Metric spaces and their various generalizations occur frequently in computer science applications. This is the reason why, in this paper, we introduced and studied the concept of fuzzy b-metric space, generalizing, in this way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek and the concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-bmetric space, extending the notion of fuzzy quasi metric space recently introduced by V. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasipseudo- b-metric into an ascending family of quasi-pseudo-b-metrics is established. The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of denotational semantics and their applications in control theory will be an important next step

Nădăban, Sorin

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INTERNATIONAL JOURNAL OF COMPUTERS COMMUNICATIONS & CONTROLISSN 1841-9836, 11(2):273-281, April 2016.Fuzzy b-Metric SpacesS. NădăbanSorin NădăbanDepartment of Mathematics and Computer ScienceAurel Vlaicu University of Arad,Elena Drăgoi 2, RO-310330 Arad, Romaniasnadaban@gmail.comAbstract: Metric spaces and their various generalizations occur frequently in com-puter science applications. This is the reason why, in this paper, we introduced andstudied the concept of fuzzy b-metric space, generalizing, in this way, both the notionof fuzzy metric space introduced by I. Kramosil and J. Michálek and the conceptof b-metric space. On the other hand, we introduced the concept of fuzzy quasi-b-metric space, extending the notion of fuzzy quasi metric space recently introduced byV. Gregori and S. Romaguera. Finally, a decomposition theorem for a fuzzy quasi-pseudo-b-metric into an ascending family of quasi-pseudo-b-metrics is established.The use of fuzzy b-metric spaces and fuzzy quasi-b-metric spaces in the study of de-notational semantics and their applications in control theory will be an importantnext step.Keywords: Fuzzy b-metric spaces, fuzzy quasi-b-metric, fuzzy quasi-pseudo-b-metric, b-metric space.1 Introduction and preliminariesThe concept of b-metric space was introduced by I.A. Bakhtin [5] and exensively used by S.Czerwic [10,11].Deﬁnition 1. [10] Let X be a nonempty set and k  1 be a given real number. A functiond : X X ! [0;1) is a b-metric on X if, for all x; y; z 2 X, the following conditions hold:(b1) d(x; y) = 0 if and only if x = y;(b2) d(x; y) = d(y; x);(b3) d(x; z)  k[d(x; y) + d(y; z)].The triple (X; d; k) will be called b-metric space.Some examples of b-metric spaces and some ﬁxed point theorems in b-metric spaces can befound in [6–8, 21]. We also note that the class of b-metric spaces is larger than that of metricspaces, since every b-metric is a metric when k = 1. In [22] an example of a b-metric space whichis not a metric space, is given.Recently, M.A. Alghamdi, N. Hussain, P. Salimi [1] introduced the notion of b-metric-likespace, which is an interesting generalization of metric-like space (introduced by A. Amini-Harandi[2]) and partial metric space (introduced by S.G. Matthews [17]). In paper [14], N. Hussain andM.H. Shah introduced the notion of cone b-metric space, generalizing