FENG-LIU TYPE FIXED POINT RESULTS FOR
MULTIVALUED MAPPINGS ON JS-METRIC SPACES
N. AL ARIFI, I. ALTUN, M. JLELI, A. LASHIN, AND B. SAMET
Abstract. In this paper, we present a …xed point theorem for multivalued mappings on generalized metric space in the sense of Jleli and
Samet [11]. In fact, we obtain as a spacial case both b-metric version
and dislocated metric version of Feng-Liu’s …xed point result.
1. Introduction and Preliminaries
Let X be any nonempty set. An element x 2 X is said to be a …xed
point of a multivalued mapping T : X ! P (X) if x 2 T x; where P (X)
denotes the family of all nonempty subsets of X. Let (X; d) be a metric
space. We denote the family of all nonempty closed and bounded subsets of
X by CB(X) and the family of all nonempty closed subsets of X by C(X).
For A; B 2 C(X), let
(
)
H(A; B) = max sup d(x; B); sup d(y; A) ;
x2A
y2B
where d(x; B) = inf fd(x; y) : y 2 Bg. Then H is called generalized PompeiHausdor¤ distance on C(X). It is well known that H is a metric on CB(X),
which is called Pompei-Hausdor¤ metric induced by d. We can …nd detailed
information about the Pompeiu-Hausdor¤ metric in [2, 10].
Let T : X ! CB(X): Then, T is called multivalued contraction if there
exists L 2 [0; 1) such that H(T x; T y)
Ld(x; y) for all x; y 2 X (see
[16]). In 1969, Nadler [16] proved that every multivalued contraction on
complete metric space has a …xed point. Then, the …xed point theory of
multivalued contraction has been further developed in di¤erent directions
by many authors, in particular, by Reich [17], Mizoguchi-Takahashi [15],
Klim-Wardowski [14], Berinde-Berinde [3], Ćirić [4] and many others [5, 6,
12, 18]. Also, Feng and Liu [8] gave the following theorem without using
generalized Pompei-Hausdor¤ distance. To state their result, we give the
following notation for a multivalued mapping T : X ! C(X): let b 2 (0; 1)
and x 2 X de…ne
Ibx (T ) = fy 2 T x : bd(x; y)
d(x; T x)g:
1991 Mathematics Subject Classi…cation. 54H25, 47H10.
Key words and phrases. Fixed point, multivalued mapping, generalized metric space,
b-metric space, dislocated metric space.
1
2
N. AL ARIFI, I. ALTUN, M. JLELI, A. LASHIN, AND B. SAMET
Theorem 1 ([8]). Let (X; d) be a complete metric space and T : X ! C(X):
If there exists a constant c 2 (0; 1) such that there is y 2 Ibx (T ) satisfying
d(y; T y)
cd(x; y);
for all x 2 X: Then T has a …xed point in X provided that c < b and the
function x ! d(x; T x) lower semicontinuous.
As mentioned in Remark 1 of [8], we can see that Theorem 1 is a real
generalization of Nadler’s.
The aim of this paper is to present Feng-Liu type …xed point results for
multivalued mappings on some generalized metric space such as b-metric
spaces and dislocated metric spaces. To do this, we will consider JS-metric
on a nonempty set.
Let X be a nonempty set and D : X X ! [0; 1] be a mapping. For
every x 2 X de…ne a set
C(D; X; x) = ffxn g
X : lim D(xn ; x) = 0g:
n!1
In this case we say that D is a generalized metric in the sense of Jleli and
Samet [11] (for short JS-metric) on X if it satis…es the following conditions:
(D1 ) for every (x; y) 2 X X, D(x; y) = 0 ) x = y;
(D2 ) for every (x; y) 2 X X, D(x; y) = D(y; x);
(D3 ) there exists c > 0 such that for every (x; y) 2 X X and fxn g 2
C(D; X; x)
D(x; y) c lim sup D(xn ; y):
n!1
In this case (X; D) is said to be JS-metric space. Note that, if C(D; X; x) = ;
for all x 2 X, then (D3 ) is trivially hold. The class of JS-metric space is
larger than many known class of metric space. For example every standard
metric space, every b-metric space, every dislocated metric space (in the
sense of Hitzler-Seda [9]) and every modular space with the Fatou property
is a JS-metric space. For more details see [11].
