(*.,f(r))20foraIIxef

202

Elena Popoviciu

6

REVUtr D'ANÂLYSE NUMÉRIQUE tìT DE TIIÉORIE DE L'APPI{OXIMATION Tome

les points L\,u2, . . . tu¡ potlrÍ ùn h, I < h < n + l, fixé, sont fixés dans X et les points uh*l, . , . , tr.,t+2 sotrtconsidérés de toutes les manières possibles dans_{ satisfont

XXVI, Nn'1-2,'1,991, pp.

203-208

l'inégalité

(19)

lur,ur,...,ilh,uh*l,...,t4¡+2iflr o,

quels que soient les point On peut énoncer le

uh*l,...,ltr*,

dans I'ensemble

TgÉonÈue 3. La propriété æpriméepar (19)

X,n ) 0 étant fixé.

èst une

EXISTENCE AND APPROXIMATION OF POSITIVE FIXED POINTS OF NONEXPANSIVE MAPS

allure et ceÍte allure

RADU PRECUP

précède I'allure de convexité d'ordre n sur X.

7. Les considérations que nous avons faites dans ce travail en résumé ont eu 1. INTRODUCTION

comme but de methe en évidence une direction de recherche qui peut êhe développée en ayant comme point de départ f idée de convexité d'ordre supérieur.

i,e,,

nÉrÉRB¡¡c

Bs

Throughout this paper E will be a real Banach space and K c. E a cone) convex set such that ÌvK c K for all À > 0. Since we do not ¿rssume

a closed

Kn(-^K) = {0},the

1. T. Popoviciu, ,Str quelques propríétes des fonctíow d'une ou de deux variables réelles, Mathernatica I (1934), 1-85. 2. T. Popoviciu, Les fonctions convães, Paris, 1945. 3. T. Popoviciu, Sur I'approxímation des fonctions convexe,t d'ordre supéríeur, Mathsmatica 10 (1934), s9-64. 4. T. Popoviciu, Remarques sur la défnitionfonctionnelle d'un polynôme d'unevariable réelle, Mathematica 12 (1936), 5-12. 5. T. Popoviciu, Notes sur les þnctions convexes d'ordre supérieur, Rovue Math. do l'Union Intorbalkanique 2 (1939), 3 l-40. 6. T. Popoviciv, Deux remarques sur lesfonctions convexes, Bull. do la Section Sc. Acad. Roumaino 20 (1938), 4549. 7. T. Popoviciv, Despre cea møi bunã aproximølie afuncliílor contínueprîn polinoame, Ed. Ardoalul, Cfuj, 1937. 8 T. Popoviciu, .Strr une généralisation des fonctíons splíne, Mathematical Structuros, Computational Mathomatics, Mathomatical Modeling (Soha) (I 975), 405-470. 9. E. Popovicitt, Teoreme de medie din analiza matemøticã ¡i legätura lor an teoria ínterpolãrii, Ed. Dacia, Chtj, 1912. 10. E. Popovicil, Sur une allure de quasi-convexité d'ordre supérieur, L'Analyso Numériquo et la Theorie do l'Approximation 11 (1982), 129-137. 11. E. Popoviciu, Sur certaines propriétës des þnctions quasi-convexes, L'Analyso Numériqræ et la Théorie de I'Approximation 12 (1983), 175-186. 12, E. Popoviciu (Moldovan), Sur une générølisation das fonctions convexes, Mathematica 1

Q4) (rese),4e-80.

13. E. Popoviciu, Despre unele momente semniJìcative în dezvoltarea teoriei convexitãlíi,

Seminarul Itinorant "Tiberiu Popoviciu", de Ecualii Functionale, Aproximare çi Convoxitato, Cluj-Napoca, 1995, p. 89-92.

cone Kcanbe, inparticulaq the entire space¿. We shall

denote by Ã^ the dual cone, i.e.,

K ={x-

e

E';(*.,r) >o forall xeK]

Also, by U and t/, *e shall denote open bounded subsets of E containing the origin; we shall assume that

UlcU K'

c. E,

of K [ì U. The following two fixed point theorems have been established in [8] by means of the continuation method, but without using the index theory. In the' : particular case when U and U, are two balls, U : B < < RQ) and U, B,(0), 0 r .R, these results have been first obtained by K. Deimhng t4l (seealso [3] and [5] for and we shall write

instead

related topics) by means of a different method. Although in [8] we havesupposed thatKn(-rK) = {0},the reader can easily see that such an assumption is not necessary.

