Volume 2011, Article ID 178306,8pages doi:10.1155/2011/178306
Research Article
Fixed Point Results in Quasimetric Spaces
Abdul Latif and Saleh A. Al-Mezel
Department of Mathematics, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia
Correspondence should be addressed to Abdul Latif,[email protected] Received 21 August 2010; Accepted 5 October 2010
Academic Editor: Qamrul Hasan Ansari
Copyrightq2011 A. Latif and S. A. Al-Mezel. 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.
In the setting of quasimetric spaces, we prove some new results on the existence of fixed points for contractive type maps with respect toQ-function. Our results either improve or generalize many known results in the literature.
1. Introduction and Preliminaries
LetXbe a metric space with metricd. We useSXto denote the collection of all nonempty subsets of X, ClX for the collection of all nonempty closed subsets of X, CBX for the collection of all nonempty closed bounded subsets of X, and H for the Hausdorff metric onCBX,that is,
HA, B max
sup
a∈Ada, B,sup
b∈B db, A
, A, B∈CBX, 1.1
whereda, B inf{da, b:b∈B}is the distance from the pointato the subsetB.
For a multivalued mapT:X → CBX, we say
aT is contraction1if there exists a constantλ∈0,1, such that for allx, y∈X, H
Tx, T y
≤λd x, y
, 1.2
bT is weakly contractive 2if there exist constantsh, b ∈0,1, h < b, such that for anyx∈X, there isy∈Ibxsatisfying
d y, T
y
≤hd x, y
, 1.3
whereIbx{y∈Tx:bdx, y≤dx, Tx}.
A pointx∈Xis called a fixed point of a multivalued mapT :X → SXifx∈Tx.
We denote FixT {x∈X:x∈Tx}.
A sequence{xn}inX is called anorbitofT atx0 ∈ X ifxn ∈ Txn−1for all integer n≥1. A real valued functionfonXis called lower semicontinuous if for any sequence{xn} ⊂X withxn → x∈Ximplies thatfx≤lim infn→ ∞fxn.
Using the Hausdorffmetric, Nadler Jr.1has established a multivalued version of the well-known Banach contraction principle in the setting of metric spaces as follows.
Theorem 1.1. LetX, dbe a complete metric space, then each contraction mapT : X → CBX has a fixed point.
Without using the Hausdorff metric, Feng and Liu 2 generalized Nadler’s contraction principle as follows.
Theorem 1.2. LetX, dbe a complete metric space and letT :X → ClXbe a weakly contractive map, thenT has a fixed point inX provided the real valued functionfx dx, TxonX is a lower semicontinuous.
In3, Kada et al. introduced the concept ofw-distance in the setting of metric spaces as follows.
A functionω:X×X → 0,∞is called aw-distance onXif it satisfies the following:
w1ωx, z≤ωx, y ωy, z,for allx, y, z∈X;
w2ωis lower semicontinuous in its second variable;
w3for any ε > 0, there existsδ > 0, such that ωz, x ≤ δ and ωz, y ≤ δ imply dx, y≤ε.
Note that in general forx, y ∈X,ωx, y/ωy, xand not either of the implications ωx, y 0⇔xynecessarily holds. Clearly, the metricdis aw-distance onX. Many other examples and properties ofw-distances are given in3.
In4, Suzuki and Takahashi improved Nadler contraction principleTheorem 1.1as follows.
Theorem 1.3. LetX, dbe a complete metric space and let T : X → ClX. If there exist aw- distance ω onX and a constantλ ∈ 0,1, such that for eachx, y ∈ X and u ∈ Tx, there is v∈Tysatisfying
ωu, v≤λω x, y
, 1.4
thenT has a fixed point.
Recently, Latif and Albar5generalizedTheorem 1.2with respect tow-distancesee, Theorem 3.3 in5, and Latif6proved a fixed point result with respect tow-distancesee, Theorem 2.2 in6which containsTheorem 1.3as a special case.
A nonempty set X together with a quasimetric d i.e., not necessarily symmetric is called a quasimetric space. In the setting of a quasimetric spaces, Al-Homidan et al.7 introduced the concept of aQ-function on quasimetric spaces which generalizes the notion of aw-distance.
A functionq:X×X → 0,∞is called aQ-function onXif it satisfies the following conditions:
Q1qx, z≤qx, y qy, z,for allx, y, z∈X;
Q2If{yn}is a sequence inXsuch thatyn → y∈ Xand forx∈X,qx, yn ≤Mfor someMMx>0, thenqx, y≤M,
Q3for any ε > 0,there exists δ > 0, such that qx, y ≤ δ and qx, z ≤ δ imply dy, z≤ε.
