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 multi- valued mappings on generalized metric space in the sense of Jleli and Samet [11]. In fact, we obtain as a spacial case bothb-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 ofX byC(X).
ForA; B 2C(X), let
H(A; B) = max (
sup
x2A
d(x; B);sup
y2B
d(y; A) )
;
whered(x; B) = inffd(x; y) :y 2Bg. ThenH is called generalized Pompei- Hausdor¤ distance onC(X). It is well known thatH is a metric onCB(X), which is called Pompei-Hausdor¤ metric induced byd. 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], ´Ciri´c [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 b2(0;1) and x2X de…ne
Ibx(T) =fy2T 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
Theorem 1([8]). Let(X; d)be a complete metric space andT :X!C(X):
If there exists a constant c2(0;1)such that there is y 2Ibx(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 functionx!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 everyx2X de…ne a set
C(D; X; x) =ffxng X: lim
n!1D(xn; x) = 0g:
In this case we say that D is a generalized metric in the sense of Jleli and Samet [11] (for short JS-metric) onX if it satis…es the following conditions:
(D1) for every(x; y)2X X,D(x; y) = 0)x=y;
(D2) for every(x; y)2X X,D(x; y) =D(y; x);
(D3) there exists c > 0 such that for every (x; y) 2 X X and fxng 2 C(D; X; x)
D(x; y) clim sup
n!1
D(xn; y):
In this case(X; D)is said to be JS-metric space. Note that, ifC(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 fxng be a sequence in X. If fxng 2 C(D; X; x), then fxng is said to be converges to x. If limn;m!1D(xn; xn+m) = 0, then fxng 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 fxng 2 C(D; X; x)\C(D; X; y), thenx=y.
After the introducing the JS-metric space, Jleli and Samet [11] presented some …xed point results including Banach contraction and ´Ciri´c type quasi- contraction mappings.
2. Main result
Let(X; D)be a JS-metric space andU X. We say thatU is sequentially open if for each sequencefxnginXsuch thatlimn!1D(xn; x) = 0for some x2U is eventually inU, that is, there existsn0 2Nsuch thatxn2U for all
n n0. Let J S be the family of all sequentially open subsets ofX, then it is easy to see that(X; J S) is a topological space. Further, a sequencefxng is convergent to xin (X; D) if and only if it is convergent tox in(X; J S).
Let C(X) be the family of all nonempty closed subsets of (X; J S) and let be the family of all nonempty subsets A of X satisfying the following property: for allx2X,
D(x; A) = 0)x2A;
where D(x; A) = inffD(x; y) : y 2Ag. In this case C(X) = . Indeed, let A2C(X)andx2X. IfD(x; A) = 0, then there exists a sequencefxnginA such thatlimn!1D(x; xn) = 0. Therefore, by the de…nition of the topology
J S, for any U 2 J S including the point x, there existsnU 2N such that xn2U for alln nU. In this case, we haveU \A6=;, that is,x2A=A.
Hence C(X) . Now, let A 2 . We will show that A 2 C(X). Let x 2XnA and fxng be a sequence inX such that limn!1D(xn; x) = 0. If there exists a subsequencefxnkgof fxng such thatfxnkg A, then we get D(x; A) = 0. SinceA2 , then x2A. This is a contradiction. Therefore, there exists n0 2 N such that xn 2 XnA for all n n0. This shows that XnA2 J S, and soA2C(X). As a consequence we getC(X) = .
Now we will consider the following special cases for J S:
Case 1. Let (X; D) be a metric space. Then it is clear that J S coincides with the metric topology D.
Case 2. Let(X; D)be ab-metric space. In this case, there are three topolo- gies 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 ofX in the usual sense, that is, a subsetU of X is open if for anyx2U, there exists" >0 such that
B(x; ") :=fy2X:D(x; y)< "g U:
Third is the D topology, which the family of all …nite intersections of C=fB(x; ") :x2X; r >0g
satis…es conditions (B1)-(B2) of ([7], Proposition 1.2.1) is a base of D. By Proposition 3.3 of [1], we know that s = D D. Also by De…nition 2.1 and Theorem 3.4 of [1], we can see that J S = 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 onX if it satis…es the following property: for all x2X andA; B 2P(X),
x^A and A B implies x^A.
LetUxbe a nonempty collection of subsets ofXfor eachx2X. If the follow- ing conditions are satis…ed, then the pair (Ux;^) is called d-neighbourhood system for x:
i) ifU 2 Ux, then x^U;
ii) ifU; V 2 Ux, then U \V 2 Ux;
iii) if U 2 Ux, then there isV U withV 2 Ux such that for ally^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 =fUx:x2Xg.
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;^)isd-neighbourhood system for x for each x 2X, 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 J S.
