R E S E A R C H
Open Access
Fixed point theorems for generalized
contractive type multivalued maps
Buthinah A Bin Dehaish
1*and Abdul Latif
2*Correspondence:
1Department of Mathematics
Faculty of Science For Girls, King Abdulaziz University, P.O. Box 53909, Jeddah, 21593, Saudi Arabia Full list of author information is available at the end of the article
Abstract
Without using the concept of Hausdorff metric, we prove some results on the existence of fixed points for generalized contractive multivalued maps with respect to u-distance. Consequently, several known fixed point results are either generalized or improved.
MSC: 47H10; 54H25
Keywords: metric space;u-distance; contractive map; fixed point
1 Introduction
Using the concept of Hausdorff metric, Nadler [] introduced the notion of multivalued contraction maps and proved a multivalued version of the well-known Banach contraction principle, which states that each closed bounded valued contraction map on a complete metric space has a fixed point. Since then various fixed point results concerning multival-ued contractions have appeared.
Without using the concept of Hausdorff metric, most recently Feng and Liu [] extended Nadler’s fixed point result, while Klim and Wardowski [] generalized their corresponding fixed point result in [].
In [] Kadaet al.introduced the notion ofw-distance and improved several classical re-sults including Caristi’s fixed point theorem. Suzuki and Takahashi [] introduced single-valued and multisingle-valued weakly contractive maps with respect tow-distance and proved fixed point results for such maps. Consequently, they generalized the Banach contraction principle and Nadler’s fixed point result. Generalizing the concept ofw-distance, Suzuki [] introduced the notion ofτ-distance on a metric space and improved several classi-cal results including the corresponding results of Suzuki and Takahashi []. In literature, several other kinds of distances and various versions of known results have appeared. For example, see [–, ] and references therein. Most recently, Ume [] generalized the notion ofτ-distance by introducing the concept ofu-distance.
In this paper, first we prove our key lemma for multivalued general contractive maps with respect tou-distance and then prove some results on the existence of fixed points for such multivalued maps. Consequently, several known fixed point results get either improved or generalized including the corresponding results of Feng and Liu [], Klim and Wardowski [], Latif and Albar [], Suzuki and Takahashi [], Latif and Abdou [], and Nadler [].
Let (X,d) be a metric space. We denote the collection of nonempty subsets of X, nonempty closed subsets ofXand nonempty closed bounded subsets ofXby X,Cl(X)
andCB(X) respectively. LetHbe the Hausdorff metric with respect tod, that is,
H(A,B) =max
sup
x∈A
d(x,B),sup
y∈B
d(y,A)
,
for everyA,B∈CB(X), whered(x,B) =infy∈Bd(x,y).
A pointx∈Xis called a fixed point ofT:X→Xifx∈T(x). We denoteFix(T) ={x∈
X:x∈T(x)}.
A sequence{xn}inXis called anorbitofTatx∈Xifxn∈T(xn–) for alln≥. A map
φ:X→Ris calledT-orbitally lower semicontinuousif, for any orbit{xn}ofTandx∈X,
xn→ximply thatφ(x)≤lim infn→∞φ(xn).
Most recently, Ume [] generalized the notion ofτ-distance by introducingu-distance as follows.
A functionp:X×X→R+ is called a u-distance onX if there exists a functionθ :
X×X×R+×R+→R+such that the following hold forx,y,z∈X:
(u) p(x,z)≤p(x,y) +p(y,z);
(u) θ(x,y, , ) = andθ(x,y,s,t)≥min{s,t}for eachs,t∈R+, and for every> , there existsδ> such that|s–s|<δ,|t–t|<δ,s,s,t,t∈R+andy∈Ximply
θ(x,y,s,t) –θ(x,y,s,t)<; ()
(u)
lim
n→∞xn=x,
lim
n→∞sup
θwn,zn,p(wn,xm),p(zn,xm)
:m≥n=
()
imply
p(y,x)≤lim inf
n→∞ p(y,xn); ()
(u)
lim
n→∞sup
p(xn,wm) :m≥n
= ,
lim
n→∞sup
p(yn,zm) :m≥n
= ,
lim
n→∞θ(xn,wn,sn,tn) = , lim
n→∞θ(yn,zn,sn,tn) =
()
imply
lim
or
lim
n→∞sup
p(wn,xm) :m≥n
= ,
lim
n→∞sup
p(zm,yn) :m≥n
= ,
lim
n→∞θ(xn,wn,sn,tn) = , lim
n→∞θ(yn,zn,sn,tn) =
()
imply
lim
n→∞θ(wn,zn,sn,tn) = ; ()
(u)
lim
n→∞θ
wn,zn,p(wn,xn),p(zn,xn)
= ,
lim
n→∞θ
wn,zn,p(wn,yn),p(zn,yn)
=
()
imply
lim
n→∞d(xn,yn) = ; ()
or
lim
n→∞θ
an,bn,p(xn,an),p(xn,bn)
= ,
lim
n→∞θ
an,bn,p(yn,an),p(yn,bn)
=
()
imply
lim
n→∞d(xn,yn) = . ()
Remark .[]
(a) Suppose thatθ fromX×X×R+×R+intoR+is a mapping satisfying(u)(u). Then there exists a mappingηfromX×X×R+×R+intoR+such thatηis nondecreasing in its third and fourth variable respectively, satisfying(u)η(u)η, where(u)η(u)ηstand for substitutingηforθ in(u)(u)respectively. (b) In the light of (a), we may assume thatθ is nondecreasing in its third and fourth
variables, respectively, for a functionθ fromX×X×R+×R+intoR+satisfying
(u)(u).
