1 Introduction and Preliminaries

J. COMPUTATIONAL ANALYSIS AND APPLICATIONS, VOL. 24, NO.4, 2018, COPYRIGHT 2018 EUDOXUS PRESS, LLC

Common Fixed Point Results for the Family of Multivalued Mappings Satisfying Contractions on a Sequence in Hausdor¤ Fuzzy Metric Space Abdullah Shoaib1 , Akbar Azam2 , and Aqeel Shahzad3 Abstract: The aim of this paper is to establish common …xed point results on a sequence contained in a closed ball for family of multivalued mapping in complete fuzzy metric space. Simple and di¤erent technique has been used. Example has been constructed to demonstrate the novelty of our results. Our results unify, extend and generalize several results in the existing literature. _____________________ 2010 Mathematics Subject Classi…cation: 46S40; 47H10; 54H25. Keywords and Phrases: common …xed point; complete fuzzy metric space; closed ball; family of multivalued mappings; Hausdor¤ fuzzy metric space. _____________________

1

Introduction and Preliminaries

The notion of fuzzy sets was …rst introduced by Zadeh [5]. Kramosil et al. [10] introduced the concept of fuzzy metric space and obtained many …xed point results. Later on many authors [7, 8, 9, 11] used this concept and prove many …xed point results using the di¤erent contractive conditions. Lopez et al. [11] discuss the method for constructing a Hausdor¤ fuzzy metric on nonempty compact subsets of a given fuzzy metric space. Sometimes, it happens that the …xed point of a mapping exists, but the contraction does not hold. Recently, Shoaib et al. [1, 2, 3, 4, 6, 13] obtained the necessary and su¢ cient conditions for the existence of a …xed point of such self mapping. In this paper, we prove the existence of a common …xed point of a family of such multivalued mappings which are contractive on a sequence contained in a closed ball instead of the whole space, by using the concept of Hausdor¤ fuzzy metric space. We also present an example to support our results. De…nition 1.1 [7] A binary operation : [0; 1] [0; 1] ! [0; 1] is said to be a continuous t-norm if it is satis…es the following conditions: i) is associative and commutative; ii) is continuous; iii) a 1 = a for all a 2 [0; 1]; iv) a b c d whenever a c and b d for each a; b; c; d 2 [0; 1]. De…nition 1.2 [10] The 3-tuple (X; M; ) is said to be a fuzzy metric space if X is an arbitrary set, is a continuous t-norm, and F is a fuzzy set on X 2 [0; 1), satisfying the following conditions for all x; y; z 2 X and t; s > 0: 1

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Abdullah Shoaib et al 692-699


F1) F (x; y; 0) = 0; F2) F (x; y; t) = 1 if and only if x = y; F3) F (x; y; t) = F (y; x; t); F4) F (x; z; t + s) F (x; y; t) F (y; z; s); F5) F (x; y; :) : (0; 1) ! [0; 1] is left-continuous. Example 1.3 [7] Let (X; d) be a metric space. De…ne a b = ab and F (x; y; t) =

ktn ; ktn + md(x; y)

for all x; y 2 X and k; m; n 2 R+ . Then (X; F; ) is a fuzzy metric space. De…nition 1.4 [9] Let (X; F; ) be a fuzzy metric space. Then, we have i) A sequence fxn g in X is said to be convergent to a point x 2 X denoted xn ! x; if lim F (xn ; x; t) = 1 for each t > 0. n!1

ii) A sequence fxn g in X is said to be a Cauchy sequence, if lim F (xn ; xn+p ; t) = n!1 1 for each t > 0, p > 0. iii) A fuzzy metric space (X; F; ) in which every Cauchy sequence is convergent is called a complete fuzzy metric space. De…nition 1.5 [11] Let (X; F; ) be a fuzzy metric space. De…ne a function HF M on C^0 (X) C^0 (X) (0; 1) by HF M (A; B; t) = min

