The Methods of Hilbert Spaces and Structure of the Fixed-Point Set of Lipschitzian Mapping

Volume 2009, Article ID 586487,12pages doi:10.1155/2009/586487

Research Article

The Methods of Hilbert Spaces and Structure of

the Fixed-Point Set of Lipschitzian Mapping

Jarosław G ´ornicki

Department of Mathematics, Rzesz´ow University of Technology, P.O. Box 85, 35-595 Rzesz´ow, Poland

Correspondence should be addressed to Jarosław G ´ornicki,[email protected]

Received 16 May 2009; Accepted 25 August 2009

Recommended by William A. Kirk

The purpose of this paper is to prove, by asymptotic center techniques and the methods of Hilbert spaces, the following theorem. LetHbe a Hilbert space, letC be a nonempty bounded closed convex subset ofH, and letM an,kn,k≥1be a strongly ergodic matrix. IfT : C → C is a lipschitzian mapping such that lim infn→ ∞infm0,1,...∞k1an,k· Tkm

2

< 2, then the set of fixed points Fix T {x ∈ C : Tx x} is a retract of C. This result extends and improves the corresponding results of7, Corollary 9and8, Corollary 1.

Copyrightq2009 Jarosław G ´ornicki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

LetEbe a Banach space and letCbe a nonempty bounded closed convex subset ofE. We say that a mappingT:C → Cis nonexpansive if

Tx−Tyx−y for everyx, y∈C. 1.1

The result of Bruck 1 asserts that if a nonexpansive mapping T : C → C has a fixed point in every nonempty closed convex subset ofCwhich is invariant underT and ifCis convex and weakly compact, then Fix T {x ∈ C : Tx x}, the set of fixed points, is nonexpansive retract ofCi.e., there exists a nonexpansive mappingR:C → Fix Tsuch that

R|FixT I. A few years ago, the Bruck results were extended byT.Dom´ınguez Benavides and Lorenzo Ram´ırez2to the case of asymptotically nonexpansive mappings if the spaceE

was sufficiently regular.

Example 1.1Goebel3,4. LetFbe a nonempty closed subset ofC. Fixz∈F, 0< ε <1 and put

Txxε·distx, F·z−x, x∈C. 1.2

It is not difficult to see that Fix T Fand the Lipschitz constant ofT tends to 1 ifε↓0.

For more information on the structure of fixed point sets see4,5 and references therein.

In 1973, Goebel and Kirk 3 introduced the class of uniformly k-lipschitzian mappings, recall that a mappingT :C → Cisuniformlyk-lipschitzian,k1, if

Tnx−Tnykx−y for everyx, y∈C, n∈N, 1.3

and proved the following theorem.

Theorem 1.2. LetEbe a uniformly convex Banach space with modulus of convexityδEand letCbe

a nonempty bounded closed convex subset ofE. Suppose thatT :C → Cis uniformlyk-lipschitzian and

k

1−δE

1

k

<1. 1.4

ThenT has a fixed point inC.Note that in a Hilbert space,k <1/2√5.

Recently Se¸dłak and Wi´snicki 6 proved that under the assumptions of Theorem 1.2,

Fix T is not only connected but even a retract of C, and next the author proved the following theorem7, Corollary 9.

Theorem 1.3. Let H be a Hilbert space,C a nonempty bounded closed convex subset ofH, and

T : C → Ca uniformlyk-lipschitzian mapping withk < √2. Then T has a fixed point inCand

Fix Tis a retract ofC.

In this paper we shall continue this work. Precisely, by means of techniques of asymptotic centers and the methods of Hilbert spaces, we establish some result on the structure of fixed point sets for mappings with lipschitzian iterates in a Hilbert space. The class of mappings with lipschitzian iterates is importantly greater than the class of uniformly lipschitzian mappings; see8, Example 1.

2. Asymptotic Center

Denote byTthe Lipschitz norm ofT:

TsupTx −Ty

x−y :x, y∈C, x /y

Lifshitz9significantly extended Goebel and Kirk’s result and found an example of a fixed point free uniformly π/2−lipschitzian mapping which leaves invariant a bounded closed convex subset ofl2_{. The validity of Lifshitz’s Theorem in a Hilbert space for}√_{2 k < π/2}

remains open.

A more general approach was proposed by the present author using the methods of Hilbert spaces, asymptotic techniques, and strongly ergodic matrix. We recall that a matrix

M an,kn,k1is calledstrongly ergodicif

ifor alln, k an,k0,

iifor allk limn→ ∞an,k0,

iiifor alln ∞_k1an,k1,

ivlimn→ ∞∞k1|an,k1−an,k|0.

Then we have the following theorem.