both notions of b-metricspaces and cone metric spaces.The concept of quasi-b-metric space was introduced by M.H. Shah and N. Hussain [20] in2012. In this paper we adopt a slight modiﬁcation of their deﬁnition.Deﬁnition 2. Let X be a nonempty set. A real valued function d : X X ! [0;1) is said to bea quasi-b-metric with constant k  1 if the following conditions are satisﬁed:Copyright© 2006-2016 by CCC Publications274 S. Nădăban(qb1) d(x; y) = d(y; x) = 0 if and only if x = y;(qb3) d(x; z)  k[d(x; y) + d(y; z)]; (8)x; y; z 2 X.The triple (X; d; k) will be called quasi-b-metric space.On the other hand, after L.A. Zadeh has introduced in his famous paper [23] the concept offuzzy set, one of the important problems is to obtain an adequate notion of fuzzy metric space. I.Kramosil and J. Michálek [16] reformulated successfully the notion of probabilistic metric space,introduced by K. Menger in 1942, in fuzzy context.Deﬁnition 3. [19] A binary operation : [0; 1] [0; 1]! [0; 1]is called triangular norm (t-norm) if it satisﬁes the following condition:1. a  b = b  a; (8)a; b 2 [0; 1];2. a  1 = a; (8)a 2 [0; 1];3. (a  b)  c = a  (b  c); (8)a; b; c 2 [0; 1];4. If a  c and b  d, with a; b; c; d 2 [0; 1], then a  b  c  d.Example 4. Three basic examples of continuous t-norms are ^; ; L, which are deﬁned by a^b =minfa; bg, a b = ab (usual multiplication in [0; 1]) and aL b = maxfa+b 1; 0g (the Lukasiewiczt-norm).Deﬁnition 5. [16] The triple (X;M; ) is said to be a fuzzy metric space if X is an arbitrary set, is a continuous t-norm and M is a fuzzy metric, i.e. a fuzzy set in X X  [0;1) such thatfor all x; y; z 2 X we have:(M1) M(x; y; 0) = 0;(M2) [M(x; y; t) = 1; (8)t > 0] if and only if x = y;(M3) M(x; y; t) = M(y; x; t); (8)t  0;(M4) M(x; z; t+ s) M(x; y; t) M(y; z; s); (8)t; s  0;(M5) M(x; y; ) : [0;1)! [0; 1] is left continuous and limt!1M(x; y; t) = 1.We note that A. George and P. Veeramani [12] modiﬁed the concept of fuzzy metric spaceintroduced by I. Kramosil and J. Michálek and deﬁned a Hausdorﬀ topology on this fuzzy space.Another approach for fuzzy metric spaces was introduced by O. Kaleva and S. Seikkala in paper[15], by setting the distance between two points to be a non-negative, upper semicontinuous,normal and convex fuzzy number.In recent years, diﬀerent types of fuzzy generalized metric spaces was considered by diﬀerentauthors in diﬀerent approaches. Thus, V. Gregori and S. Romaguera introduced in paper [13]the concept of fuzzy quasi-metric space, generalizing in this way the notions of fuzzy metricintroduced by I. Kramosil and J. Michálek and by A. George and P. Veeramani to the quasi-metric setting.On the other hand, the idea of fuzzy cone metric space has been introduced in [3] and somebasic properties and ﬁxed point theorems for diﬀerent types of contraction mappings have beendeveloped in fuzzy cone metric spaces. In paper [4], T. Bag introduced the concept of fuzzyFuzzy b-Metric Spaces 275cone b-metric space and some ﬁxed point theorems are established in such spaces for contractionmappings. We must note that Bag’s deﬁnitions for fuzzy cone metric space and for fuzzy