TsroReu

1.1 [8].

Let

f :Eu --> E be u-condensing

following conditíons hold:

(1,1) (*.,f(r))20foraIIxef¡ìaf

and x*

eK'wíth (x-,r)

(w e a k inw ar dnes s con dit i on); Reçu 1o 15 mai 1996

Str. RoSiori, nr. 40

3400 Cluj-Napoca Românía

and suppose that the

1991 AMS Subjoct Classification: 47}J09, 41H10, 47H06

=O

f(r) + ?lx for all x

(r,2) Then

2

Radu PrecuP

')04

f

has a fixed Point

x e Ku

e

KìõU

and

'

Theorem 3'1 in [8]' Theorem 1.1 is a simple conseqlrence of (1'1) triviatttllg:^?'d Theorem 1'1 E, condition In particular, for a-condensrng maps' the weli-known continuationprinciple

f*?:

,"0rr".. to Thenexttheoremisusefulwhen/(0)=0andfixedpointsinK\{0}areof interest. THEOREM 1'2

t8l' Let

f

:

Ku

+

E

/ is Moreover, these cond x,, . Ku off, that is, since

À > 1'

be anu-conrlensingmap satísfying

(l't)

is convex bounded closed, Since any uniformly convex space is reflexive and Ru (x,) weakly convergent to some thçre is a subsequence of (x,) (also denoted by obtain that x e Ku, Further, f (Kr) being bounded, from Q'z)we

*^

+

Thenf has a.fixed Point in U '

f :U -+ E is ø

space TneoReN4 1'4 tgl' Suppose E is a Hilbert (where (J is not necessarily convex) ' Thenf has nonexpansive map satisfying Q .a) afixed Point in U .

lll.

fniKu

-->

E, f,(x)

I - l-- n f( x)

stronglY'

Suppose

2(arx n

- d,nx,n, x n - x,r) = (o, * o,n)l*, - *,rl'

+

(o,

TseoRBtr¡ 2.3. Suppose E is a Hil

-

o

^)(l"l'

- l' *l')'

entire sequence (x')

with an: ll(n- 1), we c is strongly convergent, .pur.., î! additio-nalþ More exactlY, we have

satisfyi;g?i \ and (l '2 'rh:," r"qu"ri. (t,,j'-k) gíu." øy Q.

Proof' For each n e N, n 2 2, define the nap

0

" "-f(*)=Y.r t" gitU..t spaces, by Q.2) and the identity

pansive map

2,POSITIVEFIXEÐPOTNTSOFWEÀKI'YINWARDNONEXPANSIVEMAPS that' ín addítion' U is Tneonelvr 2,1' Suppose E g(1'1) and(l'2)' Then convex.- Let .f tKu -+ E beano f has af'xed Point in Ku,

(2,r)

-+

E is uniþrmly convex' LetJ':D -> E be a bounded closed set' If for a nonexpansiye map, where D c. E is a convex then ,"qr"i". (r,) - D onehas xn -+ x wealcly and xn - fQ") --> y strongþ'

LBvlue 2'2

\x for att x e ô(I and î' > 1' and

- f(*,)

Now the conclusion follows bY

Tmoneul.3t9]'SupposeEísunifurmlyconvexandthat,inaddítion,Uis map such that convex. Let f :U -+ E be a nonexpansive

f(*)

by Theorem 1'1, there exists a (unique) fixed pomt

['-1)r{',)=a

Û

ForanexampleillustratingTheoreml'2wereferto[3,Exanrple20,ll'naps, results for nonexpansive The aim of trri, paps. i, io obtain similar the following continuation theorems for Moreover, we shall g.igå".r¿izations of nonexpansive maps recentlyproved in [9]:

(1'4)

is a conhaction and, consequently, u-condensing, 1'1)and (1.2),iteasilyfollows that:f"{t-::itTli:Ì

Q.2)

and (1.2). In addition, suPPose

*- f(*) + Xe J'or all x eKÀôU' and l' > (1,3) potnt in f t-t(u t ut)' þr some e e k \ {o}. fhenf has afixed

205

Positivo Fixod Points

)

U' Thus' in Hilbert a fixed point of/

r Let

f :Ku -+ E be a nonex-

not necessaríly convex)' Then onverges to a fixed poínt of f'

Remarlc,ForK:E,Theorems2.land2.3reducetoTheoremsl'3and1.4, respectivelY.

