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R E S E A R C H

Open Access

Fixed point results for multivalued contractive

maps

B. A. Bin Dehaish

1*

and Abdul Latif

2

* Correspondence: bbindehaish@yahoo.com 1Department of Mathematics, Faculty of Science For Girls, King Abdulaziz University, P.O. Box 53909, Jeddah 21593, Saudi Arabia Full list of author information is available at the end of the article

Abstract

Using the concept ofu-distance, we prove a fixed point theorem for multivalued contractive maps. We also establish a multivalued version of the Caristi’s fixed point theorem and common fixed point result. Consequently, several known fixed point results are either improved or generalized including the corresponding fixed point results of Caristi, Mizoguchi-Takahashi, Kada et al., Suzuki-Takahashi, Suzuki, and Ume.

Mathematics Subject Classification (2000):47H10; 54H25.

Keywords:metric space, u-distance, contractive map, fixed point

1 Introduction

LetX be a complete metric space with metric d. We denote the collection of none-mpty subsets of X, nonempty closed subsets of Xand nonempty closed bounded sub-sets ofX by 2X, Cl(X), CB(X), respectively LetHbe the Hausdorff metric with respect tod, that is,

H(A,B)= max

sup xA

d(x,B), sup yB

dy,A

,

for everyA, BÎ CB(X), whered(x, B) = infyÎBd(x, y).

A point xÎ X is called a fixed point of T : X ®2X if xÎ T(x). A pointx Î X is called a common fixed point off : X®Xand Tiff(x) =xÎT(x).

A sequence {xn} inX is called anorbitofTatx0Î XifxnÎ T(xn-1) for alln≥1. A

mapj: X® ℝis called lower semicontinuousif for any sequence {xn}⊂Xwithxn ®

xÎXimply thatφ (x)≤lim inf n→∞ φ (xn).

The well known Banach contraction principle, which asserts that“each single-valued contraction selfmap on a complete metric space has a unique fixed point” has been generalized in many different directions. Among these generalizations, the following Caristi’s fixed point theorem [1] may be the most valuable one and has extensive appli-cations in the field of mathematics.

Theorem 1.1.Let X be a complete metric space and let ψ: X®(-∞, ∞]be a proper, lower semicontinuous bounded below function. Let f be a single-valued selfmap of X. If for each xÎ X

dx,f(x)ψ (x)ψf(x),

(2)

then f has a fixed point.

Investigations on the existence of fixed points for multivalued maps in the setting of metric spaces was initiated by Nadler [2]. Using the concept of Hausdorff metric, he generalized Banach contraction principle which states that each multivalued contrac-tion map T : X ® CB(X) has fixed point provided X is complete. Since then, many authors have used the Hausdorff metric to obtain fixed point results for multivalued maps. For example, see [3-6], and references therein.

Kada et al. [7] introduced the notion ofw-distance on a metric space as follows: A functionω :X ×X ® ℝ+ is called w-distance onX if it satisfies the following for

x, y, zÎX:

(w1)ω(x, z)≤ω(x, y) +ω(y, z);

(w2) the mapω(x,.) :X® ℝ+ is lower semicontinuous; i.e., for {yn} inXwithyn®y Î X,ωx,y≤lim inf

n→∞ ω

x,yn

;

(w3) for anyε> 0, there existsδ> 0 such thatω(z, x)≤δandω(z, y)≤δimplyd(x,

y)≤ε.

Note that, in general forx, y ÎX,ω(x, y)≠ω(y, x), andω(x, y) = 0 ⇔x=y does not necessarily hold. Clearly, the metricdis aw-distance onX. Examples and properties of aw-distance can be found in [7,8]. For single valued maps, Kada et al. [7] improved several classical results including the Caristi’s fixed point theorem by replacing the involved metric with a generalized distance. Using this generalized distance, Suzuki and Takahashi [9] have introduced notions of single-valued and multivalued weakly contractive maps and proved fixed point results for such maps. Consequently, they generalized the Banach contraction principle and Nadler’s fixed point result. Recent fixed point results concerningw-distance can be found [4,8,10-13].

Recently, Susuki [14] generalized the concept of w-distance by introducing the fol-lowing notion ofτ-distance on metric space (X, d).

