Common Fixed Point of Multivalued Generalized -Weak Contractive Mappings

Volume 2010, Article ID 708984,13pages doi:10.1155/2010/708984

Research Article

Common Fixed Point of Multivalued Generalized

ϕ

-Weak Contractive Mappings

Behzad Djafari Rouhani

_{and Sirous Moradi}

1_{Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968, USA} 2_{Department of Mathematics, Faculty of Science, University of Arak, Arak 38156-879, Iran}

Correspondence should be addressed to Behzad Djafari Rouhani,[email protected]

Received 18 September 2009; Accepted 10 January 2010

Academic Editor: Mohamed A. Khamsi

Copyrightq2010 B. Djafari Rouhani and S. Moradi. 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.

Fixed point and coincidence results are presented for multivalued generalizedϕ-weak contractive mappings on complete metric spaces, whereϕ : 0,∞ −→ 0,∞is a lower semicontinuous function withϕ0 0 andϕt>0 for allt >0. Our results extend previous results by Zhang and Song2009, as well as by Rhoades2001, Nadler1969, and Daffer and Kaneko1995.

1. Introduction

Let X, d be a metric space. We denote the family of all nonempty closed and bounded subsets ofXbyCBX.

A mapping T : X → X is said to be ϕ-weak contractive if there exists a map ϕ : 0,∞ → 0,∞withϕ0 0 andϕt>0 for allt >0 such that

dTx, Ty≤dx, y−ϕdx, y 1.1

for allx, y∈X.

Also two mappingsT, S:X → Xare called generalizedϕ-weak contractions if there exists a mapϕ:0,∞ → 0,∞withϕ0 0 andϕt>0 for allt >0 such that

for allx, y∈X, where

Mx, y:max

dx, y, dx, Tx, dy, Sy,d

x, Sydy, Tx 2

. 1.3

A mappingT :X → CBXis said to be a weak contraction if there exists 0≤α <1 such that

HTx, Ty≤αNx, y, 1.4

for allx, y∈X, whereHdenotes the Hausdorffmetric onCBXinduced byd, that is,

HA, B:max

sup

x∈Adx, B, sup

y∈Bd

y, A

, 1.5

for allA, B∈CBX, and where

Nx, y:max

dx, y, dx, Tx, dy, Ty,d

x, Tydy, Tx 2

. 1.6

A mapping T : X → CBX is said to be ϕ-weak contractive if there exists a map ϕ:0,∞ → 0,∞withϕ0 0 andϕt>0 for allt >0 such that

HTx, Ty≤dx, y−ϕdx, y, 1.7

for allx, y∈X.

The concepts of weak andϕ-weak contractive mappings were defined by Daffer and Kaneko1in 1995.

Many authors have studied fixed points for multivalued mappings. Among many others, see, for example,1–4, and the references therein.

In the following theorem, Nadler3 extended the Banach Contraction Principle to multivalued mappings.

Theorem 1.1. LetX, dbe a complete metric space. SupposeT : X → CBXis a contraction mapping in the sense that for some0≤α <1,

HTx, Ty≤αdx, y, 1.8

for allx, y∈X. Then there exists a pointx∈Xsuch thatx∈Tx(i.e.,xis a fixed point ofT).

Daffer and Kaneko1proved the existence of a fixed point for a multivalued weak contraction mapping of a complete metric spaceXintoCBX.

Theorem 1.2. LetX, dbe a complete metric space andT :X → CBXbe such that

for some 0 ≤ α < 1 and for all x, y ∈ X (i.e., weak contraction). If x → dx, Tx is lower semicontinuous (l.s.c.), then there existsx0 ∈Xsuch thatx0∈Tx0.

In Section 3 we extend Nadler and Daffer-Kaneko’s theorems to multivalued generalized weak contraction mappingsseeDefinition 2.1.

Rhoades 5, Theorem 2 proved the following fixed point theorem for ϕ-weak contractive single valued mappings, giving another generalization of the Banach Contraction Principle.

Theorem 1.3. LetX, dbe a complete metric space, and letT :X → Xbe a mapping such that

dTx, Ty≤dx, y−ϕdx, y, 1.10

for everyx, y ∈ X (i.e., ϕ-weak contractive), where ϕ : 0,∞ → 0,∞ is a continuous and nondecreasing function withϕ0 0andϕt>0for allt >0. ThenThas a unique fixed point.

