Volume 2009, Article ID 131068,9pages doi:10.1155/2009/131068
Research Article
Common Fixed Point Theorem of
Two Mappings Satisfying a Generalized Weak
Contractive Condition
M. Abbas and M. Ali Khan
Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan
Correspondence should be addressed to M. Abbas,[email protected]
Received 6 August 2009; Accepted 11 November 2009
Recommended by Evgeny Korotyaev
Existence of common fixed point for two mappings which satisfy a generalized weak contractive condition is established. As a consequence, a common fixed point result for mappings satisfying a contractive condition of integral type is obtained. Our results generalize, extend, and unify several well-known comparable results in literature.
Copyrightq2009 M. Abbas and M. A. Khan. 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 and Preliminaries
LetXbe a metric space andT :C → Ca mapping. Recall thatTis contraction ifdTx, Ty≤ kdx, y for allx, y ∈ X,where 0 ≤ k < 1.A pointx ∈ Cis a fixed point of T provided
Tx x.If a mapT satisfiesFT FTnfor eachn ∈ N,whereFTdenotes the set of
all fixed points ofT, then it is said to have propertyP.Banach contraction principle which gives an answer on existence and uniqueness of a solution of an operator equationTx x is the most widely used fixed point theorem in all of analysis. Branciari1obtained a fixed point theorem for a mapping satisfying an analogue of Banach’s contraction principle for an integral type inequality. Akgun and Rhoades2have shown that a map satisfying a Meir- Keeler type contractive condition of integral type has a propertyP.Rhoades and Abbas
3extended4, Theorem 1for mappings satisfying contractive condition of integral type. They also studied several results for maps which have propertyP,defined on a metric space satisfying generalized contractive conditions of integral type. Rhoades5proved two fixed point theorems involving more general contractive condition of integral typesee, also6,7. If mapsSand T satisfyFS∩FT FSn∩FTnfor eachn ∈N,then they are said to
Recently Dutta and Choudhury 9 gave a generalization of Banach contraction principle, which in turn generalize4, Theorem 1and corresponding result of 10. Sessa
11defined the concept of weakly commuting to obtain common fixed point for pairs of maps. Jungck generalized this idea, first to compatible mappings 12and then to weakly compatible mappings13. There are examples that show that each of these generalizations of commutativity is a proper extension of the previous definition. The aim of this paper is to present a common fixed point theorem for weakly compatible maps satisfying a generalized weak contractive condition which is more general than the corresponding contractive condition of integral type. Our results substantially extend, improve, and generalize comparable results in literature3,14,15.
The following definitions and results will be needed in the sequel.
Definition 1.1. LetX be a set, andf, gselfmaps ofX. A pointxinX is called a coincidence point offandgif and only iffxgx. We will callwfxgxa point of coincidence off
andg.
Definition 1.2. Two mapsf andgare said to be weakly compatible if they commute at their coincidence points.
Lemma 1.3see16. Letf andg be weakly compatible self maps of a setX. Iff andg have a unique point of coincidencew(say), thenwis the unique common fixed point offandg.
2. A Common Fixed Point Theorem
Set {φ : R → R : φ is a Lebesgue integrable mapping which is summable and nonnegative and satisfiesε0φtdt > 0,for eachε > 0} andG {ψ : 0,∞ → 0,∞ : ψ
is continuous and nondecreasing mapping withψt 0 if and only ift0}.
The following is the main result of this paper.
Theorem 2.1. Letf, gbe two self maps of a metric spaceX, dsatisfying
ψdfx, fy≤ψdgx, gy−ϕdgx, gy 2.1
for allx, y∈X, whereψ, ϕ∈G.If range ofgcontains the range offandgXis a complete subspace ofX, thenfandghave a unique point of coincidence inX.Moreover iffandgare weakly compatible,
fandghave a unique common fixed point.