Let (X; D) be a JS-metric space, x 2 X and fxn g be a sequence in
X. If fxn g 2 C(D; X; x), then fxn g is said to be converges to x. If
limn;m!1 D(xn ; xn+m ) = 0, then fxn g is said to be Cauchy sequence. If
every Cauchy sequence in (X; D) is convergent, then (X; D) is said to be
complete. By Proposition 2.4 of [11], we see that every convergent sequence
in (X; D) has a unique limit. That is, if fxn g 2 C(D; X; x) \ C(D; X; y),
then x = y.
After the introducing the JS-metric space, Jleli and Samet [11] presented
some …xed point results including Banach contraction and Ćirić type quasicontraction mappings.
2. Main result
Let (X; D) be a JS-metric space and U X. We say that U is sequentially
open if for each sequence fxn g in X such that limn!1 D(xn ; x) = 0 for some
x 2 U is eventually in U , that is, there exists n0 2 N such that xn 2 U for all
FENG-LIU TYPE FIXED POINT RESULTS
3
n n0 . Let JS be the family of all sequentially open subsets of X, then it
is easy to see that (X; JS ) is a topological space. Further, a sequence fxn g
is convergent to x in (X; D) if and only if it is convergent to x in (X; JS ).
Let C(X) be the family of all nonempty closed subsets of (X; JS ) and let
be the family of all nonempty subsets A of X satisfying the following
property: for all x 2 X,
D(x; A) = 0 ) x 2 A;
where D(x; A) = inffD(x; y) : y 2 Ag. In this case C(X) = . Indeed, let
A 2 C(X) and x 2 X. If D(x; A) = 0, then there exists a sequence fxn g in A
such that limn!1 D(x; xn ) = 0. Therefore, by the de…nition of the topology
JS , for any U 2 JS including the point x, there exists nU 2 N such that
xn 2 U for all n nU . In this case, we have U \ A 6= ;, that is, x 2 A = A.
Hence C(X)
. Now, let A 2 . We will show that A 2 C(X). Let
x 2 XnA and fxn g be a sequence in X such that limn!1 D(xn ; x) = 0. If
there exists a subsequence fxnk g of fxn g such that fxnk g A, then we get
D(x; A) = 0. Since A 2 , then x 2 A. This is a contradiction. Therefore,
there exists n0 2 N such that xn 2 XnA for all n
n0 . This shows that
XnA 2 JS , and so A 2 C(X). As a consequence we get C(X) = .
Now we will consider the following special cases for JS :
Case 1. Let (X; D) be a metric space. Then it is clear that
with the metric topology D .
JS
coincides
Case 2. Let (X; D) be a b-metric space. In this case, there are three topologies on X as follows: First is sequential topology s , which is de…ned as in
De…nition 3.1 (3) of [1]. Second is the D topology [13], which is the family
of all open subsets of X in the usual sense, that is, a subset U of X is open
if for any x 2 U , there exists " > 0 such that
Third is the
D
B(x; ") := fy 2 X : D(x; y) < "g
U:
topology, which the family of all …nite intersections of
C = fB(x; ") : x 2 X; r > 0g
satis…es conditions (B1)-(B2) of ([7], Proposition 1.2.1) is a base of D . By
D . Also by De…nition 2.1
Proposition 3.3 of [1], we know that s = D
and Theorem 3.4 of [1], we can see that JS = s .
Case 3. Let (X; D) be a dislocated metric space in the sense of Hitzler and
Seda [9]. In this case, the set of balls does not in general yield a conventional
topology. However, by de…ning a new membership relation, which is more
general than the classical membership relation from set theory, Hitzler and
Seda [9] constructed a suitable topology on dislocated metric space as follows:
Let X be a set. A relation ^ X P (X) is called d-membership relation
on X if it satis…es the following property: for all x 2 X and A; B 2 P (X),
x^A and A
B implies x^A.
4
N. AL ARIFI, I. ALTUN, M. JLELI, A. LASHIN, AND B. SAMET
Let Ux be a nonempty collection of subsets of X for each x 2 X. If the following conditions are satis…ed, then the pair (Ux ; ^) is called d-neighbourhood
system for x:
i) if U 2 Ux , then x^U;
ii) if U; V 2 Ux , then U \ V 2 Ux ;
iii) if U 2 Ux , then there is V
U with V 2 Ux such that for all y^V we
have U 2 Uy ;
iv) if U 2 Ux and U V , then V 2 Ux :
The d-neighbourhood system (Ux ; ^) generates a topology on X. This
topological space is called d-topological space and indicated as (X; U; ^),
where U = fU x : x 2 Xg.
