Volume 2008, Article ID 891375,8pages doi:10.1155/2008/891375
Research Article
Common Fixed Point Theorems for
Weakly Compatible Maps Satisfying
a General Contractive Condition
Cristina Di Bari and Calogero Vetro
Dipartimento di Matematica ed Applicazioni, Universit`a degli Studi di Palermo, Via Archirafi 34, 90123 Palermo, Italy
Correspondence should be addressed to Cristina Di Bari,[email protected]
Received 21 May 2008; Accepted 3 September 2008
Recommended by Wolfgang Kuehnel
We introduce a new generalized contractive condition for four mappings in the framework of metric space. We give some common fixed point results for these mappings and we deduce a fixed point result for weakly compatible mappings satisfying a contractive condition of integral type.
Copyrightq2008 C. Di Bari and C. Vetro. 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
The study of common fixed point of mappings satisfying contractive type conditions has been a very active field of research during recent years. The most general of the common fixed point theorems pertaining to four mappings,A, B, S, andT of a metric spaceX, d, uses either a Banach-type contractive condition of the form
dAx, By≤kMx, y, 0≤k <1, 1.1
where
Mx, y max
dSx, Ty, dAx, Sx, dBy, Ty,
dSx, By dAx, Ty 2
, 1.2
or a Meir-Keeler-typeε, δ-contractive condition, that is, givenε >0, there exists aδ >0 such that
or aψ-contractive condition of the form
dAx, By≤ψMx, y, 1.4
involving a contractive gauge functionψ :0,∞→0,∞such thatψt< tfor eacht >0. Clearly, Banach-type contractive condition is a special case of both conditions Meir-Keeler-typeε, δ-contractive andψ-contractive. A ψ-contractive condition does not guarantee the existence of a fixed point unless some additional condition is assumed. Moreover, a ψ-contractive condition, in general, does not imply the Meir-Keeler-type ε, δ-contractive condition1, Example 1.1.
Recently, some fixed point results for mappings satisfying an integral-type contractive condition are obtained in2–5. Suzuki6showed that Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Zhang7introduced a generalized contractive-type condition for a pair of mappings in metric space and proved common fixed point theorems that extend results in 3–5. In this paper, we give a new generalized contractive-type condition for four mappings in metric space and prove some common fixed point results for these mappings. The results obtained extend well-known comparable results in2–5,7.
Lemma 1.1see8. For every functionψ:0,∞→0,∞, letψnbe thenth iterate ofψ. Then
the following hold:
iifψis nondecreasing, then for eacht >0, limn→∞ψnt 0 impliesψt< t;
iiifψis right continuous withψt< tfort >0, then limn→∞ψnt 0.
2. Common fixed points
In this section, we give our main result. Two self-mappingsAandSof a metric spaceX, d are called weakly compatible if they commute at their coincidence points. LetA, B, S,andT be self mappings of a metric spaceX, d. In the sequel, we set
Mx, y max
dSx, Ty, dAx, Sx, dBy, Ty,
dSx, By dAx, Ty 2
. 2.1
Lemma 2.1. Let A, B, S, and T be self-mappings of a metric space X, d such that AX ⊂ TX, BX⊂SX. Assume that there existF, ψ:0,∞→0,∞such that
iFis nondecreasing, continuous, andF0 0< Ftfor everyt >0;
iiψis nondecreasing, right continuous, andψt< tfor everyt >0. If for allx, y∈X,
FdAx, By≤ψFMx, y, 2.2
then for eachx0∈X, the sequenceynof points ofXdefined by the rule
y2nAx2nTx2n1, y2n−1Bx2n−1Sx2n 2.3
Proof. We have
Mx2n, x2n1
max
dy2n−1, y2n
, dy2n, y2n−1
, dy2n1, y2n
,
dy2n−1, y2n1
dy2n, y2n
2
max
dy2n, y2n−1
, dy2n, y2n1
, d
y2n−1, y2n1
2
maxdy2n, y2n−1
, dy2n, y2n1 .