coneb-metric spaces generalized the notion of fuzzy metric introduced by Kaleva and Seikkala.In this paper we introduced and studied the concept of fuzzy b-metric space, generalizing, inthis way, both the notion of fuzzy metric space introduced by I. Kramosil and J. Michálek andthe concept of b-metric space. On the other hand, we introduced the concept of fuzzy quasi-b-metric space, extending the notion of fuzzy quasi-metric space recently introduced by V. Gregoriand S. Romaguera. Finally, a decomposition theorem for a fuzzy quasi-pseudo-b-metric into anascending family of quasi-pseudo-b-metrics is established.2 Fuzzy b-metric spacesDeﬁnition 6. Let X be a nonempty set, let k  1 be a given real number and  be a continuoust-norm. A fuzzy set M in X  X  [0;1) is called fuzzy b-metric if, for all x; y; z 2 X, thefollowing conditions hold:(bM1) M(x; y; 0) = 0;(bM2) [M(x; y; t) = 1; (8)t > 0] if and only if x = y;(bM3) M(x; y; t) = M(y; x; t); (8)t  0;(bM4) M(x; z; k(t+ s)) M(x; y; t) M(y; z; s); (8)t; s  0;(bM5) M(x; y; ) : [0;1)! [0; 1] is left continuous and limt!1M(x; y; t) = 1.The quadruple (X;M; ; k) is said to be a fuzzy b-metric space.Remark 7. The class of fuzzy b-metric spaces is larger than the class of fuzzy metric spaces,since a fuzzy b-metric space is a fuzzy metric space when k = 1.Example 8. Let (X; d; k) be a b-metric space. LetMd : X X  [0;1)! [0; 1];Md(x; y; t) =(tt+d(x;y) if t > 00 if t = 0:Then (X;Md;^; k) is a fuzzy b-metric space. Md will be called standard fuzzy b-metric.Proof: We check only (bM4), because verifying the other conditions is standard.Let x; y; z 2 X and t; s > 0. Without restraining the generality we assume that Md(x; y; t) Md(y; z; s). Thus tt+d(x;y)  ss+d(y;z) , i.e. td(y; z)  sd(x; y).On the other handMd(x; z; k(t+ s)) =k(t+ s)k(t+ s) + d(x; z) k(t+ s)k(t+ s) + k[d(x; y) + d(y; z)]=t+ st+ s+ d(x; y) + d(y; z):We will prove thatt+ st+ s+ d(x; y) + d(y; z) tt+ d(x; y):Hence we will obtain that Md(x; z; k(t + s))  Md(x; y; t) = Md(x; y; t) ^Md(y; z; s), what hadto be veriﬁed. We remark thatt+ st+ s+ d(x; y) + d(y; z) tt+ d(x; y),276 S. Nădăbant2 + st+ td(x; y) + sd(x; y)  t2 + st+ td(x; y) + td(y; z), sd(x; y)  td(y; z) ;which is true. 2Deﬁnition 9. Let k  1 be a real given number. A function f : R ! R will be called k-nondecreasing if for t < s we will have that f(t)  f(ks).Proposition 10. For all x; y 2 X the mapping M(x; y; ) : [0;1)! [0; 1] is k-nondecreasing.Proof: For 0 < t < s we haveM(x; y; ks) M(x; x; s  t) M(x; y; t) = 1 M(x; y; t) = M(x; y; t) :2Theorem 2.1. Let (X;M; ; k) be a fuzzy b-metric space. For x 2 X; r 2 (0; 1),t > 0 we deﬁne the open ballB(x; r; t) := fy 2 X : M(x; y; t) > 1  rg :ThenTM := fT  X : x 2 T iﬀ (9)t > 0; r 2 (0; 1) : B(x; r; t)  Tgis a topology on X.Proof: It is obvious that ; and X belong to TM .Let fTigi2I  TM and T =Si2ITi. We will show that T 2 TM . Let x 2 T . Then there existsi0 2 I such that x 2 Ti0 . As Ti0 2 TM , there exist t > 0, r 2 (0; 1) such that B(x; r; t)  Ti0 .Thus B(x; r; t)  Si2ITi = T .Let now fTigni=1  TM and T =nTi=1Ti. We will show that T 2 TM . Let x 2 T . We obtainthat x 2 Ti; (8)i = 1; n. Thus(9)ti > 0; ri 2 (0; 1) : B(x; ri; ti)  Ti; (8)i = 1; n :Letr = minfri; i = 1; ng; t = mintik; i = 1; n:We have that B(x; r; t)  B(x; ri; ti); (8)i = 1; n. Indeed, for y 2 B(x; r; t), we haveM(x; y; t) > 1   r  1   ri; (8)i = 1; n. As t  tik ; (8)i = 1; n, we obtain thatM(x; y; t) M(x; y; ti). Thus M(x; y; ti) > 1  ri; (8)i = 1; n. Hence y 2 B(x; ri; ti); (8)i = 1; n.Therefore B(x; r; t)  Ti; (8)i = 1; n. Thus B(x; r; t) nTi=1Ti = T . 