ON A FIXED POINT THEOREM OF GREGU

(1)

Vol. 9 No. (1986) 23-28

ON A FIXED POINT THEOREM OF GREGU

BRIANFISHER

Department

of Mathematics

University of Leicester Leicester

LEI

7RH, England

and SALVATORE SESSA Istituto Matematico Facolta’ Di Architettura

Univesita’ Di Napoli

Via Vonteoliveto 3 80134 Naples, Italy

(Received May

18,

1984)

ABSTRACT.

We censider two selfmaps

T

and of a closed convex subset

C

of a Bar.ach space

X

which are weakly commuting in

X,

i.e.

!IT

^x ^I

xll

^<

llx Txll

^for ^any ^x ⁱⁿ

^X,

and satisfy the inequality

’ITx ^Ty!l

^_<

alllx ^lyll

⁺

^(I ^a)

^max

^{llTx ^Ixll, ^llTy ^lyll}

for all x,) ir. C, where 0 < a

I. It

is proved that if is linear and non-expansive in C and such that IC contains TC, then

T

and have a unique common fixed point in

C. KEY

WORDS

AND

PHRASES. Conon

fixed

^point, ^Banach ^space.

1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 54H25, ^47HI0.

I. INTRODUCTION.

The second author

[I],

generalizing a result of

Das

and Naik

[2],

defined two mappings

T

and of a metric

space (X,d)

^into itself to be weakly commuting if

d(Tlx,ITx)

^<_

d(Ix,Tx) (I.I)

for all x in

X. Two

commuting mappings clearly satisfy

(I.I)

but the

converse

is not generally true as is shown with the following example:

EXAMPLE I.

et

X [0,i]

with the Euclidean metric and efine

T

and by

Tx x/(x+4), Ix

x/2

for all x in X. Then

x x x2

d(Tlx, ITx)

x + 8 2x + 8

2(x ,)(x

+

4)

x2 + 2x x x

d(Ix,Tx)

<-

2(x

+’4) -

^{x +} ⁴

for all x in X but for any x

#

^O"

Tix

x/(x+8)

>

x/(2x+8) ITx.

(2)

24 B. FISHER AND S. SESSA

From

now on, C denotes a closed convex subset of a Banach space

X. In

a recent paper

Gregus [3]

proved the folloing theorem:

THEOREI. i.

Let T

be a mapping of C into itself satisfying the inequality

IITx ^Tyll

^-<.

^allx Yll

⁺

bllTx xll

⁺

^cllTy Yll

for all x,y in

C,

where 0 ^< a < 1, b ^>

C,

c ^> 0 and a + b + c

I.

Then T has a unique fixed point.

Mappings satisfying inequality

(1.2)

with a

I

and b c 0 are called nonexpansive and were considered by Kirk

[4].

Nong [5]

studied mappings satisfying inequality

(1.2)

with a 0 and b=c=1/2.

2.

MAIN RESULTS.

We now prove the following generalization of Theorem I:

THEOREM 2. Let

T

and be two weakly cor:nuting mappings of C into itself satisfying the inequal

IITx ^Ty!I

^-<-

ail ^lyll

⁺

⁽ ^a)

^max

^{IITx xll ^,IITy ^lYll} ^(2.1)

for all x,y in C, where 0 ^< a < 1. If is linear, nonexpansive in C and such that IC contains TC, then

T

and have a unique common fixed point in

C. PROOF.

Let x x be an arbitrary point in C and choose points

x

_I, x 2, x

3 in C such that

Ix Tx, Ix

2

Tx

_1,

Ix

3

Tx?.

This can be done since IC contains

TC.

Then for r 1,2,3 we have on using inequality

(2.1)

ITx

_r

IXrll ^lTx

^r

^TXr_lll

-<

alllx

_r

IXr_ll

⁺

^(l-a)max ^{IITx

_r

Xrll, llTXr_ _I- IXr_lll.

al ITx

r-1

mx,. -I ^II

⁺

^(I-a)

^max

^{I ^Tx

r

^Ix

r

^II ^lTx

r-1

^Ix

and so

lTx

_r

iXrll

^-<

^lTXr_ I IXr_ll

I

follows that for r 1,2,3.

Further

llTx

₂

Txll

^-<

alllx

₂

Ixll

⁺

^(l-a)max ^{llTx

₂

Ix211, ^llTx ^Ixll}

(2.2)

-<

a(l ITx _I IXlll

⁺

^ITx ^Ixll)

⁺

^(l-a) ^ITx ^Ixll

<-

(1+a) lTx Ixll

(3)

on using inequality (2.2). Thus

llTx

₂

IXll

^(l+a)

^llrx

We will now define a point z by

(2.3)

z x2

+- >(3"

Since_ C is convex th polnt z is in C and being lit,ear, we have

1/2

z

2 Ix

⁺

^Ix

³

- ^Tx ^Tx

^2.

