Ž .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 222, 494]504 1998ARTICLE NO. AY985947

On Characterizations of Fixed and CommonFixed Points

Zeqing Liu

Department of Mathematics, Liaoning Normal Uni ersity, Dalian, Liaoning 116029,People’s Republic of China

Yuguang Xu

Department of Mathematics, Kunming Junior Normal College, Kunming, Yunnan650031, People’s Republic of China

and

Yeol Je Cho*

Department of Mathematics, Gyeongsang National Uni ersity, Chinju 660-701, Korea

Submitted by L. Debnath

Received November 24, 1997

In this paper we give necessary and sufficient conditions for the existence offixed and common fixed points of self-mappings of metric spaces. Our resultsgeneralize, improve, and unify some results of Chang, Jungck, Fisher, and Dien.Q 1998 Academic Press

Key Words: fixed point; common fixed point; periodic point; compatible map-pings; C-mapping; L-mapping.

1. INTRODUCTION AND PRELIMINARIES

w xJungck 6, 7 first gave two necessary and sufficient conditions for thew xexistence of fixed points of continuous mappings. Afterwards, Chang 1 ,

w x w x w x w x w xFisher 4 , Guo 5 , Kang and Ryu 9 , Khan and Fisher 10 , Kubiak 11 ,

* E-mail: yjcho@nongae.gsnu.ac.kr.

494

0022-247Xr98 $25.00Copyright Q 1998 by Academic PressAll rights of reproduction in any form reserved.

FIXED AND COMMON FIXED POINTS 495

w x w x w xLiu 12]14 , Park 15 , Sessa, Mukherjee and Som 18 , Singh, Tiwari, andw xGupta 19 , and others generalized Jungck’s results in various directions.

w xChang 2 introduced the concept of C-mappings and proved that theŽ . Ž . w xmappings numbered 1 ] 25 by Rhoades 17 are all particular cases of

C-mappings and gave a necessary and sufficient condition for the existenceof fixed points of a C-mapping.

w xRecently, Dien 3 obtained some fixed point theorems, which deal withCaristi’s condition and the Banach contraction principle.

In this paper we introduce the concept of L-mappings, which is equiva-lent to the concept of C-mappings, and establish criteria for the existenceof fixed points of L-mappings. By using the ideas of Jungck and Dien, wealso obtain necessary and sufficient conditions for the existence of fixedand common fixed points of continuous mappings. Our results generalize,

w x w ximprove, and unify the corresponding results of Chang 2 , Dien 3 , Fisherw x w x4 , and Jungck 6 .

w xDEFINITION 1.1 8 . Let f and g be self-mappings of a metric spaceŽ .X, d . The mappings f and g are said to be compatible if

lim d fgx , gfx s 0,Ž .n nnª`

� 4whenever x is a sequence in X such that lim fx s lim gxn ns1 nª` n nª` ns t for some x g X.

w x Ž .DEFINITION 1.2 2 . Let T be a self-mapping of a metric space X, d .

Ž .1 The mapping T is called a C-mapping if, for any x g X and anyinteger n G 2 with

1.1 T i x / T j x , 0 F i - j F n y 1,Ž .1.2 d T n x , T i x - max d x , T m x : 1 F m F n� 4Ž . Ž . Ž .

for i s 1, 2, . . . , n y 1.Ž . Ž .2 The mapping T is called a Rhoades mapping if d Tx, Ty -� Ž . Ž . Ž . Ž . Ž .4max d x, y , d x, Tx , d y, Ty , d x, Ty , d y, Tx for any distinct x, y g X.

w xChang 2 proved that every Rhoades mapping is a C-mapping. But wecan give an example that the converse is not true:

w x � 4EXAMPLE. Let X s 0, 1 j 3, 6, 10 with the usual metric. Define T :X ª X by

w xTx s x for x g 0, 1 , T 10 s 6, T 6 s 3, T 3 s 1.Ž . Ž . Ž .

