Volume 2009, Article ID 769467,7pages doi:10.1155/2009/769467
Research Article
Fixed Points for
w
-Contractive Multimaps
Abdul Latif and Marwan A. Kutbi
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Abdul Latif,latifmath@yahoo.com
Received 3 February 2009; Accepted 4 March 2009
Recommended by Jerzy Dydak
Using the generalized Caristi’s fixed point theorems we prove the existence of fixed points for self and nonself multivalued weaklyw-contractive maps. Consequently, Our results either improve or generalize the corresponding fixed point results due to Latif2007, Bae2003, Suzuki, and Takahashi1996and others.
Copyrightq2009 A. Latif and M. A. Kutbi. 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.
1. Introduction
It is well known that Caristi’s fixed point theorem 1 is equivalent to Ekland variational principle 2, which is nowadays is an important tool in nonlinear analysis. Most recently,
many authors studied and generalized Caristi’s fixed point theorem to various directions. For example, see3–6and references therein.
Using the concept of Hausdorffmetric, Nadler Jr.7has proved multivalued version of the Banach contraction principle which states that each closed bounded valued contraction map on a complete metric space, has a fixed point. Recently, Bae 4 introduced a notion of multivalued weakly contractive maps and applying generalized Caristi’s fixed point theorems he proved several fixed point results for such maps in the setting of metric and Banach spaces. Many authors have been using the Hausdorffmetric to obtain fixed point results for multivalued maps on metric spaces, but, in fact for most cases the existence part of the results can be proved without using the concept of Hausdorffmetric.
Recently, using the concept ofw-distance8, Suzuki and Takahashi9introduced a notion of multivalued weakly contractivein short,w-contractive maps and improved the Nadler’s fixed point result without using the concept of Hausdorff metric. Most recently, Latif10generalized the fixed point result of Suzuki and Takahashi9, Theorem 1. Some
interesting examples and fixed point results concerningw-distance can be found in6,11–15
and references therein.
improve or generalize the corresponding results due to Latif10, Bae4, Mizoguchi and
Takahashi16, Suzuki and Takahashi 9, Husain and Latif17, Kaneko 18 and many others.
2. Preliminaries
LetX be a metric space with metricd. We use 2X to denote the collection of all nonempty subsets ofX and ClX for the collection of all nonempty closed subsets ofX. Recall that a real-valued function ϕ defined on X is said to be lower (upper) semicontinuous if for any sequence {xn} ⊂ X with xn → x ∈ X imply that ϕx ≤ lim infn→ ∞ϕxn ϕx ≥ lim supn→ ∞ϕxn.
Introducing the following notion ofw–distance, Kada et al.8improved the Caristi’s fixed point theorem, Ekland variational principle, and Takahashi existence theorem.
A functionω :X×X → 0,∞is called aw-distance onXif it satisfies the following for anyx, y, z∈X:
w1ωx, z≤ωx, y ωy, z;
w2a mapωx,·:X → 0,∞is lower semicontinuous;
w3for any >0,there existsδ >0 such thatωz, x≤δandωz, y≤δ
implydx, y≤.
Note that, in general forx, y∈X,ωx, y/ωy, xand not either of the implications
ωx, y 0⇔xynecessarily hold. Clearly, the metricdis aw-distance onX. LetY, ·
be a normed space. Then the functionsω1, ω2 : Y ×Y → 0,∞defined byω1x, y y
andω2x, y x y for allx, y∈Yarew-distances8.
Let Mbe a nonempty subset of X. A multivalued map T : M → 2X is called w
-contractive9if there exist aw-distanceωonXand a constanth∈ 0,1such that for any
x, y∈Xandu∈Txthere isv∈Tysatisfying
ωu, v≤hωx, y. 2.1
In particular, if we takeωd, thenw-contractive map is a contractive type map17.
We sayT is weaklyw-contractive if there exists aw-distanceωonXsuch that for any
x, y∈Xandu∈Txthere isv∈Tywith
ωu, v≤ωx, y−ϕωx, y, 2.2
whereϕis a function from0,∞to0,∞such thatϕis positive on0,∞andϕ0 0.