A function p:X ×X ®ℝ+is aτ-distance onXif it satisfies the following conditions

for any x, y, zÎ X:

(τ1)p(x, z)≤p(x, y) +p(y, z);

(τ2)h(x, 0) = 0 and h(x, t)≥tfor allxÎX andt≥0, andhis concave and continu-ous in its second variable;

(τ3) limn xn =xand limn sup{h(zn, p(zn, xm)) : m≥ n} = 0 imply p(u, x)≤ limn inf

p(u, xn) for alluÎX;

(τ4) limnsup{p(xn, ym) :m≥n} = 0 and limnh(xn, tn) = 0 imply limnh(yn, tn) = 0;

(τ5) limnh(zn, p(zn, xn)) = 0 and limnh(zn, p(zn, yn)) = 0 imply limnd(xn, yn) = 0.

Examples and properties of τ-distance are given in [14]. In [14], Suzuki improved several classical results including the Caristi’s fixed point theorem for single-valued maps with respect to τ-distance.

In the literature, several other kinds of distances and various versions of known results are appeared. For example, see [15-19], and references therein. Most recently, Ume [20] generalized the notion ofτ-distance by introducingu-distance as follows:

A function p:X×X®ℝ+is calledu-distance onXif there exists a functionθ:X×

X ×ℝ+×ℝ+ ®ℝ+such that the following hold forx, y, zÎX:

(u1)p(x, z)≤p(x, y) +p(y, z).

(u2)θ(x, y, 0, 0) = 0 andθ(x, y, s, t)≥min{s, t} for eachs,tÎ ℝ+, and for everyε> 0,

(3)

θ

x,y,s,tθx,y,s0,t0< ε. (1)

(u3) limn®∞xn=x

lim n→∞ sup

θwn,zn,p(wn,xm),p(zn,xm)

:mn= 0 (2)

imply

py,x, lim inf n→∞ p

y,xn

(3)

(u4)

lim n→∞sup

p(xn,wm)

:mn= 0,

lim n→∞sup

pyn,zm,

:mn= 0,

lim

n→∞θ (xn,wn,sn,tn)= 0,

lim n→∞θ

yn,zn,sn,tn

= 0

(4)

imply

lim

n→∞θ (wn,zn,sn,tn)= 0 (5)

or

lim n→∞sup

p(wn,xm)

:mn= 0,

lim n→∞sup

pzm,yn,

:mn= 0,

lim

n→∞θ (xn,wn,sn,tn)= 0,

lim n→∞θ

yn,zn,sn,tn

= 0

(6)

imply

lim

n→∞θ (wn,zn,sn,tn)= 0; (7)

(u5)

lim n→∞θ

wn,zn,p(wn,xn),p(zn,xn)

= 0,

lim n→∞θ

wn,zn,p

wn,yn

,pzn,yn

= 0 (8)

imply

lim n→∞ d

xn,yn

= 0 (9)

or

lim n→∞θ

an,bn,p(xn,an),p(xn,bn)

= 0,

lim n→∞θ

an,bn,p

yn,an

,pyn,bn

= 0 (10)

imply

lim

(4)

Remark 1.1. [20] (a) Suppose thatθ fromX ×X ×ℝ+ ×ℝ+ into ℝ+ is a mapping

satisfying (u2) ~ (u5). Then there exists a mapping hfromX ×X ×ℝ+×ℝ+ intoℝ+

such thathis nondecreasing in its third and fourth variable, respectively satisfying (u2)

h ~ (u5)h, where (u2)h ~ (u5)h stand for substituting h for θ in (u2) ~ (u5), respectively

(b) In the light of (a), we may assume that θis nondecreasing in its third and fourth variables, respectively, for a functionθ fromX ×X ×ℝ+×ℝ+intoℝ+ satisfying (u2)

~ (u5).

(c) Eachτ-distancepon a metric space (X, d) is also au-distance onX.

We present some examples of u-distance which are notτ-distance. (For the detail, see [20]).

Example 1.1. Let X = ℝ+ with the usual metric. Define p: X × X ® ℝ+ by

px,y=14x2. Thenpis au-distance onXbut not aτ-distance onX.

Example 1.2. LetX be a normed space with norm ||.||. Then a functionp:X×X® ℝ+ defined by p(x, y) = ||x|| for every x, y Î X is a u-distance on X but not aτ

-distance.

It follows from the above examples and Remark 1.1(c) that u-distance is a proper extension ofτ-distance. Other useful examples are also given in [20]).

Let X be a metric space with a metric d and let pbe a u-distance onX. Then a sequence {xn} inX is calledp-Cauchy [20] if there exists a functionθfromX×X ×ℝ+

×ℝ+intoℝ+ satisfying (u2) ~ (u5) and a sequence {zn} of Xsuch that

limn→∞sup

θzn,zn,p(zn,xm),p(zn,xm)

:mn= 0, (12)

or

limn→∞sup

θzn,zn,p(xm,zn),p(xm,zn)

:mn= 0. (13)

The following lemmas concerningu-distance are crucial for the proofs of our results.

