Iranian Journal Of Fuzzy Systems

Archive of SID58S. B. Hosseini, D. O‘Regan and R. Saadati(Ns (ε), ε) L Pµ,ν (ym yk , t)Pµ,ν ( nX(m)(αj(k) αj )xj , t)j 1 L (µ0 (nX(m) αj(k) αj c, t), ν0 (j 1 L (m)j 1 αj(µ0 (1, (µ0 (αj(m) αj(k) αj d, t))j 1t/ c (µ0 (1, Pn L nX t/ c (m) αj(m) (k)αj (k)αj ), ν0 (1, Pn), ν0 (1,(k)j 1t/ d t/ d (m) αj(m) αj , t/ c ), ν0 (αj(k)(m) αj(k))) αj )) αj (k) αj , t/ d )).(m)This shows that each of the n sequences {αj } where j 1, 2, 3, ., n is Cauchy inR. Hence it converges and let αj denote the limit. Using these n limits α1 , ., αn ,we define,y α1 x1 . αn xn .Clearly, y W . FurthermorePµ,ν (ym y, t) Pµ,ν (nX(m)(αj αj )xj , t)j 1 L (m)T n 1 [Pµ,ν (α1(m) α1 )x1 , t/n), · · · , Pµ,ν (αn αn )xn , t/n)] 1L whenever m and every t 0. This shows that an arbitrary sequence {ym }is convergent in W . Hence W is complete. Corollary 2.3. Every finite dimensional subspace W of an intuitionistic fuzzynormed space (V, Pµ,ν , T ) is closed in V .Theorem 2.4. In a finite dimensional intuitionistic fuzzy normed space (V, Pµ,ν , T ),any subset K V is compact if and only if K is closed and IF-bounded.Proof. By Theorem 1.17, compactness implies closedness and IF-boundedness. Wemust prove the converse. Let K be closed and IF-bounded. Let dim V n and{x1 , ., xn } be a linearly independent set of V . We consider any sequence {xm } inK. Each xm has a representation,(m)xm α1 x1 . αn(m) xn .Since K is IF-bounded, so is {xm }, and therefore there are t 0 and 0 r 1such that Pµ,ν (xm , t) L (Ns (r), r) for all m N. On the other hand by Theorem2.1 there are c, d 6 0 and an intuitionistic fuzzy norm Pµ0 ,ν0 such thatwww.SID.ir

Archive of SIDSome Results on Intuitionistic Fuzzy Spaces(Ns (r), r) L 59Pµ,ν (xm , t)nX(m)Pµ,ν (αj xj , t)j 1 L (µ0 (cnX(m) αj , t), ν0 (dnXj 1 L (µ0 (1, L (µ0 (1, (µ0 (αjt c (m) j 1 αjPnt(m) c αj(m)(m) αj , t))j 1 ), ν0 (1,), ν0 (1, d t(m) d αj(m), t/ c ), ν0 (αjt Pnj 1(m) αj )))), t/ d )).(m)Hence the sequence of {αj }, (j fixed), is IF-bounded and by Theorem 1.20 it has alimit pointPn αj , (1 j n). Let {zm } be the subsequence of {xm } which convergesto z j 1 αj xj . Since K is closed, z K. This shows that an arbitrary sequence{xm } in K has a subsequence which converges in K. Hence K is compact. Definition 2.5. A sequence {xm } in an intuitionistic fuzzy normed space (V, Pµ,ν , T )is said to be weakly convergent if there is an x V such that for every Ṽ and everyf Ṽ and t 0,Pµ0 ,ν0 (f (xm ) f (x), t) 1L .We write:Wxm x.Theorem 2.6. Let (V, Pµ,ν , T ) be an intuitionistic fuzzy normed space and {xm }be a sequence in V . Then:(i) Convergence implies weak convergence with the same limit.(ii) If dim V , then weak convergence implies convergence.Proof. (i) Let xm x then for every t 0 we havePµ,ν (xm x, t) 1L .By Definition 1.21 for every f Ṽ we have,Pµ0 ,ν0 (f (xm ) f (x), t) L Pµ0 ,ν0 (f (xm x), t)(µ(xm x, t/c), ν(xm x, t/d))Wfor c, d 6 0. Then xm x.W(ii) Let xm x and dim V n. Let {x1 , ., xn } be a linearly independent setof V . Then(m)xm α1 x1 . αn(m) xn .and,x α1 x1 . αn xn .www.SID.ir