Proof. Letx0be an arbitrary point ofX. Choose a pointx1inXsuch thatfx0 gx1.This
can be done, since the range ofg contains the range off.Continuing this process, having chosenxn inX, we obtainxn1inX such thatfxn gxn1, n 0,1,2, . . . .Suppose for
anyn, gxn/gxn1,since, otherwise,f andg have a point of coincidence. From2.1, we
have
ψdgxn1, gxn
ψdfxn, fxn−1
≤ψdgxn, gxn−1
−ϕdgxn, gxn−1
< ψdgxn, gxn−1
,
that is,ψdgxn1, gxn< ψdgxn, gxn−1, and hence
dgxn, gxn1
≤dgxn, gxn−1
. 2.3
It follows that {dgxn, gxn1} is monotone decreasing sequence of numbers and
conse-quently there exists r ≥ 0 such that dgxn, gxn1 → r asn → ∞.Suppose that r > 0,
then
0< ψr≤ψdgxn1, gxn
ψdfxn, fxn−1
≤ψdgxn, gxn−1
−ϕdgxn, gxn−1
, 2.4
which on taking limit asn → ∞yields
ψr≤ψr−ϕr< ψr, 2.5
which is a contradiction. Thereforer 0.Now we prove that{gxn}is a Cauchy sequence.
If not, then there exist someε > 0 and subsequences{gxnk}and{gxmk}of{gxn}withk < nk< mksuch thatdgxnk, gxmk≥3εfor eachk.Asdgxnk1, gxnk → 0 ask → ∞, for large
enoughk, we havedgxnk1, gxnk< εanddgxmk1, gxmk< ε. Thus we obtain
dgxnk1, gxmk
≥dgxnk, gxmk
−dgxnk1, gxnk
> ε,
dgxnk1, gxmk−1
≥dgxnk, gxmk
−dgxmk−1, gxmk
−dgxnk1, gxnk
> ε.
2.6
We may assume thatnkare even andmkare odd and thatdgxnk, gxmk> εfor allk. Put
rkmin
mk:d
gxnk, gxmk
> ε. 2.7
Now,
ε < dgxnk, gxrk
≤dgxnk, gxrk−2
dgxrk−2, gxrk−1
dgxrk−1, gxrk
2.8
implies thatdgxnk, gxrk → εask → ∞. Furthermore
dgxnk, gxrk
−dgxnk, gxnk1
−dgxrk, gxrk1
≤dgxnk1, gxrk1
≤dgxnk, gxrk
dgxnk, gxnk1
dgxrk, gxrk1
givesdgxnk1, gxrk1 → ε, ask → ∞. Therefore
ψdgxnk1, gxrk1
ψdfxnk, fxrk
≤ψdgxnk, gxrk
−ϕdgxnk, gxrk
. 2.10
Taking limit ask → ∞yields
ψε≤ψε−ϕε, 2.11
which is a contradiction. Hence {gxn}is a Cauchy sequence. From completeness of gX,
there exists a pointqingXsuch thatgxn → qasn → ∞.Consequently, we can findpin
Xsuch thatgp q.Now
ψdgxn1, fp
ψdfxn, fp
≤ψdgxn, gp
−ϕdgxn, gp
2.12
on taking limit asn → ∞implies
ψdq, fp≤ψ0−ϕ0, 2.13
ψdq, fp 0, andfp q.Henceqis the point of coincidence off andg.Assume that there is another point of coincidentr inX such thatr /q.Then there existssinX such that
fs gs r. Using2.1, we have
ψdgp, gsψdfp, fs
≤ψdgp, gs−ϕdgp, gs
< ψdgp, gs,
2.14
which is a contradiction which proves the uniqueness of point of coincidence; the result now follows fromLemma 1.3
Corollary 2.2. Letf, gbe two self maps of a metric spaceX, dsatisfying
ψdfx,fy
0
φtdt≤
ψdgx, gy
0
φtdt−
ϕdgx,gy
0
φtdt 2.15
for allx, y ∈ X, whereφ ∈ and ψ, ϕ ∈ G.If range ofg contains the range off and gXis a complete subspace ofX, thenfandg have a unique point of coincidence inX.Moreover iff andg
Proof. DefineΦ:R → RbyΦx x0φtdt,thenΦ∈Gand2.15becomes
Φψdfx, fy≤Φψdgx, gy−Φϕdgx, gy, 2.16
which further can be written as
ψ1
dfx, fy≤ψ1
dgx, gy−ϕ1
dgx, gy, 2.17
whereψ1 Φ◦ψandϕ1 Φ◦ϕ∈G.Clearlyψ1, ϕ1∈G.Hence byTheorem 2.1fandghave
unique common fixed point.
Now we present two examples in the support ofTheorem 2.1.
Example 2.3. LetX 0,1∪ {2,3,4, . . .},
dx, y
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩
x−y, ifx, y∈0,1, x /y,
xy, if at least one ofxory /∈0,1, x /y,
0, ifxy.