By choosingψt t−ϕt,ϕ-weak contractions become mappings of Boyd and Wong

type6, and by definingkt 1−ϕt/tfort >0 andk0 0, thenϕ-weak contractions

become mappings of Reich type7.

Recently Zhang and Song 8 proved the following theorem on the existence of a common fixed point for two single valued generalizedϕ-weak contraction mappings.

Theorem 1.4. LetX, dbe a complete metric space, and letT, S: X → Xbe two mappings such that for allx, y∈X

dTx, Sy≤Mx, y−ϕMx, y, 1.11

(i.e., generalized ϕ-weak contractions), where ϕ : 0,∞ → 0,∞ is an l.s.c. function with

ϕ0 0andϕt>0for allt >0. Then there exists a unique pointx∈Xsuch thatxTxSx.

In Section 4, we extend Theorem 1.3 by assuming ϕ to be only l.s.c., and extend Theorem 1.4to multivalued mappings.

2. Preliminaries

In this paper,X, ddenotes a complete metric space andHdenotes the Hausdorffmetric on CBX.

Definition 2.1. Two mappingsT, S :X → CBXare called generalized weak contractions if there exists 0≤α <1 such that

HTx, Sy≤αMx, y, 2.1

Definition 2.2. Two mappingsT, S:X → CBXare called generalizedϕ-weak contractive if there exists a mapϕ:0,∞ → 0,∞withϕ0 0 andϕt>0 for allt >0 such that

HTx, Sy≤Mx, y−ϕMx, y 2.2

for allx, y∈X.

In the proof of our main results, we will use the following well-known lemma, and refer to Nadler3or Assad and Kirk9for its proof.

Lemma 2.3. IfA, B∈CBXanda∈A, then for eachε >0, there existsb∈Bsuch that

da, b≤HA, B ε. 2.3

3. Extension of Nadler and Daffer-Kaneko’s Theorems

The following theorem extends Nadler and Daffer-Kaneko’s Theorems to a coincidence theorem, without assumingx→dx, Txto be l.s.c.

Theorem 3.1. LetX, dbe a complete metric space, and letT, S:X → CBXbe two multivalued mappings such that for allx, y∈X,

HTx, Sy≤αMx, y, 3.1

where0 ≤ α <1(i.e., multivalued generalized weak contractions). Then there exists a pointx ∈ X

such thatx∈Txandx∈Sx(i.e.,TandShave a common fixed point). Moreover, if eitherT orSis single valued, then this common fixed point is unique.

Proof. ObviouslyMx, y 0 if and only ifxyis a common fixed point ofTandS. Letε > 0 be such thatβ αε < 1. Letx0 ∈ X andx1 ∈Sx0. ByLemma 2.3, there existsx2 ∈Tx1such thatdx2, x1 ≤ HTx1, Sx0 εMx1, x0. Again by usingLemma 2.3, there existsx3∈Sx2such thatdx3, x2≤HSx2, Tx1 εMx2, x1. By induction and using Lemma 2.3, we can find in this way a sequence{xn}inXsuch thatx2k1 ∈Sx2kand

dx2k1, x2k≤HSx2k, Tx2k−1 εMx2k, x2k−1 3.2

andx2k2 ∈Tx2k1and

It follows that

dx2n1, x2n

≤HTx2n−1, Sx2n εMx2n−1, x2n

≤βMx2n−1, x2n

βmax

dx2n−1, x2n, dx2n−1, Tx2n−1, dx2n, Sx2n,dx2n−1, Sx2n ₂dx2n, Tx2n−1

≤βmax

dx2n−1, x2n, dx2n−1, x2n, dx2n, x2n1,dx2n−1, x₂2n1 0

βmax{dx2n−1, x2n, dx2n, x2n1}

βdx2n−1, x2n,

3.4

since if otherwise dx2n, x2n1 > dx2n, x2n−1, then dx2n, x2n1 ≤ βdx2n, x2n1 and so

dx2n, x2n1 0. Hence 0dx2n, x2n1> dx2n, x2n−1and this is a contradiction. Similarly,

dx2n2, x2n1≤βdx2n1, x2n. 3.5

From3.4and3.5, we conclude that

dxk1, xk≤βdxk, xk−1, 3.6

for allk∈N. Sinceβ <1 and3.6holds,{xn}is a Cauchy sequence. SinceX, dis complete, there existsx∈Xsuch that limn→ ∞xnx.