inf F (a; B; t); inf F (A; b; t) ;

a2A

b2B

for all A; B 2 C^0 (X) and t > 0, where C^0 (X) is the collection of all nonempty compact subsets of X. De…nition 1.6 [7] Let (X; F; ) be a fuzzy metric space. Then, BF (x; r; t) = fy 2 X : F (x; y; t) > 1

rg

BF (x; r; t) = fy 2 X : F (x; y; t)

rg

and 1

are called open and closed balls respectively, with centre x 2 X and radius r for 0 < r < 1, t > 0. Lemma 1.7 [11] Let (X; F; ) be a complete fuzzy metric space. Then, for each a 2 X, B 2 C^0 (X) and for all t > 0 there is bo 2 B such that F (a; bo ; t) = F (a; B; t): Lemma 1.8 [12] Let (X; F; ) be a complete fuzzy metric space. (C^0 (X); HF M ; ) is a hausdor¤ fuzzy metric space on C^0 (X). Then, for all A; B 2 C^0 (X), for each a 2 A and for all t > 0 there exists ba 2 B; satis…es F (a; B; t) = F (a; ba ; t); then HF M (A; B; t) F (a; ba ; t):

2

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2

Main Results

Let (X; F; ) be a fuzzy metric space, x0 2 X and let fS : 2 g be a family of multivalued mappings from X to C^0 (X). Then, there exists x1 2 Sa x0 for some a 2 , such that F (x0 ; Sa x0 ; t) = F (x0 ; x1 ; t); for all t > 0: Let x2 2 Sb x1 be such that F (x1 ; Sb x1 ; t) = F (x1 ; x2 ; t): Continuing this process, we construct a sequence xn of points in X such that xn+1 2 S xn , F (xn ; S xn ; t) = F (xn ; xn+1 ; t); for all t > 0: We denote this iterative sequence fXS (xn ) : 2 g and say that fXS (xn ) : 2 g is a sequence in X generated by x0 . Theorem 2.1 Let (X; F; ) be a complete fuzzy metric space, where be a continuous t-norm, de…ned as a a a or a b = minfa; bg. Let (C^0 (X); HF M ; ) be a Hausdor¤ fuzzy metric space on C^0 (X), fS : 2 g be a family of multivalued mappings from X to C^0 (X) and fXS (xn ) : 2 g be a sequence in X generated by x0 . Assume that, for some 0 < i;j k < 1; for all t > 0, x0 2 X, for all x; y 2 BF (x0 ; r; t) \ fXS (xn ) : 2 g; with x 6= y and for all i; j 2 with i 6= j; we have HF M (Si x; Sj y;

i;j t)

F (x; y; t)

(2.1)

and, for some t > 0 F (x0 ; x1 ; (1

k)t))

1

r:

(2.2)

Then, fXS (xn ) : 2 g is a sequence in BF (x0 ; r; t) and fXS (xn ) : 2 g ! z 2 BF (x0 ; r; t): Also, if (2.1) holds for z; then there exists a common …xed point for the family of multivalued mappings fS : 2 g in BF (x0 ; r; t). Proof: Let fXS (xn ) : 2 g be a sequence in X generated by x0 . If x0 = x1 , then x0 is a common …xed point of Sa for all a 2 . Let x0 6= x1 and by Lemma 1:8; we have F (x1 ; x2 ; t) HF M (Sa x0 ; Sb x1 ; t): By induction, we have by Lemma 1:8; we have F (xn ; xn+1 ; t)

HF M (Si xn

1; S

xn ; t):

(2.3)

First, we will show that xn 2 BF (x0 ; r; t). By (2:2); we get F (x0 ; x1 ; t) = F (x0 ; Sa x0 ; t) > F (x0 ; x1 ; (1 F (x0 ; x1 ; t) > 1 r:

k)t)

1

r

3

694



This shows that x1 2 BF (x0 ; r; t): Let x2 ; F (xj ; xj+1 ; t)