Lemma 1.1. LetXbe a metric space with a metricdand letpbe au-distance on X. If {xn} is ap-Cauchy sequence, then {xn} is a Cauchy sequence.

Lemma 1.2. LetXbe a metric space with a metricdand letpbe au-distance onX. (1) If sequences {xn} and {yn} ofXsatisfy limn®∞p(z, xn) = 0, and limn®∞p(z, yn) = 0

for some zÎX, then limn®∞d(xn, yn) = 0.

(2) Ifp(z, x) = 0 andp(z, y) = 0, thenx=y.

(3) Suppose that sequences {xn} and {yn} ofXsatisfy limn®∞p(xn, z) = 0, and limn®∞

p(yn, z) = 0 for some zÎX. Then limn®∞d(xn, yn) = 0.

(4) Ifp(x, z) = 0 andp(y, z) = 0, thenx=y.

Lemma 1.3. LetXbe a metric space with a metricdand letpbe au-distance on X. Suppose that a sequence {xn} ofXsatisfies

limn→∞ supp(xn,xm,)

:m>n= 0, (14)

or

limn→∞ supp(xm,xn)

:m>n= 0. (15)

Then {xn} is a p-Cauchy sequence.

(5)

Theorem 1.2. Let X be a complete metric space with metric d, letj: X®(-∞,∞]be a proper lower semicontinuous function which is bounded from below. Let p be a u-dis-tance on X. Suppose that f is a single-valued selfmap of X such that

φfx+px,fxφ (x),

for all xÎX. Then there exists x0 ÎX such that fx0=x0,and p(x0,x0) = 0.

We say a multivalued mapT:X ®2Xis contractive with respect tou-distancepon X (in short, p-contractive) if there exist a u-distanceponX and a constantr Î (0, 1) such that for any x, yÎX anduÎT(x), there isυÎT(y) satisfying

p(u,v)rpx,y.

In particular, a single-valued map g: X ®X isp-contractive if there exist au -dis-tance ponXand a constantrÎ(0, 1) such that for eachx, yÎ X

pg(x),gyrpx,y.

In this article, using the concept of u-distance, first we prove a useful lemma for multivalued mappings in metric spaces. Then using our lemma we prove a fixed point result for closed valued p-contraction mappings. Also, we prove multivalued version of the Caristi’s fixed point theorem and then applying this result we establish common fixed point theorem. Consequently, several known fixed point results are either improved or generalized.

2 The results

Using Lemma 1.3, we prove the following key lemma in the setting of metric spaces.

Lemma 2.1. Let X be a metric space with metric d.Let T :X ® Cl(X)be a p-con-tractive map. Then, there exists an orbit {un}of T at u0 such that {un}is a Cauchy

sequence.

Proof. Letu0 be an arbitrary but fixed element ofXand letu1Î Tu0 be fixed. Since

T isp-contractive, there existsu2 ÎTu1 such that

p(u1,u2)rp(u0,u1),

wherer Î(0, 1). Continuing this process, we get a sequence {un} inXsuch thatun+1 Î Tunand

p(un,un+1)rp(un−1,un),

for all nÎN. Thus for anynÎ N, we have

p(un,un+1)rp(un−1,un)≤ · · · ≤rnp(u0,u1)

Now, for any n,mÎNwithm>n,

p(un,um)p(un,un+1)+p(un+1,un+2)+· · ·+p(um−1,um)

rn 1 +r+r2+· · ·+rmn−1p(u0,u1)

rn

(6)

and hence

lim n→∞ sup

p(un,um):m>n

= 0.

By Lemma 1.3, {un} is ap-Cauchy sequence and hence by Lemma 1.1, {un} is a

Cau-chy sequence.

Now, applying Lemma 2.1 we prove the following fixed point result for multivalued p-contractive maps.

Theorem 2.2.Let X be a complete metric space with metric d and let T: X®Cl(X) be p-contractive map. Then there exists x0 ÎX such that x0 ÎTx0and p(x0,x0) = 0.

Proof. By Lemma 2.1, there exists a Cauchy sequence {un} inXsuch thatunÎTun-1

for each nÎN. SinceX is complete, {un} converges to someυ0Î X. FornÎ N, from

(u3) and the proof of Lemma 2.1, we have

p(un,v0)≤lim inf

m→∞ p(un,um)rn

1−rp(u0,u1)

Since unÎTun-1 andTis p-contractive, there existwnÎTv0such that

p(un,wn)rp(un−1,v0).

