Some fixed fuzzy point results using Hausdorff metric in fuzzy metric spaces

Transcription

1 Annals of Fuzzy Mathematics and Informatics Volume 13, No 5, (May 017), pp ISSN: (print version) ISSN: (electronic version) c Kyung Moon Sa Co Some fixed fuzzy point results using Hausdorff metric in fuzzy metric spaces Ferhan Şola Erduran, Cemil Yıldız Received 4 April 016; Revised 4 September 016; Accepted 13 October 016 Abstract In this paper, we obtain coupled fixed fuzzy point theorem for a fuzzy mapping in Hausdorff fuzzy metric space and using it, we obtain a common fixed fuzzy point for a hybrid fuzzy pair Also we give an auxiliary example to support our main theorem 010 AMS Classification: 54H5, 47H10 Keywords: Fuzzy mapping, Fixed fuzzy point, Hausdorff fuzzy metric Corresponding Author: Ferhan Şola Erduran (ferhansola@yahoocomtr) 1 Introduction The theory of fuzzy sets was introduced by Zadeh [1] in 1965 Since then, to use this concept in topology and analysis, many authors have developed the theory of fuzzy sets and applications Especially, Deng [3], Erceg [5], Kaleva and Seikkala [11], Kramosil and Michalek [1], Georege and Veeramani [7] have introduced the concept of fuzzy metric space in different ways Grabiec [8] extended classical fixed point theorems of Banach and Edelstein to complete and compact fuzzy metric spaces After that, fuzzy metric spaces became one of the most popular area for study of fixed point theory In recent years, many mathematicians such as [4, 13, 15] etc, established several fixed point theorems in fuzzy metric spaces In 004, Rodríguez-López and Romaguera [14] introduced Hausdorff fuzzy metric on the set of the non-empty compact subsets of a given fuzzy metric space Later, several authors proved some fixed point theorems for multivalued maps in fuzzy metric spaces Heilpern [10] first introduced the concept of fuzzy mappings and proved a fixed point theorem for fuzzy contraction mappings Many authors extended the results of Heilpern and proved several fixed fuzzy point theorems (see [6, 17, 19, 0]) Recently, Phiangsungnoen et al [16] have studied fuzzy fixed point theory for fuzzy mappings in Hausdorff fuzzy metric spaces and then, Abbas et al [1] have obtained fixed fuzzy points of fuzzy mappings in Hausdorff fuzzy metric spaces under generalized contractive conditions

2 On the other hand, Bhaskar and Lakshmikantham [] introduced the notion of a coupled fixed point in partially ordered metric spaces, also discussed some problems of the uniqueness of a coupled fixed point, and applied their results to the problems of the existence and uniqueness of a solution for the periodic boundary value problems In this paper, we establish the existence of coupled fixed fuzzy point theorem for fuzzy mappings in Hausdorff fuzzy metric space and give an auxiliary example to support our theorem As an application, we obtain a common fixed fuzzy point for a hybrid fuzzy pair Our results can be applied in theoretical computer science In the sequel, we need the following Preinaries Definition 1 ([18]) A binary operation : [0, 1] [0, 1] [0, 1] is called a t-norm if it satisfies the following conditions: (i) is associative and commutative, (ii) a 1 = a for every a [0, 1], (iii) a b c d whenever a c and b d, for a, b, c, d [0, 1] If in addition, is continuous, then is called a continuous t-norm Typical examples of a continuous t-norms are a b = min {a, b}, a b = ab/ max {a, b, λ} for 0 < λ < 1, a b = ab, a b = max {a + b 1, 0} Definition ([9]) For each a [0, 1], the sequence { n a} n=1 is defined by 1 a = a and n a = ( n 1 a ) a A t-norm is said to be of H-type, if the sequence of functions { n a} n=1 is equicontinuous at a = 1, ie, for any λ (0, 1), there exist δ (λ) (0, 1) such that a > 1 δ implies n a > 1 λ, for all n N The concept of fuzzy metric space is defined by George and Veeramani as follows Definition 3 ([7]) A fuzzy metric space is a triple (X, M, ), where X is a nonempty set, is a continuous t-norm and M is a fuzzy set on X X (0, ), satisfying the following conditions: for all x, y, z X, s, t > 0, (GV 1 ) M(x, y, t) > 0, (GV ) M(x, y, t) = 1 iff x = y, (GV 3 ) M(x, y, t) = M(y, x, t), (GV 4 ) M(x, y, ) : (0, ) (0, 1] is continuous, (GV 5 ) M(x, z, t + s) M(x, y, t) M(y, z, s) It is known that M is continuous on X (0, ) and M(x, y, ) is non-decreasing on (0, ) Definition 4 ([7]) Let (X, M, ) be a fuzzy metric space (i) A sequence {x n } is said to be convergent to a point x X, if n M(x n, x, t) = 1 for all t > 0 (ii) A sequence {x n } is said to be a Cauchy sequence, if for each 0 < ε < 1 and t > 0, there exist n 0 N such that M(x n, x m, t) > 1 ε for each n, m n 0 (iii) A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete 64