Ž . Ž . w xIt is easy to see that T does not satisfy conditions 1 ] 25 in Rhoades 16 .Ž .We now show that T is a C-mapping. In order to ensure that 1.1 holds,

LIU, XU, AND CHO496

� 4we have the following cases for x g 3, 6, 10 :In case x s 3: For n s 2, we obviously have

d T 2 3 , T 3 s 0 - 2 s max d T 2 3 , 3 , d T 3 , 3 .Ž . Ž . Ž . Ž .� 4Ž .Ž . Ž .

� 4In case x s 6: For n g 2, 3 , we have

d T 2 6 , T 6 s 2 - 5 s max d T 2 6 , 6 , d T 6 , 6Ž . Ž . Ž . Ž .� 4Ž .Ž . Ž .

and

3 3 2 � 4max d T 6 , T 6 , d T 6 , T 6 smax 0, 2 s2Ž . Ž . Ž . Ž .� 4Ž . Ž .- 5 s max d T 3 6 , 6 , d T 2 6 , 6 , d T 6 , 6 .Ž . Ž . Ž .� 4Ž .Ž . Ž .

� 4In case x s 10: For n g 2, 3, 4 , we have

d T 2 10 , T 10 s 3 - 7 s max d T 2 10 , 10 , d T 10 , 10 ,Ž . Ž . Ž . Ž .� 4Ž .Ž . Ž .3 3 2 � 4max d T 10 , T 10 , d T 10 , T 10 smax 5, 2 s5Ž . Ž . Ž . Ž .� 4Ž . Ž .

- 9 s max d T 3 10 , 10 , d T 2 10 , 10 , d T 10 , 10 ,Ž . Ž . Ž .� 4Ž .Ž . Ž .and

max d T 4 10 , T 10 , d T 4 10 , T 2 10 , d T 4 10 , T 3 10Ž . Ž . Ž . Ž . Ž . Ž .� 4Ž . Ž . Ž .� 4s max 5, 2, 0 s 5 - 9

s max d T 4 10 , 10 , d T 3 10 , 10 , d T 2 10 , 10 , d T 10 , 10 .Ž . Ž . Ž . Ž .� 4Ž .Ž . Ž . Ž .

Ž .Thus condition 1.2 is satisfied. Hence T is a C-mapping.From the above example, we have the following:

PROPOSITION 1.1. E¨ery Rhoades mapping is a C-mapping, but thecon¨erse is not true.

Ž .DEFINITION 1.3. Let T be a self-mapping of a metric space X, d . Themapping T is called an L-mapping if, for any x g X and any integer n G 2

Ž .with condition 1.1 ,

1.3 d T n x , T i x - max d T p x , T q x : 0 F p - q F n� 4Ž . Ž . Ž .

for i s 1, 2, . . . , n y 1.

PROPOSITION 1.2. C-mapping and L-mapping are equi alent.

Proof. It is obvious that every C-mapping is a L-mapping.

FIXED AND COMMON FIXED POINTS 497

Ž .Conversely, let T be a L-mapping and let x g X. Suppose that 1.1 issatisfied for some n G 2. Then we have

min d T i x , T j x : 0 F i - j F k y 1 ) 0,� 4Ž .where 2 F k F n. Put

A s max d T n x , T i x : 1 F i F n y 1 ,� 4Ž .n

B s max d x , T i x : 1 F i F n .� 4Ž .n

Ž .From 1.3 , it follows that

A - max B , A , max d T p x , T q x : 1 F p - q - n y 1� 4� 4Ž .n n n

s max B , max d T p x , T q x : 1 F p - q F n y 1� 4� 4Ž .n

s max B , A , max d T p x , T q x : 1 F p - q F n y 2� 4� 4Ž .n ny1

F max B , B , max d T p x , T q x : 1 F p - q F n y 2� 4� 4Ž .n ny1

s max B , max d T p x , T q x : 1 F p - q F n y 2� 4� 4Ž .n

F ???

F max B , max d T p x , T q x : 1 F p - q F 2� 4� 4Ž .n

s max B , d Tx , T 2 xŽ .� 4n

F max B , max d x , Tx , d x , T 2 x� 4Ž . Ž .� 4n

s B .n

Ž .Therefore, 1.2 is satisfied and so T is a C-mapping. This completes theproof.