2.4
Similarly
Mx2n, x2n−1
maxdy2n, y2n−1
, dy2n−1, y2n−2 . 2.5
If for somenwe have eithery2n y2n−1 ory2n y2n1, then by condition2.2we
obtain that the sequenceynis definitely constant and thus is a Cauchy sequence. Suppose yn/yn−1for eachn.
From
Fdy2n, y2n1
FdAx2n, Bx2n1
≤ψFMx2n, x2n1
ψFdy2n, y2n−1
< Fdy2n, y2n−1
,
Fdy2n, y2n−1
FdAx2n, Bx2n−1
≤ψFMx2n, x2n−1
ψFdy2n−1, y2n−2
< Fdy2n−1, y2n−2
,
2.6
we deduce
Fdyn1, yn
< Fdyn, yn−1
, 2.7
for alln∈N. Now, from
Fdyn1, yn
≤ψFdyn, yn−1
≤ · · · ≤ψnFdy0, y1
2.8
andiiofLemma 1.1, we obtain limn→∞Fdyn1, yn 0, which implies
lim
n→∞d
yn1, yn
0. 2.9
We prove thatynis a Cauchy sequence. Suppose not, then there existsε > 0 such thatdyn, ym≥2εfor infinite values ofmandnwithm < n. This assures that there exist two
sequencesmk,nkof natural numbers, withmk< nk, such that
dy2mk, y2nk1
It is not restrictive to suppose thatnkis the least positive integer exceedingmkand satisfying 2.10. We have
ε < dy2mk, y2nk1
≤dy2mk, y2nk−1
dy2nk−1, y2nk
dy2nk, y2nk1
≤εdy2nk−1, y2nk
dy2nk, y2nk1
.
2.11
Thendy2mk, y2nk1→ε. We note
dy2mk, y2nk1
−dy2mk, y2mk1
−dy2nk2, y2nk1
≤dy2mk1, y2nk2
≤dy2mk, y2nk1
dy2mk, y2mk1
dy2nk2, y2nk1
,
2.12
and thusdy2mk1, y2nk2→εask→∞. We have
Mx2nk2, x2mk1
max
dy2mk, y2nk1
, dy2nk1, y2nk2
, dy2mk, y2mk1
,d
y2mk1, y2nk1
dy2mk, y2nk2
2
dy2mk, y2nk1
dk,
2.13
wheredk→0 ask→∞anddk≥0 for allk. Then from
Fdy2mk1, y2nk2
FdAx2nk2, Bx2mk1
≤ψFMx2nk2, x2mk1
ψFdy2mk, y2nk1
dk
,
2.14
ask→∞,Fbeing continuous andψright continuous, we get
Fε≤ψFε< Fε. 2.15
This is a contradiction. Thereforeynis a Cauchy sequence.
Lemma 2.2. LetX, dbe a metric space and letA, B, S, T, F, andψ be as inLemma 2.1. If one of AX, TX, BX,andSXis a complete subspace ofX, then the following hold:
iAandShave a coincidence point;
iiT andBhave a coincidence point.
Proof. Fixx0 ∈Xand letynbe the sequence defined inLemma 2.1. Ify2n y2n−1 for some
for somen, thenAx2n Tx2n1 Bx2n1 Sx2n2, andT andBhave a coincidence point.
Assume thatyn/yn1 for everynandTXis complete. ByLemma 2.1, the sequenceynis
Cauchy; asy2n⊂TX, there existsu∈TXsuch thatyn →u. Letv∈Xbe such thatTvu.
To prove thatBvu. We have
Mx2n, v max
dy2n−1, u
, dy2n, y2n−1
, dBv, u,
dy2n−1, Bv d
y2n, u
2
.
2.16
IfBv /u, thenMx2n, v du, Bvdefinitely and consequently for largen,
FdAx2n, Bv
≤ψFMx2n, v
ψFdu, Bv. 2.17
Fbeing continuous, asn→∞, we obtain
Fdu, Bv≤ψFdu, Bv< Fdu, Bv. 2.18
This is a contradiction, thereforeBv uand v is a coincidence point forT and B. From BX⊂SX, which givesu∈SX, we deduce that there existsw∈Xsuch thatSwu. To prove thatAwu. We have
Mw, v max
du, u, dAw, u, du, u,
du, u dAw, u 2
dAw, u, 2.19
and hence
FdAw, Bv≤ψFMw, uψFdAw, u< FdAw, u, 2.20
which givesAwu.