3 (iv) A subset A X is said to be closed, if for each convergent sequence {x n } with x n A and x n x, we have x A (v) A subset A X is said to be compact, if each sequence in A has a convergent subsequence to a point in A Rodríguez-López and Romaguera [14] introduced Hausdorff fuzzy metric of a given fuzzy metric space (X, M, ) on K M (X), where K M (X) denotes the family of all non-empty compact subsets of X Definition 5 ([14]) Let (X, M, ) be a fuzzy metric space For A, B K M (X), x X and t > 0, define and M (x, A, t) = sup M (x, a, t) a A { } H M (A, B, t) = min inf M(a, B, t), inf M(A, b, t) a A b B The 3-tuple (K M (X), H M, ) is a fuzzy metric space and the fuzzy metric H M is called Hausdorff fuzzy metric induced by the fuzzy metric M Lemma 6 Let X be a non-empty set and g : X X Then there exist a subset E X such that g(e) = g(x) and g : E X is one-to-one Now, we recall some definitions and properties of fixed fuzzy point Let X be an arbitrary non-empty set A fuzzy set in X is a function with domain X and values in [0, 1] A fuzzy set A in X is characterized by a mapping A : X [0, 1] A (x) is called the grade of membership of x in A Let A be a fuzzy set in X If X endowed with a topology, for α (0, 1], the α-level set of A is denoted by [A] α and is defined as follows: [A] α = {x X : A (x) α}, [A] 0 = {x X : A (x) > 0}, where B denotes the closure of B in X Corresponding to each α (0, 1] and x X, the fuzzy point (x) α of X is a fuzzy set (x) α : X [0, 1] given by { α, if y = x, (x) α (y) = 0, otherwise Definition 7 ([19]) Let X be a non-empty set I = [0, 1] and I X stands for the collection of all fuzzy sets in X A mapping F : X I X is said to be fuzzy mapping Definition 8 A fuzzy point x α in X is called a fixed point of a fuzzy mapping F, if x α F x, ie, α F x (x) or x [F x] α, ie, the fixed degree of x in F is at least α If x 1 F x, then x is called a fixed point of the fuzzy mapping F In 014, Phiangsungnoen et al [16] proved the following theorem Theorem 9 Let (X, M, ) be a complete fuzzy metric space and α : X (0, 1] be a mapping such that [T x] α(x) is a non-empty compact subset of X for all x X Suppose that T : X I X is a fuzzy mapping such that ) H M ([T x] α(x), [T y] α(y), kt M(x, y, t) 643

