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f-Contractive multivalued maps and coincidence points

Abstract

In this paper, we prove a result on the existence of an f-orbit for generalized f-contractive multivalued maps. Then, we establish main results on the existence of coincidence points and common fixed points for generalized f-contractive maps not involving the extended Hausdorff metric and the continuity condition. Our results either generalize or improve a number of metric fixed point results.

MSC: 47H10, 47H09, 54H25.

1 Introduction

Let (X,d) be a metric space. Let 2 X , Cl(X) and CB(X) denote the collection of nonempty subsets of X, nonempty closed subsets of X, and nonempty closed bounded subsets of X, respectively. Let H be the Hausdorff metric with respect to d, that is,

H(A,B)=max { sup x A d ( x , B ) , sup y B d ( y , A ) }

for every A,BCB(X), where d(x,B)= inf y B d(x,y).

Let f:XX be a single-valued map, and let T:X 2 X be a multivalued map. A point xX is called a fixed point of T if xT(x), and the set of fixed points of T is denoted by Fix(T). A point xX is called a coincidence point of f and T if f(x)T(x). We denote by C(fT) the set of coincidence points of f and T.

We say a sequence { x n } in X is an f-orbit of T at x 0 X if f x n T x n 1 for all n1. We say that f and T weakly commute if fTxTfx for all xX. Clearly, commuting maps f and T weakly commute.

A multivalued map T:XCB(X) is called

(i) contraction [1] if for a fixed constant λ(0,1) and for each x,yX,

H ( T ( x ) , T ( y ) ) λd(x,y).

(ii) f-contraction [2] if for a fixed constant λ(0,1) and for each x,yX,

H ( T ( x ) , T ( y ) ) λd ( f ( x ) , f ( y ) ) .

Using the concept of Hausdorff metric, Nadler [1] established the following fixed point result for multivalued contraction maps, which in turn is a generalization of the well-known Banach contraction principle.

Theorem 1.1 [1]

Let (X,d) be a complete metric space, and let T:XCB(X) be a contraction map. Then Fix(T).

This result has been generalized in many directions. Kaneko [2] extended the corresponding results of Jungck [3], Nadler [1] and others as follows.

Theorem 1.2 [2]

Let (X,d) be a complete metric space, and let T:XCB(X) be a multivalued f-contraction map which commutes with a continuous map f. Then C(fT).

This result has been generalized in different directions. For example, see [410].

On the other hand, Kada et al. [11] introduced the concept of w-distance on a metric space as follows:

Let (X,d) be a metric space. A function ω:X×X[0,) is called a w-distance on X if it satisfies the following for each x,y,zX:

( w 1 ) ω(x,z)ω(x,y)+ω(y,z);

( w 2 ) a map ω(x,):X[0,) is lower semicontinuous;

( w 3 ) for any ϵ>0, there exists δ>0 such that ω(z,x)δ and ω(z,y)δ imply d(x,y)ϵ.

Note that, in general, for x,yX, ω(x,y)ω(y,x) and not either of the implications ω(x,y)=0x=y necessarily hold. We say the w-distance ω on X is a w 0 -distance if x=y implies ω(x,y)=0. Clearly, the metric d is a w-distance on X. Let (Y,) be a normed space. Then the functions ω 1 , ω 2 :Y×Y[0,) defined by ω 1 (x,y)=y and ω 2 (x,y)=x+y for all x,yY are w-distances [11]. Many other examples and properties of the w-distance can be found in [11, 12].

The following useful lemma concerning a w-distance is given in [11].

Lemma 1.1 [11]

Let (X,d) be a metric space, and let ω be a w-distance on X. Let { x n } and { y n } be sequences in X, and let { α n } and { β n } be sequences in [0,) converging to zero. Then, for the w-distance ω on X, the following hold for every x,y,zX:

(a) if ω( x n ,y) α n and ω( x n ,z) β n for any nN, then y=z; in particular, if ω(x,y)=0 and ω(x,z)=0, then y=z;

(b) if ω( x n , y n ) α n and ω( x n ,z) β n for any nN, then { y n } converges to z;

(c) if ω( x n , x m ) α n for any n,mN with m>n, then { x n } is a Cauchy sequence;

(d) if ω(y, x n ) α n for any nN, then { x n } is a Cauchy sequence.

For xX and A 2 X , we denote, ω(x,A)= inf y A ω(x,y). Now, let T:XCl(X) be a multivalued map, and let f:XX be a single-valued map. We say

(iii) T is w-contractive [12] if there exist a w-distance ω on X and λ(0,1) such that for any x,yX and uT(x), there is vT(y) with

ω(u,v)λω(x,y).