(c) Eachτ-distancepon a metric space(X,d)is also au-distance onX.
We present some examples ofu-distances which are notτ-distances. For details, see [].
Example . LetX=R+with the usual metric. Definep:X×X→R+byp(x,y) = ()x.
Example . LetXbe a normed space with norm · . Then a functionp:X×X→R+
defined byp(x,y) =xfor everyx,y∈Xis au-distance onXbut not aτ-distance.
It follows from the above examples and Remark .(c) thatu-distance is a proper exten-sion ofτ-distance. Other useful examples onu-distance are also given in [].
Let (X,d) be a metric space and let pbe a u-distance onX. A sequence{xn} inXis
calledp-Cauchy [] if there exists a functionθ fromX×X×R+×R+intoR+satisfying
(u)(u) and a sequence{zn}ofXsuch that
lim
n→∞sup
θzn,zn,p(zn,xm),p(zn,xm)
:m≥n= , ()
or
lim
n→∞sup
θzn,zn,p(xm,zn),p(xm,zn)
:m≥n= . ()
The following lemmas concerningu-distance are crucial for our results.
Lemma .[] Let(X,d)be a metric space and let p be a u-distance on X. If{xn}is a
p-Cauchy sequence in X, then{xn}is a Cauchy sequence.
Lemma . [] Let(X,d)be a metric space and let p be a u-distance on X. If{xn} is a
p-Cauchy sequence and{yn}is a sequence satisfying
lim
n→∞sup
p(xn,ym) :m≥n
= ,
then{yn}is also a p-Cauchy sequence and d(xn,yn) = .
Lemma .[] Let(X,d)be a metric space and let p be a u-distance on X. Suppose that a sequence{xn}of X satisfies
lim
n→∞sup
p(xn,xm) :m>n
= , ()
or
lim
n→∞sup
p(xm,xn) :m>n
= . ()
Then{xn}is a p-Cauchy sequence.
We say that a multivalued mapT:X→Xis generalizedp-contractive if there exist a
u-distanceponXand a constantb∈(, ) such that, for anyx∈X, there isy∈Jbxsatisfying py,T(y)≤kp(x,y)p(x,y),
whereJx
b={y∈T(x) :bp(x,y)≤p(x,T(x))}andkis a function from [,∞) to [,b) with
lim supr→t+k(r) <bfor eacht∈[,∞).
In particular, if we takeu=dand a constant mapk=h<b,h∈(, ), then the generalized p-contractive mapTreduces to the definition of contractive maps due to Feng and Liu [].
2 The results
First, we prove our key lemma in the setting of metric spaces.
Lemma . Let(X,d)be a metric space. Let T:X→Cl(X)be a generalized p-contractive map. Then, there exists an orbit{un}of T at u∈X such that the sequence of nonnegative
real numbers{p(un,T(un))}is decreasing to zero and{un}is a Cauchy sequence.