In particular, if we takeϕt 1−htfor a constanthwith 0 < h <1,then a weakly
w-contractive map isw-contractive. If we definekt 1−ϕt/tfort >0 andk0 0, then
kis a function from0,∞to0,1with lim supr→tkr<1,for everyt∈0,∞. Also we get
ωu, v≤kωx, yωx, y, 2.3
We say a multivalued mapT : M → 2X is w-inward if for each x ∈ M,Tx ⊂ w-IMx, where w-IMxis thew-inward set ofMatx, which consists all the elementsz∈X
such that eitherzxor there existsy∈Mwithy /xandωx, z ωx, y ωy, z.
In particular, if we take ω d, then w-inward set is known as metrically inward set4.
A pointx∈Mis called a fixed point ofT :M → 2Xifx∈Txand the set of all fixed points ofTis denoted by FixT.
In the sequel, otherwise specified, we will assume that ψ : X → 0,∞ is lower semicontinuous function, ϕ : 0,∞ → 0,∞is positive function on0,∞and ϕ0 0 andωis aw-distance onX.
Using the concept ofw-distance, Kada et al.8have generalized Caristi’s fixed point theorem as follows.
Theorem 2.1. LetX, dbe a complete metric space. Letf :X → X be a map such that for each
x∈X,
ψfxωx, fx≤ψx. 2.4
Then, there existsxo∈Xsuch thatfxo xoandωxo, xo 0.
Now, we state generalized Caristi’s fixed point theorems which are variant to the results of Bae4, Theorem 2.1 and Corollary 2.5.
Theorem 2.2. LetX, dbe a complete metric space. Letf :X → X be a map such that for each
x∈X,
ωx, fx≤maxcψx, cψfxψx−ψfx, 2.5
wherec:0,∞ → 0,∞is an upper semicontinuous function from the right. Then,f has a fixed pointx0∈Xsuch thatωx0, x0 0.
Theorem 2.3. LetX, dbe a complete metric space. Letϕ : 0,∞ → 0,∞be lower semicon-tinuous function such thatϕt>0 fort >0 and
lim sup t→0
t
ϕt <∞. 2.6
Letf:X → Xbe a map such that for eachx∈X,ωx, fx≤ψxand
ϕωx, fx≤ψx−ψfx. 2.7
Then,fhas a fixed pointx0∈Xsuch thatωx0, x0 0.
Suzuki and Takahashi9 have proved the following fixed point result which is an improved version of the multivalued contraction principle due to Nadler Jr.7.
Theorem 2.4. LetX, dbe a complete metric space. Then each multivaluedw-contractive mapT :
3. Main Results
Without using the Hausdorff metric, we prove the following fixed point result for multivalued self map.
Theorem 3.1. Let X, d be a complete metric space and let T : X → ClX be a weakly w -contractive map for whichϕis lower semicontinuous from the right and lim supt→0t/ϕt <∞. ThenT has a fixed point.
Proof. LetG {x, y :x ∈ X, y ∈ Tx}be the graph ofT. Clearly,Gis a closed subset of
X×X. Define a metricρonGby
ρx, y,u, vmaxdx, u, dy, v. 3.1
ThenG, ρis a complete metric space andρisw-distance onG. Now, defineψ:G → 0,∞
byψx, y ωx, y dx, yfor allx, y∈Gandc:0,∞ → 0,∞by
ct
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
t
ϕt, ift >0,
lim sup t→0
t ϕt
, ift0. 3.2
Thenψ is lower semicontinuous andcis upper semicontinuous from the right becauseϕis lower semicontinuous from the right. Definep:G×G → 0,∞by
px, y,u, vmaxψx, y, ρx, y,u, v. 3.3
Thenpis aw-distance onGsee14, page 47. Now, suppose FixT∅. Then for eachx, y∈ G, we havex /y. Sincey∈Txthere isz∈Tysuch that
ωy, z≤ωx, y−ϕωx, y. 3.4
Sincex, y,y, z∈G, we have
px, y,y, zρx, y,y, zωx, y ψx, y, 3.5
also, note that
px, y,y, zωx, y≤ ωx, y ϕωx, y
ωx, y−ωy, z. 3.6
Define a functionf:G → Gbyfx, y y, z, then we get
Thus, byTheorem 2.2,f has a fixed point, which is impossible. Hence,T must has a fixed point. This completes the proof.