Let(X; D) be a generalized metric space andT :X!C(X) be a multi- valued mapping. For a constant b2 (0;1) and x 2X; we will consider the following set in our main result:
Ibx(T) =fy2T 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 constantc > 0 such that for any x2X there is y2Ibx(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 con- structed a sequencefxng in X satisfying
(i) xn+1 2T xn; (ii) D(xn; xn+1)<1;
(iii)bD(xn+1; xn+2) cD(xn; xn+1)andbD(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, sinceD has triangular inequality, form; n2
N withm > n, we get from (iii),
D(xn; xm) D(xn; xn+1) + +D(xm 1; xm) c
b
n
D(x0; x1) + + c b
m 1
D(x0; x1) (c=b)n
1 (c=b)D(x0; x1):
Sincec < b, then fxng 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; n2Nwith m > n, we get from (iii), D(xn; xm) sD(xn; xn+1) + +sm nD(xm 1; xm)
s c b
n
D(x0; x1) + +sm n c b
m 1
D(x0; x1)
= s c
b
n1 (sc=b)m n
1 (sc=b) D(x0; x1):
Sincesc < b, then fxng 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. Letx0 2X be such that D(x0; T x0) <1:Then, from (2.2) there existsx1 2Ibx0(T) such that
D(x1; T x1) cD(x0; x1):
Note that, since x1 2Ibx0(T), thenx12T x0 and bD(x0; x1) D(x0; T x0)<1: Forx12X, there existsx2 2Ibx1(T) such that
D(x2; T x2) cD(x1; x2):
By the way, we can construct a sequencefxnginXsuch thatxn+1 2Ibxn(T) and
D(xn+1; T xn+1) cD(xn; xn+1) (2.3) for all n2N. Note that, since D(x0; T x0) <1, then D(xn; xn+1)<1 for all n2N.
Again, sincexn+1 2Ibxn(T), we have xn+12T xn and
bD(xn; xn+1) D(xn; T xn) (2.4) for all n2N. Therefore from (2.3) and (2.4), we get
bD(xn+1; xn+2) D(xn+1; T xn+1) cD(xn; xn+1) (2.5) and
D(xn+1; T xn+1) cD(xn; xn+1) c
bD(xn; T xn): (2.6)
Hence (i), (ii) and (iii) hold. Furthermore, from (2.5) and (2.6), we get
nlim!1D(xn; xn+1) = lim
n!1D(xn; T xn) = 0:
Now, if fxng is Cauchy sequence then by the completeness of (X; D), there existsz2X such thatxn2C(D; X; z), that is limn!1D(xn; z) = 0.
Therefore, by the lower semicontinuity of the functionf(x) =D(x; T x), we get
0 D(z; T z) =f(z) lim inf
n!1 f(xn) = lim inf
n!1 D(xn; T xn) = 0:
Since T z2C(X), we get z2T 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 constantc >0 such that for anyx2X there isy 2Ibx(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 onb-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 x2X there is y2Ibx(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 x2X there is y2Ibx(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 Distin- guished Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).
References
[1] T. V. An, L. Q. Tuyen and N. V. Dung, Stone-type theorem onb-metric spaces and applications, Topology and its Applications, 185-186 (2015) 50-64.
[2] V. Berinde and M. P¼acurar, The role of the Pompeiu-Hausdor¤ metric in …xed point theory, Creat. Math. Inform., 22 (2) (2013), 35-42.
[3] M. Berinde and V. Berinde, On a general class of multi-valued weakly Picard map- pings, J. Math. Anal. Appl., 326 (2007), 772-782.
[4] Lj. B. ´Ciri´c, Multi-valued nonlinear contraction mappings, Nonlinear Anal., 71 (2009), 2716-2723.
[5] Lj. B. ´Ciri´c and J. S. Ume, Common …xed point theorems for multi-valued nonself mappings, Publ. Math. Debrecen, 60 (2002), 359-371.
[6] P. Z. Da¤er and H. Kaneko, Fixed points of generalized contractive multivalued mappings, J. Math. Anal. Appl., 192 (1995), 655-666.
[7] R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol.6, Helder- mann Verlag, Berlin, 1988.
[8] Y. Feng and S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl., 317 (2006), 103-112.
[9] P. Hitzler and A.K. Seda, Dislocated topologies, J. Electr. Eng., 51(12) (2000), 3-7.
[10] V. I. Istr¼a¸tescu, Fixed Point Theory an Introduction, Dordrecht D. Reidel Publishing Company 1981.
[11] M. Jleli and B. Samet, A generalized metric space and related …xed point theorems, Fixed Point Theory and Applications (2015) 2015:61.
[12] T. Kamran and Q. Kiran, Fixed point theorems for multi-valued mappings obtained by altering distances, Math. Comput. Modelling, 54 (2011), 2772-2777.
[13] M. A. Khamsi and N. Hussain, KKM mappings in metric type spaces, Nonlinear Anal., 7 (9) (2010), 3123-3129.
[14] D. Klim and D. Wardowski, Fixed point theorems for set-valued contractions in com- plete metric spaces, J. Math. Anal. Appl., 334 (2007), 132-139.
[15] N. Mizoguchi and W. Takahashi, Fixed point theorems for multivalued mappings on complete metric spaces, J. Math. Anal. Appl.,141 (1989), 177-188.
[16] S.B. Nadler, Multi-valued contraction mappings, Paci…c J. Math., 30 (1969), 475-488.
[17] S. Reich, Some problems and results in …xed point theory, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), 179-187, Contemp. Math., 21, Amer. Math. Soc., Providence, RI, 1983.
[18] T. Suzuki, Mizoguchi-Takahashi’s …xed point theorem is a real generalization of Nadler’s, J. Math. Anal. Appl., 340 (2008), 752-755.
King Saud University, College of Science, Geology and Geophysics Depart- ment, P.O. Box 2455, Riyadh 11451, Saudi Arabia
E-mail address: [email protected]
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: [email protected]
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
E-mail address: [email protected]
Address 1: King Saud University, College of Engineering, Petroleum and Gas Engineering Department, P.O. Box 800, Riyadh 11421, Saudi Arabia, Ad- dress 2: Benha University, Faculty of Science, Geology Department, P.O. Box 13518, Benha, Egypt
E-mail address: [email protected]
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia
E-mail address: [email protected]