HF M (S xj

1; S

; xj 2 BF (x0 ; r; t): Now, we have xj ; t)

F (xj

1 ; xj ;

t

)

;

HF M (S xj

2; S

xj

t

1;

)

(by Lemma 1:8)

;

F (xj F (xj F (xj ; xj+1 ; t)

t ;m ;

2 ; xj 1 ; 2 ; xj 1 ;

F (x0 ; x1 ;

t ) k2

) ;

::::

F (x0 ; x1 ;

t ) kj

t ) kj

(2.4)

Now, F (x0 ; xj+1 ; t)

F (x0 ; xj+1 ; t)

F (x0 ; xj+1 ; (1 k j+1 )t) F (x0 ; x1 ; (1 k)t) F (x1 ; x2 ; (1 k)kt) :::: F (xj ; xj+1 ; (1 k)k j t) F (x0 ; x1 ; (1 k)t) F (x0 ; x1 ; (1 k)t) :::: F (x0 ; x1 ; (1 k)t) (by (2.4)) 1 r 1 r :: 1 r = 1 r 1 r:

This implies that xj+1 2 BF (x0 ; r; t): Now, inequality (2.4) can be written as F (xn ; xn+1 ; t)

F (x0 ; x1 ;

t ): kn

(2.5)

Let n; m 2 N with m > n: Assume that m = n + p; we have F (xn ; xn+p ; t)

F (xn ; xn+1 ; (1 F (xn ; xn+1 ; (1 F (xn ; xn+1 ; (1

k)t) F (xn+1 ; xn+p ; kt) k)t) HF M (Sj xn ; Sk xn+p 1 ; kt) kt k)t) F (xn ; xn+p 1 ; ) j;k

F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 k)t) F (xn+1 ; xn+p 1 ; (1 F (xn ; xn+1 ; (1 k)t) HF M (Sj xn ; Sl xn+p F (xn ; xn+1 ; (1 k)t) kt F (xn ; xn+p 2 ; )

F (xn ; xn+p 1 ; t) F (xn ; xn+1 ; (1 k)t) k)t) F (xn ; xn+1 ; (1 k)t) 2 ; kt) F (xn ; xn+1 ; (1 k)t)

j;l

F (xn ; xn+p ; t)

F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 F (xn ; xn+p 2 ; t):

k)t)

4

695



Using the above, we have F (xn ; xn+p ; t)

F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 k)t) ::: F (xn ; xn+1 ; t) F (xn ; xn+1 ; (1 k)t) F (xn ; xn+1 ; (1 k)t) ::: F (xn ; xn+1 ; (1 k)t) (1 k)t (1 k)t F (x0 ; x1 ; ) F (x0 ; x1 ; ) ::: n k kn (1 k)t F (x0 ; x1 ; ) (by (2.5)) kn (1 k)t F (x0 ; x1 ; ): kn

F (xn ; xn+p ; t)

F (xn ; xn+p ; t) As, we have

lim F (x; y; t) = 1 for all x; y 2 X:

t!1

In particular F (x0 ; x1 ;

(1

k)t ) = 1 as n ! 1: kn

By using above, we get F (xn ; xm ; t) = 1 as n ! 1: Hence, fXS (xn )g is a Cauchy sequence in BF (x0 ; r; t): As every closed ball in a complete fuzzy metric space is complete. So, BF (x0 ; r; t) is complete. Then, there exists z 2 BF (x0 ; r; t), such that xn ! z as n ! 1: Now, for some q 2 ; we have F (z; Sq z; t) F (z; xn ; (1 k)t) F (xn ; Sq z; kt): By Lemma 1:8, we have F (z; Sq z; t)

F (z; xn ; (1

k)t) HF M (Sr xn

F (z; xn ; (1

k)t) F (xn

1 ; Sq z; kt) kt ) 1 ; z;

F (z; xn ; (1

k)t) F (xn

1 ; z; t):

F (z; Sq z; t)

