B @(1 - m>4 x . . . x (rIB ‘d3(1 - r$)xc,

xA) < 6

for each integer n 2 no and each x E B(s) . The paper is organized as follows. Theorems 1.1 and 1.2 are established in Section 2. In Section 3, we prove auxiliary results needed for the proof of Theorem 1.3 which is given in Section 4. 2.

PROOFS

OF THEOREMS

1.1 AND

1.2

1.1. We may assume that E E (0,l). Recall that XA is the unique fixed point of A. There exists an integer no 2 4 such that for each x E B(2s + 2 + 2p(zA, 0)) and each integer n 2 no, ~(xA,A~z) 5 8-le. (2.1) PROOF

OF THEOREM

Set U = {B E A : p(Ax, Bx) 5 (8no)-‘e,

x E B(8s

+ 8 + 8p(x~,

0)))

.

(2.2)

Let B E U. It is easy to see that for each x E K and all integers n 2 1, p(AQ,

Bnx)

5 p (A“x, AB+‘x)

+ p (AB+‘xc,

5 p (in--5,

+ p (A~n-lx,

B+-~x)

Bnx)

BOX) ,

(2.3)

and p(B”x,

xA) I p(Bnx,

Px)

I p(B”x, Using (2.2)-(2.4), n=1,2 ,..., no,

+ p(Anx, xA) I p(Bnx, Anx) + ~(x, xA)

A”x) + P(G 0) + ~(6 xA)*

(2.4)

we can show by induction that for all CCE B(4s + 4 + 4p(z~, d)), and for all p(A%,

Bnz)

(2.5)

5 (8no)-‘en,

and p(B%,

0) 5 2P(xA, ‘3) + P(x, 0) + ;.

Let y E B(S). We intend to show that p(XA, B”y) ~(8, Bmy)

I E for all integers n 2 no. Indeed, by (2.5),

5 ; + 2p(xA, d) + s,

m=l,...,no.

(2.6)

By (2.5) and (2.1), P (ZA, Bnoy) I :.

(2.7)

Now, we are ready to show by induction that for all integers m 2 no, P(xA,Bmy)

By (2.7), inequality (2.8) is valid for m = no.

5 f-

(2.8)

S. REICH AND A. J. ZASLAVSKI

1426

Assume that an integer k L no and that (2.8) is valid for all integers m E [72c,k]. Together with (2.6) this implies that

P

(0,h)

i= l,...,k.

2 f + 3(x,4,6) + S,

(2.9)

Set j=l+k-no, By (2.9),(2.10),(2.1),

x = Bjy.

(2.10)

and (2.5),

p(A’%,Px)

2 ;,

P(zA,A~‘x)

I

and

i,

P (XA,

B”+‘Y)

5

f.

This completes the proof of Theorem 1.1.

I

PROOF OF THEOREM 1.2. For each A E A and y E (0, l), define A, : K + K by A+ Let A E A and y E (0,l).

= (1 - y)Ax @ 70,

x E K.

Clearly,

p(A,z,A,y)

I (1 - y)p(A~Ay)

5

(1 -

Y)P&Y),

x,y~K.

Therefore, there exists x(A, y) E K such that

A,Wt

r>>= x(4 r>.

Evidently, A, is super-regular and the set {A, : A E A, y E (0,1)) is everywhere dense in A. By Theorem 1.1 for each A E A, each y f (0, l), and each integer i 2 1, there exist an open neighborhood U(A, y, i) of Ay in A and an integer n(A, y, i) 2 2 such that the following property holds: (i) for each B E U(A,r,i),

each 2 E B(4i+‘), &(A,

and each n L n(A,y,i),

y), Bnx) 5 4+‘.

Define FO

=

fi

U{U(A, y, i) : A E U, y E (0,1), z=q,q+l,...}.

q=l

Clearly, & is a countable intersection of open everywhere dense sets in A. Let A E 30. There exist sequences {A,},“=, c A, {rq},“=r c (0, l), and a strictly increasing sequence of natural numbers i, + cc as q -+ cm such that

A E U(Aq7~q,iq),

q=1,2,....

(2.11)

By Property (i) and (2.11) for each 2 E B(4iq+1) and each integer n 2 n(A,, rq, iq), p(a:(A,,y,),A”z)

< 4-+l.

This implies that A is super-regular. The theorem is proved.

I

Krasnoselskii-Mann Iterations

3. AUXILIARY

1427

LEMMAS

Let i’nE(O,l),

n=1,2

,...,

&-nr~~=O,

gPa=l.

(3.1)

n=l

LEMMA 3.1.

Let A E d, Sr > 0, and let no > 2 be an integer. Then, there exists a neighbor-

hood U ofA in A and a number S, > Sr such that for each B E U, each sequence {ri}yz;’

c (0, l]

and each sequence {xi)yI?r c K satisfying Xl

E B(S),

xi+1

=

.i=l,...,?zs-1,

(3.2)

no - 1 and S, = S,,.