Proof SinceT(x)∈Cl(X) for anyx∈X,Jbxis nonempty for any constantb∈(, ). Letu
be an arbitrary but fixed element ofX, there existsu∈Jbu⊆T(u) such that
pu,T(u)
≤kp(u,u)
p(u,u), k
p(u,u)
<b, ()
bp(u,u)≤p
u,T(u)
. ()
Using () and (), we have
pu,T(u)
–pu,T(u)
≥bp(u,u) –k
p(u,u)
p(u,u)
= b–kp(u,u)
p(u,u) > . ()
Similarly, there isu∈Jbu⊆T(u) such that
pu,T(u)
≤kp(u,u)
p(u,u), k
p(u,u)
<b, ()
bp(u,u)≤p
u,T(u)
. ()
Using () and (), we have
pu,T(u)
–pu,T(u)
≥bp(u,u) –k
p(u,u)
p(u,u)
= b–kp(u,u)
p(u,u) > . ()
From () and (), it follows that
p(u,u)≤
bp
u,T(u)
≤
bk
p(u,u)
p(u,u)≤p(u,u). ()
Continuing this process, we get an orbit{un}ofT inXsuch thatun+∈Jbun⊆T(un),
bp(un,un+)≤p
un,T(un)
, ()
pun+,T(un+)
≤kp(un,un+)
p(un,un+), k
p(un,un+)
<b. ()
Using () and (), we get
pun,T(un)
–pun+,T(un+)
≥bp(un,un+) –k
p(un,un+)
p(un,un+)
= b–kp(un,un+)
and thus for eachn
pun,T(un)
>pun+,T(un+)
, ()
p(un,un+)≤p(un–,un). ()
Note that the sequences{p(un,T(un))}and{p(un,un+)}are decreasing and bounded, thus
convergent. Now, by definition of the functionkthere existsα∈[,b) such that
lim sup
n→∞
kp(un,un+)
=α. ()
Thus for anyb∈(α,b), there existsn∈Nsuch that for eachn>n
kp(un,un+)
<b, ()
and thus for eachn>n, we have
kp(un,un+)
× · · · ×kp(un+,un+)
<bn–n
. ()
Also, it follows from () that for eachn>n,
pun,T(un)
–pun+,T(un+)
≥βp(un,un+), ()
whereβ=b–b. Using () and (), for eachn>n, we have
pun+,T(un+)
≤kp(un,un+)
p(un,un+)
≤
bk
p(un,un+)
pun,T(un)
≤
bk
p(un,un+)
kp(un–,un)
pun–,T(un–)
.. .
≤
bn k
p(un,un+)
× · · · ×kp(u,u)
pu,T(u)
= k(p(un,un+))× · · · ×k(p(un+,un+)) bn–no
×k(p(un,un+))× · · · ×k(p(u,u))p(u,T(u))
bn ,
and thus
pun+,T(un+)
<
b
b
n–nk(p(u
n,un+))× · · · ×k(p(u,u))p(u,T(u))
bn . ()
Now, since b <b, we have limn→∞(bb)n–n = , and hence the decreasing sequence
{p(un,T(un))}converges to . Now, we show that{un}is a Cauchy sequence. Note that
for eachn>n
whereγ =b
b < . Thus, for anyn,m∈N,m≥n>n, we get
p(un,um)≤ m–
j=n
p(uj,uj+)
≤γn+γn++· · ·+γm–p(u,u)
≤ γn
–γp(u,u),
()
and hence
lim
n→∞sup
p(un,um) :m≥n
= .
Thus, by Lemma ., {un}is a p-Cauchy sequence, and hence by Lemma .,{un} is a
Cauchy sequence.
Applying Lemma ., we obtain the following main fixed point result.
Theorem . Let(X,d)be a complete metric space and let T:X→Cl(X)be a generalized p-contractive map. Suppose that a real valued function f on X defined by f(u) =p(u,T(u)) is T -orbitally lower semicontinuous. Then there exists v∈X such that v∈Fix(T)and
p(v,v) = .
Proof By Lemma ., there exists a Cauchy sequence{un}inXsuch that the decreasing
sequence{f(un)}={p(un,T(un))}converges to . Due to the completeness ofX, there
ex-ists somev∈Xsuch thatlimn→∞un=v. Sincef isT-orbitally lower semicontinuous,
we have
≤f(v)≤lim inf
n→∞ f(un) = , ()
that is, f(v) = p(v,T(v)) = . Thus there exists a sequence {vn} ⊂T(v) such that limn→∞p(v,vn) = . It follows that
≤ lim
n→∞sup
p(un,vm) :m≥n
≤ lim
n→∞sup
p(un,v) +p(v,vm) :m≥n
= . ()
Since{un}is ap-Cauchy sequence, it follows from () and Lemma . that{vn}is also
ap-Cauchy sequence andlimn→∞d(un,vn) = . Thus, by Lemma .,{vn}is a Cauchy
se-quence in the complete space. Due to the closedness ofT(v), there existsz∈Xsuch that limn→∞vn=z∈T(v). Consequently, using (u) we get
p(v,z)≤lim inf
n→∞ p(v,vn)≤,
and thusp(v,z) = . But, sincelimn→∞un=v,limn→∞vn=zandlimn→∞d(un,vn) = ,
we havev=z. Hencev∈Fix(T) andp(v,v) = .
