Volume 2010, Article ID 718340,8pages doi:10.1155/2010/718340
Research Article
Common Fixed Points for Generalized ϕ-Pair Mappings on Cone Metric Spaces
F. Sabetghadam and H. P. Masiha
Department of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran 15418, Iran
Correspondence should be addressed to H. P. Masiha,[email protected] Received 19 October 2009; Revised 12 December 2009; Accepted 10 January 2010 Academic Editor: Massimo Furi
Copyrightq2010 F. Sabetghadam and H. P. Masiha. 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.
We define the concept of generalizedϕ-pair mappings and prove some common fixed point theorems for this type of mappings. Our results generalize some recent results.
1. Introduction
Huang and Zhang1recently introduced the concept of cone metric spaces and established some fixed point theorems for contractive mappings in these spaces. Afterwards, Rezapour and Hamlbarani 2 studied fixed point theorems of contractive type mappings by omitting the assumption of normality in cone metric spaces. Also, other authors proved the existence of points of coincidence, common fixed point, and coupled fixed point for mappings satisfying different contraction conditions in cone metric spaces see 1–
12. In 6 Di Bari and Vetro introduced the concept of ϕ-map and proved a main theorem generalizing some known results. We define the concept of generalizedϕ-mappings and prove some results about common fixed points for such mappings. Our results generalize some results of Huang and Zhang 1, Di Bari and Vetro 6, and Abbas and Jungck 3. First, we recall some standard notations and definitions in cone metric spaces.
LetE be a real Banach space and letθ denote the zero element inE. A coneP is a subset ofEsuch that
iPis closed, nonempty, andP /{θ},
iiifa, bare nonnegative real numbers andx, y∈P, thenaxby∈P, iiiP∩−P {θ}.
For a given coneP ⊂ E, the partial ordering≤with respect toP is defined byx≤ y if and only ify−x ∈ P. The notation x < y will stand forx ≤ ybut x /y. Also, we will usex yto indicate thaty−x ∈ intP where intP denotes the interior ofP. Using these notations, we have the following definition of a cone metric space.
Definition 1.1see1. LetX be a nonempty set and letEbe a real Banach space equipped with the partial ordering ≤with respect to the coneP ⊂ E. Suppose that the mappingd : X×X → Esatisfies the following conditions:
d1θ≤dx, yfor allx, y∈Xanddx, y θif and only ifxy, d2dx, y dy, xfor allx, y∈X,
d3dx, y≤dx, z dz, yfor allx, y, z∈X.
Thendis called a cone metric onX, andX, dis called a cone metric space.
The conePis called normal if there exists a constantK >0 such that for everyx, y∈E ifθ≤ x≤ y,thenx ≤Ky. The least positive number satisfying this inequality is called the normal constant ofP. The coneP is called regular if every increasingdecreasingand bounded abovebelowsequence is convergent inE. It is known that every regular cone is normal1 see also2, Lemma 1.1.
Definition 1.2see1. LetX, dbe a cone metric space, let{xn}be a sequence inX,and let x∈X.
i{xn}is said to be Cauchy sequence if for everyc∈Ewithθcthere existsN∈N such that for alln, m≥N,dxn, xmc.
ii{xn}is said to be convergent tox, denoted by limn→ ∞xnxorxn → xasn → ∞ if for everyc∈Ewithθcthere existsN∈Nsuch that for alln≥N,dxn, xc.
iiiXis said to be complete if every Cauchy sequence inXis convergent inX.
ivX is said to be sequentially compact if for every sequence{xn}inX there exists a subsequence{xni}of{xn}such that{xni}is convergent inX.
Clearly, every sequentially compact cone metric space is complete see 1–12 for more related results about complete cone metric spaces. We also note that the relations PintP⊆intPandλintP ⊆intPλ >0always hold true.
Definition 1.3see13. LetTandSbe self-mappings of a cone metric spaceX, d. One says thatSandT are compatible if limn→ ∞dSTxn, TSxn θ, whenever{xn}is a sequence inX such that limn→ ∞Txnlimn→ ∞Sxntfor somet∈X.
The concept of weakly compatible mappings is introduced as follows.
Definition 1.4see13. The self-mappingsT andSof a cone metric spaceX, dare said to be weakly compatible if they commute at their coincidence points, that is, ifTuSufor some u∈X, thenTSuSTu.