(3.3)

?-jBXj @ (1 - 7-&i,

the following relations hold: xi E B(S*),

i=l,...,na.

PROOF. Set &+I = 2Sj + 2 + 2p(B, A@,

i=l,...,

Set U = {B E A : p(Az, BE) 5 1, ICE B(S,)}.

(3.4)

Assume that B E CT,{T~}~~;’ c (0, I], {zi)~~r c K and that (3.2) holds. We will show that

p(@,x:i> 5 si,

i=l,...,ns.

(3.5)

Clearly, for i = 1, equation (3.5) is valid. Assume that the integer m E [l, no - I] and that (3.5) holds for all integers i = 1, . . . , m. Then, by (3.5) with i = m, (3.2), (3.4), and (3.3),

~(6,xm+l) = ~(6,~mB(xm)@(1 - ~mhn> I p(r,B(B) @(I- ~m>xm>, ~mB(4 @(1 - ~mhn) + ~(6,rmB(6) @(I- ~n&c,> 5 ~m~(e,xrn) I

+ de, B(e)) + P(B(@, ~mB(e)@Cl- ~mh) s,,,+ p(e, A(6)) + ,44(e), B(6)) + ,@(Qxm) 5 Sm+ ~(0,A(e)) + I+ p(x,, 6) + ~(6, A@ + p@(e), B(e)) I 2% + Me, A(e)) + 2 = %+I.

The lemma is proved.

I

For each A E A and each y E (0, l), define A, : K + K by

A,x = (1 - y)Ax CDye, Let A E A and y E (0,l).

x E K.

Clearly,

P&X, A,Y) L Cl- Y)P(GY), There exists o(A,r)

(3.6)

x,y~K.

(3.7)

E K such that

A,(44 Y)) = 4% 7). Clearly, A, is super-regular and the set {A, : A E A, y E (0, 1)) is everywhere dense in A.

(3.8)

1428

S. REICH

LEMMA 3.2.

AND

A. J. ZASLAVSKI

Let A E A, y E (0, l), T E (0,11, and 2, y E X.

Then,

p(rA,a:@Cl- r)x>T&Y @(1 - r)y) I (1 - Y~)P(x,Y). PROOF. By (3.7), p(rA,a:

@ (1 - r)z, rA,y

@ (I-

T)Y) I r&Q,

A,Y) + (1 - T)P(Z, Y)

I (1 - r)P(?

Y) + T(l - Y)P(? Y) = P(? Y)U - YT).

The lemma is proved.

I

LEMMA 3.3. Let A E A, y E (0, l), and S, S > 0. Then, there exist a neighborhood U of A, in A and an integer no > 4 such that for each B E U, each sequence of numbers ri E [Fi, 11, i = l,... , n,-, - 1, and each 2, y E B(S), the following inequality holds:

p((r,,--1B @(1 - r,,-111) x .-. x (rlB @(1 - r$)z,

(m,,-1B (3 (1 - rn,-l)l) x 3-. x (rtB @ (1 - rl)I)Y)

< 6.

PROOF. Choose a number 70 E (O,Y). Clearly nz-,(l

(3.9)

- Tori) + 0, as n -+ co. Therefore, there exists an integer no 2 4 such that

(3.10) By Lemma 3.1 there exist a neighborhood VI of A, in B E VI, each sequence {ri}Fz;’ XI E

B(S),

A and a number S, > 0 such that for each

c (0, 11,and each sequence {zi}y& xi+1

=

riBxi

i=l

$ (1 - ri)xi,

c X satisfying

,...,no-1,

(3.11)

the following relation holds: i=l

zi E I$%),

l”‘.,

(3.12)

720.

Choose a natural number ml such that ml

>

25%+ 2,

8ml'

i=l

< S(y - To)ri,

I**‘, 120- 1,

(3.13)

.

(3.14)

and define U = {B E VI : p(A+,

Bz)

x E B(ml)}

< ml’,

Assume that B E U, ri E [Vi, 11, i = 1,. . . , no - 1, and (3.15)

x, Y E B(S). Set

x1 = 2,

Yl = Y,

Q+I = riBzi

@ (1 - r:)xi,

yi+l = riByi

i=l,...,no-1.

@ (1 - ri)yi,

(3.16)

It follows from the definition of VI (see (3.11) and (3.12)) that yi,xi

E

B(S),

i=l,...,n~.

(3.17)

To prove the lemma, it is sufficient to show that (3.18)

Krasnoselskii-Mann

Iterations

1429

Assume the contrary. Then, P(%Yi)

i=l,...,rlc.