2. CHARACTERIZATIONS OF FIXED POINTSOF L-MAPPINGS

w xFrom Theorem 1 of Chang 2 and Proposition 1.2, we have the follow-ing:

Ž .THEOREM 2.1. Let T be an L-mapping from a metric space X, d intoitself. Then the following conditions are equi alent:

Ž .i T has a fixed point in X.Ž .ii T has a periodic point in X.Ž .iii There exist integers m, n with m ) n G 0 and a point x in X such

that T m x s T n x.n Ž .Moreover, T x is a fixed point of T provided that iii holds.

LIU, XU, AND CHO498

From Theorem 2.1, we immediately obtain the following:

THEOREM 2.2. Let T be a continuous L-mapping from a metric spaceŽ . � n 4X, d into itself. Suppose that there exists x g X such that T x has ans0cluster point u. Then the following conditions are equi alent:

Ž . � n 4iv T has a fixed point in T u .ns0

Ž . � n 4v T has a periodic point in T u .ns0

Ž . m nvi There exist integers m, n with m ) n G 0 such that T u s T u.

Remark 2.1. Theorems 2.1 and 2.2 extend and improve Theorems 1 andw x2 of Chang 2 , respectively.

THEOREM 2.3. Let FF be a family of L-mappings from a metric spaceŽ .X, d into itself. Then the following conditions are equi alent:

Ž .vii FF has a fixed point in X.Ž .viii FF has a periodic point in X.Ž .ix There exist x, y g X such that for each T g FF there exist integers

Ž . Ž . Ž . Ž . nŽT . mŽT .m T , n T with m T ) n T G 0 such that T x s T x s y.

QUESTION. If T is a continuous L-mapping from a compact metric spaceŽ .X, d into itself, does T possess a fixed point in X ?

w xRemark 2.2. Park 16 proved that T has a unique fixed point in X if TŽ .is a continuous Rhoades mapping from a compact metric space X, d into

itself.

3. CHARACTERIZATIONS OF FIXED POINTS OFCONTINUOUS MAPPINGS

THEOREM 3.1. Let f be a continuous self-mapping of a metric spaceŽ .X, d . Then the following statements are equi alent:

Ž .3.1 f has a fixed point in X.Ž .3.2 There exist z g X, a mapping g : X ª X, and a function F from

w . Ž . Ž .X into 0, ` such that f and g are compatible, g X ; f X , g is continuous,and

a d gx , z F r d fx , z q F fx y F gxŽ . Ž . Ž . Ž . Ž .

w .for all x g X and some r g 0, 1 .Ž .3.3 There exist z g X, a mapping g : X ª X, and a continuous

w . Ž . Ž .function F from X into 0, ` such that f and g are compatible, g X ; f X ,Ž .and a holds.

FIXED AND COMMON FIXED POINTS 499

Ž . Ž . Ž .Proof. 3.1 « 3.2 and 3.3 Let w be a fixed point of f. TakeŽ .r s 1r2, gx s w s z, and F x s 1 for all x g X. It is easy to verify that

Ž . Ž .3.2 and 3.3 hold.Ž . Ž . Ž . Ž .3.2 « 3.1 Take any point x g X. Since g X ; f X , there ex-0

Ž .ists x g X such that gx s fx for n G 1. It follows from a thatn ny1 n

d fx , z s d gx , zŽ .Ž .jq1 j

F r d fx , z q F fx y F gxŽ .Ž . Ž .j j j

s r d fx , z q F fx y F fx ,Ž . Ž . Ž .j j jq1

which implies that

nq1 n n

d fx , z F r d fx , z q F fx y F fx ,Ž . Ž . Ž . Ž .Ý Ý Ýj j j jq1js1 js0 js0

that is,

n r 1d fx , z F d fx , z q F fx y F fxŽ . Ž . Ž .Ž .Ý j 0 0 nq11 y r 1 y rjs1

r 1F d fx , z q F fx ,Ž . Ž .0 01 y r 1 y r

` Ž . Ž .which means that the series Ý d fx , z is convergent and so d fx , zns1 n nª 0 as n ª `. The continuity of f and g implies that fgx s ffx ª fzn nq1and gfx ª gz as n ª `. The compatibility of f and g ensures thatn

d fz , gz F d fz , fgx q d fgx , gfx q d gfx , gz ª 0Ž . Ž . Ž . Ž .n n n n

Ž .as n ª `. Hence fz s gz. From a , it follows that

d gz , z F r d fz , z q F fz y F gz s r d gz , z ,Ž . Ž . Ž . Ž . Ž .