Corollary . Let(X,d)be a complete metric space and let T:X→Cl(X)be a multi-valued map such that a real-multi-valued function f on X defined by f(x) =d(x,T(x))is lower semi-continuous. If there exists b∈(, )such that for any x∈X there is y∈Ibxsatisfying
dy,T(y)≤kd(x,y)d(x,y),
where Ibx={y∈T(x) :bd(x,y)≤d(x,T(x))}; and k is a function from[,∞)to[,b)with
lim supr→t+k(r) <b, for every t∈[,∞). ThenFix(T)=∅.
Remarks .
(a) Theorem . also generalizes fixed point theorems of Latif and Abdou [, Theorem .], Suzuki [, Theorem ], Suzuki and Takahashi [, Theorem ]. (b) It is worth mentioning that in the proofs of [, Theorem .], [, Theorem .], and
[, Theorem .] a full force of the lower semicontinuity of the real-valued function
f is not used, but in factT-orbitally lower semicontinuity off is enough to obtain the conclusions.
Applying Lemma ., we also obtain the following fixed point result for a multivalued generalizedp-contractive map where we use another suitable condition.
Theorem . Let X be a complete metric space with metric d and let T:X→Cl(X)be a generalized p-contractive map. Assume that
infp(u,v) +pu,T(u):u∈X> , ()
for every v∈X with v∈/T(v). ThenFix(T)=φ.
Proof By Lemma ., there exists an orbit{un}ofTwhich is a Cauchy sequence in a
com-plete metric spaceX, so there exists somev∈Xsuch thatlimn→∞un=v. Thus, using
(u) and (), we have
p(un,v)≤lim inf
m→∞p(un,um)≤
γn
–γp(u,u), ()
whereγ =b
b < . Also, we get
pun,T(un)
≤p(un,un+)≤γnp(u,u). ()
Assume thatv∈/T(v). Then we have
< infp(u,v) +p
u,T(u):u∈X
≤infp(un,v) +p
un,T(un)
:n>n
≤inf
γn
–γp(u,u) +γ
np(u
,u) :n>n
=
–γ
–γp(u,u)
infγn:n>n
= ,
Remarks . Theorem . generalizes [, Theorem .] and [, Theorem .].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally. The authors read and approved the final manuscript.
Acknowledgements
The authors thank the referees for their valuable comments. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No.9-843-D1432. The authors, therefore, acknowledge with thanks technical and financial support of DSR.
Author details
1Department of Mathematics Faculty of Science For Girls, King Abdulaziz University, P.O. Box 53909, Jeddah, 21593, Saudi
Arabia. 2Department of Mathematics Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi
Arabia.
Received: 6 April 2012 Accepted: 6 August 2012 Published: 31 August 2012 References
1. Al-Homidan, S, Ansari, QH, Yao, J-C: Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory. Nonlinear Anal.69, 126-139 (2008)
2. Ansari, QH: Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory. J. Math. Anal. Appl.334, 561-575 (2007)
3. Ansari, QH (ed.): Topics in Nonlinear Analysis and Optimization. World Education, Delhi (2012)
4. Feng, Y, Liu, S: Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings. J. Math. Anal. Appl.317, 103-112 (2006)
5. Hirunworkakit, S, Petrot, N: Some fixed point theorem for contractive multi-valued mappings induced by generalized distance in metric spaces. Fixed Point Theory Appl.2011, 78 (2011). doi:10.116/1687-1812-2011-78
6. Kada, O, Susuki, T, Takahashi, W: Nonconvex minimization theorems and fixed point theorems in complete metric spaces. Math. Jpn.44, 381-391 (1996)
7. Klim, D, Wardowski, D: Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl.334, 132-139 (2007)
8. Latif, A, Abdou, AAN: Fixed points of generalized contractive maps. Fixed Point Theory Appl.2009, Article ID 487161 (2009)
9. Latif, A, Albar, WA: Fixed point results in complete metric spaces. Demonstr. Math.41, 145-150 (2008)
10. Lin, L-J, Wang, S-Y, Ansari, QH: Critical point theorems and Ekeland type variational principle with applications. Fixed Point Theory Appl.2011, Article ID 914624 (2011)
11. Nadler, SB: Multivalued contraction mappings. Pac. J. Math.30, 475-488 (1969)
12. Suzuki, T, Takahashi, W: Fixed point theorems and characterizations of metric completeness. Topol. Methods Nonlinear Anal.8, 371-382 (1996)
13. Suzuki, T: Generalized distance and existence theorems in complete metric spaces. J. Math. Anal. Appl.253, 440-458 (2001)
14. Ume, JS: Existence theorems for generalized distance on complete metric space. Fixed Point Theory Appl.2010, Article ID 397150 (2010)
doi:10.1186/1687-1812-2012-135