4 for all t > 0, where k (0, 1) If there exist x 0 X and x 1 [T x 0 ] α(x0) such that n i=nm ( x 0, x 1, th i) = 1 for all t > 0 and h > 1, then T has a fixed fuzzy point Let S be the class of functions ψ : (0, 1] (0, 1] satisfying sup ψ (x n ) < 1 n whenever {x n } (0, 1] is non-decreasing or x n = 1 Define n W α (X) = { A I X : A α is non-empty and compact } For A, B W α (X) and α (0, 1], define { Mα (x, B, t) = sup y B α M(x, y, t), H Mα (A, B, t) = H M (A α, B α, t) Note that M α is a non-increasing function of α and H Mα metric induced by the fuzzy metric M on W α (X) is the Hausdorff fuzzy Definition 10 ([1]) Let Y be an arbitrary subset of a fuzzy metric space (X, M, ) A mapping F : Y W α (X) is called a fuzzy mapping over the set Y, ie, F y W α (X), for each y in Y Lemma 11 ([1]) Let (X, M, ) be a fuzzy metric space, x, y X and A, B W α (X) Then the following conditions hold: (1) for each x X, B W α (X) and t > 0, there is (b 0 ) α B such that M α (x, B, t) = M (x, b 0, t), () M α (x, A, t) = 1 implies that (x) α A, (3) for (y x ) α B with M(x, y x, t) = M α (x, B, t), we have M α (x, C, t + s) M α (x, B, t) M α (y x, C, s) Abbas et al [1] generalized Theorem 9 as follow: Theorem 1 ([1]) Let (X, M, ) be a complete fuzzy metric space and F : X W α (X) a fuzzy mapping Suppose that, for all x, y X and t > 0, the following condition holds: H Mα (F x, F y, ψ(m(x, y, t))t) M(x, y, t), where ψ S Then F has a fixed fuzzy point 3 Fixed fuzzy point theorems In this section we prove a coupled fixed fuzzy point theorem for fuzzy mappings and a common coupled fixed fuzzy point theorem for hybrid fuzzy pair in fuzzy metric spaces Let g : X X be a mapping and F : X X W α (X) be a fuzzy mapping and denote [F (X X)] α = [F (x, y)] α We call (g, F ) a hybrid fuzzy pair (x,y) X X 644

5 Definition 31 An element ((x) α, (y) α ) X X is called (i) a coupled fixed fuzzy point of the mapping F : X X W α (X), if (x) α F (x, y), (y) α F (y, x), ie, x [F (x, y)] α, y [F (y, x)] α, (ii) a coupled coincidence fuzzy point of the mappings F : X X W α (X) and g : X X, if (gx) α F (x, y), (gy) α F (y, x), ie, gx [F (x, y)] α, gy [F (y, x)] α, (iii) a common coupled fixed fuzzy point of the mappings F : X X W α (X) and g : X X, if (x) α = (gx) α F (x, y), (y) α = (gy) α F (y, x), ie, x = gx [F (x, y)] α, y = gy [F (y, x)] α Definition 3 An element (x) α X is called a common fixed fuzzy point of the mappings F : X X W α (X) and g : X X, if (x) α = (gx) α F (x, x), ie, x = gx [F (x, x)] α Definition 33 The mappings F : X X W α (X) and g : X X are called weakly fuzzy compatible maps, if g [F (x, y)] α [F (gx, gy)] α, g [F (y, x)] α [F (gy, gx)] α, whenever gx [F (x, y)] α, gy [F (y, x)] α Definition 34 Let F : X X W α (X) and g : X X The mapping g is called F -weakly commuting at some point (x, y) X X, if ( g x ) F (gx, gy) α and ( g y ) F (gy, gx), ie, α g x [F (gx, gy)] α and g y [F (gy, gx)] α Theorem 35 Let (X, M, ) be a complete fuzzy metric space such that is a t- norm of H-type and a b ab for all a, b [0, 1] and M is with t M(x, y, t) = 1 for all x, y X Let F : X X W α (X) fuzzy mapping such that (31) H Mα (F (x, y), F (u, v), kt) [M(x, u, t)] 1/ [M(y, v, t)] 1/ for all x, y, u, v X, where 0 < k < 1 Then F has a fixed fuzzy point Proof Let x 0, y 0 are given points in X Since [F (x 0, y 0 )] α and [F (y 0, x 0 )] α are nonempty we can choose x 1, y 1 X such that x 1 [F (x 0, y 0 )] α and y 1 [F (y 0, x 0 )] α Since [F (x 1, y 1 )] α is compact, from Lemma 11, we can choose x [F (x 1, y 1 )] α such that M(x 1, x, kt) = sup z [F (x 1,y 1)] α M (x 1, z, kt) H Mα (F (x 0, y 0 ), F (x 1, y 1 ), kt) [M (x 0, x 1, t)] 1/ [M(y 0, y 1, t)] 1/ Similarly, since [F (y 1, x 1 )] α is compact, from Lemma 11, we can choose y [F (y 1, x 1 )] α such that M(y 1, y, kt) = sup z [F (y 1,x 1)] α M (y 1, z, kt) H Mα (F (y 0, x 0 ), F (y 1, x 1 ), kt) [M(y 0, y 1, t)] 1/ [M (x 0, x 1, t)] 1/ Continuing this way, we can obtain two sequences {x n } and {y n } such that x n+1 [F (x n, y n )] α, y n+1 [F (y n, x n )] α for all n 0 and (3) M (x n, x n+1, kt) [M (x n 1, x n, t)] 1/ [M(y n 1, y n, t)] 1/, 645