2Remark 11. Previous theorem extends to fuzzy b-metric space a similar result obtained by A.George and P. Veeramani [12] in the context of fuzzy metric space. The deﬁnitions for convergentsequence and Cauchy sequence given by A. George and P. Veeramani [12] in the context of fuzzymetric space can be translated in the context of fuzzy b-metric space, as follows.Deﬁnition 12. Let (X;M; ; k) be a fuzzy b-metric space and (xn) be a sequence in X. Thesequence (xn) is said to be convergent if there exists x 2 X such that M(xn; x; t) = 1; (8)t > 0.In this case, x is called the limit of the sequence (xn) and we note limn!1xn = x, or xn ! x.Remark 13. Let (X;M; ; k) be a fuzzy b-metric space. A sequence (xn) is convergent to x ifand only if (xn) is convergent to x in topology TM .Fuzzy b-Metric Spaces 277Indeed,xn ! x in topology TM ,, (8)r 2 (0; 1); (8)t > 0; (9)n0 2 N : xn 2 B(x; r; t); (8)n  n0 ,, (8)r 2 (0; 1); (8)t > 0; (9)n0 2 N : M(xn; x; t) > 1  r; (8)n  n0 ,, limn!1M(xn; x; t) = 1; (8)t > 0 :Deﬁnition 14. Let (X;M; ; k) be a fuzzy b-metric space and (xn) be a sequence in X. Thesequence (xn) is said to be a Cauchy sequence if(8)r 2 (0; 1); (8)t > 0; (9)n0 2 N : M(xn; xm; t) > 1  r; (8)n;m  n0 :A fuzzy b-metric space in which every Cauchy sequence is convergent is called complete fuzzyb-metric space.3 Fuzzy quasi-b-metric spacesDeﬁnition 15. A fuzzy quasi-b-metric space is a quadruple (X;M; ; k), where X is a nonemptyset,  is a continuous t-norm, k  1 is a given real number andM is a fuzzy set in XX [0;1)such that for all x; y; z 2 X we have:(qbM1) M(x; y; 0) = 0;(qbM2) [M(x; y; t) = M(y; x; t) = 1; (8)t > 0] if and only if x = y;(qbM3) M(x; z; k(t+ s)) M(x; y; t) M(y; z; s); (8)t; s  0;(qbM4) M(x; y; ) : [0;1)! [0; 1] is left continuous and limt!1M(x; y; t) = 1.Remark 16. V. Gregori and S. Romaguera [13] also gave this deﬁnition in the particular casek = 1 and the triple (X;M; ) is called fuzzy quasi-metric space.Proposition 17. If Q is a fuzzy quasi-b-metric, then Q 1 deﬁned by Q 1(x; y; t) = Q(y; x; t) isalso a fuzzy quasi-b-metric (called the conjugate of Q).Proof: We have to check only (qbM3).Q 1(x; z; k(t+ s)) = Q(z; x; k(s+ t))  Q(z; y; s) Q(y; x; t) = Q 1(x; y; t) Q 1(y; z; s) :2Deﬁnition 18. [18]. Let ;  be two t-norms. We say that  dominates  and we denote    if(x1  x2)  (y1  y2)  (x1  y1)  (x2  y2); (8)x1; x2; y1; y2 2 [0; 1]:Remark 19. [18]. For any t-norm  we have ^  .Proposition 20. Let (X;Q; ; k) be a fuzzy quasi-b-metric space and  be a continuous t-normsuch that   . Let M be a fuzzy set in X X  [0;1) deﬁned byM(x; y; t) = Q(x; y; t) Q 1(x; y; t) :Then (X;M; ; k) is a fuzzy b-metric space.278 S. NădăbanProof: It is easy to check (bM1)  (bM3) and (bM5). We prove (bM4).M(x; z; k(t+ s)) = Q(x; z; k(t+ s)) Q 1(x; z; k(t+ s))  [Q(x; y; t) Q(y; z; s)]  [Q 1(x; y; t) Q 1(y; z; s)]  [Q(x; y; t) Q 1(x; y; t)]  [Q(y; z; s) Q 1(y; z; s)] = M(x; y; t) M(y; z; s) :2Corollary 21. Let (X;Q; ; k) be a fuzzy quasi-b-metric space andM(x; y; t) = minfQ(x; y; t); Q(y; x; t)g :Then (X;M; ; k) is a fuzzy b-metric space.Proof: We apply previous proposition for  = ^  . 