We have

dx2n2, Sx≤HTx2n1, Sx

≤αMx2n1, x

αmax

dx2n1, x, dx2n1, Tx2n1, dx, Sx,dx2n1, Sx ₂dx, Tx2n1

≤αmax

dx2n1, x, dx2n1, x2n2, dx, Sx,dx2n1, Sx ₂ dx, x2n2

.

3.7

Lettingn → ∞in the above inequality, we conclude thatdx, Sx≤αdx, Sx. Sodx, Sx 0. SinceSx∈CBX, we havex∈Sx.

Furthermore, ifT is single valued, then this common fixed point is unique. In fact, ifx andyare two common fixed points forTandS, then

dx, y≤H{x}, Sy H{Tx}, Sy

≤αMx, y

αmax

dx, y, dx, Tx, dy, Sy,d

x, Sydy, Tx 2

≤αmax

dx, y,0,0,d

x, ydy, x 2

αdx, y.

3.8

Since 0≤α <1,dx, y 0, and soxy.

Remark 3.2. The last part of the proof ofTheorem 3.1shows that if S, T : X → CBXare multivalued andx0is a common fixed point, andTx0orSx0is a singleton, then the common fixed point ofT andSis unique.

By takingTSinTheorem 3.1, we get the following corollary that extends the Daffer and Kaneko theoremTheorem 1.2.

Corollary 3.3. LetX, dbe a complete metric space and letT :X → CBXbe such that

HTx, Ty≤αNx, y, 3.9

for some0 ≤α < 1and for allx, y ∈X(i.e., weak contraction). Then there existsx0 ∈X such that x0∈Tx0.

Example 3.4. LetX 0,1be endowed with the Euclidean metric. LetS, T :X → CBXbe defined byTx 0, x/4andSy{y/4}. Obviously,

HTx, Symax y 4 −

x 4 ,y₄

≤ 1

2max y−x , y−y

4

1

2max

dx, y, dy, Sy

≤ 1

2M

x, y.

SoT andShave a common fixed pointx0, and sinceSis single valued, this fixed point is unique.

4. Extension of Rhoades and Zhang-Song’s Theorems

First we extend Zhang and Song’s theorem Theorem 1.4 to the case where one of the mappings is multivalued.

Theorem 4.1. LetX, dbe a complete metric space and letT :X → X andS: X → CBXbe two mappings such that for allx, y∈X,

H{Tx}, Sy≤Mx, y−ϕMx, y, 4.1

(i.e., generalizedϕ-weak contractive) where ϕ : 0,∞ → 0,∞ is l.s.c. withϕ0 0 and

ϕt>0for allt >0. Then there exists a unique pointx∈Xsuch thatTxx∈Sx.

Proof. Unicity of the common fixed point follows from4.1.

ObviouslyMx, y 0 if and only ifxyis a common fixed point ofTandS. Letx0 ∈Xandx1∈Sx0. Letx2:Tx1. ByLemma 2.3, there existsx3∈Sx2such that

dx3, x2≤HSx2,{Tx1} 1

2ϕMx2, x1. 4.2

We letx4:Tx3. Inductively, we letx2n:Tx2n−1, and byLemma 2.3, we choosex2n1 ∈Sx2n such that

dx2n1, x2n≤HSx2n,{Tx2n−1} 1₂ϕMx2n, x2n−1. 4.3

We break the argument into four steps.

Step 1. limn→ ∞dxn1, xn 0.

Proof. Using4.1and4.3,

dx2n1, x2n≤H{Tx2n−1}, Sx2n 1₂ϕMx2n−1, x2n

≤Mx2n−1, x2n−₂1ϕMx2n−1, x2n,

where

dx2n−1, x2n

≤Mx2n−1, x2n

max

dx2n−1, x2n, dx2n−1, Tx2n−1, dx2n, Sx2n,dx2n−1, Sx2n dx2n, Tx2n−1 2

≤max

dx2n−1, x2n, dx2n−1, x2n, dx2n, x2n1,dx2n−1, x₂2n1 0

max{dx2n−1, x2n, dx2n, x2n1}

dx2n−1, x2n by4.4.