which implies that z s gz s fz.Ž . Ž . � 4 Ž . Ž .3.3 « 3.1 Let x and z be as in the proof of 3.2 « 3.1 .n ns1

Then fgx s ffx ª fz as n ª `. From the compatibility of f and g, wen nq1have

d gfx , fz F d gfx , fgx q d fgx , fz ª 0Ž . Ž . Ž .n n n n

LIU, XU, AND CHO500

Ž .as n ª `, that is, gfx ª fz as n ª `. It follows from a thatn

d gfx , z F r d ffx , z q F ffx y F gfxŽ . Ž . Ž . Ž .n n n n

for n G 1. Note that F is continuous. Letting n tend to infinity in theŽ . Ž .above inequality, we have d fz, z F r d fz, z , which means that fz s z.

This completes the proof.

THEOREM 3.2. Let f be a continuous self-mapping of a metric spaceŽ . Ž .X, d . If 3.2 is satisfied, then f and g ha¨e a unique common fixed pointin X.

Ž . Ž .Proof. From the proof of 3.2 « 3.1 , we have z s fz s gz. SupposeŽ .that w is also a common fixed point of f and g. It follows from a that

d w , z s d gw , z F r d fw , z q F fw y F gw s r d w , z ,Ž . Ž . Ž . Ž . Ž . Ž .

which implies that w s z. Hence f and g have a unique common fixedpoint in X. This completes the proof.

Remark 3.1. Theorem 3.2 generalizes and improves Theorem 2.2 ofw xDien 3 .

THEOREM 3.3. Let f be a continuous self-mapping of a complete metricŽ . Ž .space X, d . Then 3.1 is equi alent to the following:

Ž .3.4 There exist a mapping g : X ª X and functions F, C from X intow . Ž . Ž .0, ` such that f and g are compatible, g X ; f X , g is continuous, and

bŽ .d gx , gy F a d fx , fy q a d fx , gx q a d fy , gyŽ . Ž . Ž . Ž .1 2 3

q a d fx , gy q a d fy , gxŽ . Ž .4 5

q F fx y F gx q C fy y C gyŽ . Ž . Ž . Ž .

w .for all x, y g X, where a , a , a , a , and a are in 0, 1 with a q a q a1 2 3 4 5 1 4 5- 1 and a q a q a q 2 a - 1.1 2 3 4

Ž . Ž .Proof. 3.1 « 3.4 Let w be a fixed point of f. Take r s a s 1r2,1a s a s a s a s 0, gx s w s z for all x g X. Let F and C be con-2 3 4 5

Ž .stant. Then condition 3.4 is satisfied.Ž . Ž .3.4 « 3.1 Let x be an arbitrary point of X. Define a sequence0

� 4 Ž . Ž .x in X by gx s fx for n G 1. This is possible since g X ; f X .n ns1 ny1 n

FIXED AND COMMON FIXED POINTS 501

Ž . Ž .Put d s d fx , fx for n G 0. From b , we haven n nq1

d s d gx , gxŽ .jq1 j jq1

F a d fx , fx q a d fx , gx q a d fx , gxŽ . Ž . Ž .1 j jq1 2 j j 3 jq1 jq1

q a d fx , gx q a d fx , gxŽ . Ž .4 j jq1 5 jq1 j

q F fx y F gx q C fx y C gxŽ . Ž .Ž . Ž .j j jq1 jq1

F a q a q a d q a q a dŽ . Ž .1 2 4 j 3 4 jq1

m k

q F fx y F fx q C fx y C fx ,Ž . Ž . Ž . Ž .Ý Ýi j i jq1 i jq1 i jq2is1 is1

which implies that

d F a d q b F fx y F fx q b C fx y C fx ,Ž . Ž . Ž . Ž .jq1 j j jq1 jq1 jq2

where

a q a q a 11 2 4a s , b s .