Now, let (X; D) be a dislocated metric space in the sense of Hitzler and
Seda [9]. De…ne a membership relation ^ as the relation
f(x; A) : there exists " > 0 for which B(x; ")
Ag:
(2.1)
In this case, by Proposition 3.5 of [9], we know that (Ux ; ^) is d-neighbourhood
system for x for each x 2 X, where Ux be the collection of all subsets A of
X such that x^A. By taking into account the De…nition 2.2, De…nition 3.8
and Proposition 3.9 of [9] we can see that the d-topology generated by (2.1)
on (X; D) coincides with the topology JS .
Let (X; D) be a generalized metric space and T : X ! C(X) be a multivalued mapping. For a constant b 2 (0; 1) and x 2 X; we will consider the
following set in our main result:
Ibx (T ) = fy 2 T x : bD(x; y)
D(x; T x)g:
Theorem 2. Let (X; D) be a complete generalized metric space and T :
X ! C(X) be multivalued mapping. Suppose there exists a constant c > 0
such that for any x 2 X there is y 2 Ibx (T ) satisfying
D(y; T y)
cD(x; y):
(2.2)
If there exists x0 2 X such that D(x0 ; T x0 ) < 1, then it can be constructed a sequence fxn g in X satisfying
(i) xn+1 2 T xn ;
(ii) D(xn ; xn+1 ) < 1;
(iii) bD(xn+1 ; xn+2 ) cD(xn ; xn+1 ) and bD(xn+1 ; T xn+1 ) cD(xn ; T xn ):
If this constructed sequence is Cauchy and the function f (x) = D(x; T x)
is lower semicontinuous, then T has a …xed point.
Now consider the following important remarks, before giving the proof of
Theorem 2.
Remark 1. If (X; D) is a metric space (or dislocated metric space in the
sense of Hitzler and Seda [9]) and c < b, then the mentioned sequence in
Theorem 2 is Cauchy. Indeed, since D has triangular inequality, for m; n 2
FENG-LIU TYPE FIXED POINT RESULTS
5
N with m > n, we get from (iii),
D(xn ; xm )
D(xn ; xn+1 ) +
+ D(xm
c n
c
D(x0 ; x1 ) +
+
b
b
(c=b)n
D(x0 ; x1 ):
1 (c=b)
1 ; xm )
m 1
D(x0 ; x1 )
Since c < b, then fxn g is Cauchy sequence.
Remark 2. If (X; D) is a b-metric space with b-metric constant s and
sc < b, then the mentioned sequence in Theorem 2 is Cauchy. Indeed, in
this case we have
D(x; y) s[D(x; z) + D(z; y)]:
Therefore, for m; n 2 N with m > n, we get from (iii),
D(xn ; xm )
sD(xn ; xn+1 ) +
+ sm n D(xm 1 ; xm )
c m 1
c n
s
D(x0 ; x1 ) +
+ sm n
D(x0 ; x1 )
b
b
c n 1 (sc=b)m n
D(x0 ; x1 ):
= s
b
1 (sc=b)
Since sc < b, then fxn g is Cauchy sequence.
Proof of Theorem 2. First observe that, since T x 2 C(X) for all x 2 X,
Ibx (T ) is nonempty. Let x0 2 X be such that D(x0 ; T x0 ) < 1: Then, from
(2.2) there exists x1 2 Ibx0 (T ) such that
D(x1 ; T x1 )
Note that, since x1 2
Ibx0 (T ),
then x1 2 T x0 and
bD(x0 ; x1 )
For x1 2 X, there exists x2 2
cD(x0 ; x1 ):
D(x0 ; T x0 ) < 1:
Ibx1 (T )
D(x2 ; T x2 )
such that
cD(x1 ; x2 ):
By the way, we can construct a sequence fxn g in X such that xn+1 2 Ibxn (T )
and
D(xn+1 ; T xn+1 ) cD(xn ; xn+1 )
(2.3)
for all n 2 N. Note that, since D(x0 ; T x0 ) < 1, then D(xn ; xn+1 ) < 1 for
all n 2 N.