1 1 = 1:

r;q

Letting n ! 1; we have

This implies that z 2 Sq z: Hence, z 2 \ Sq z. This completes the proof. q2

Let (X; F; ) be a fuzzy metric space, x0 2 X and let S be a multivalued mapping from X to C^0 (X). Then, there exists x1 2 Sx0 , such that F (x0 ; Sx0 ; t) = F (x0 ; x1 ; t); for all t > 0: Let x2 2 Sx1 be such that F (x1 ; Sx1 ; t) = F (x1 ; x2 ; t): 5

696



Continuing this process, we construct a sequence xn of points in X such that xn+1 2 Sxn , F (xn ; Sxn ; t) = F (xn ; xn+1 ; t); for all t > 0: We denote this iterative sequence fXS(xn )g and say that fXS(xn )g is a sequence in X generated by x0 . Corollary 2.2 Let (X; F; ) be a complete fuzzy metric space, where be a continuous t-norm, de…ned as a a a or a a = minfa; bg. Let (C^0 (X); HF M ; ) is Hausdor¤ fuzzy metric space on C^0 (X), x0 2 X, S : X ! C^0 (X) be a multivalued mapping and fXS(xn )g be a sequence in X generated by xo . Assume that for some k 2 (0; 1) t > 0, and xo 2 X, we have HF M (Sx; Sy; kt)

F (x; y; t) for all x; y 2 BF (x0 ; r; t) \ fXS(xn )g

(2.6)

and F (x0 ; Sx0 ; (1

k)t))

1

r

Then, fXS(xn )g is a sequence in BF (x0 ; r; t) and fXS(xn )g ! z 2 BF (x0 ; r; t): Also, if (2.6) holds for z; then there exists a …xed point for S in BF (x0 ; r; t). Proof: By using the similar steps as we have used in Theorem 2.1, it can be proved easily. Corollary 2.3 Let (X; F; ) be a complete fuzzy metric space, where be a continuous t-norm, de…ned as a a a or a a = minfa; bg. Let x0 2 X and S : X ! X be a self mapping. Assume that for some k 2 (0; 1), t > 0 and xo 2 X, we have F (Sx; Sy; kt)

F (x; y; t) for all x; y 2 BF (x0 ; r; t)

and F (x0 ; Sx0 ; (1

k)t))

1

r:

Then S has a …xed point in BF (x0 ; r; t). Example 2.4 Let X = [0; 5] and d : X X ! R be a complete metric space de…ned by, d(x; y) = jx yj for all x; y 2 X t Denote a b = ab or a b = minfa; bg for all a; b 2 [0; 1] and F (x; y; t) = t+d(x;y) for all x; y 2 X and t > 0. Then, we can …nd that (X; F; ) is a complete fuzzy metric space. Consider the multivalued mappings S : X ! C^0 (X) where = a; 1; 2; 3; : de…ned as, 8 x x ; 2n ] if x 2 [0; 72 ] < [ 3n ; where n = 1; 2; ; Sn x = : [2nx; 3nx] if x 2 ( 72 ; 5]

and

Sa x =

8 x 5x < [ 3 ; 12 ] if x 2 [0; 27 ] :

:

[2x; 3x] if x 2 ( 72 ; 5]

6

697



Considering, x0 =

1 2

and r = 34 . then, BF (x0 ; r; t) = [0; 72 ]. Now,

F (x0 ; Sa x0 ; t) F (x1 ; S1 x1 ; t)

1 1 1 5 = F ( ; Sa ; t) = F ( ; ; t) 2 2 2 24 5 5 5 5 = F ( ; S1 ; t) = F ( ; ; t) 24 24 24 48

5 5 5 ; 48 ; 192 ; :::g in X generated by So, we obtain a sequence fXS (xn )g = f 21 ; 24 5 x0 : Now, for x = 4, y = 5, k = 1;a = 6 and t = 1; we have