2. Main Results
In this section, we introduce the notation of generalized ϕ-mapping and a contractive condition called generalizedϕ-pair. We prove some results on common fixed points of these mappings on cone metric spaces.
LetPbe a cone. A nondecreasing mappingϕ:P → Pis called aϕ-mapping6if ϕ1ϕθ θandθ < ϕw< wforw∈P\ {θ},
ϕ2w−ϕw∈intPfor everyw∈intP, ϕ3limn→ ∞ϕnw θfor everyw∈P\ {θ}.
Definition 2.1. LetPbe a cone and let{wn}be a sequence inP. One says thatwn −−→ θif for every∈Pwithθthere existsN∈Nsuch thatwnfor alln≥N.
For a nondecreasing mappingF :P → P,we define the following conditions which will be used in the sequal:
F1Fw θif and only ifwθ,
F2for everywn∈P,wn−−→ θif and only ifFwn−−→ θ, F3for everyw1, w2∈P,Fw1w2≤Fw1 Fw2.
Definition 2.2. The self-mappingsf, g: X → Xare called generalizedϕ-pair if there exist a ϕ-mapping and a mappingFsatisfying the conditionsF1,F2,andF3such that
F d
fx, fy
≤ϕ F
d
gx, gy
, 2.1
for everyx, y∈X.
Now, we are in the position to state the following theorem.
Theorem 2.3. Let X, d be a cone metric space and let f, g : X → X be a generalized ϕ-pair.
Suppose thatfandgare weakly compatible withfX⊂gXsuch thatfXorgXis complete. Then the self-mappingsfandghave a unique common fixed point inX.
Proof. Letx0 ∈ X and choosex1 ∈ X such that fx0 gx1. This can be done, sincefX ⊂ gX. Continuing this process, after choosingxn ∈ X, we choosexn1 ∈X such thatgxn1 fxn. Sincef andg are generalized ϕ-pair, byDefinition 1.2, there exist aϕ-mapping and a mappingFsatisfying the conditionsF1–F3and the inequality of2.1. By2.1, we deduce
F d
fxn1, fxn
≤ϕ F
d
gxn1, gxn
ϕ F
d
fxn, fxn−1
≤ϕ2 F
d
gxn, gxn−1
≤ · · · ≤ϕn F
d
fx1, fx0
. 2.2
Let∈intP,then, byϕ2,0−ϕ∈intP. Byϕ3,
nlim→ ∞ϕn F
d
fx1, fx0
θ. 2.3
Therefore, one can find thatN∈Nsuch that, for allm≥N,Fdfxm, fxm1−ϕ. We show that
F d
fxm, fxn1
, 2.4
for a fixedm ≥N andn≥ m. This holds whenn m. Now let2.4hold for somen≥ m, then we have
F d
fxm, fxn2
≤F d
fxm, fxm1 F
d
fxm1, fxn2 −ϕ ϕ
F d
gxm1, gxn2 −ϕ ϕ
F d
fxm, fxn1 −ϕ ϕ .
2.5
Therefore, by induction andF2we deduce that{fxn}is a Cauchy sequence. Suppose that fXis a complete subspace ofX, then there existsy∈fX⊂gXsuch thatfxn → yand also gxn → yThis holds also ifgXis complete withy ∈gX.. Letz ∈ Xbe such thatgz y.
We show that fz gz. ByF2for θ one can choose a natural numberN such that Fdy, fxn/2 andFdgxn, gz/2 for alln≥N. Then,
F d
y, fz
≤F d
y, fxn F
d
fxn, fz
≤F d
y, fxn ϕ
F d
gxn, gz
< F d
y, fxn F
d
gxn, gz
2
2 . 2.6
Thus,/m−Fdy, fz ∈ P for every m ∈ N. This implies that−Fdy, fz ∈ P,and hence,Fdy, fz θ. So applyingF1,we getdy, fz θwhich implies thatyfzgz, that is,yis a point of coincidence offandg. Now, we use the hypothesis thatf andgare weakly compatible to deduce thatyis a common fixed point offandg. Fromfz gzy, by compatibility offandg, it follows thatfyfgzgfzgy. Ifgy /y, then we have
F d
fy, fz
≤ϕ F
d
gy, gz
< F d
gy, gz F
d
fy, fz
, 2.7
which implies thatfyygy. Soyis a common fixed point offandg. The uniqueness of the common fixed point is clear.