Note that everyw-distance is aQ-function, but the converse is not true in general7.
Now, we state some useful properties ofQ-function as given in7.
Lemma 1.4. LetX, dbe a complete quasimetric space and letqbe aQ-function onX. Let{xn}and {yn}be sequences inX. Let{αn}and{βn}be sequences in0,∞converging to 0, then the following hold for anyx, y, z∈X:
iifqxn, y≤αnandqxn, z≤βnfor alln≥1,theny z; in particular, ifqx, y 0 andqx, z 0, thenyz;
iiifqxn, yn≤αnandqxn, z≤βnfor alln≥1,then{yn}converges toz;
iiiifqxn, xm≤αnfor anyn, m≥1 withm > n,then{xn}is a Cauchy sequence;
ivifqy, xn≤αnfor anyn≥1,then{xn}is a Cauchy sequence.
Using the conceptQ-function, Al-Homidan et al.7recently studied an equilibrium version of the Ekeland-type variational principle. They also generalized Nadler’s fixed point theoremTheorem 1.1in the setting of quasimetric spaces as follows.
Theorem 1.5. LetX, dbe a complete quasimetric space and letT : X → ClX. If there exist Q-functionqonX and a constantλ ∈ 0,1, such that for eachx, y ∈ X andu ∈ Tx, there is v∈Tysatisfying
qu, v≤λq x, y
, 1.5
thenT has a fixed point.
In the sequel, we considerXas a quasimetric space with quasimetricd.
Considering a multivalued mapT :X → SX, we say
cTis weaklyq-contractive if there existQ-functionqonXand constantsh, b∈0,1, h < b, such that for anyx∈X, there isy∈Jbxsatisfying
q y, T
y
≤hq x, y
, 1.6
whereJbx{y∈Tx:bqx, y≤qx, Tx}andqx, Tx inf{qx, y:y∈Tx};
dT is generalizedq-contractive if there exists aQ-functionqonX, such that for each x, y∈Xandu∈Tx, there isv∈Tysatisfying
qu, v≤k q
x, y q
x, y
, 1.7
wherekis a function of0,∞to0,1, such that lim supr→tkr<1 for allt≥0.
Clearly, the class of weakly q-contractive maps contains the class of weakly contractive maps, and the class of generalized q-contractive maps contains the classes of generalized ω-contraction maps6,ω-contractive maps4, andq-contractive maps7.
In this paper, we prove some new fixed point results in the setting of quasimetric spaces for weakly q-contractive and generalized q-contractive multivalued maps. Conse- quently, our results either improve or generalize many known results including the above stated fixed point results.
2. The Results
First, we prove a fixed point theorem for weakly q-contractive maps in the setting of quasimetric spaces.
Theorem 2.1. Let X be a complete quasimetric space and let T : X → ClX be a weakly q- contractive map. If a real valued function fx qx, Txon X is lower semicontinuous, then there existsvo∈X, such thatqvo, Tvo 0.Further, ifqvo, vo 0,thenv0 is a fixed point of T.
Proof. Letxo∈X.SinceT is weakly contractive, there isx1∈Jbxo ⊆Txo, such that
qx1, Tx1≤hqxo, x1, 2.1
whereh < b.Continuing this process, we can get an orbit{xn}ofT atxosatisfyingxn1∈Jbxn and
qxn1, Txn1≤hxn, xn1, n0,1,2, . . . . 2.2
Sincebqxn, xn1≤qxn, Txnandh < b <1,thus we get
qxn1, Txn1≤qxn, Txn. 2.3
If we putah/b, then also we have
qxn1, Txn1≤aqxn, Txn. 2.4
Thus, we obtain
qxn, Txn≤anqxo, Tx0, n0,1,2, . . . , 2.5
and since 0 < a < 1, hence the sequence {fxn} {qxn, Txn}, which is decreasing, converges to 0. Now, we show that{xn}is a Cauchy sequence. Note that
qxn, xn1≤anqxo, x1, n0,1,2, . . . . 2.6
Now, for any integern, m≥1 withm > n, we have
qxn, xm≤qxn, xn1 qxn1, xn2 · · ·qxm−1, xm
≤anqxo, x1 an1qxo, x1 · · ·am−1qxo, x1
≤ an
1−a qxo, x1,
2.7
and thus byLemma 1.4,{xn}is a Cauchy sequence. Due to the completeness ofX, there exists somev0∈X, such that limn→ ∞xnvo.Now, sincefis lower semicontinuous, we have
0≤fvo≤lim inf
n→ ∞ fxn 0, 2.8
and thus, fvo qvo, Tvo 0.It follows that there exists a sequence {vn} in Tv0, such thatqv0, vn → 0.Now, ifqvo, vo 0,then byLemma 1.4,vn → v0. SinceTv0is closed, we getv0∈Tv0.