Again, since xn+1 2 Ibxn (T ), we have xn+1 2 T xn and
bD(xn ; xn+1 )
D(xn ; T xn )
for all n 2 N. Therefore from (2.3) and (2.4), we get
bD(xn+1 ; xn+2 )
D(xn+1 ; T xn+1 )
and
D(xn+1 ; T xn+1 )
cD(xn ; xn+1 )
cD(xn ; xn+1 )
c
D(xn ; T xn ):
b
(2.4)
(2.5)
(2.6)
6
N. AL ARIFI, I. ALTUN, M. JLELI, A. LASHIN, AND B. SAMET
Hence (i), (ii) and (iii) hold. Furthermore, from (2.5) and (2.6), we get
lim D(xn ; xn+1 ) = lim D(xn ; T xn ) = 0:
n!1
n!1
Now, if fxn g is Cauchy sequence then by the completeness of (X; D),
there exists z 2 X such that xn 2 C(D; X; z), that is limn!1 D(xn ; z) = 0.
Therefore, by the lower semicontinuity of the function f (x) = D(x; T x), we
get
0
D(z; T z) = f (z)
lim inf f (xn ) = lim inf D(xn ; T xn ) = 0:
n!1
n!1
Since T z 2 C(X), we get z 2 T z.
By taking into account Remark 1 and Remark 2, we obtain the following
results from Theorem 2.
Corollary 1 (Feng-Liu’s …xed point theorem). Let (X; d) be a complete
metric space and T : X ! C(X) be multivalued mapping. Suppose there
exists a constant c > 0 such that for any x 2 X there is y 2 Ibx (T ) satisfying
d(y; T y)
cd(x; y):
Then T has a …xed point provided that c < b and the function f (x) =
d(x; T x) is lower semicontinuous.
Corollary 2 (Feng-Liu’s …xed point theorem on b-metric space). Let (X; d)
be a complete b-metric space with b-metric constant s and T : X ! C(X)
be multivalued mapping. Suppose there exists a constant c > 0 such that for
any x 2 X there is y 2 Ibx (T ) satisfying
d(y; T y)
cd(x; y):
Then T has a …xed point provided that sc < b and the function f (x) =
d(x; T x) is lower semicontinuous.
Corollary 3 (Feng-Liu’s …xed point theorem on dislocated metric space).
Let (X; d) be a complete dislocated metric space and T : X ! C(X) be
multivalued mapping. Suppose there exists a constant c > 0 such that for
any x 2 X there is y 2 Ibx (T ) satisfying
d(y; T y)
cd(x; y):
Then T has a …xed point provided that c < b and the function f (x) =
d(x; T x) is lower semicontinuous.
Acknowledgement 1. The authors extend their appreciation to Distinguished Scientist Fellowship Program (DSFP) at King Saud University (Saudi
Arabia).
FENG-LIU TYPE FIXED POINT RESULTS
7
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8
N. AL ARIFI, I. ALTUN, M. JLELI, A. LASHIN, AND B. SAMET
King Saud University, College of Science, Geology and Geophysics Department, P.O. Box 2455, Riyadh 11451, Saudi Arabia
E-mail address: nalarifi@ksu.edu.sa
Adress 1: King Saud University, College of Science, Riyadh, Saudi Arabia,
Adress 2: Department of Mathematics, Faculty of Science and Arts, Kirikkale
University, 71450 Yahsihan, Kirikkale, Turkey
E-mail address: ishakaltun@yahoo.com
Department of Mathematics, College of Science, King Saud University,
P.O. Box 2455, Riyadh, 11451, Saudi Arabia
E-mail address: jleli@ksu.edu.sa
Address 1: King Saud University, College of Engineering, Petroleum and
Gas Engineering Department, P.O. Box 800, Riyadh 11421, Saudi Arabia, Address 2: Benha University, Faculty of Science, Geology Department, P.O. Box
13518, Benha, Egypt
E-mail address: arlashin@ksu.edu.sa
Department of Mathematics, College of Science, King Saud University,
P.O. Box 2455, Riyadh, 11451, Saudi Arabia
E-mail address: bsamet@ksu.edu.sa