6 (33) M (y n, y n+1, kt) [M (y n 1, y n, t)] 1/ [M(x n 1, x n, t)] 1/ We prove that {x n } and {y n } are Cauchy sequences Let δ n (t) = [M (x n, x n+1, t)] 1/ [M(y n, y n+1, t)] 1/ Then, by operating t-norm on (3) and (33), from a b ab we obtain δ n (kt) δ n 1 (t) This implies that ( ) ( ) t t δ n (t) δ n 1 δ 0 k k n Since n δ 0 ( t k n ) = 1, for all t > 0, we have (34) n δ n (t) = 1 for all t > 0 We show that, for any p N (35) M (x n, x n+p, t) p δ n 1 (t kt) and M (y n, y n+p, t) p δ n 1 (t kt) It is obvious for p = 1 Assume that (35) holds for some p By (3), we have M (x n, x n+1, t) M (x n, x n+1, kt) δ n 1 (t), and so M (x n, x n+1, t kt) δ n 1 (t kt) From (31), (35) and a b ab, we have M (x n+1, x n+p+1, kt) [M (x n, x n+p, t)] 1/ [M (y n, y n+p, t)] 1/ p δ n 1 (t kt) Then by the (GV 5 ), we have M (x n, x n+p+1, t) = M (x n, x n+p+1, t kt + kt) M (x n, x n+1, t kt) M (x n+1, x n+p+1, kt) δ n 1 (t kt) ( p δ n 1 (t kt)) = p+1 δ n 1 (t kt) Similarly, we have M (y n, y n+p+1, t) p+1 δ n 1 (t kt) Then (35) holds, for all p Suppose that t > 0 and ε (0, 1] are given By hypothesis, is a t-norm of H-type, there exist η > 0 such that (36) p (s) > 1 ε, for all s (1 η, 1] and for all p By (34), there exist n 0 such that δ n 1 (t kt) > 1 η, for all n n 0 Thus, from (35) and (36), we get M (x n, x n+p, t) > 1 ε and M (y n, y n+p, t) > 1 ε, for all n n 0 and for all p So {x n } and {y n } are Cauchy sequences Since X is complete, x, y X such that n x n = x and n y n = y Now we show that x [F (x, y)] α and y [F (y, x)] α We write x = x n, y = y n, u = x, v = y in (31) H Mα (F (x n, y n ), F (x, y), kt) [M(x n, x, t)] 1/ [M(y n, y, t)] 1/ On taking it as n, we get which implies that n H M α (F (x n, y n ), F (x, y), kt) = 1 n sup M(x n+1, x, t) = 1 x [F (x,y)] α Hence there exist a sequence {x n} with x n [F (x, y)] α such that n M (x n, x n, t) = 1, 646