Now, we prove the following useful lemma.
Lemma 2.2. LetX, dbe a complete quasimetric space and letT : X → ClXbe a generalized q-contractive map, then there exists an orbit{xn}ofT atx0, such that the sequence of nonnegative numbers{qxn, xn1}is decreasing to zero and{xn}is a Cauchy sequence.
Proof. Letxobe an arbitrary but fixed element ofXand letx1 ∈Tx0. SinceT is generalized as aq-contractive, there isx2∈Tx1, such that
qx1, x2 ≤k
qxo, x1
qxo, x1. 2.9
Continuing this process, we get a sequence{xn}inX, such thatxn1 ∈Txnand qxn, xn1≤k
qxn−1, xn
qxn−1, xn. 2.10
Thus, for alln≥1, we have
qxn, xn1< qxn−1, xn. 2.11
Writetnqxn, xn1. Suppose that limn→ ∞tnλ >0, then we have
tn≤ktn−1tn−1. 2.12
Now, taking limits asn → ∞on both sides, we get λ≤lim sup
n→ ∞ ktn−1λ < λ, 2.13
which is not possible, and hence the sequence of nonnegative numbers {tn}, which is decreasing, converges to 0. Finally, we show that {xn} is a Cauchy sequence. Let α lim supr→0kr < 1. There exists real numberβsuch thatα < β < 1. Then for sufficiently largen,ktn < β, and thus for sufficiently largen, we havetn < βtn−1.Consequently, we obtaintn< βnt0, that is,
qxn, xn1< βnqxo, x1, n0,1,2, . . . . 2.14
Now, for any integersn, m≥1, m > n,
qxn, xm≤qxn, xn1 qxn1, xn2 · · ·qxm−1, xm
< βnqxo, x1 βn1qxo, x1 · · ·βm−1qxo, x1
< βn
1−βqxo, x1,
2.15
and thus byLemma 1.4,{xn}is a Cauchy sequence.
ApplyingLemma 2.2, we prove a fixed point result for generalizedq-contractive maps.
Theorem 2.3. LetX, dbe a complete quasimetric space then each generalized q -contractive map T :X → ClXhas a fixed point.
Proof. It follows fromLemma 2.2that there exists a Cauchy sequence{xn}inXsuch that the decreasing sequence{qxn, xn1}converges to 0. Due to the completeness ofX, there exists somev0 ∈X such that limn→ ∞xn vo.Letnbe arbitrary fixed positive integer then for all positive integersmwithm > n, we have
qxn, xm≤ βn
1−βqxo, x1. 2.16
LetM βn/1−βqx0, x1, thenM≥0. Now, note that
qxn, xm≤M⇒qxn, v0≤M. 2.17
Sincenwas arbitrary fixed, we have
qxn, v0≤ βn
1−βqxo, x1, for all positive integern. 2.18
Note that qxn, vo converges to 0. Now, since xn ∈ Txn−1 and T is a generalized q- contractive map, then there isun∈Tv0, such that
qxn, un≤k
qxn−1, v0
qxn−1, v0. 2.19
And for largen, we obtain
qxn, un≤k
qxn−1, v0
qxn−1, v0< βqxn−1, v0, 2.20
thus, we get
qxn, un< βqxn−1, v0≤ βn
1−βqxo, x1. 2.21
Thus, it follows fromLemma 1.4thatun → v0. SinceTv0is closed, we getv0∈Tv0. Corollary 2.4. LetX, dbe a complete quasimetric space andqaQ-function onX. LetT :X → ClXbe a multivalued map, such that for anyx, y∈Xandu∈Tx, there isv∈Tywith
qu, v≤k q
x, y q
x, y
, 2.22
wherekis a monotonic increasing function from0,∞to0,1, thenThas a fixed point.
Finally, we conclude with the following remarks concerning our results related to the known fixed point results.
Remark 2.5. 1Theorem 2.1generalizesTheorem 1.2according to Feng and Liu2and Latif and Albar5, Theorem 3.3.
2Theorem 2.3generalizes Theorem 1.3according to Suzuki and Takahashi4and Theorem 1.5according to Al-Homidan et al.7and contains Latif’s Theorem 2.2 in6.
3Theorem 2.3also generalizes Theorem 2.1 in8in several ways.
4Corollary 2.4improves and generalizes Theorem 1 in9.
Acknowledgments
The authors thank the referees for their kind comments. The authors also thank King Abdulaziz University and the Deanship of Scientific Research for the research Grant no. 3- 35/429.
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