Fixed point theorems for compatible mappings with applications to the solutions of functional equations arising in dynamic programmings

Internat. J. Math. & Math. Sci. VOL. 20 NO. 4 (1997) 673-680

673

FIXED

POINT

THEOREMS

FOR COMPATIBLE

MAPPINGS

WITH APPLICATIONS TO THE SOLUTIONS

OF

FUNCTIONAL

EQUATIONS

ARISING IN DYNAMIC PROGRAMMINGS

NAN-JING HUANG

Department

ofMathematics SichuanUniversity Chengdu,Sichuan 610064

P.R CHINA

BYUNGSOOLEE

Department

ofMathernatics

Kyungsung

University

Pusan608-736

KOREA

MEEKWANG KANG

Department

ofMathematics Dongeui University

Pusan614-714

KOREA

(ReceivedOctober4, 1995and in revisedform February21, 1996)

ABSTRACT. Some common fixed point theorems for compatible mappings are shown As an application, theexistenceanduniqueness of commonsolutionsforaclass of functional equations arising indynamic programmingsare discussed.

KEYWORDS ANDPItRASES: Commonfixedpoint, compatible mapping, dynamic programming 1991AMSSUBJECTCLASSI:FICATION CODES: 541-125,47H10

1. INTRODUCTION

In [1] the concept of compatible mappings was introduced as a generalization of commuting mappings and further investigationwas given in

[2-9]

Thepurpose ofthispaperis toprovesome common fixedpointtheoremsfor compatible mappings, whichgeneralized somerecentresults of[4, 10-13] Asanapplication,weusetheresults presentedto study theexistence anduniquenessproblem ofacommon solutionforaclassof functional equations arisingindynamic programmings,whichgeneralized the corresponding results of 14,15].

2. FIXED POINT THEOREMS

DEFIIIITION 2.1. Self mappings

A

and S ofa metric space

(X, d)

are called compatible, if

lirnd(ASzn,

SAzn)

0 whenever

{z,}

is a sequencein X suchthat lim,

Az

lim,

Szn

for some inX

Itis clearthat commuting mappings andweaklycommuting mappingsareallcompatiblemappings, butthe converseisfalse(see 1,

4]).

LEMMA

2.2 [1,4] If

A

and S are compatible sdf mappings ofa metric space

(X, d)

and lirn,

Szn

lim,

Az

tforsometin

X,

thenlim,ASz, StifSis continuous.

Thefollowingtheorem can be obtainedfromTheorem 8 in 16].

TIIEOREM2.3. Let

(X, d)

be acompletemetricspace and

A,

B, SandTareself mappingsof

X. SupposethatSand

T

arecontinuous,

A(X) c

T(X), B(X)

C

S(X),

and that

A,

Sand

B, T

are compatible and satisfythefollowingcondition:

d(Az,

By)

<_

(max{d(Sx,

Ty),

d(Sz,

Ax),

d(Ty, By),

1

[d(Sx,

By)

+

d(Ty,

Ax)]}),

V x,y6

X,

(2 l) 2

LEE AND M K

Then

A,

B,

SandThaveauniquecommon fixed point inX

We merelystatethe prooffor convenience

PROOF. Since

A(X)

C

T(X)

and

B(X) c S(X),

wecanchooseasequence

(z,,}

inXsuch that

Sx2, Bx2,_Iand

Tx,_I

Ax2,_2 forallnin thesetNofallpositive integers Let

6

N).

(2.2)

Asin 10],wecanprovethat yn

}

is aCauchysequencein

X

Letting y,,

.

X

(n

oo),we knowthat

{y2,,

}

and

{y_,_l

}

convergetoy.too.

SinceAand

S,

BandTarebothcompatible,itfollows fromthecontinuity ofSand

T,

(2.2)and

Lemma2.2that

Ty2,.,_

Ty.,

By2,- Ty., Sy2,., Sy., Ay2,., Sy.. (2.3)

By (2.1)and(2.2)wehave

d(Ay2,,By2,.,_

<_

’(max{d(Sv2,,

Ty2r,-1),d(Sy2,.,, Ay2,.,),

1

[d(Sy2,

By2,.,-)

+

d(Ty2,-1

Ay2n)]}).

d(Ty2,.,-,By2,,-I

),

Bytheuppersemicontinuity of

(t)

and

(2.3)

wehave

d(Sy.,

Ty.