7 for each t > 0 From (GV 5 ), we have M (x n, x, t) M (x n, x n, t/) M (x n, x, t/) and let n, we obtain M n (x n, x, t) = 1, ie, n x n = x Since [F (x, y)] α is compact, x [F (x, y)] α By the same way, we can show that y [F (y, x)] α Therefore (x, y) is a coupled fixed fuzzy point of F Example 36 Let X = {1,, 3}, a b = ab, for all a, b [0, 1] and { x M (x, y, t) =, x y y x, y x, for all x, y X, t > 0 Let α ( 0, 3) 1, define a fuzzy mapping F : X X Wα (X) as follows: for x = 1, y X, for x =, y X, for x = 3, y X, F (x, y)(z) = α, z = 1 0, z = α, z = 3 α 3, z = 1 F (x, y)(z) = α, z = 0, z = 3 F (x, y)(z) = α, z = 1 α 3, z = 0, z = 3 Note that [F (x, y)] α = {1}, for x = 1, y X; [F (x, y)] α = {}, for x =, y X; [F (x, y)] α = {1} for x = 3, y X Thus x = u = 1, y, v X, x = u =, y, v X, and x = u = 3, y, v X, we have H Mα (F (x, y), F (u, v), kt) = 1 Also we consider the following cases: Case 1: If x = 1, u =, y, v X, we obtain H Mα (F (x, y), F (u, v), kt) = 1 [M(x, u, t)]1/ [M (y, v, t)] 1/ Case : If x = 1, u = 3, y, v X, we obtain H Mα (F (x, y), F (u, v), kt) = 1 3 [M(x, u, t)]1/ [M (y, v, t)] 1/ Case 3: If x =, u = 3, y, v X, we obtain H Mα (F (x, y), F (u, v), kt) = 1 [M(x, u, t)]1/ [M (y, v, t)] 1/ Consequently, for each k (0, 1), x, y, u, v X and t > 0, we have H Mα (F (x, y), F (u, v), kt) [M(x, u, t)] 1/ [M (y, v, t)] 1/ 647

8 Thus all conditions of Theorem 35 are satisfied So (1, ) is a coupled fixed fuzzy point of F As an application of the Theorem 35, we obtain a coupled coincidence and common fixed fuzzy point theorem for a hybrid fuzzy pair We denote the set of coupled coincidence points of mappings F and g by C α (F, g) Note that if (x, y) C α (F, g), then (y, x) is also in C α (F, g) Theorem 37 Let (X, M, ) be a complete fuzzy metric space such that is a t-norm of H-type and a b ab, for all a, b [0, 1] and M is with t M(x, y, t) = 1, for all x, y X and (g, F ) a hybrid fuzzy pair such that [F (X X)] α g(x) Suppose that, for all x, y X, 0 < k < 1 and t > 0, the following condition holds: (37) H Mα (F (x, y), F (u, v), kt) [M(gx, gu, t)] 1/ [M(gy, gv, t)] 1/ Then F and g have a coupled coincidence fuzzy point Moreover F and g have a coupled common fixed point if one of the following conditions holds: (1) F and g are weakly fuzzy compatible, n g n x = u and n g n y = v, for some (x, y) C α (F, g), u, v X, and g is continuous at uand v, () g is F -weakly commuting, for some(x, y) C α (F, g), gx and gy are fixed points of g, ie,, g x = gx and g y = gy, (3) g is continuous at x, y, for some (x, y) C α (F, g) and for some u, v X, n g n u = x and n g n v = y Proof By Lemma 6, there exist E X such that g : E X is one-to-one and g(e) = g(x) Now, define a mapping Λ : g(e) g(e) W α (X) by Λ (gx, gy) = F (x, y) for all gx, gy g (E) Since g is one-one on E, Λ is well defined Since H Mα (Λ (gx, gy), Λ (gu, gv), kt) = H Mα (F (x, y), F (u, v), kt) [M(gx, gu, t)] 1/ [M(gy, gv, t)] 1/, the mapping Λ satisfies (31) and thus all conditions of Theorem 35 By using Theorem 35 with a mapping Λ, it follows that Λ has a coupled fixed fuzzy point (u, v) g(e) g(e) such that u [Λ (u, v)] α and v [Λ (v, u)] α Since [F (X X)] α g(x), there exist u, v X such that (gu ) α = (u) α and (gv ) α = (v) α So (gu ) α Λ (gu, gv ) = F (u, v ) and (gv ) α Λ (gv, gu ) = F (v, u ), ie, gu [Λ (gu, gv )] α = [F (u, v )] α and gv [Λ (gv, gu )] α = [F (v, u )] α This implies that (u, v ) X X is a coupled coincidence fuzzy point of the mappings F and g Hence C α (F, g) is nonempty Suppose that (1) holds Then for some (x, y) C α (F, g), n gn x = u and n gn y = v, where u, v X Since g is continuous at u and v, we have that gu = u and gv = v As F and g are weakly fuzzy compatible (g n x, g n y) C α (F, g), for all n 1,ie, for all n 1, we have (g n x) α F ( g n 1 x, g n 1 y ) and (g n y) α F ( g n 1 y, g n 1 x ) 648