It follows that

llTz Izll

^-<-

1 ^llTz Txill ,llTz _Tx211

-<

1/2 ^[alllz- _Xlll ^+(I-a)max {llTz- zll, llTx xl}]

+

[alllz mll

⁺

^(l-a)max ^{llTz- ^Izll ^IlTx

2

Ix21l}

a(llz Xlll

⁺

^llIz ^Ix211)

⁺

^(i-a)max ^{llTz ^zll, ^IITx ^xll}

on using inequalities

(2.1)

and

(2.2). Now

IIz xill

^_<

. ^I>’2 ^xill

⁺

^1/2 ^lzx

³

^Ixill

2

IIT>- xlll +1/211Tx 2- xil

(i +

a) llTx- xll

from inequalities

(2.2)

and

(2.3)

and

lllz ^Ix

₂

’I 1/2 ^lllx

3

Ix211 ^llTx

2

Ix211

^_<

1/2 ^llTx ^Ixll.

!t

follows that

IITz Izll

^-<

-a(3

⁺

^a) llTx zxll

⁺

^(l-a)max ^{llTz zll, IITx ^xll}

and so

llTz-Zz II

⁺

x.ltTx- Zxll

where

(4

a +

e.2)14 I.

We therefore have

inf

{I lTz Izll

^z

:

^x2 + x

3}

^-< ^X.inf

^{llTx ^Txl]

^x ^C}

and since e obviously have

inf

{I ITz Izll

^z

=

^x² ⁺

^x3}

^>_ ^inf

^{llTx- ^Ixll

^x

^C},

(4)

26 B. FISHER AND S. SESSA it follows that

inf

{II Tx Ixll

^x ^C}

^O.

Ea(!, of tle sets

K

_n {x C

lTx Ixll

^I/n},

^H

_n ^{x ^C

lTx Ixll ^(a

⁺

^l)/an}

(foF n 1,2 )must therefore be non-empty and obviously

KID ^KoD ^DK

^I"1

hus each of the sets

TKn,

^where

TK

_n denotes the closure of

TK

_n, must be hop-empty for n 1,2 and

TKI

ⁿ

Further, for arbitrary x, y in Kn,

IITx- ^Ty!I

^<-a

lllx- ^lyll ^+(l-a)max

<_

a([[Tx- I[[

⁺

[[Tx- Tyl[

+

IITY- ^IYlI) ^(1-a)/n

_<

(a+l)/n

+

al ITx

and so

lTx TYll -<(l-a)n

a+l Thus

im diam

(TK n)

^im ^diam

(-n)

⁰

It

follows, by a well ^knowp result of

Cantor (see,

for example

[6],

p.

156),

that the intersection

r,__C%1 ^TK

n contains exactly one point w.

Now let y be an arbitrary point in

TK n.

^Then for arbitrary ^> 0 there exists a point

y’

in

KF.

^{such that}

ITy’ Yll

^<

^(2.4)

and so, using the weak commutativity of

T

and and the nonexpansiveness of

I,

we have from

(2.1)

and

(2.4)"

I[Ty- lYll

^<

I[Ty- TIy’I[

+

fITly’ ITy’[I

+ -<

a[IIy IZy’ll

+

(l-a)

max

+lily’ Ty’II

+

I[Ty’

<-ally- ly’II

+

(l-a)max {]ITy- lyI[, fiTly’-

+ I/n + e

<-

a(llY TY’II

⁺

^l/n)

⁺

^(l-a)

^max

<-

(l+a)

⁺

(a+l)/n

⁺

(l-a)

max

{lily lyIl,

2/n}

Since e is arbitrary it follows that

ITY IYll

^<

^(a+l)/n

⁺

^(l-a)

^max

^{IITY ^IYlI,

^Z/n}

If

ITY ^ly]l

2/n, then we have

ITy lyll

^< 2/n ^<

(a+l)/an.

(2.5)

(5)

If

l]ly- lyi]

^> 21n,

(2.5)

implies

]ITy IYll

^-<

^(a+i).n

⁺

^(1-a).IITy ^lyll.

c Hn

^and so the point w must be So in both cases y lies in

H

n Thus

TK

_{n_}

in H for n 1,2 It follows that

n

lmw lwl!

^s

(a+l)/an

for n 1,2 and so

Tw

lw.

Since

(I.I)

holds, we also have

ITw TIw.

Thus

IT2w ^Tw!i

^<_ ^a

If

^ITw

lwI!

⁺

^(l-a)

^max

^{I T2w ITwlI, II ^Tw ^lwlI}

alIT2w

and it follows that

Tw w’

is a fixed point of

T

since a 1. Further lw’

ITw

⁺

TIw TTw Tw’ w’

and so

w’

is also a fixed point of I. Now suppose that T and have a second common fixed pcint

w".