1 y a y a 1 y a y a3 4 3 4

Thus it follows that

nq1 n n

d F a d q b F fx y F fxŽ . Ž .Ý Ý Ýj j j jq1js1 js0 js0

n

q b C fx y C fx ,Ž . Ž .Ý jq1 jq2js0

which means that

n

d F c d q b F fx y F fxŽ . Ž .Ý j 0 0 nq1js1

q b C fx y F fxŽ . Ž .1 nq2

F c d q bF fx q bC fxŽ . Ž .0 0 1

Ž . `for all n G 1, where c s ar 1 y a . Therefore, the series Ý d isns1 nŽ . nqpy1convergent. For any n, p G 1, we have d fx , fx F Ý d . Thisn nqp isn i

� 4means that fx is a Cauchy sequence in X. The completeness of Xn ns1ensures that there exists t g X such that fx ª t as n ª `. Since f and gnare continuous, fgx s ffx ª ft and gfx ª gt as n ª `. Noting that fn nq1 n

LIU, XU, AND CHO502

Ž . Ž .and g are compatible, d fgx , gfx ª d ft, gt s 0 as n ª `, that is,n nŽ .ft s gt. It follows from b that

d ft , fx s d gt , gxŽ . Ž .nq1 n

F a d ft , fx q a d ft , gt q a d fx , gxŽ . Ž . Ž .1 n 2 3 n n

q a d ft , gx q a d fx , gtŽ . Ž .4 n 5 n

q F ft y F gt q C fx y C gxŽ . Ž . Ž . Ž .n n

s a q a d ft , fx q a d q a d ft , fxŽ . Ž . Ž .1 5 n 3 n 4 nq1

q C fx y C fx ,Ž . Ž .n nq1

which implies that

n ny1 ny1

1 y a d ft , fx F a q a d ft , fx q a dŽ . Ž .Ž . Ž .Ý Ý Ý4 j 1 5 j 3 jjs1 js0 js0

ny1

q C fx y C fx .Ž . Ž .Ý j jq1js0

Therefore, we have

ny1

1 y a y a y a d ft , fxŽ . Ž .Ý1 4 5 jjs1

ny1

F a q a d ft , fx q a d q C fx .Ž . Ž . Ž .Ý1 5 0 3 j 0js0

Since the series Ý` d is convergent and a q a q a - 1, it followsns1 n 1 4 5` Ž .that the series Ý d ft, fx is also convergent. This implies that fx ª ftns1 n n

as n ª `, that is, t s ft s gt. This completes the proof.

Remark 3.2. Theorem 3.3 extends, improves, and unifies Theorem ofw x w xJungck 6 and Theorem 2 of Fisher 4 .

THEOREM 3.4. Let f be a continuous self-mapping of a complete metricŽ . Ž .space X, d . If 3.4 is satisfied, then f and g ha¨e a unique common fixed

point in X.

FIXED AND COMMON FIXED POINTS 503

Ž . Ž .Proof. It follows from the proof of 3.4 « 3.1 that f and g have acommon fixed point t in X. If ¨ is also a common fixed point of f and g,

Ž .from b , we have

d ¨ , t sd g¨ , gt F a d ¨ , t q a d ¨ , ¨ a d t , t q a d ¨ , tŽ . Ž . Ž . Ž . Ž . Ž .1 2 3 4

q a d t , ¨ q F ¨ y F ¨ q C t y C tŽ . Ž . Ž . Ž . Ž .5

s a q a q a d ¨ , t ,Ž . Ž .1 4 5

which implies that r s t since a q a q a - 1; that is, f and g have a1 4 5unique common fixed point. This completes the proof.