Thus for anynÎN

p(un,wn)rp(un−1,v0)

rn

1−rp(u0,u1),

and so nlim→∞p(un,wn)= 0. Now, since nlim→∞p(un,v0)= 0it follows from Lemma 1.2 that

lim

n→∞d(wn,v0)= 0

Since the sequence {wn}⊂Tυ0andTυ0 is closed, we getυ0Î Tυ0. Since Tisp

-con-tractive map so for suchυ0there isυ1ÎTυ0such that

p(v0,v1)rp(v0,v0)

Thus, we also have a sequence {υn} inXsuch thatυn+1Î Tυnand

p(v0,vn+1)rp(v0,vn),

for all nÎN. Now, as in the proof of Lemma 2.1 we getυnis a p-Cauchy sequence

in X and thus it converges to some x0 Î X. Moreover, we have

p(v0,x0)≤lim inf

n→∞ p(v0,vn)≤0, which impliesp(v0,x0) = 0. So for anynÎN we have

p(un,x0)p(un,v0)+p(v0,x0)

rn

1−rp(u0,u1)

Now, sincenlim→∞p(un,v0)= 0andnlim→∞p(un,x0)= 0, so by Lemma 1.2, it follows that

d(x0,υ0) = 0. Hence we getx0 =υ0and p(υ0, v0) = 0.

A direct consequence of Theorem 2.2 is the following generalization of the Banach contraction principle.

Corollary 2.3.Let X be a complete metric space with metric d. If a single-valued map T : X ®X is p-contractive, then T has a unique fixed point x0 Î X. Further, such x0

(7)

Proof. By Theorem 2.2, it follows that there exists x0 ÎXwithTx0=x0andp(x0,x0)

= 0 For the uniqueness of x0we let y0 =Ty0. Then by the definition ofTthere existr Î (0, 1) such thatp(x0, y0) =p(Tx0, Ty0)≤rp(x0,y0), and p(y0,y0) = p(Ty0,Ty0)≤rp

(y0,y0). Thus

px0,y0

=py0,y0

= 0,

and hence by Lemma 1.2, we havex0 =y0.

Remark 2.4. Sincew-distance andτ-distance areu-distance, Theorem 2.2 is a gener-alization of [[9], Theorem 1], while Corollary 2.3 contains [[9], Theorem 2] and [[14], Theorem 2].

We now prove a multivalued version of the Caristi’s fixed point theorem with respect to u-distance.

Theorem 2.5. Let X be a complete metric space and letj: X® (-∞, ∞]be proper, lower semicontinuous bounded below function. Let T :X®2X. Assume that there exists a u-distance p on X such that for every x ÎX, there exists yÎTx satisfying

φy+px,yφ (x).

Then T has a fixed point x0Î X such that p(x0,x0) = 0.

Proof. For eachxÎX, we putf(x) =y, whereyÎT(x)⊂Xandj(y) +p(x, y)≤j(x). Note thatfis a selfmap ofXsatisfying

φf(x)+px,f(x)φ (x),

for everyxÎ X. Since the mapjis proper, there existsuÎ Xwithj(u) <∞and so we getp(u, u) = 0. Put

M=xX:φ (x)φ (u)p(u,x),

and assume that for a sequence {xn} in Meithernlim→∞ sup

p(xn,xm):m>n

= 0or

lim n→∞ sup

p(xm,xn):m>n

= 0. Note thatM is nonempty because uÎ M. Now, we

show that the set Mis closed. Let {xn} be a sequence inMwhich converges to somex Î X. Then {xn} is a p-Cauchy sequence and thus it follows from (u3) that

p(u,x)≤lim inf

n→∞ p(u,xn). (16)

Using the lower semicontinuity jit is easy to show that the setM is closed inX. Thus Mis a complete metric space. Now, we show that the setM is invariant underf. Note that for eachxÎM, we have

φf(x)+px,f(x)φ (x)φ (u)p(u,x)

and thus

φf(x)φ (u)p(u,x)+px,f(x)φ (s)ps,f(x).

It follows that f(x)ÎM and hencefis a selfmap ofM. Applying Theorem 1.2, there existsx0Î Msuch thatf(x0) =x0ÎT(x0) andp(x0,x0) = 0.

(8)

Further, Theorem 2.5 contains [[7], Theorem 2] and [[14], Theorem 3] which are sin-gle-valued generalizations of the Caristi’s fixed point theorem.

Theorem 2.7. LetX be a complete metric space, fbe a single-valued selfmap ofX withf(X) =M complete and letT: X®2Xbe such thatT(X)⊂M. Assume that there exists au-distanceponX such that for everyxÎ X, there existsyÎTx satisfying

px,fyφ (x)φfy,

where j: M® (-∞,∞] is proper, lower semicontinuous, and bounded from below. Then, there exits a pointx0 ÎMsuch thatx0Î fT(x0).