Theorem 2.1see8. LetCbe a nonempty bounded closed convex subset of a Hilbert space and let

M an,kn,k1be a strongly ergodic matrix. IfT :C → Cis a mapping such that

glim inf n→ ∞ minf0,1,...

∞

k1

an,k·Tkm

2

<2, 2.2

thenT has a fixed point inC.

This result generalizes Lifshitz’s Theoremin case of a Hilbert spaceand shows that the theorem admits certain perturbations in the behavior of the norm of successive iterations in infinite sets; see8, Example 1.

LetE be a Banach space. Recall that the modulus of convexityδE is the functionδE : 0,2 → 0,1defined by

δEε inf 1−1

2xy:x1,y1,x−yε

2.3

anduniform convexitymeansδEε>0 forε >0. A Hilbert spaceHis uniformly convex. This fact is a direct consequence of parallelogram identity.

Now we prove some version of Se¸dłak and Wi´snicki’s result 6, Lemma 2.1. LetC

be a nonempty bounded closed convex subset of a real Hilbert spaceH, letM an,kn,k1 be a strongly ergodix matrix, and letT : C → Cbe a mapping such that Tk _{1 for all}

k1,2, . . ., and

lim sup n→ ∞

∞

k1

an,k·Tk

2

Letx, y∈Cwe use

ry,Tkxlim sup n→ ∞

∞

k1

an,k·y−Tkx

2 ,

rC,Tkxinf

y∈Cr

y,Tkx

2.5

to denote the asymptotic radius of {Tk_{x} at y and the asymptotic radius of {T}k_{x} in C,} respectively. It is well known in a Hilbert space8that the asymptotic center of {Tk_{x} in}

C:

AC,Tkxy∈C:ry,TkxrC,Tkx 2.6

is a singleton.

Let A : C → Cdenote a mapping which associates with a given x ∈ Ca unique

z∈AC,{Tk_{x}, that is,zAx. The following Lemma is a crucial tool to proveTheorem 4.1.}

Lemma 2.2. LetHbe a Hilbert space and letCbe a nonempty bounded closed convex subset ofH.

Then the mappingA:C → Cis continuous.

Proof. On the contrary, suppose that there existsx0∈Candε0>0 such that for allη >0 there

existsx1 ∈Csuch thatx1−x0 < ηandz1−z0ε0, where{z0} AC,{Tkx0},{z1} AC,{Tk_x1}.

Fixη >0 and takex1 ∈Csuch that

x1−x0< η, z1−z0ε0. 2.7

LetR0rC,{Tk_x0},R1rC,{Tk_x1}andRlim

n→ ∞∞k1an,k· z1−Tkx02. Notice that

R0< R. 2.8

Chooseε >0. Then

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

z1−Tkx0 <

√

Rε,

z0−Tk_{x0 <R0ε <}√_Rε,

z0−z1 ≥ε0

2.9

for all but finitely manyk.

If, for example,z1−Tkx0

√

Rεfor all everyonek, then

z1−Tkx0 2

Multiplying both sides of this inequalityfor fixedkby suitable element of the matrixM

and summing up such obtained inequalities fork1, we have forn1,2, . . . ,

∞

k1

an,k·z1−Tkx0

2

Rε. 2.11

Taking the limit superior asn → ∞on each side, we get

Rlim sup n→ ∞

∞

k1

an,k·z1−Tkx0

2

Rε > R, 2.12

which is contradiction.

It follows by2.9and the properties ofδHthat

Tkx0−z1z0

2

1−δH

ε0

√

Rε

_√

Rε,

Tk_x

0−

z1z0

2 2

1−δH

ε0

√

Rε

2

Rε.

2.13

Multiplying both sides of this inequality by suitable elements of the matrixMand summing up such obtained inequalities fork1, taking the limit superior asn → ∞on each side, we get

R0<lim sup n→ ∞

∞

k1

an,k·Tkx0−

z1z0

2 2

1−δH

ε0

√

Rε

2

Rε.

2.14

Moreover,

Tkx0−z12Tkx0−Tkx1Tkx1−z12

Tk_x

0−Tkx122Tkx0−Tkx1·Tkx1−z1Tkx1−z12

Tk2_{· x}

0−x122Tk· x0−x1 ·Tkx1−z1Tkx1−z12

Tk2_2Tk_{· x}

0−x1 ·diamCTkx1−z1 2

3Tk2_{·diamC· x0−x1T}k_x1−z12_.

Multiplying both sides of this inequality by suitable elements of the matrixMand summing up such obtained inequalities fork1, taking the limit superior asn → ∞on each side, we get

Rlim sup n→ ∞

∞

k1

an,k·Tkx0−z1

2

3·diam C· x0−x1 ·lim sup

n→ ∞

∞

k1

an,k·Tk

2

lim sup n→ ∞

∞

k1

an,k·Tkx1−z1 2

3·B·diam C·ηR1ε.