2Example 22. Let (X; d; k) be a quasi-b-metric space. LetMd : X X  [0;1)! [0; 1];Md(x; y; t) =(tt+d(x;y) if t > 00 if t = 0:Then (X;Md;^; k) is a fuzzy quasi-b-metric space. Md will be called standard fuzzy quasi-b-metric.Proof: The proof is standard. 2Proposition 23. If (X;M; ; k) is a fuzzy quasi-b-metric space, then the relation M on X deﬁnedbyx M y if and only ifM(x; y; t) = 1; (8)t > 0is a partial ordering.Proof. It is easy to check.4 Fuzzy quasi-pseudo-b-metric spacesDeﬁnition 24. Let X be a nonempty set. A function d : X X ! [0;1) is called quasi-pseudo-b-metric with constant k  1 if the following conditions are satisﬁed:(qpb1) d(x; x) = 0;(qpb3) d(x; z)  k[d(x; y) + d(y; z)]; (8)x; y; z 2 X.The triple (X; d; k) will be called quasi-pseudo-b-metric space.Deﬁnition 25. A fuzzy quasi-pseudo-b-metric space is a quadruple (X;M; ; k), where X is anonempty set,  is a continuous t-norm, k  1 is a given real number and M is a fuzzy set inX X  [0;1) such that for all x; y; z 2 X we have:(qpbM1) M(x; y; 0) = 0;(qpbM2) [M(x; x; t) = 1; (8)t > 0];(qpbM3) M(x; z; k(t+ s)) M(x; y; t) M(y; z; s); (8)t; s  0;Fuzzy b-Metric Spaces 279(qpbM4) M(x; y; ) : [0;1)! [0; 1] is left continuous and limt!1M(x; y; t) = 1.Theorem 4.1. Let (X;M;^; k) be a fuzzy quasi-pseudo-b-metric space andd(x; y) := infft > 0 : M(x; y; t) > g;  2 (0; 1) :Then D = fdg2(0;1) is an ascending family of quasi-pseudo-b-metrics on X.Proof: (qp1) d(x; x) = infft > 0 : M(x; x; t) > g = 0.(qp2)k[d(x; y) + d(y; z)] = k[infft > 0 : M(x; y; t) > g+ inffs > 0 : M(y; z; s) > g] == k[infft+ s > 0 : M(x; y; t) > ;M(y; z; s) > g] == inffk(t+ s) > 0 : M(x; y; t) ^M(y; z; s) > g  inffk(t+ s) > 0 : M(x; z; k(t+ s)) > g = d(x; z) :It remains to prove that D = fdg2(0;1) is an ascending family. Let 1  2. Thenft > 0 : M(x; y; t) > 2g  ft > 0 : M(x; y; t) > 1g :Thusinfft > 0 : M(x; y; t) > 2g  infft > 0 : M(x; y; t) > 1g ;namely d2(x; y)  d1(x; y); (8)(x; y) 2 X X. 25 Conclusions and further worksIn this paper we introduce the notions of fuzzy b-metric space and fuzzy quasi-b-metric space.Thus, we have built a fertile ground to study, in further papers, some ﬁxed point theorems in thesespaces. The ﬁrst problem is to established fuzzy versions of Banach contraction mapping principlein fuzzy b-metric spaces. From here we will obtain a lot of applications both in Mathematics aswell as in Engineering and Computer Science. The second issue is to study set-valued contractionsin fuzzy b-metric spaces and their applications in control theory and convex optimization. A realchallenge is to extend the results of C. Chifu and G. Petruşel [9] in fuzzy b-metric spaces. Weintend to obtain some ﬁxed point theorems for multivalued operators