3. NONZETìO FIXED POINTS

of fixed points in This section deals with the existence and approximation 0 as a fixed point' K \ {0} of weakly inwarcl nonexpansive maps which may have 3'I' Suppose E is un.¡formly CTnyex. In additioll, assume that THBoREN4

Raclu Precup

206

(3,1) (1,2). Al,so, ,rltppo,te

Ku -> E be a nonexpansive tlrut there is e e,K \ {0} such that

(3.2) f

Let

f

{x- f(x);x €KnôUl}nR*e

has a Jìxed point

in

map satisfying

'.

k,

=

(LI)

x¡r)

1

n¡

Ø.

(4.f)

Lt +

It

fl0,ürereexists'x,, e

f[ì(7\U,)

a fixed point of fn. Further, as in the proof of The otem2.l, there is a subsequence of (xr) weakly convergent to some x e Ku, Since x, É(Jr, by (3.1), we see that ¡ * 0. Finally, by Lemma 2.2,we obtzinf(x): x. ø Remarlc. Condition (3.1) implies

tl;x,t

thatKis normal, i.e.,

(u,*)*>-0 for all x eK with lxl> R and u eA(x)

E

be a nonex-

and Q.2)' Then the s equence (x u),,>,,,

c

f

given by (2.2) strongly converges to afi'xed point

f

:

forsomeR>0. Thenthereexists x eK with lxl o,

In Hilbert spaces we have a more precise result.

p ans ive map s atßfying

a

(coerciveness with respect to ze.ro),

which contradicts (3,2),

inf{lx +

.,t(,t +

- f(xt) -+ Xse,

n)

n,f(r)=(A+¡)-t(r).

TgBoneu 4.1, Suppose E is uníþrmly convex rtnd A:E -+28 is

(4,3)

Therefore,accordingtoTheorcml.2,foreach

:E -+

In this section we deal with the solvability of the inclusion 0 e A(x), ot, equivalentþ, of the equation (,1 + t)-t(x) = x, where A is uhyperaccretive map, The results are direct consequences of the theorems of Sections 2 and3,

(4.2) Ào for some l"o e f{.*.

follows

x¡

f

lryperaccretivemap. In addition, assume that

= Ìr¡e for all k,

Clearly, (l"o) is bounded and so we may suppose

(,t+t)(n)=n, where (r, y) , = lyl tinq ,-'(l y n r, | - lyl) For a hyperu"'Ci3tiu. map A one considers the nonexpansive tnap

Proof. For each n e I'{, n ) 2, the mapf,,given by (2.1) satisfies (1.1), (1.2) and also ( 1 .3) for n large enough, say n > no , Indeed, otherwise it would exist the sequences ("à - N,(rr) c K[ì ôU, and (fo) . Ri such that no -+ æ and

x¡-

aIl x.yeD(A), ueA(x) and veA(y),

and

t {o}

1

207

Positive Fixed Points

5

(r-u,r_ I)*>0for

o øconr,(KnôBr(o))

ancl that U is convex.

Then

4

þrsome e e K\{0} and 0 e A(x).

,t(a+ r)1(KìAB,(o))[ìR*" = Ø and r e ]0, nl.fhenthereexists x e K\{0} with |tl = n

Proof, Apply Theorem 3.1to U

=

Bo(0),

U, = B,(0)and/given by (a.1)' r

208

Radu Precup

RI,VUE D'ÂNALYSE NUMÉRIQUE ET DD THÉORIE DE L'APPROXIM.{TION

6

Tome

Let Ai E -> 2E be a hyperaccretíve map satisfying (4.2), (4.3) and (4.6). Then the sequenc. (*,),r,,0 - K, THEoREM 4.3. Suppose E is a Hilbert space.

'