Example 2.4. LetE Rand letP {x ∈R:x ≥0}be a normal cone. LetX 1,∞with usual metricdx, y |x−y|. Definef, g:X → Xbyfxxandgx2x−1, for allx∈X.
Also, defineF, ϕ:P → Pbyϕw 2/3wandFw 1/2w, for allw∈P. Then 1fandgare weakly compatible,
2fX⊂gX,
3we haveFdfx, fy≤ϕFdgx, gy, 4f1g11.
Example 2.5. LetER2and letP{x, y∈R2 :x, y≥0}be a normal cone. LetX 1,∞ with metricdx, y |x−y|,1/2|x−y|. Definef, g : X → X byfx x1/2 and gx2x−1, for allx∈X. Also, defineF, ϕ:P → P byϕw1, w2 1/2w1,1/3w2and Fw1, w2 w2, w1w2, for allw1, w2∈P. Then
1fandgare weakly compatible, 2fX⊂gX,
3we haveFdfx, fy≤ϕFdgx, gy, 4f1g11.
Example 2.6. LetER2and letP{x, y∈R2 :x, y≥0}be a normal cone. LetX 1,∞ with metricdx, y |x−y|,2|x−y|. Definef, g:X → Xbyfx 1/2x1 andgxx, for allx∈X. Also, defineF, ϕ:P → Pbyϕw1, w2 1/2w1,2/3w2andFw1, w2
w1, w1w2, for allw1, w2∈P. Then 1fandgare weakly compatible, 2fX⊂gX,
3we haveFdfx, fy≤ϕFdgx, gy, 4f2g22.
If we let the mappingF be the identity mapping inTheorem 2.3, then we obtain the following corollary.
Corollary 2.7. LetX, dbe a cone metric space. Suppose that the mappingsf, g:X → Xsatisfy d
fx, fy
≤ϕ d
gx, gy
, 2.8
for allx, y∈X. IffX⊂gX,f andgare weakly compatible, andfXorgXis complete, thenfand ghave a unique common fixed point inX.
Remark 2.8. Corollary 2.7 generalizes Theorem 1 in 6. Also, if we choose theϕ-mapping defined byϕw kw, wherek∈0,1is a constant, thenTheorem 2.3generalizes Theorem 2.1 in3. Furthermore, if we letgbe the identity map ofX, then we obtain Theorem 1 in1, that is, the extension of the Banach fixed point theorem for cone metric spaces.
If we replace the conditionϕ1with the following condition:
ϕ1there existsk∈0,1/2such thatϕw≤kwforw\{θ}andϕθ θ, then we have the following theorems.
Theorem 2.9. LetX, dbe a cone metric space and letf, g:X → Xbe self-mappings such that F
d
fx, fy
≤ϕ F
d fx, gx
d
fy, gy
, 2.9
for allx, y ∈X whereϕis a nondecreasing mapping fromP intoP satisfying the conditionsϕ1, ϕ2,andϕ3,andF:P → Pis a nondecreasing mapping satisfying the conditionsF1–F3. Suppose thatfandgare weakly compatible,fX⊂gX,andfXorgXis complete. Then the mappingsfand ghave a unique common fixed point inX.
Proof. Letx0be an arbitrary point inX. Choose a pointx1 ∈Xsuch thatfx0gx1. This can be done sincefX⊂gX. Continuing this process, after choosingxn∈Xwithfxn gxn1, by 2.9andϕ1,we have
F d
fxn1, fxn
≤ϕ F
d
fxn1, gxn1 d
fxn, gxn
≤k F
d
fxn1, fxn F
d
fxn, fxn−1
. 2.10
Consequently,
F d
fxn1, fxn
≤hF d
fxn, fxn−1
, 2.11
wherehk/1−k. Forn > mwe have F
d
fxn, fxm
≤F d
fxn, fxn−1 F
d
fxn−1, fxn−2 · · ·F
d
fxm1, fxm
≤
hn−1hn−2· · ·hm F
d
fx1, fx0
≤ hm 1−hF
d
fx1, fx0
.
2.12
ThenFdfxn, fxm −−→ θ asn, m → ∞,and hence, byF2,{fxn}is a Cauchy sequence.