5 HF M (S1 4; Sa 5; ) 6

=

F (4; 5; 1)

=

5 5 inf F (a; Sa 5; ); inf F (S1 4; b; ) a2S1 4 6 b2Sa 5 6 1 1 = = 0:5 1 + j4 5j 2

min

= 0:22

So, we have 5 HF M (S1 4; Sa 5; ) 6

F (4; 5; 1)

So, the contractive condition does not hold on X. Now, for all x; y 2 BF (x0 ; r; t)\ fXS (xn )g, we have HF M (Sn x; Sa y; kt)

=

min

inf F (a; Sa y; kt); inf F (Sn x; b; kt)

a2Sn x

b2Sa y

y 5y 5 x x 5 inf F (a; [ ; ]; t); inf F ([ ; ]; b; t) 3 12 6 b2Sa y 3n 2n 6 x 5y 5 x y 5 = min F ( ; ; t); F ( ; ; t) 2n 12 6 3n 3 6 (5=6)t (5=6)t = min ; (5=6)t + jx=3 y=3j (5=6)t + jy=3 x=3j t (5=6)t = F (x; y; t) HF M (Sx; Sy; kt) = (5=6)t + jx=3 y=3j t + jx yj =

min

a2Sn x

So, the contractive condition holds on BF (x0 ; r; t) \ fXS (xn )g. Also, for t = 1; we have F (x0 ; x1 ; (1

1 5 1 = F( ; ; ) 2 24 6 4 1 = > =1 11 4

k)t))

r

Hence, all the conditions of above theorem are satis…ed. Now, we have fXS (xn )g is a sequence in BF (x0 ; r; t), and fXS (xn )g ! 0 2 BF (x0 ; r; t). Moreover, fS : = a; 1; 2 g has a common …xed point 0. Competing interests The authors declare that they have no competing interests. 7

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References [1] M. Arshad, A. Shoaib, I. Beg, Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered complete dislocated metric space, Fixed Point Theory and Appl. (2013), 2013:115. [2] M. Arshad, A. Shoaib, and P. Vetro, Common Fixed Points of a Pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Dislocated Metric Spaces, Journal of Function Spaces, 2013 (2013), Article ID 63818. [3] M. Arshad , A. Shoaib, M. Abbas and A. Azam, Fixed Points of a pair of Kannan Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, Miskolc Mathematical Notes, 14(3), 2013. [4] M. Arshad, A. Azam, M. Abbas and A. Shoaib, Fixed point results of dominated mappings on a closed ball in ordered partial metric spaces without continuity, U.P.B. Sci. Bull., Series A, 76(2), 2014. [5] L. A. Zadeh, Fuzzy Sets. Information and Control, 8(3), 1965. [6] I. Beg, M. Arshad , A. Shoaib, Fixed Point on a Closed Ball in ordered dislocated Metric Space, Fixed Point Theory, 16(2), 2015. [7] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64(3), 1994. [8] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Sets and Systems, 90(3), 1997. [9] D. Gopal, C. Vetro, Some New Fixed Point Theoerms In Fuzzy Metric Spaces, Iranian Journal of Fuzzy Systems, 11(3), 2014. [10] I. Kramosil, J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika 11(5) (1975). [11] J. Rodriguez-Lopez, S. Romaguera, The Hausdor¤ fuzzy metric on compact sets, Fuzzy Sets and Systems 147(2), 2004. [12] A. Shoaib, Ph.D thesis, Fixed points theorems for locally and globally contractive mappings in ordered spaces, 2016. [13] A. Shoaib, M. Arshad and M. A. Kutbi, Common …xed points of a pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Partial Metric Spaces, J. Comput. Anal. Appl., 17(2014), 255-264. 1;3

Department of Mathematics and Statistics, Riphah International University, Islamabad - 44000, Pakistan. 1 E-mail address: [email protected]. 3 E-mail address: [email protected]. 2 Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad - 44000, Pakistan. E-mail address:[email protected]. 8

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