Then

llw’ w"ll IITw’

-<

ailzw’ lw"II

⁺

^(-a)

^max

^{IITw’ ^zw’II, IImw"- ^.w"II}

and the uniqueness of the common fixed point follows since a 1. This completes the poof of the theorem.

EXAIIPLE

2. Let X R and C

[0,1]

^with the usual norm. Let T and be as in example 1. is clearly linear and nonexpansive and further

TC

[0,1/5]

C

[0,1/2]

IC Thus

411x Yll

^I

llx Yll

ITx TYll _(x+4)(y+4)

^-< ₂

II ^Ix ^IYll

for all x,y in C and inequality

(2.1)

^is satisfied for a 1/2.

So ^all the assumptions of Theorem 2 hold and w 0 is the unique common fixed point of T and

I.

Letting be the identity mapping in Theorem 2, we have the following corollary which extends Theorem i:

COROLI_ARY. Let

T

be a mapping of C into itself satisfying the inequality

llTx ^Tyl! allx ^yll

⁺

^(l-a)

^max

^{lITx ^xll, ^lITy ^yll}

for all x,) in

C,

where 0 a 1. Then

T

has a unique fixed point.

The result of this corollary was given in

[7].

We note that the weak commutativity in Theorem 2 is a necessary condition. It suffices to consider the following example:

EXAMPLE

3. Let X R and let C

[0,1]

with the usual norm.

Define T and by

Tx

1/3,

Ix

x/2 for any x in

C.

(6)

28 B. FISHER AND S. SESSA

It

is easily seen that all the conditions of Theorem 2 are satisfied except that of weak commutativity since with x I/2

fiT" ^{I/2) ^IT(1/2)II

^1/6 ^>

¹¹¹² ^IT(I2) ^I(112)II

However T

and do not have a common fixed polnt.

We conclude that although the mappings

T

and in Theorem 2 have a unique c:_mmon fixed point in

C,

it is possible for them to have other fixed points, as proved in the next example:

Example

.

^Let

^X

^C

^R

² ^with ^norm

ll(:’.,Y)ll

^max

{Ixl, IYl}

for all

(x,y)

in

R 2.

Define mappings

T

and on

R

² by

T(x,y) (O,y), l(x,y): (x,-y)

R

²

R

²

for all

(x,y)

in

R

² lhen for all

(x,y) (x’,y) IIT(x,Y) T(x’,Y’)II IY

and

alll(x,y)- l(x’,y’)ll

+

(I-a)max {llT(x,y)- l(x,y)!l, llT(x’,y’)- l(x’,y’)ll}

a max

{Ix-x’l,ly-y’l}

⁺

(l-a)max {Ixl, 21y I, Ix’l, 21y’ I}

>

elY Y’l

⁺

^2(l-a)

^max

{IYl, lY’l}

>

alY- Y’l

⁺

^(I-a)(IYl

⁺

^IY’I}

>_

if 0 < a < 1. Since

T

commutes with and is a linear isometry, it follows that all the conditiors of Theorem 2 are satisfied but

T

and each have an infinite number of fixed, points.

REFERENCES

1.

SESSA, S. On

a leak Commutativity Condition of Fappings in Fixed Point

Con-

siderations, Publ.

Inst.

Math., 32

(46) (1982),

149-153.

2.

DAS, K.M.

and

NAIK, K.V. Corm.ton

Fixed Point Theorems for Commuting

Maps

on

a

Metric

Space, Proc. Amer.

Math.

Soc.,

77

(1979),

369-373.

v

3.

GREGUS, Jr., M. A

Fixed Point Theorem in Banach

Space,

Boil.

Un. Mat.

Ital.,

(5_)

^17-a

^(1980),

^193-198.

4.

KIRK, W.A. A

Fixed Point Theorem for Mappings Which do not

Increase

Distances,

Amer.

Math

Monthlj/, 7__2(1965),

^1004-1006.

5. IiONG, Ch.

S. On Kannan Maps, Proc. Amer.

Math.

Soc.,

47

(1975),

105-111.

6.

DUGUNDIJ, J.

and

GRANAS, A.

Fixed Point

Theor), ^I,

Polish Scientific Publishers,

Warsawa (1982).

7.

FISHER, B.

Common Fixed Points on a Banach

Space, C.huni ^{Juan J.,} X_._l (1982),

12-15.

(7)

Mathematical Problems in Engineering

Special Issue on

Time-Dependent Billiards

Call for Papers

This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.

This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under diﬀerent approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many diﬀerent fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).

We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.

To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.

Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://

mts.hindawi.com/according to the following timetable:

Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009

Guest Editors

Edson Denis Leonel,Departamento de Estatística, Matemática Aplicada e Computação, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil ; [email protected]

Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;

[email protected]

Hindawi Publishing Corporation http://www.hindawi.com

ON A FIXED POINT THEOREM OF GREGU