<_ ,(max{d(Sy., Ty. ),

O, O, d(Sy.,

Ty.

)}

(d(S.,

T.)).

ThisimpliesthatSy. Ty.

Similarly, from(2.1), (2.2)and

(2.3)

wecan obtain

Sy. By., Ty. Ay..

Hencewehave

Ay. By. Sy. Ty..

From (2.1)and(2.2)wehave

d(Ax2,,By.) <_ (max{d(Sx_,,,Ty.), d(Sx2n, Ax2,),

1

d(Ty., By.),

-

[d(Sx2,.,,

By.)

+

d(Ty.,

Ax2n)]}),

and then

d(y.,By.)

<_ (d(y,,By.)).

(2 4)

LEMMA

2.5

17].

Let

K

beaclosed subset ofacompleteconvex metricspace X. Ifx

K

and y

K,

then thereexists apointz OKsuchthat

d(x, z)

+

d(z, V)

d(x,

y).

DEFINITION 2.6. Let

(X,d)

be a metric space, KC

X

and A,S"K--,X. The pair of mappings

A

andSiscalledcompatible,if

lira. d(ASx.,SAz.)

0whenever

{x,,}

is asequenceinK

such that

Axe,

Sx.

Kand

lim.

Axn

lim.

Sx.

K Hencewehave V.

By.

Ay.

Sy.

Ty.

Theuniquenessis obvious. Thiscompletestheproof

DEFINITION 2.4. A metricspace

(X, d)

is(metrical)convex, ifforeachx,y Xwith x

#

y, themexists az

X,

x z

=

y, such that

FIXED POINT THEOREMS FOR COMPATIBLE MAPPINGS 67 5

LEMMA 2.7. Let

(X,d)

be a metric space, KCX and A,S’K-X

IrA

and S are compatible mappings,

Azn,

Szn

6K and lim,Azn

limnSz

for some 6

K,

then

limASzn

StifSis continuous.

PROOF. ItisobviousfromDefinition 2 6

TItEOREM2.8. Let

(X, d)

be acompleteconvexmetricspaceandKa nonemptyclosedsubsetof

X SupposethatSandTare continuousmappings fromXintoXwithOKC

S(K)

N

T(K)

and that

A, B"

K

X are continuous mappings with

A(K)

NKC

S(K),

B(K)

tKC

T(K)

Suppose

furtherthatthe pairs of mappings

A, T

and

B,

Sarecompatibleandsatisfying

d(Az,

By)

<_ (d(Tz,

Sy)),

gz,

K,

(2 5)

where(I)

[0,

oo)

-,

[0,

oo)

_{isnondecreasinguppersemi-continuous}and

(I,(t) <

ooforall

>_

0 If forx6

K,

Tx OKimplies

Ax,

Bx KandSz6OKimpliesAz,

Bx

6K,then there exists azEKsuchthat

z Tz Sz

Az

Bz.

IfTv Sv Av By,thenTz Tv

PROOF. Letp6OK. Using Lemma 2 5 andtheproof of I] we canchoose two sequences

{P-}.N

and

{id.}.z

N such that for any n6N,p6

K,

._,

Ap2n,

p.

Bp2n-1 and the following implicationshold

If

,

6

K,

then

n

Tp2,if

p.

K,

thenTp2.6OKand (i)

d(Sp2.-1, Tp2,)

+

d(Tp2,,Bp2,-I d(Sp2,.,-,Bp2,-,

If,+

6

K,

then

id2,,+

Sp2,+l,

ifid2,+

6

K,

thenSp2,+l 6OKand

(ii)

d(T,S+)

+

d(S.+,A)

d(T.,A.)