2.18
ThenX, dis a complete metric space17. Considerf : X → X, andψ, ϕ∈Gas given in
9:
fx
⎧ ⎨ ⎩
x−1
2x
2, if 0≤x≤1,
x−1, ifx >1,
2.19
ψt
⎧ ⎨ ⎩
t, if 0≤t≤1,
t2, ift≥1, 2.20
ϕt
⎧ ⎪ ⎨ ⎪ ⎩
1 2t
2, if 0≤t≤1,
1
2, if t≥1.
2.21
Letg:X → Xbe defined as
gx
⎧ ⎨ ⎩
x, if 0≤x≤1,
x1, ifx >1.
Assume thatx > yand discuss the following cases. Whenx∈0,1,then
ψdfx, fy
x− 1
2x
2
−
y−1
2y
2
≤x−y−1
2
x−y2
ψdgx, gy−ϕdgx, gy.
2.23
Takingxin{3,4, . . .},andyin0,1,we obtain
ψdfx, fy
x−1y−1
2y
2
2
≤xy−12,
ψdgx, gy xy12,
ϕdgx, gy 1
2.
2.24
Hence
ψdfx, fy≤ψdgx, gy−ϕdgx, gy. 2.25
Now, whenx∈ {3,4, . . .},andy /∈0,1, then
ψdfx, fyx−1y−12
<xy−12,
ψdgx, gyxy22,
ϕdgx, gy 1
2.
2.26
Obviously2.31holds. Finally whenx2,we havey∈0,1,fx1,and
dfx, fy1−
y−1
2y
2
so thatψdfx, fy≤1,then
ψdgx, gy−ϕdgx, gy3y2−1
2
>1≥ψdfx, fy.
2.28
Thus all conditions ofTheorem 2.1are satisfied. Moreoverf andg have a unique common fixed point.
Example 2.4. LetX 0,1and f, g:X → Xbe given as
fx 2
5x
23
5, gx 2 3x
21
3. 2.29
Considerψ, ϕ∈Gasψt 1/2tandϕt 1/10t.Then we have
ψdfx, fy 2
10
x2−y2
≤ 1
2 2 3
x2−y2− 1
10 2 3
x2−y2
ψdgx, gy−ϕdgx, gy.
2.30
Note that x 1 is the unique coincidence point of f and g,and f and g are commuting at x 1.Hence all conditions ofTheorem 2.1are satisfied. Moreover,x 1 is the unique common fixed point offandg.
Following theorem can be viewed as generalization and extension of3, Theorem 3.
Theorem 2.5. Letfbe a self map of a complete metric spaceX, dsatisfying
ψdfx,fy
0
φtdt≤
ψdx,y
0
φtdt−
ϕdx,y
0
φtdt 2.31
for allx, y∈X, whereφ∈andψ, ϕ∈G.Thenfhas a unique fixed point. Moreoverfhas property
Proof. Existence and uniqueness of fixed point offfollows fromCorollary 2.2. Now we prove thatfhas propertyP.Letu∈Ffn. We shall always assume thatn >1, since the statement
forn1 is trivial. We claim thatfuu.If not, then, by2.31,
ψdu,fu
0
φtdt
ψdfnu,ffnu
0
φtdt
ψdffn−1u,ffnu
0
φtdt
≤
ψdfn−1u,fnu
0
φtdt−
ϕdfn−1u,fnu
0
φtdt
≤
ψdfn−1u,fnu
0
φtdt
ψdffn−2u,ffn−1u
0
φtdt
≤
ψdfn−2u,fn−1u
0
φtdt−
ϕdfn−2u,fn−1u
0
φtdt
≤
ψdfn−2u,fn−1u
0
φtdt.
2.32
Continuing this process we arrive at
ψdu,fu
0
φtdt≤
ψdu,fu
0
φtdt−
ϕdu,fu
0
φtdt
<
ψdu,fu
0
φtdt,
2.33
which is a contradiction. Hence the result follows.
Remarks 2.6. Existence and uniqueness of fixed point offin above theorem also follows from
9, Theorem 1.
Remarks 2.7. aIt is noted that if mapsfandginvolved inTheorem 2.1are commuting, then they have propertyQ.
bSuzuki18 observed that Branciari1, Theorem 1 is a particular case of Meir-Keeler fixed point theorem19. We pose an open problem to see if a link exists between the contractive conditions2.15and the Meir-Keeler condition.
Acknowledgment
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