9 Using (37), we obtain ( M α (gu, F (u, v), t) M gu, g n x, t ) ( M α g n x, F (u, v), t ) ( M gu, g n x, t ) ( H Mα F ( g n 1 x, g n 1 y ), F (u, v), t ) ( M gu, g n x, t ) ( H Mα F ( g n 1 x, g n 1 y ), F (u, v), k t ) ( M gu, g n x, t ) [ ( M g n x, gu, t 1/ [ ( M g )] n y, gv, )] t 1/ On taking it as n, we have M α (gu, F (u, v), t) This implies that gu [F (u, v)] α and (u) α = (gu) α F (u, v) By the same way, we obtain gv [F (v, u)] α and (v) α = (gv) α F (v, u) Therefore (u, v) is a common fixed fuzzy point of F and g Now suppose that () holds If for some (x, y) C α (F, g), g is F -weakly commuting and g x = gx and g y = gy, then gx = gx [F (gx, gy)] α and gy = gy [F (gy, gx)] α Thus, (gx, gy) is a coupled common fixed point of F and g If (3) holds, then by the continuity of g at x, y, we get x = gx [F (x, y)] α and y = gy [F (y, x)] α Thus (x, y) is a coupled common fixed point of F and g 4 Conclusions In this paper, we obtain a coupled fixed fuzzy point theorem for a fuzzy mapping in fuzzy metric space and using it, we obtain a common fixed fuzzy point for a hybrid fuzzy pair Our results can be applied in theoretical computer science Acknowledgements The authors are thankful to the referees,the editors - in - chief and managing editors for their valuable suggestions and constructive comments References [1] M Abbas, B Ali and C Vetro, Fixed fuzzy points of fuzzy mappings in Hausdorff fuzzy metric spaces with application, Iranian Journal of Fuzzy Systems 1 (015) [] T G Bhaskar and V Lakshmikantham Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal 65 (006) [3] Z Deng, Fuzzy pseudo-metric spaces, Journal of Mathematical Analysis and Applications 86 (198) [4] K S Dersanambika and M R Aswathy, Fixed point theorems in fuzzy metric spaces, Int J Contemp Math Sciences 6 () (011) [5] M A Erceg, Metric spaces in fuzzy set theory, J Math Anal Appl 69 (1979) [6] VD Estruch and A Vidal, A note on fixed fuzzy points for fuzzy mappings, Rend Ist Math Univ Trieste 3 (001) [7] A George and P Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems 64 (1994) [8] M Grabiec, Fixed points in fuzzy metric spaces, Fuzzy Sets and Systems 7 (1988)

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