Fuher,asin[3]we cprovethat

(

6

N)

(2.6)

d(S.+I,T..2)

On(r)

J

wherer

m{

d(T,

S

),

d(T,

Sp1

}.

TMs

impliesthatfor yn6

N,

d(T,, T,+2)

en-1

(r)

+

Hencethesequence

{T=

}=eN

is aCauchy

suence.

Since

X

iscompleted

K

isclose,itfollows that theestsaz Ksuchthatz

li T,

From (2 6)wehave

z lira

T

lira

S.1.

Nowweprovethatz

Tz

Sz

Az

Bz

Itisobous that there

es

a

suenc

{n

CN

suchthat

T,

B,-1,

or

S,-1

A=,-2,

Vk N. Without lossof generity,wecsuppose at

T,

B=_,

VkqN From(2.5)wehave

d(ST=,,Az)

d(SB,_,BS,_I)

+

d(BS,_,Az)

d(SB=,_,BS_)

+

O(d(SS_I,Tz)).

Since

B,

Secompatible dSiscontinuous,wehave

d(S, A,)

(d(S, T)).

(2 7)

676 LEE

d(Ap2,, Tp2,.,,) d(Ap,,,, Bp2,,_

<

b(d(Sp,2,-,Tp2,)).

Bytheupper semi-continuityof

(t),

itfollowsthat limAp2, z.

k

Using(2.5)wehave

d(Ap2,,, BSp’2r,,-

<

(d(Tp2,.,,, SSp2,..,_] ).

Since

B,

SarecompatibleandSiscontinuous,itfollows from(2.8)andLemma27 that

d(z, Sz) < (d(z, Sz)).

Thisimpliesthat d

(z, Sz)

0, e. z Sz.

Since

A, T

arecompatibleand

A

and

T

arecontinuous, from

(2.8)

andLemma2.7 we have

Az limATp,2,.,,, Tz.

k

Inviewof(2.7)wehave

d(Sz, Tz) < (d(Sz, Tz))

andso

z Sz Tz Az.

Besides, from(2.5)wehave

d(Az, Bz) <_ b(d(Sz, Tz))

(0)

O.

Hence

z Tz Sz

Az

Bz.

Finally,ifTv Sv Av

Bv,

then

d(Tv,

Sz)

d(Av, Bz) < ,(d(Sz, Tv)

(28)

where

(t)

isthesame as inTheorem2.8. If for everyn

e

11 andx

K,

Tx

OK implies A,x

.

K and Sx OK implies A,.,x K,

then there exists a z Ksuch that

z

Tz

Sz

A,.,z,

Vn N,

and ifTv Sv

Av

for everyn ll,thenTz Tv.

REMARK2.10. Theorem 2.9 is ageneralization of Theorem in 11 and soTv Sz

T

z.

Thiscompletestheproofof Theorem28.

Asanimmediateconsequencewecan obtain thefollowingresult.

THEOREM 2.9. Let

(X, d)

be acomplete convex metric space, K. anonempty closed subset of

X,

andS and

T

continuous mappings from

X

into

X

such that OKC

S(K)f’l

T(K).

Suppose

that for every heN,

A,.,

:K X is a continuous mapping with

A,.,(K)

fqKC

T(K)

and

A2r-I

(K)

NKC

S(K),

and that thepairs of mappings A2,-1,

T

andA2,Sarecompatible suchthat for anyn

FIXEDPOINTTHEOREMS FOR COMPATIBLE MAPPINGS 67 7 3. APPLICATIONS

Throughoutthis section we assumethatXand

Y

areBanachspaces, S

c X

is astatespace,

D

CY

a decision space and

R

(-

oo,

+

x)

We denote by

B(S)

the set ofall bounded real-valued functionsdefinedonS.