in fuzzy b-metric spacesendowed with a graph. This paper may be of interest for researchers working in the followingﬁelds belonging to Computer Science and Information Technology:(i) Integrated solution in computer-based control and communications(ii) Computational intelligence methods(iii) Advanced decision support systemswhere fuzzy metric spaces will be applied in dealing with the problems such as: ﬁxed pointtheorems and their applications in the semantics of programs; distance measurement betweenprograms with important results to measure the complexity of programs and algorithms; colorimage processing and image denoising; the use of some types of fuzzy metrics in cognitive infor-mation, in time series and in bioinformatics; the appplications in neural networks; data miningand web mining applications.280 S. NădăbanBibliography[1] Alghamdi, M.A., Hussain, N., Salimi, P. (2013); Fixed point and coupled ﬁxed point theo-rems on b-metric-like spaces, Journal of Inequalities and Applications, 2013:402.[2] Amini-Harandi, A. (2012). Metric-like spaces, partial metric spaces and ﬁxed points, FixedPoint Theory and Applications, 2012:204.[3] Bag, T. (2013); Fuzzy cone metric spaces and ﬁxed point theorems of contractive mappings,Annals of Fuzzy Mathematics and Informatics, 6(3): 657–668.[4] Bag, T. (2014); Some ﬁxed point theorems in fuzzy cone b-metric spaces, InternationalJournal of Fuzzy Mathematics and Systems, 4(2): 255–267.[5] Bakhtin, I.A. (1989); The contraction mapping principle in quasi-metric spaces, Funct. Anal.Unianowsk Gos. Ped. Inst., 30: 26–37.[6] Boriceanu, M., Bota, M., Petruşel, A. (2010); Multivalued fractals in b-metric spaces, Cen-tral European Journal of Mathematics, 8(2): 367–377.[7] Boriceanu, M., Petruşel, A., Rus, I.A. (2010); Fixed point theorems for some multivaluedgeneralized contraction in b-metric spaces, International J. Math. Statistics, 6: 65–76.[8] Boriceanu, M. (2009); Strict ﬁxed point theorems for multivalued operators in b-metricspaces, Intern. J. Modern Math., 4: 285–301.[9] Chifu, C., Petruşel, G. (2014); Fixed point for multivalued contraction in b-metric spaceswith applications to fractals, Taiwanese Journal of Mathematics, 18(5): 1365–1375.[10] Czerwik, S. (1993); Contraction mappings in b-metric space, Acta Math. Inf. Univ. Os-traviensis, 1: 5–11.[11] Czerwik, S. (1998); Non-linear set-valued contraction mappings in b-metric spaces, Atti.Sem. Math. Fig. Univ. Modena, 46(2): 263–276.[12] George, A., Veeramani, P. (1994); On some results in fuzzy metric spaces, Fuzzy Sets andSystems, 64: 395–399.[13] Gregori, V., Romaguera, S. (2004); Fuzzy quasi-metric spaces, Applied General Topology,5(1): 128–136.[14] Hussain, N., Shah, M.H. (2011); KKM mappings in cone b-metric spaces, Comput. Math.Appl., 61(4): 1677–1684.[15] Kaleva, O., Seikkala, S. (1984); On fuzzy metric spaces, Fuzzy Sets and Systems, 12: 215–229.[16] Kramosil, I., Michálek, J. 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Fuzzy b-Metric Spaces

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Fuzzy b-Metric Spaces

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