Proof. For eachy ÎM, define

Jy=fTy= xT(y)

f(x).

Clearly,J carriesMinto 2M. Now, for eachsÎJ(y), there exists sometÎT(y) withs =f(t) andp(y, f(t))≤j(y) -j(f(t)), that is;p(y, s)≤j(y) -j(s). Sincejis proper, there existszÎMwithj(z) < +∞.

Let

Y =yM:φyφ (z)pz,y,

and assume that for a sequence {xn} inY eithernlim→∞ sup{p(xn, xm) :m>n}= 0or

lim

n→∞ sup{p(xm, xn) :m>n}= 0. Note thatYis nonempty closed subset of a complete space M. Thus Yis a complete metric space. Now we show thatYis invariant under the map J. Now, lets ÎJ(y),y ÎY. By definition ofJ, there existstÎT(y) such thats =f(t), and

φ (s)+py,sφyφ (z)pz,y

and hence

φ (s)φ (z)p(z,s),

proving thats Î Yand henceJ(y)⊂Yfor ally Î Y. Now, Theorem 2.5 guarantees that there exitsx0 ÎMsuch thatx0Î J(x0) =fT(x0).

Finally, we obtain a common fixed point result.

Theorem 2.8. Suppose that X, M, f, andT satisfy the assumptions of Theorem 2.7. Moreover, the following conditions hold:

(a)fandTcommute weakly. (b)x∉Fix(f) impliesx∉fT(x).

Then Tandfhave a common fixed point inM.

Proof. As in the proof of Theorem 2.7, there exits x0 Î M such that x0 Î fT(x0).

Using conditions (a) and (b), we obtain

x0=f(x0)fT(x0)Tf(x0)=T(x0)

Thus, x0must be a common fixed point offandT.

Acknowledgements

(9)

Author details

1Department of Mathematics, Faculty of Science For Girls, King Abdulaziz University, P.O. Box 53909, Jeddah 21593, Saudi Arabia2Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Authors’contributions

All authors participated in the design of this work and performed equally. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Received: 18 October 2011 Accepted: 17 April 2012 Published: 17 April 2012

References

1. Caristi, J: Fixed point theorem for mapping satisfying inwardness conditions. Trans Am Math Soc.215, 241–251 (1976) 2. Nadler, SB: Multivalued contraction mappings. Pacific J Math.30, 475488 (1969)

3. Bae, JS: Fixed point theorems for weakly contractive multivalued maps. J Math Anal Appl.284, 690–697 (2003). doi:10.1016/S0022-247X(03)00387-1

4. Latif, A: Generalized Caristi’s fixed points theorems. Fixed Point Theory and Applications2009, 7 (2009). Article ID 170140

5. Mizoguchi, N, Takahashi, W: Fixed point theorems for multivalued mappings on complete metric spaces. J Math Anal Appl.141, 177–188 (1989). doi:10.1016/0022-247X(89)90214-X

6. Naidu, SVR: Fixed points and coincidence points for multimaps with not necessarily bounded images. Fixed Point Theory and Appl.3, 221–242 (2004)

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

8. Takahashi, W: Nonlinear Functional Analysis: Fixed point theory and its applications. Yokohama Publishers Inc, Japan (2000)

9. Suzuki, T, Takahashi, W: Fixed point Theorems and characterizations of metric completeness. Topol Methods Nonlinear Anal.8, 371–382 (1996)

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11. Latif, A, Abdou, AAN: Multivalued generalized nonlinear contractive maps and fixed points. Nonlinear Anal.74, 14361444 (2011). doi:10.1016/j.na.2010.10.017

12. Park, S: On generalizations of the Ekland-type variational principles. Nonlinear Anal.39, 881–889 (2000). doi:10.1016/ S0362-546X(98)00253-3

13. Ume, JS, Lee, BS, Cho, SJ: Some results on fixed point theorems for multivalued mappings in complete metric spaces. IJMMS.30, 319–325 (2002)

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

15. Al-Homidan, S, Ansari, QH, Yao, J-C: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal.69, 126–139 (2008). doi:10.1016/j.na.2007.05.004 16. Ansari, QH: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and

fixed point theory. J Math Anal Appl.334, 561–575 (2007). doi:10.1016/j.jmaa.2006.12.076

17. Ansari, QH: Metric Spaces: Including Fixed Point Theory and Set-valued Maps. Narosa Publishing House, New Delhi (2010)

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References

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