As suggested in Bellman and Lee [18], the basic form of the functional equations of dynamic programming is

f(z)

optuH(z,y,

f(T(z,y))),

wherezand y represent thestateanddecision vectorsrespectively,

T

represents the transformation of theprocess, and

f

(z)

representsthe optimalreturnfunctionwith initial state z(hereopt denotesmax or rain)

Inthissection, weshall study theexistenceand uniqueness ofacommon solutionofthefollowing functionalequations arisingindynamic programmings

f(x)

supHl(X,y,f(T(x,y))),

x6

S,

(3 1)

y6D

g(x)

supH2(x,y,g(T(x,y))),

=

e S,

(32)

y6D

p(x) supFl(x,y,p(T(c,y))), x S, (3 3)

yeD

q(x) supF2(x,y,q(T(x,y))), x S, (34)

yq.D

where

T

Sx D--,

S, H,

and

F,

S

x

D

x R-,

R,

1,2.

THEOREM3.1. Supposethat the followingconditions are satisfied

(i)

H,

and

F,

arebounded, 1, 2.

(ii) [H(x,y,h(t))- H(x,y,k(t))[

<_

(max{]T]h(t)-

T2k(t)],

ITch(t)- Alh(t)[, [T2k(t)- Ak(t)[,

1/2

[[Tlh(t)-

Ak(t)[

+

[Tk(t)-

Ah(t)[]}),

for all

(x,

y)

e

S x

D,

h,k

B(S)

and

e

S,where isthe same as in Theorem 2 3, and themappings

A,

and

T,

aredefined asfollows

A,h(x)

supH,(x,y,h(T(x,y))), x

e S,

h

B(S)

and

yD

then

T,k(x)

suplZ,(x,y,k(T(x,y))), x S, k

B(S),

i-1,2.

yD

(iii)For any

{

k,

}

C

B(S)

andk

B(S),

limsup[k,(x)-

k(x)[

0 implies limsup[T,k,,(x)-

T,k(x)]

0,

:r6S xS

i= 1,2.

(iv) Forany h

B(S),

thereexistk:l, ]’,2

B(S)

suchthat

Alh(x)

T2k(x),

A2h(x)

Tlk,2(x),

x6S.

(v)For any

{

k,

}

C

B(S),

if there exists h

B(S)

suchthat

limsup

]A,k(x)- h(x)l

limsup

]T,k,,(x)

h(x)l

O, xeS

limsuplA,

T,k.(x)

T,A,k.(x)]

0, 1,2. zES

PROOF. Forany k, k

B(S),

let

d(h, k)

sup

{Ih(x)

k(x)l

x

S},

then

(B(S),

d) is acompletemetricspace From (i)-(v)weknowthat

A,

and

T

areselfmappings of

B(S),T,

are continuous,

AI(B(S))

C

T2(B(S)),A2(B(S))

c

TI(B(S)),

and the pair of mappings

A,,

T

arecompatible, 1,2.

Lethi,

h2

beanytwopoints of

B(S),

let x Sand 7beanypositivenumber,there existYl,andy2

inDsuch that

Also wehave

A,h(x)

+

(i

,2).

where x,

T(x,

y,) (3 5)

Ah(x) > Hl(x,y:z,hl(x2)),

(3 6)

A2h2(x) >

H2(x,ya,h2(xl)). (3 7)

From (3.5), (3 7)and(ii)wehave

Alhl(x)

A2h2(x) < Hl(x,

yl,hl(X))-

H2(X,

yl,h2(Xl))

<

[H(x,y,h(x))- H2(x,y,h(x))[ +7

<_

(max{lTth(x)

T2h2(x)

I,

[Tlh(x)

Alh(x)[,

1

IT2h2(xx)-

Ah2(z)[,-

[ITh(xa)

A2h(x)l

+

[Th2(zl)-

Ah(x)[]})

+

rl

<

(max{d(Thl,Th),d(Tlh,Ah),

1

d(Th,

Ah2

),

-

[d(T

h,

Ah2

+

d(T2hg.,Ahl)]})

+

7.

Similarly from(3.5), (3.6)and(ii)wehave

Ah(x)

Ah(x)

>_

,(max{d(Tha,Th),d(Th,Alh),

1

d(T2h2,A2h2),

[d(Th,A2h2)

+

d(T2hg_,Ah)]}) rl.