(iv) T is generalized f-contractive if there exist a w 0 -distance ω on X and λ(0,1) such that for any x,yX, uT(x), there is vT(y) with

ω(u,v)λ M f (x,y),

where

M f ( x , y ) = max { ω ( f ( x ) , f ( y ) ) , ω ( f ( x ) , T ( x ) ) , ω ( f ( y ) , T ( y ) ) , 1 2 [ ω ( f ( x ) , T ( y ) ) + ω ( f ( y ) , T ( x ) ) ] } .

Using the concept of w-distance, Suzuki and Takahashi [12] improved Nadler’s fixed point result as follows.

Theorem 1.3 Let (X,d) be a complete metric space. Then for each w-contractive map T:XCl(X), the set Fix(T).

This result has been generalized by many authors, for example, see [1316]. In this paper, first we establish a lemma with respect to a w-distance, which is an improved version of the lemma given in [3], and then we prove a key lemma on the existence of an f-orbit for generalized f-contractive maps. Finally, we present our main results on the existence of coincidence points and common fixed points for generalized f-contractive maps not involving the extended Hausdorff metric. As a consequence, we obtain a fixed point result. Our results either generalize or improve a number of known results.

2 Results

Using the concept of w-distance, first we improve a corresponding result of Jungck [3] as follows.

Lemma 2.1 Let (X,d) be a complete metric space with a w-distance ω. If there exist a sequence { x n } in X and a constant λ, 0<λ<1, such that for all nN,

ω( x n , x n + 1 )λω( x n 1 , x n ),

then the sequence { x n } converges in X.

Proof It is enough to show that { x n } is a Cauchy sequence in X. Note that for each nN, we have

ω ( x n , x n + 1 ) λ ω ( x n 1 , x n ) λ 2 ω ( x n 2 , x n 1 ) λ n ω ( x 0 , x 1 ) .

Thus

ω( x n , x n + 1 ) λ n ω( x 0 , x 1 ).

Consequently, for mn, we get

ω ( x n , x m ) ω ( x n , x n + 1 ) + ω ( x n + 1 , x n + 2 ) + + ω ( x m 1 , x m ) λ n ω ( x 0 , x 1 ) + λ n + 1 ω ( x 0 , x 1 ) + + λ m 1 ω ( x 0 , x 1 ) ,

and thus

ω( x n , x m ) λ n 1 λ ω( x 0 , x 1 ).

Since 0<λ<1, we have λ n 0 as n. And thus by Lemma 1.1, { x n } is a Cauchy sequence in X. Since X is complete, the sequence { x n } converges to a point in X. □

The following lemma is crucial for our main results.

Lemma 2.2 Let (X,d) be a complete metric space, and let T:XCl(X) be a generalized f-contractive map such that T(X)f(X). Then there exists an f-orbit { x n } of T at x 0 X such that {f( x n )} converges in X.

Proof Let x 0 X and choose y 0 T( x 0 ). Since T( x 0 )f(X), then there exists x 1 X such that f( x 1 )= y 0 T( x 0 ), and thus, by the definition of T, there exists y 1 T( x 1 ) such that

ω ( f ( x 1 ) , y 1 ) λ M f ( x 0 , x 1 ),

where 0<λ<1. Since T( x 1 )f(X), there exists x 2 X such that f( x 2 )= y 1 T( x 1 ). Thus

ω ( f ( x 1 ) , f ( x 2 ) ) λ M f ( x 0 , x 1 ).

Similarly, using the definition of T and the fact that T(X)f(X), there exists x 3 X such that f( x 3 )T( x 2 ) and

ω ( f ( x 2 ) , f ( x 3 ) ) λ M f ( x 1 , x 2 ).

Continuing this process, we get a sequence { x n } in X such that for all n, f( x n + 1 )T( x n ) and

ω ( f ( x n ) , f ( x n + 1 ) ) λ M f ( x n 1 , x n ),

that is,

ω ( f ( x n ) , f ( x n + 1 ) ) λ max { ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n 1 ) , T ( x n 1 ) ) , ω ( f ( x n ) , T ( x n ) ) , 1 2 [ ω ( f ( x n 1 ) , T ( x n ) ) + ω ( f ( x n ) , T ( x n 1 ) ) ] } .