2.16

Similarly,

R1<lim sup

n→ ∞ ∞

k1

an,k·Tkx1−z0 2

3·diam C· x1−x0 ·lim sup n→ ∞

∞

k1

an,k·Tk2lim sup n→ ∞

∞

k1

an,k·Tkx0−z02

3·B·diamC·ηR0ε.

2.17

From2.16and2.17, we have

R3·B·diamC·ηR1ε <6·B·diamC·η2·εR0. 2.18

IfR0 0, then from2.18it followsR 0. This is contradiction with2.8. IfR0 > 0, then combining2.18with2.14and applying the monotonicity ofδH, we obtain

R0<

1−δH

ε0

6·B·diamC·η3·εR0

2

6·B·diamC·η3·εR0. 2.19

Lettingη, ε↓0 and using the continuity ofδH, we conclude that

1

1−δH

ε0

R0

2

<1. 2.20

3. The Methods of Hilbert Spaces

LetM,T be as above. We define functionals

du lim sup n→ ∞

∞

k1

an,k·u−Tku

2 ,

rx lim sup n→ ∞

∞

k1

an,k·x−Tku2,

3.1

whereu, x ∈ C. LetzinCbe an asymptotic center of{Tku}_k1 with respect tor·andC, which minimizes the functionalrxoverxinCfor fixu∈C.

Lemma 3.1. One hasrzdu.

Proof. It is consequence of the above definitions.

Lemma 3.2. One hasz−u2du.

Proof. For anyk∈N, we have

z−u22z−Tku2Tku−u2

−zu−2Tku2

2z−Tk_u2_2Tk_u−u2_.

3.2

Multiplying both sides of this inequality by suitable elements of the matrixMand summing up such obtained inequalities fork1, taking the limit superior asn → ∞on each side, we get

z−u2 2 lim sup n→ ∞

∞

k1

an,k·z−Tku

2

2 lim sup n→ ∞

∞

k1

an,kTku−u

2

2rz du4du.

3.3

Lemma 3.3. One hasrTs_zTs2_{·rzfor alls∈N.}

Proof. Fixs∈N, then we have

∞

k1

an,kTsz−Tku

2

s

k1

an,kTsz−Tku

2

Ts2· ∞

ks1

an,kz−Tk−su

2

s k1

an,kTsz−Tku2Ts2· _∞

k1

an,kz−Tk−su2− s k1

an,kz−Tk−su2

.

Since the matrixMis strongly ergodic,

s

k1

an,kTsz−Tku

2

−→0,

s k1

an,kz−Tk−su2−→0,

3.5

asn → ∞, we get thesis.

Lemma 3.4. One hasrz z−x2_{rxfor everyx∈C.}

Proof. Forx∈Cand 0< t <1, we have

tx 1−tz−Tku2tx−Tku2 1−tz−Tku2−t1−tx−z2. 3.6

Multiplying both sides of this inequality by suitable elements of the matrixMand summing up such obtained inequalities fork1, taking the limit superior asn → ∞on each side, we get

lim sup n→ ∞

∞

k−1

an,ktx 1−tz−Tku

2

t·lim sup n→ ∞

∞

k1

an,kx−Tku

2

1−t·lim sup n→ ∞

∞

k1

an,kz−Tku

2

−t1−tx−z2.

3.7

Sincerzrtx 1−tz, we obtain

rzt·rx 1−t·rz−t1−tx−z2,

rzrx−1−tx−z2.

3.8

Takingt↓0, we get,rz z−x2rx.

4. Main Result

We are now in position to prove our main result.

Theorem 4.1. LetCbe a nonempty bounded closed convex subset of a Hilbert space and letM

an,kn,k1be a strongly ergodic matrix. IfT :C → Cis a mapping such that

glim inf

n→ ∞ minf0,1,...

∞

k1

an,k·Tkm

2

<2, 4.1

Proof. Let{ni}and{mi}be sequences of natural numbers such that

g lim

i→ ∞

∞

k1 ani,k·

Tkmi2_{<2. 4.2}

ByTheorem 2.1, Fix T /∅. For any x ∈ Cwe can inductively define a sequence{zj}in the following manner:z1is the unique point inCthat minimizes the functional

lim sup i→ ∞

∞

k1 ani,k·

y−Tkmi_x2 4.3

overy∈C,andzj1is the unique point inCthat minimizes the functional

lim sup i→ ∞

∞

k1 ani,k·

y−Tkmi_z

j

2

4.4

overy∈C, that is,zjAjx,j1,2, . . . .First we prove the following inequality:

dzg−1 du, 4.5

where

du lim sup i→ ∞

∞

k1

u−Tkmi_u2_, 4.6

andzis the asymptotic center inCwhich minimizes the functional

rx lim sup i→ ∞

∞

k1

x−Tkmi_u2 4.7

overxinC.