Hencewehave

IAlh(x)-

A2h2(x)[ <_

(max{d(Th,T2h2),d(Thx,Ah),

1

d(T2h2, A2h2

),

-

[d(T

h,

A2h2

+

d(T2h2,Ah)]})

+

rl.

(3 8)

Since(3.8)istrueforanyx Sandr/isany positivenumber,we have

d(Al

h A2h2 <_ (max(d(T

h,

T2h:

), d(T

h,

Al hl

),

1

[d(Thl

Ah),

d(T2h2,

A2h2

),

"

+

d(T2h2,Aht)]}).

Therefore byTheorem2.3,A1,A2,

T1

and

T2

haveaunique common fixedpoint h*

B(S),

i.e h*

(x)

is auniquecommonsolutionoffunctionalequations(3.1) (3.4). Thiscompletestheproof

Thefollowing resultis animmediateconsequenceofTheorem2.3andTheorem 3

THEOREM3.2. Supposethatthe following conditionsare satisfied: (i)

Hi

isbounded, 1,2;

FIXEDPOINTTHEOREMS FOR COMPATIBLEMAPPINGS 679 for all

(z,

y)ESx

D,

h,k

B(S)

and

S,

where isthe same as inTheorem23and

At

isdefined by

Ah(x)

supH,(x,y,h(T(x,y))),

x S, h

B(S),

1,2. yED

Then the functionalequations(3.1)and(3.2)haveaunique commonsolution in

B(S)

REMARK3.3. Theorem 3.2 is ageneralizationof Theorem2.1in 15].

TtlEOREM3.4. Supposethatthefollowingconditionsare satisfied.

(i) H,and

F

arebounded, 1,2,

(ii)

[Hl(x,y,h(t))- H2(x,y,k(t))[ <

(ITh(t)- T2k(t)l)

for all (x,y)

S

D,

h,k

B(S)

and

S,

where isthe sameas inTheorem2.8 and

T,

isdefined as inTheorem3 1, 1,2;

(iii) For any

{k,,}

C

B(S)

andk

B(S),

limsup

[k,,(x)- k(z)[

0 implies limsup

[T, kn(z)- T,k(z)[

0

xES xS

and

limsuplAzk,,(x)

A,k(x)l

0, 1,2,

xES

where

Az

isdefined as inTheorem 3.1, 1,2;

(iv) For anyh

B(S)

suchthatsupxeslh(z)l 1,thereexistk,

k2

B(S)

suchthat sup

Ik,(z)l <_

1 and

k(x)

h(x),

x

S,

i=1,2"

then

(v) Foranyh

B(S)

such thatsupes

Ih(x)[ <

1, there existk,

k

B(S)

such that

supl,(x)l

_<

1,

-

1,2,

Ah(x)

T2kx(x)

and

A2h(x)

Tk2(x),

x S" (vi)For anyh

B(S)

such thatsupxes

Ih(x)] <

1,

suplZh(x)[ 1 implies

sup[Ash(x)[

< 1, i,j 1,2;

(vii)Forany

{k}

C

B(S),

ifthere exists h

B(S)

suchthatsupzes

IT(x)]

< 1and limsup

[Ak(x)- h(x)!

limsup

[T,k,(x)

h(x)[

0,

zS

limsuPlA,

T,k.(x)

T,Ak,(x)l

O,

1,2

Then the systemoffunctionalequations(3 1) (3.4)haveauniquecommon solution h"

B(S)

and

supslh’(z)l

_<

1.

PROOF. Letusconsider

B(S)

as aBanachspaceof all bounded real-valued functionsdefined onS

with a supremum norm, and

K

the closed unit ball in

B(S).

By conditions (i)-(vii) we know that

At

K

B(S)

and

T,

B(S)

B(S),

1,2, satisfy all of theconditionsofTheorem 2.8 and have auniquecommonfixedpoint h" K,i.e, h*

(x)

isaunique commonsolutionoffunctional equations

(3.1) (3 4).

REMARK3.5. Theorem 3 4 is ageneralization ofTheorem 3.2 in 14].

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