Note that

ω ( f ( x n ) , f ( x n + 1 ) ) λ max { ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n ) , f ( x n + 1 ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n + 1 ) ) + ω ( f ( x n ) , f ( x n ) ) ] } = λ max { ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n ) , f ( x n + 1 ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n + 1 ) ) ] } ,

and we get

ω ( f ( x n ) , f ( x n + 1 ) ) λmax { ω ( f ( x n 1 ) , f ( x n ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n + 1 ) ) ] } .

Also, note that

ω ( f ( x n ) , f ( x n + 1 ) ) λ max { ω ( f ( x n 1 ) , f ( x n ) ) , 1 2 [ ω ( f ( x n 1 ) , f ( x n ) ) + ω ( f ( x n ) , f ( x n + 1 ) ) ] } λ max { [ ω ( f ( x n 1 ) , f ( x n ) ) , ω ( f ( x n ) , f ( x n + 1 ) ) ] } .

Thus, for each nN, we get

ω ( f ( x n ) , f ( x n + 1 ) ) λω ( f ( x n 1 ) , f ( x n ) ) .
(1)

Since the sequence {f( x n )} is in the complete metric space X satisfying the inequality (1), it follows from Lemma 2.1 that {f( x n )} converges in X. □

Remark 2.1 Since for each nN we have

ω ( f ( x n ) , f ( x n + 1 ) ) λω ( f ( x n 1 ) , f ( x n ) ) ,

following the proof of Lemma 2.1, we obtain the following two useful inequalities.

ω ( f ( x n ) , f ( x n + 1 ) ) λ n ω ( f ( x 0 ) , f ( x 1 ) )
(2)

and for mn

ω ( f ( x n ) , f ( x m ) ) λ n 1 λ ω ( f ( x 0 ) , f ( x 1 ) ) .
(3)

Without using the extended Hausdorff metric and continuity conditions, we prove a coincidence result which improves many known results including Theorem 1.2 due to [2], Theorem 3.3 in [17] and Theorem 2 in [7].

Theorem 2.1 Suppose that all the hypotheses of Lemma  2.2 hold. Furthermore, if for every yX with f(y)T(y)

inf { ω ( f ( x ) , y ) + ω ( f ( x ) , T ( x ) ) : x X } >0.

Then C(fT).

Proof By Lemma 2.2, there exists an f-orbit { x n } of T at x 0 X such that {f( x n )} converges in X. Also note that for each nN, we have

ω ( f ( x n ) , f ( x n + 1 ) ) λω ( f ( x n 1 ) , f ( x n ) ) ,

where 0<λ<1. Let f( x n )yX. Now since ω(f( x n ),) is lower semicontinuous, from Remark 2.1 (2), we have

ω ( f ( x n ) , y ) lim m inf ω ( f ( x n ) , f ( x m ) ) λ n 1 λ ω ( f ( x 0 ) , f ( x 1 ) ) .

Since λ<1, we get ω(f( x n ),y)0 as n. Assume that f(y)T(y), then from the hypothesis and Remark 2.1, we get

0 < inf { ω ( f ( x ) , y ) + ω ( f ( x ) , T ( x ) ) : x X } inf { ω ( f ( x n ) , y ) + ω ( f ( x n ) , T ( x n ) ) : n N } inf { ω ( f ( x n ) , y ) + ω ( f ( x n ) , f ( x n + 1 ) ) : n N } inf { λ n 1 λ ω ( f ( x o ) , f ( x 1 ) ) + λ n ω ( f ( x 0 ) , f ( x 1 ) ) : n N } = { 2 λ 1 λ } ω ( f ( x 0 ) , f ( x 1 ) ) inf { λ n : n N } = 0 ,

which is impossible, and thus f(y)T(y), that is, y is a coincidence point of f and T. □

If we take f=I (an identity map on X) in Theorem 2.1, we obtain the following improved version of the corresponding fixed point results in [12, 17, 18].

Corollary 2.1 Let (X,d) be a complete metric space, let ω be a w-distance on X, and let T:XCl(X) be a multivalued map satisfying the following:

(I) for fixed λ(0,1), for each x,yX and uT(x), there exists vT(y) such that

ω(u,v)λ M ω (x,y),

where

M ω (x,y)=max { ω ( x , y ) , ω ( x , T ( x ) ) , ω ( y , T ( y ) ) , 1 2 [ ω ( x , T ( y ) ) + ω ( y , T ( x ) ) ] } ,

(II) inf{ω(x,y)+ω(x,T(x)):xX}>0.

Then Fix(T).

Finally, we obtain a common fixed point result.