4.5

SoMx2n−1, x2n dx2n−1, x2n. Hence by4.4,

dx2n1, x2n≤dx2n, x2n−1. 4.6

Also

dx2n2, x2n1 dTx2n1, x2n1 ≤H{Tx2n1}, Sx2n

≤Mx2n1, x2n−ϕMx2n1, x2n,

4.7

where

dx2n1, x2n

≤Mx2n1, x2n

max

dx2n1, x2n, dx2n1, Tx2n1, dx2n, Sx2n,dx2n1, Sx2n dx2n, Tx2n1 2

≤max

dx2n1, x2n, dx2n1, x2n2, dx2n, x2n1,

0dx2n, x2n2 2

max{dx2n1, x2n, dx2n1, x2n2} dx2n1, x2n

by4.7.

4.8

SoMx2n1, x2n dx2n1, x2n. Hence by4.7,

dx2n2, x2n1≤dx2n1, x2n. 4.9

dxk1, xk≤dxk, xk−1, 4.10

for allk∈N.

Therefore, the sequence{dxk1, xk}is monotone nonincreasing and bounded below. So there existsr ≥0 such that

lim

n→ ∞dxn1, xn nlim→ ∞Mxn1, xn r. 4.11

Sinceϕis l.s.c.,

ϕr≤lim inf

n→ ∞ ϕMxn, xn−1≤lim infn→ ∞ ϕMx2n−1, x2n. 4.12

By4.4, we conclude that

r≤r−1

2ϕr, 4.13

and soϕr 0. Hencer0.

Step 2. {xn}is a bounded sequence.

Proof. If{xn}were unbounded, then byStep 1,{x2n}and{x2n−1}are unbounded. We choose the sequence{nk}∞k1such thatn1 1,n2> n1is even and minimal in the sense that dxn2, xn1>1, anddxn2−2, xn1≤1, and similarlyn3> n2is odd and minimal in the

sense thatdxn3, xn2>1, anddxn3−2, xn2≤1, . . . , n2k> n2k−1is even and minimal in the sense thatdxn2k, xn2k−1 > 1 anddxn2k−2, xn2k−1 ≤ 1, andn2k1 > n2kis

odd and minimal in the sense thatdxn2k1, xn2k>1 anddxn2k1−2, xn2k≤1.

Obviouslynk ≥ k for everyk ∈ N. ByStep 1, there existsN0 ∈ Nsuch that for all k≥N0we havedxk1, xk<1/4. So for everyk≥N0, we havenk1−nk≥2 and

1< dxnk1, xnk ≤dxnk1, xnk1−1

dxnk1−1, xnk1−2

dxnk1−2, xnk

≤dxnk1, xnk1−1dxnk1−1, xnk1−21.

4.14

Hence limk→ ∞dxnk1, xnk 1.Also

1< dxnk1, xnk ≤dxnk1, xnk11

dxnk11, xnk1

dxnk1, xnk ≤dxnk1, xnk11

dxnk11, xnk1

dxnk1, xnk

dxnk, xnk1dxnk1, xnk ≤2dxnk1, xnk11

dxnk1, xnk2dxnk1, xnk,

4.15

So ifnk1is odd, then

dxnk11, xnk1

≤Mxnk1, xnk−ϕMxnk1, xnk, 4.16

where

1< dxnk1, xnk≤Mxnk1, xnk

max

dxnk1, xnk, dxnk1, Txnk1, dxnk, Sxnk,

dxnk1, Sxnkdxnk, Txnk1

2

≤max

dxnk1, xnk, dxnk1, xnk11

, dxnk, xnk1

,

dxnk1, xnk1

dxnk, xnk11

2

≤max

dxnk1, xnk, dxnk1, xnk11

, dxnk, xnk1

,

2dxnk1, xnkdxnk1, xnkdxnk11, xnk1

2

,

4.17

and this shows that limk→ ∞Mxnk1, xnk 1.Sinceϕis l.s.c. and4.16holds, we have

1≤1−ϕ1. Soϕ1 0 and this is a contradiction.

Step 3. {xn}is Cauchy.

Proof. Let Cn sup{dxi, xj : i, j ≥ n}. Since {xn} is bounded,Cn < ∞for all n ∈ N. Obviously{Cn}is decreasing. So there existsC ≥ 0 such that limn→ ∞Cn C. We need to show thatC0.