Archive of SID60S. B. Hosseini, D. O‘Regan and R. SaadatiBy assumption, for every f Ṽ and t 0 we havePµ0 ,ν0 (f (xm ) f (x), t) 1L .We take in particular f1 , ., fn , defined by fj xj 1 and fj xi 0, (i 6 j). Therefore(m)(m)fj (xm ) αj and fj (x) αj . Hence, fj (xm ) fj (x) implies αj αj .From this we obtain, for each t 0Pµ,ν (xm x, t) nX(m)Pµ,ν ( (αj αj )xj , t)j 1 L n 1(m)(m) α1 )x1 , t/n), ., Pµ,ν ((αn αn )xn , t/n)]ttn 1), ., Pµ,ν (xn ,)]T[Pµ,ν (x1 ,(m)(m)n α1 α1 n αn αn 1L T[Pµ,ν ((α1as m . This shows that {xm } converges to x. 3. Some Fundamental Theorems in Intuitionistic Fuzzy FunctionalAnalysisDefinition 3.1. Let (V, Pµ,ν , T ) be an intuitionistic fuzzy normed space and Wbe a linear manifold in V . Let Q : V V /W be the natural map, Qx x W .We define:Pµ̄,ν̄ (x W, t) {Pµ,ν (x y, t) : y W }, t 0.Theorem 3.2. If W is a closed subspace of the intuitionistic fuzzy normed space(V, Pµ,ν , T ) and the intuitionistic fuzzy norm Pµ̄,ν̄ is defined as above, then:(a) Pµ̄,ν̄ is a fuzzy norm on V /W ;(b) Pµ̄,ν̄ (Qx, t) L Pµ,ν (x, t) ;(c) If (V, Pµ,ν , T ) is a complete intuitionistic fuzzy normed space (intuitionisticfuzzy Banach space) and for every a, b in [0, 1], a b a · b and a b max(a, b),then so is (V /W, Pµ̄,ν̄ , T ).Proof. By Definition 1.3, Pµ̄,ν̄ L and the proof follows as in [12, 18]. Theorem 3.3. [12, 18] Let W be a closed subspace of an intuitionistic fuzzy normedspace (V, Pµ,ν , T ). If two of the spaces V , W , V /W are complete so is the thirdone.Theorem 3.4. (Open mapping theorem) [12, 18] If T is a continuous linear operator from the intuitionistic fuzzy Banach space (V, Pµ,ν , T ) onto the intuitionisticfuzzy Banach space (V 0 , P 0 µ0 ,ν 0 , T ) and for every a, b in [0, 1], a b a · b anda b max(a, b), then T is an open mapping.Theorem 3.5. (Closed graph theorem) [12, 18] Let T be a linear operator from theintuitionistic fuzzy Banach space (V, Pµ,ν , T ) into the intuitionistic fuzzy Banachspace (V 0 , P 0 µ0 ,ν 0 , T ). Suppose for every sequence {xn } in V such that xn xand T xn y for some elements x V and y V 0 it follows T x y. Then T iscontinuous.www.SID.ir

Archive of SIDSome Results on Intuitionistic Fuzzy Spaces61Theorem 3.6. An intuitionistic fuzzy normed space (V, Pµ,ν , T ) where T (a, b) (min(a1 , b1 ), max(a2 , b2 )), is locally convex; here a (a1 , a2 ) and b (b1 , b2 ).Proof. It suffices to consider the family of neighborhoods of the origin, B(0, r, t),with t 0 and 0 r 1. Let t 0, 0 r 1, x, y B(0, r, t) and α [0, 1].ThenPµ,ν (αx (1 α)y, t) L L T (Pµ,ν (αx, αt), Pµ,ν ((1 α)y, (1 α)t))T (Pµ,ν (x, t), Pµ,ν (y, t))(min(µ(x, t), µ(y, t)), max(ν(x, t), ν(y, t)))(Ns (r), r).Thus αx (1 α)y belongs to B(0, r, t) for every α [0, 1]. 4. Approximation Theory in Intuitionistic Fuzzy Metric SpacesDefinition 4.1. Let (X, MM,N , T ) be an intuitionistic fuzzy metric space andA, B X. We defineMM,N (A, B, t) {MM,N (a, b, t) : a A and b B}.For a X, we write MM,N (a, B, t) instead of MM,N ({a}, B, t).Definition 4.2. A sequence converges sub-sequentially if it has a convergent subsequence. In the above notation xn xn0 x0 identifies the subsequence and thepoint to which it converges. Recall that a subset C of an intuitionistic fuzzy metricspace is compact if every sequence in C converges sub-sequentially to an elementof C. Also, given two sequences xn and yn , and a subsequence xn0 of the first sequence, the corresponding subsequence of the second is denoted yn0 . A subset ofan intuitionistic fuzzy metric space is IF-boundedly compact if every IF-boundedsequence in the subset is sub-sequentially convergent. In the above notation, Y isIF-boundedly compact if for any IF-bounded sequence yn in Y , there is a point x0(not necessarily in Y ) for which yn yn0 x0 .Definition 4.3. For an intuitionistic fuzzy metric space X and nonempty subsetsB and C, a sequence bn B is said to converge in distance to C iflim MM,N (bn , C, t) MM,N (B, C, t).n The subset B is approximately compact relative to C if every sequence bn B whichconverges in distance to C is sub-sequentially convergent to an element of B. Wecall B (a subset of X) approximately compact provided that B is approximatelycompact relative to each of the singletons of X; B is proximinal if for every x Xsome element b in B satisfies the equation MM,N (x, b, t) MM,N (x, B, t).The following theorem says that points can be replaced by compact subsets inthe definition of approximate compactness.Theorem 4.4. Let B and C be nonempty subsets of an intuitionistic fuzzy metricspace (X, MM,N , T ). If B is approximately compact and C is compact, then B isapproximately compact relative to C.www.SID.ir