Theorem 2.2 Suppose that all the hypotheses of Theorem  2.1 hold. Further, if the maps f and T commute weakly and satisfy the condition that f(x) f 2 (x), which implies f(x)T(x), then f and T have a common fixed point.

Proof From Theorem 2.1 we have f(y)T(y), and thus we get f(y)= f 2 (y). Note that

f(y)=f ( f ( y ) ) f ( T ( y ) ) T ( f ( y ) ) ,

that is, f(y) is a fixed point of T. Also note that f(y) is a fixed point of f and thus f(y) is a common fixed point of T and f. □

References

  1. Nadler SB: Multivalued contraction mappings. Pac. J. Math. 1969, 30: 475–488. 10.2140/pjm.1969.30.475

    MathSciNet  Article  Google Scholar 

  2. Kaneko H: Single-valued and multivalued f -contractions. Boll. Unione Mat. Ital. 1985, 6: 29–33.

    Google Scholar 

  3. Jungck G: Commuting mappings and fixed points. Am. Math. Mon. 1976, 83: 261–263. 10.2307/2318216

    MathSciNet  Article  Google Scholar 

  4. Abbas M, Hussain N, Rhoades BE: Coincidence point theorems for multivalued f -weak contraction mappings and applications. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. a Mat. (Ed. Impr.) 2011, 105(2):261–272. doi:10.1007/s13398–011–0036–4

    MathSciNet  Article  Google Scholar 

  5. Daffer PZ, Kaneko H: Multivalued f -contractive mappings. Boll. Unione Mat. Ital. 1994, 8-A(7):233–241.

    MathSciNet  Google Scholar 

  6. Hussain N, Alotaibi A: Coupled coincidences for multi-valued nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 81 (18 November 2011)

    Google Scholar 

  7. Kaneko H, Sessa S: Fixed point theorems for compatible multi-valued and single-valued mappings. Int. J. Math. Math. Sci. 1989, 12(2):257–262. 10.1155/S0161171289000293

    MathSciNet  Article  Google Scholar 

  8. Latif A, Tweddle I: Some results on coincidence points. Bull. Aust. Math. Soc. 1999, 59: 111–117. 10.1017/S0004972700032652

    MathSciNet  Article  Google Scholar 

  9. Pathak HK: Fixed point theorems for weak compatible multivalued and single-valued mappings. Acta Math. Hung. 1995, 67(1–2):69–78. 10.1007/BF01874520

    Article  Google Scholar 

  10. Pathak HK, Khan MS: Fixed and coincidence points of hybrid mappings. Arch. Math. 2002, 3: 201–208.

    MathSciNet  Google Scholar 

  11. Kada O, Susuki T, Takahashi W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn. 1996, 44: 381–391.

    Google Scholar 

  12. Suzuki T, Takahashi W: Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal. 1996, 8: 371–382.

    MathSciNet  Google Scholar 

  13. Bin Dehaish BA, Latif A: Fixed point results for multivalued contractive maps. Fixed Point Theory Appl. 2012., 2012: Article ID 61

    Google Scholar 

  14. Latif A, Abdou AAN: Fixed points of generalized contractive maps. Fixed Point Theory Appl. 2009., 2009: Article ID 487161. doi:10.1155/2009/487161

    Google Scholar 

  15. Latif A, Abdou AAN: Multivalued generalized nonlinear contractive maps and fixed points. Nonlinear Anal. 2011, 74: 1436–1444. 10.1016/j.na.2010.10.017

    MathSciNet  Article  Google Scholar 

  16. Suzuki T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl. 2001, 253: 440–458. 10.1006/jmaa.2000.7151

    MathSciNet  Article  Google Scholar 

  17. Daffer PZ, Kaneko H: Fixed points generalized contractive multi-valued mappings. J. Math. Anal. Appl. 1995, 192: 655–666. 10.1006/jmaa.1995.1194

    MathSciNet  Article  Google Scholar 

  18. Kaneko H: A general principle for fixed points of contractive multivalued mappings. Math. Jpn. 1986, 31(3):407–422.

    Google Scholar 

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Acknowledgements

This paper is funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. 491/130/1432. The author therefore acknowledges with thanks DSR for technical and financial support.

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Kutbi, M.A. f-Contractive multivalued maps and coincidence points. J Inequal Appl 2013, 141 (2013). https://doi.org/10.1186/1029-242X-2013-141

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Keywords

  • metric space
  • fixed point
  • multivalued contractive map
  • coincidence point