Suppose that fX is a complete subspace of X, then there exists q ∈ fX ⊂ gX such that fxn → qand alsogxn → qthis holds ifgXis complete. Letp ∈X be such thatgp q.
By F2, for a fixed θ c and every m ∈ N there exists a natural numberN such that Fdgxn1, gxnc1−k/2kandFdgxn1, gpc1−k/2 for alln≥N. Hence,
F d
gp, fp
≤F d
gp, fxn F
d
fxn, fp
≤F d
gp, fxn ϕ
F d
fxn, gxn d
fp, gp
≤F d
gp, fxn
k F
d
fxn, gxn
F d
gp, fp ,
2.13
which implies that
F d
gp, fp
≤ 1 1−kF
d gp, fxn
k 1−kF
d
fxn, gxn
c 2 c
2 c.
2.14
Thus,Fdgp, fp c/mfor allm ≥ 1. This implies thatFdgp, fp θ,and therefore, gpfp. Sincefandgare weakly compatible,fqfgpgfpgq. Ifq /gq, then
F d
fq, fp
≤ϕ F
d fq, gq
d fp, gp
≤k F
d
fq, gq F
d
fp, gp
, 2.15
which givesFdfq, fp θ,and hence,gqfqq. Soqis a common fixed point forfand g. The uniqueness of common fixed point is clear.
If inTheorem 2.9we letF be IdX and let theϕ-mapping be ϕw kw, wherek ∈ 0,1/2is a constant, then we obtain the following corollary.
Corollary 2.10. LetX, dbe a cone metric space and letf, g:X → Xbe self-mappings such that d
fx, fy
≤k d
fx, gx d
fy, gy
, 2.16
for allx, y ∈X, wherek ∈ 0,1/2is a constant. Suppose thatf andgare weakly compatible, the range ofg contains the range off,and fXorgXis complete. Then the mappingsf and g have a unique common fixed point inX.
Remark 2.11. Corollary 2.10generalizes Theorem 2.3 of3. If inCorollary 2.10we letgbe the identity map onX, then we obtain Theorem 3 of1.
Theorem 2.12. LetX, dbe a cone metric space and letf, g:X → Xbe self-mappings such that F
d
fx, fy
≤ϕ F
d fx, gy
d
fy, gx
, 2.17
for allx, y∈X. Suppose thatfandgare weakly compatible, the range ofgcontains the range off, andfXorgXis complete. Then the mappingsfandghave a unique common fixed point inX.
Proof. Letx0be an arbitrary point inX. Choose a pointx1inXsuch thatfx0 gx1. This can be done sincefX⊂gX. Continuing this process having chosenxninXsuch thatfxngxn1, we have
F d
fxn1, fxn
≤ϕ F
d
fxn1, gxn d
fxn, gxn1
≤ϕ F
d
fxn1, gxn F
d
fxn, gxn1
≤k F
d
fxn1, fxn−1
≤k F
d
fxn1, fxn F
d
fxn, fxn−1 .
2.18
So,
F d
fxn1, fxn
≤hF d
fxn, fxn−1
, 2.19
wherehk/1−k<1. Now letm, n∈Nwithn > m. Then, F
d
fxn, fxm
≤F d
fxn, fxn−1 F
d
fxn−1, fxn−2
· · ·F d
fxm1, fxm
≤
hn−1· · ·hm F
d
fx1, fx0
≤ hm 1−hF
d
fx1, fx0
.
2.20
Following an argument similar to that one given inTheorem 2.9, we obtain a unique common fixed point offandg.
If inTheorem 2.12we letFbe the identity map onXand let theϕ-map beϕw kw, wherek∈0,1/2is a constant, then we obtain the following corollary.
Corollary 2.13. LetX, dbe a cone metric space and letf, g:X → Xbe self-mappings such that
d fx, fy
≤k d
fx, gy d
fy, gx
, 2.21
for allx, y ∈X, wherek ∈ 0,1/2is a constant. Suppose thatf andgare weakly compatible, the range ofg contains the range off,and fXorgXis complete. Then the mappingsf and g have a unique common fixed point inX.
Remark 2.14. Corollary 2.13generalizes Theorem 2.4 of3and if inCorollary 2.13we letgbe the identity map onX, then we obtain Theorem 4 of1.
Acknowledgment
The authors would like to thank the referees for their valuable and useful comments.
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