By using a method similar to that in the proofs of Theorems 3.3 and 3.4,we have the following:

THEOREM 3.5. Let f and g be continuous self-mappings of a completeŽ .metric space X, d . Then the following statements are equi alent:

Ž .3.5 f and g ha¨e a common fixed point.Ž .3.6 There exist mappings S, T : X ª X and functions F, C from Xw . Ž . Ž .into 0, ` such that the pairs f , S and g, T are compatible, S X ; f X ,

Ž . Ž .T X ; g X , S and T are continuous, and

c d Sx , Ty F r d fx , gy q F fx y F SxŽ . Ž . Ž . Ž . Ž .q C gy y C TyŽ . Ž .

w .for all x, y g X and some r g 0, 1 .

THEOREM 3.6. Let f and g be continuous self-mappings of a completeŽ . Ž .metric space X, d . If 3.6 is satisfied, then f , g, S, and T ha¨e a unique

common fixed point in X.

w xRemark 3.3. Theorem 2.1 of Dien 3 is a special case of Theorem 3.6.

Remark 3.4. The functions F and C in Theorems 3.1 ; 3.6 can be� 4m � 4kreplaced by the finite families F and C of functions from X intoi is1 j js1

w .0, ` , respectively, and then we have the same conclusions in Theorems3.1 ; 3.6.

ACKNOWLEDGMENT

The authors would like to express their deep thanks to the referee for his helpfulsuggestions. Further, the third author was supported by the Academic Research Fund ofMinistry of Education, Korea, 1997, Project No. BSRI-97-1405.

LIU, XU, AND CHO504

REFERENCES

1. S. S. Chang, On common fixed point theorems for a family of F-contraction mappings,Ž .Math. Japon. 29 1984 , 527]536.

2. S. S. Chang, On Rhoades’ open questions and some fixed point theorems for a class ofŽ .mappings, Proc. Amer. Math. Soc. 97 1986 , 343]346.

3. N. H. Dien, Some remarks on common fixed point theorems, J. Math . Anal. Appl. 187Ž .1994 , 76]90.

Ž .4. B. Fisher, Mappings with a common fixed point, Math. Sem. Notes 7 1979 , 81]84.5. B. Guo, New common fixed point theorems for commuting map pair with the sequence of

Ž .mappings on a metric space, Math. Japon. 33 1988 , 361]374.Ž .6. G. Jungck, Commuting mappings and fixed point, Amer. Math. Monthly 83 1976 ,

261]263.Ž .7. G. Jungck, An iff fixed point criterion, Math. Mag. 49 1976 , 32]34.

8. G. Jungck, Compatible mappings and common fixed point, Internat. J. Math. Math. Sci. 4Ž .1986 , 771]779.

9. S. M. Kang and J. W. Ryu, A common fixed point theorem for compatible mappings,Ž .Math. Japon. 35 1990 , 153]157.

10. M. S. Khan and B. Fisher, Some fixed point theorems for commuting mappings, Math.Ž .Nachr. 106 1982 , 323]326.

11. T. Kubiak, Common fixed points of pairwise commuting mappings, Math. Nachr. 118Ž .1984 , 123]127.

Ž .12. Z. Liu, On characterizations of fixed points, Jnanabha 23 1993 , 123]127.˜Ž .13. Z. Liu, Characterizations of fixed and common fixed points, Acta Sci. Math. Szeged 59

Ž .1994 , 579]584.Ž .14. Z. Liu, Families of mappings and fixed points, Publ. Math. Debrecen 47 1995 , 161]166.Ž .15. S. Park, Fixed points of f-contractive maps, Rocky Mountain J. Math. 8 1978 , 743]750.Ž .16. S. Park, On general contractive type conditions, J. Korean Math. Soc. 17 1980 , 131]140.

17. B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans.Ž .Amer. Math. Soc. 226 1977 , 257]290.

18. S. Sessa, R. N. Mukherjee, and T. Som, A common fixed point theorem for weaklyŽ .commuting mappings, Math. Japon. 31 1986 , 235]245.

19. S. L. Singh, B. M. L. Tiwari, and V. K. Gupta, Common fixed points of commutingŽ .mappings in 2-metric spaces and an application, Math. Nachr. 95 1980 , 293]297.