> 6,

(3.19)

Fix i E (1,. . . ,720 - 1). It follows from (3.16), (3.17), (3.13), (3.14), and (3.7) that

P(~~+I,Y~+I) =p(riB% Cl3(1 -ri)xi,riByi Cl3(1 -ri)y+) 5 /driAyxi @(1 - ribi, riA,yi @(1 - ri)Yi) + /$A-+i, BQ) + I rip(A,zi,

&z/i) + (1 - r&(zi,

I 2m,1 f (l -ri)p(Si,Yi) 5 2m,1

+p(Zcij/i)(l

+ri(l

Byi) (3.20)

yi) + 2m,*

+rip(Si,yi)(l

-Ti

p(A,yi,

-7))

-7)

= 2mlr +p(Xi,yi)(l

-yri).

By (3.20), (3.13), and (3.19),

P(Xci+l9 Yi+l) 5 Ptxi7Yi)(l -

70ri),

and since this inequality holds for all i E { 1, . . . , no - l}, it follows from (3.15) and (3.10) that 720-1

5 2S

P(xn07YnO)

n (1 -7cyOri) < $. i=l

This contradicts (3.19). This proves the lemma.

PROOF

4.

I

OF THEOREM

1.3

Let lim +=n= 0, n+oo

{~n’n)zLc (O,l), By Theorem 1.2, there exists a set &

C%=W.

(4.1)

n=l

A which is a countable intersection of open everywhere

c

dense sets such that each A E 30 is super-regular. For each A E d and each y > 0, define A, E A by

A,x = (1 - y)Ax @ 70, Clearly, A, is super-regular and for A E

x E K.

A and y E (0,l) there exists z(A, y) E K for which

A,(+4

r>>= x(A r>-

(4.2)

Let A E d, y E (0, l), and let i > 1 be an integer. By Lemma 3.3 there exist an open neighborhood Ur (A, 7, i) of A, in A and an integer no(A, 7, i) 2 4 such that the following property holds: (a) for each B E U1(A, y, i), each sequence of numbers j=l

rj E [i’j,il, and each pair of sequences {“~i}~$“‘~),

~1,YIE B (gi+‘(4

+ 4M-4,

Z~+I = riBzi @ (1 - r+)ti,

r...,no(A,y,i)

{yi)~~~‘y’i)

c X

- 1,

satisfying

r), 0))) ,

(4.3)

yi+r = riByi @ (1 - ri)yi,

i=

1 ,.-*,

no(A,y,i) - 1,

(4.4)

the following inequality holds: P (xno(A,7.i), ?&(A,y,i))

5 8-i-1*

(4.5)

s. REICH AND A. J. ZASLAVSKI

1430

Since A, is super-regular, by Theorem 1.1 there is an open neighborhood and an integer n(A, y, i), such that WA, Y, 9 c VI (A, Y, i),

U(A, y, i) of A, in A

n(A, Y, i) 2 no@, Y, i),

(4.6)

and the following property holds: (b) for each B E U(A, y, i), each z E B(@‘(2+2p(s(A,

&(A,

y), 0))) and each integer m 2 n(A, y, i),

r), Bmx) 5 8-1-i.

(4.7)

Define F=Fsfl

fi u{U(A, y, i) : A E A, y 1q=l

E (0, l), z=q,q+l,...} ’

Clearly, F is a countable intersection of open everywhere dense sets in Let A E 3. Then, A E Fo and it is super-regular. There exists z(A)

1 *

A. E K such that

A@(A)) = z(A). There exist sequences {Aq}gl C A, {r,}~r natural numbers {iq}~~I such that

c

(O,l),

(4.8) and a strictly increasing sequence of

q= 1,2 ) . . . .

A E WA,,~q,~q),

(4.9)

Let 6, s > 0. Choose a natural number q such that 2q > 16(s + l),

2-9 < .3-r&

(4.10)

and consider the open set U(A,, y,,i,). Let rj E [Fj, 11, j = 1,2,. . . , and B E U(A,,%, theorem (Assertion (i)) is valid.

iq). By Property (a), the first part of the

To prove Assertion (ii), assume in addition that B is regular. Then, there is e(B)

E K such

that B@(B))

= z(B).

(4.11)

By Property (b), ~(a:(Aq,yq),a(A)),p(s(Aq,yq),z(B)) Let 51 E B(s)


(4.12)

and ~~j+r = TjBxj

$ (1 - rj)xj,

j = 1,2,....

It follows from Property (a) and (4.11) that P (Zj,

X(B))

5

8-i’-1,

for all integers j 1 n(A,, 7qr iq).

Together with (4.12) and (4.10), this implies that for all integers j 2 n(A,, y,, iq),

p(q,x(A)) This completes the proof of Theorem 1.3.

5 3. 8-iq-1

< b.

I

Krasnoselskii-Mann

1431

Iterations

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