For everyk∈N, there existsnk, mk∈Nsuch thatmk> nk≥kand

Ck−1 k ≤d

xmk, xnk≤Ck. 4.18

By4.18, we conclude that

lim k→ ∞d

xmk, xnkC. 4.19

FromStep 1and4.19, we have

lim k→ ∞d

xmk1, xnk1

lim

k→ ∞d

xmk1, xnk

lim

k→ ∞d

xmk, xnk1

lim

k→ ∞d

xmk, xnkC.

So we may assume that for everyk∈N,mkis odd andnkis even. Hence

dxmk1, xnk1

dTxmk, xnk1

≤HTxmk, Sxnk ≤Mxmk, xnk

−ϕMxmk, xnk

,

4.21

where

dxmk, xnk

≤Mxmk, xnk

max

dxmk, xnk, dxmk, Txmk, dxnk, Sxnk,

dxmk, Sxnk

dxnk, Txmk

2

≤max

dxmk, xnk, dxmk, xmk1

, dxnk, xnk1

,

dxmk, xnk1

dxnk, xmk1

2

.

4.22

This inequality shows that limk→ ∞Mxmk, xnk C.Sinceϕis l.s.c. and4.21holds, we

haveC≤C−ϕC. HenceϕC 0 and soC0. Therefore,{xn}is a Cauchy sequence.

Step 4. T andShave a common fixed point.

Proof. SinceX, dis complete and{xn}is Cauchy, there existsx∈Xsuch that limn→ ∞xnx. For everyn∈N

dx2n2, Sx dTx2n1, Sx≤H{Tx2n1}, Sx

≤Mx2n1, x−ϕMx2n1, x,

4.23

where

Mx2n1, x

max

dx2n1, x, dx2n1, Tx2n1, dx, Sx,dx2n1, Sx ₂dx, Tx2n1

max

dx2n1, x, dx2n1, x2n2, dx, Sx,dx2n1, Sx ₂ dx, x2n2

,

4.24

and this shows that limn→ ∞Mx2n1, x dx, Sx.

Sinceϕis l.s.c. and4.23holds, lettingn → ∞in4.23we get

Soϕdx, Sx 0 and hencedx, Sx 0. SinceSx∈CBX, thenx∈Sx.Also

dTx, x≤H{Tx}, Sx≤Mx, x−ϕMx, x, 4.26

where

Mx, x max

dx, x, dx, Tx, dx, Sx,dx, Sx dx, Tx 2

dx, Tx. 4.27

So from4.26, we have

dTx, x≤dTx, x−ϕdTx, x. 4.28

ThusϕdTx, x 0, and hencedTx, x 0. Therefore,xTx.

Remark 4.2. In the proof of Theorem 2.1 in Zhang and Song 8, the boundedness of the sequence{Cn}is used, but not proved. Also, for the proof that{xn}is a Cauchy sequence, the monotonicity ofϕis used, without being explicitly mentioned.

In our proof ofTheorem 4.1, which is different from8, Theorem 2.1,ϕis not assumed to be nondecreasing.

The following theorem extends Rhoades’ theorem by assumingϕto be only l.s.c..

Theorem 4.3. LetX, dbe a complete metric space, and letT :X → Xbe a mapping such that

dTx, Ty≤dx, y−ϕdx, y, 4.29

for everyx, y∈X(i.e.,ϕ-weak contractive), whereϕ:0,∞ → 0,∞is an l.s.c. function with

ϕ0 0andϕt>0for allt >0. ThenThas a unique fixed point.

Proof. The proof is similar to the proof ofTheorem 4.1, by takingST, and replacingMx, y withdx, y.

Remark 4.4. With a similar proof as in Theorem 4.1, in Theorem 4.3 we can replace the inequality 4.29by the following inequality 4.30 for two single valued mappingsT, S : X → X.

dTx, Sy≤Mx, y−ϕdx, y. 4.30

5. Conclusion and Future Directions

We have extended Nadler and Daffer-Kaneko’s theorems to a coincidence theorem without assuming the lower semicontinuity of the mappingx→dx, Tx.

Acknowledgment

This work is dedicated to Professor W. A. Kirk for his 70th birthday

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