In fact, we put inLemma 3.4xTp_{z. Then byLemma 3.3, we get}

rz z−Tp_z2_rTp_zTp2_·rz,

z−Tp_z2_Tp2_−1·rz. 4.8

Forpmkiwe have

z−Tkmi_z2

Tkmi2−₁

and hence

dz lim sup i→ ∞

∞

k1 ani,k·

z−Tkmi_z2

lim i→ ∞

∞

k1 ani,k·

Tkmi2−₁

·rz

g−1·rz by Lemma 3.1

g−1·du.

4.10

Next byLemma 3.2and inequality4.5, we have

_zj1−zjAj1_x−Aj_x2g−1j

dx2·αj·diamC, 4.11

whereαg−1<1 forx∈C,j1,2, . . . .Thus

sup x∈C

Apx−Ajx α

j

1−α·2·

diam C−→0 ifp, j−→ ∞, 4.12

which implies that the sequence{Aj_{x}converges uniformly to a function}

Rx lim

j→ ∞A

j_{x, x∈C.}

4.13

It follows fromLemma 2.2thatR:C → Cis continuous. Moreover,

Rx−Tkmi_Rx2_2Rx−_Aj_x2_Aj_x−_Tkmi_Rx2

−RxTkmi_Rx−_2Aj_x2

2Rx−Ajx22Ajx−Tkmi_Rx2

2Rx−Aj_x2_22Aj_x−Tkmi_Aj_x2_2Tkmi_Aj_x−_Tkmi_Rx2

24Tkmi2

·Rx−Aj_x2_4Aj_x−Tkmi_Aj_x2_.

Multiplying both sides of this inequalities by suitable elements of the matrixMand summing up such obtained inequalities fork1, taking the limit superior asi → ∞on each side, we get

dRx lim sup i→ ∞

∞

k1 ani,k·

Rx−Tkmi_Rx2

24 lim i→ ∞

∞

k1 ani,k·

Tkmi2

Rx−Ajx2

4 lim sup i→ ∞

∞

k1

ani,k·A

j_x−Tkmi_Aj_x2

24gRx−Aj_x2_4dAj_{x by 4.5}

24gRx−Aj_x2_4g−1j_{·dx → 0 ifj → ∞.}

4.15

Thus,dRx 0. This implies thatRx TRx; see8for details. ThusRx TRxfor every

x∈CandRis a retraction ofConto Fix T.

IfM an,kn,k1is the Cesaro matrix, that is, forn1,2, . . . ,

an,k ⎧ ⎨ ⎩ 1

n for k1,2, . . . , n,

0 for kn1,

4.16

then we have the following corollary.

Corollary 4.2. LetCbe a nonempty bounded closed convex subset of a Hilbert space. IfT :C → C

is a mapping such that

glim inf n→ ∞ minf0,1,...

1

n

n

k1

Tkm2<2, 4.17

thenFix T {x∈C:Txx}is a retract ofC.

References

1 R. E. Bruck Jr., “Properties of fixed-point sets of nonexpansive mappings in Banach spaces,”

Transactions of the American Mathematical Society, vol. 179, pp. 251–262, 1973.

2 T. Dom´ınguez-Benavides and P. Lorenzo Ram´ırez, “Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings,”Proceedings of the American Mathematical Society, vol. 129, no. 12, pp. 3549–3557, 2001.

3 K. Goebel and W. A. Kirk, “A fixed point theorem for transformations whose iterates have uniform Lipschitz constant,”Studia Mathematica, vol. 47, pp. 135–140, 1973.

5 R. E. Bruck, “Asymptotic behavior of nonexpansive mappings,” in Fixed Points and Nonexpansive Mappings, R. C. Sine, Ed., vol. 18 of Contemp. Math., pp. 1–47, American Mathematical Society, Providence, RI, USA, 1983.

6 E. Se¸dłak and A. Wi´snicki, “On the structure of fixed-point sets of uniformly Lipschitzian mappings,”

Topological Methods in Nonlinear Analysis, vol. 30, no. 2, pp. 345–350, 2007.

7 J. G ´ornicki, “Remarks on the structure of the fixed-point sets of uniformly Lipschitzian mappings in uniformly convex Banach spaces,”Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 303–310, 2009.

8 J. G ´ornicki, “A remark on fixed point theorems for Lipschitzian mappings,”Journal of Mathematical Analysis and Applications, vol. 183, no. 3, pp. 495–508, 1994.

9 E. A. Lifshitz, “A fixed point theorem for operators in strongly convex spaces,” Voronezhsk˘ı