The same result holds if we suppose that one ofSX, AX, BXis complete.
Theorem 2.3. LetA, B, S, andT be self-mappings of a metric spaceX, dsuch thatAX ⊂ TX, BX⊂SX. Assume that there existF, ψ:0,∞→0,∞such that
iFis nondecreasing, continuous, andF0 0< Ftfor everyt >0;
iiψis nondecreasing, right continuous, andψt< tfor everyt >0;
iiiFdAx, By≤ψFMx, yfor allx, y∈X.
If one ofAX, TX, BX,andSXis a complete subspace ofX, then the following hold:
ivAandShave a coincidence point;
vT andBhave a coincidence point.
Proof. Fix x0 ∈ X and let yn be the sequence defined inLemma 2.1. Assume that TX is
complete and letu, v,andwbe as inLemma 2.2. IfAandSare weakly compatible, then
AuASwSAwSu, 2.21
therefore u is a coincidence point of A and S. To prove that dAu, u 0. Suppose that dAu, u/0. We have
Mu, v max
dSu, u, dAu, Su, du, u,
dSu, u dAu, u 2
dAu, u
FdAu, BvFdAu, u≤ψFMu, vψFdAu, u< FdAu, u.
2.22
This is a contradiction, and thusAu u. SinceAuSu u, we obtain thatuis a common fixed point forAandS.
Similarly, ifBandTare weakly compatible, we deduce thatuis a common fixed point forBandT. Now ifAandSas well asBandT are weakly compatible, thenuis a common fixed point forA, B, S, andT. Ifz∈X is also a common fixed point forA, B, S, andT with u /z, then
FdAu, Bz≤ψFMu, zψFdAu, Bv< FdAu, Bv, 2.23
which givesuz.
Letϕ:0,∞→0,∞be a Lebesgue integrable function which is nonnegative and such that
ε
0
ϕtdt >0, for everyε >0. 2.24
The functionF :0,∞→0,∞, withFs s0ϕtdtsatisfies conditioniofLemma 2.1
and fromTheorem 2.3we deduce the following theorem.
Theorem 2.4see2, Theorem 2.1. LetA, B, S, andTbe self-mappings of a metric spaceX, d such thatAX⊂TX,BX⊂SX. Assume that there exists a nondecreasing right continuous function ψ:0,∞→0,∞, withψt< tfor allt >0, such that
dAx,By
0
ϕtdt≤ψ
Mx,y
0
ϕtdt
, 2.25
whereϕ:0,∞→0,∞is a Lebesgue integrable function which is nonnegative and such that
ε
0
If one ofAX, TX, BX, andSXis a complete subspace ofX, then the following hold:
iAandShave a coincidence point;
iiT andBhave a coincidence point.
Further, ifAandS as well asB andT are weakly compatible, thenA, B, S, and T have a unique common fixed point.
Remark 2.5. Theorem 2.4is a generalization of the main theorem in3, of4, Theorem 2, and of5, Theorem 2.
If inTheorem 2.3, we assumeST IX, whereIXis the identity map onX, we obtain
the following theorem.
Theorem 2.6. LetAandBbe self-mappings of a metric spaceX, d. Assume that there existF, ψ :
0,∞→0,∞such that
iFis nondecreasing, continuous, andF0 0< Ftfor everyt >0;
iiψis nondecreasing, right continuous, andψt< tfor everyt >0;
iiiFdAx, By≤ψFmx, yfor allx, y∈X,
where
mx, y max
dx, y, dAx, y, dBy, y,
dAx, y dx, By 2
. 2.27
If one ofAXandBXis a complete subspace ofX, thenAandShave a unique common fixed point. Moreover, for each x0 ∈ X, the iterated sequence xn with x2n1 Ax2n and x2n2 Bx2n1 converges to the common fixed point ofAandB.
Theorem 2.6includes7, Theorem 1.
Acknowledgment
The authors are supported by the University of PalermoR.S. ex 60%.
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