As a consequence, we obtain the following recent fixed point result of Latif 10, Theorem 2.2.
Corollary 3.2. LetX, dbe a complete metric space. LetT :X → ClXbe a map such that for any
x, y∈Xandu∈Txthere isv∈Tywith
ωu, v≤kωx, yωx, y, 3.8
wherekis function from0,∞to0,1with lim supr→tkr<1,for everyt∈0,∞.ThenT has a fixed point.
Proof. Defineϕ:0,∞ → 0,∞by
ϕt mint1−kt,lim inf r→t r
1−kr ∀t≥0. 3.9
Thenϕt>0 for allt >0,ϕis lower semicontinuous from the rightsee19. Also note that
lim sup t→0
t
ϕt <∞, 3.10
and for eachx, y∈X, we have
ϕωx, y≤ωx, y1−kωx, y. 3.11
It follows from3.8and3.11that
ωu, v≤ωx, y−ϕωx, y. 3.12
ThusTis weaklyw-contractive map for whichϕis lower semicontinuous from the right and lim supt→0t/ϕt<∞. Therefore, byTheorem 3.1,Thas a fixed point.
Remark 3.3. aTheorem 3.1generalizes Theorem 2.4 of Suzuki and Takahashi9. Indeed,
consider ϕωx, y 1−hωx, y for a constant h with 0 < h < 1. Theorem 3.1also generalizes and improves the fixed point result of Bae4, Theorem 3.1.
bCorollary 3.2generalizes fixed point result of Husain and Latif17, Theorem 2.3
and improves16, Theorem 5. Moreover, it improves and generalizes18, Theorem 1.
Without using the Hausdorff metric, we prove the following fixed point result for nonself multivalued maps with respect tow-distance.
Theorem 3.4. LetMbe a closed subset of a complete metric spaceX, dand letT :M → ClX
Proof. LetG,ρ,p, andψ be the same as in the proof ofTheorem 3.1. Suppose FixT ∅. Then, for eachx, y∈Gwe havex /y. Sincey∈Tx⊂ w-IMxthere existsu∈Mwith
u /xand
ωx, y ωx, u ωu, y. 3.13
Since the mapT is weaklyw-contractive, there existsv∈Tusuch that
ωy, v≤ωx, u−ϕωx, u, 3.14
whereϕis lower semicontinuous and lim supt→0t/ϕt < ∞. From3.13and3.14, we
get
ϕωx, u≤ωx, u−ωy, v ωx, y−ωu, y ωy, v. 3.15
Thus,
ϕωx, u≤ωx, y−ωu, v. 3.16
Sincex, y,u, v∈G, we have
ρx, y,u, vmaxωx, u, ωy, v, 3.17
and hence, we get
px, y,u, vωx, u≤ωx, y ψx, y. 3.18
Now, define a functionf :G → Gbyfx, y u, v. Then from3.18we get
px, y, fx, y≤ψx, y, 3.19
and using3.16, we obtain
ϕpx, y, fx, y≤ψx, y−ψfx, y. 3.20
Thus byTheorem 2.3,fhas a fixed point, which is impossible. Hence, it follows thatTmust has a fixed point.
Using the same method as in the proof ofCorollary 3.2, we can obtain the following fixed point result for nonself generalizedw-contractions.
Corollary 3.5. LetMbe a closed subset of a complete metric spaceX, dand letT :M → ClX
be a map satisfying inequality3.8for whichk :0,∞ → 0,1is upper semicontinuous. ThenT
Remark 3.6. aOurTheorem 3.4andCorollary 3.5improve the results of Bae4, Theorem 3.3 and Corollary 3.4, respectively.
bThe analogue of all the results of this section can be established with respect to
τ-distance20.
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