Archive of SID62S. B. Hosseini, D. O‘Regan and R. SaadatiProof. Let bn B be any sequence converging in distance to C and let the sequencecn C satisfy(4.1)lim MM,N (bn , cn , t)n MM,N (B, C, t).Since C is compact, cn cn0 c0 C. Hence, for every ε 0 there exists n0such that for n0 n0MM,N (B, C, t) L L L MM,N (bn0 , c0 , t)T (MM,N (bn0 , cn0 , t ε), MM,N (cn0 , c0 , ε))T (MM,N (B, C, t ε), (Ns (ε), ε)).Since ε 0 was arbitrary, then limn MM,N (bn0 , c0 , t) MM,N (B, C, t). Therefore, bn0 converges in distance to c0 so, since B is approximately compact, bnbn0 b0 B, that is, bn converges sub-sequentially to an element of B. Theorem 4.5. Let B and C be nonempty subsets of an intuitionistic fuzzy metricspace (X, MM,N , T ). If B is approximately compact and IF-bounded, and C isIF-boundedly compact, then B is approximately compact relative to C.Proof. Let bn B be any sequence converges in distance to C and let cn Csatisfy (2.1). As bn is IF-bounded, so is cn . Since C is IF-boundedly compact,cn cn0 c0 X. Proceed as in the proof of last theorem. Theorem 4.6. Let B and C be nonempty subsets of an intuitionistic fuzzy metricspace (X, MM,N , T ). If B is closed and IF-boundedly compact and C is IF-bounded,then B is approximately compact relative to C.The proof is the same as the classically case (see [14]).Lemma 4.7. [19] Let (X, MM,N , T ) and (Y, MM,N , T ) be intuitionistic fuzzy metric spaces. If we defineM((x, y), (x0 , y 0 ), t) T (MM,N (x, x0 , t), MM,N (y, y 0 , t)).then (X Y, M, T ) is an intuitionistic fuzzy metric space and the topology inducedon X Y is the product topology.Theorem 4.8. Let S and P be nonempty subsets of intuitionistic fuzzy metricspaces (X, MM,N , Tn ) and (Y, MM,N , Tn ), respectively. Suppose that P is compact.If S is IF-boundedly compact or approximately compact, then so is S P .Proof. If S is IF-boundedly compact, we show that any sequence (sn , pn ) in S Pwhich is IF-bounded has a convergent subsequence. Indeed, by definition of theproduct intuitionistic fuzzy metric, sn is IF-bounded and since S is IF-boundedlycompact, sn sn0 s0 X. By compactness of P , pn pn0 p0 P . Hence,(sn , pn ) (sn00 , pn00 ) (s0 , p0 ) X Y .If S is approximately compact, let (x, y) be any element in X Y and suppose that(sn , pn ) is a sequence in S P which converges in distance to (x, y), that is,lim M((sn , pn ), (x, y), t) M(S P, (x, y), t).n www.SID.ir

Archive of SIDSome Results on Intuitionistic Fuzzy Spaces63By compactness of P , pn pn0 p0 P . Hence, limn M((sn0 , p0 ), (x, y), t) M(S P, (x, y), t) solim Tn (MM,N (sn0 , x, t), MM,N (p0 , y, t)) Tn (MM,N (S, x, t), MM,N (P, y, t)).n0 Since MM,N (p0 , y, t) L MM,N (P, y, t) then limn0 MM,N (sn0 , x, t) L MM,N (S, x, t)which implies limn0 MM,N (sn0 , x, t) MM,N (S, x, t). Hence, sn0 converges indistance to x and since S is approximately compact, sn0 sn00 s0 S. Therefore, (sn , pn ) (sn00 , pn00 ) (s0 , p0 ) S P , i.e., S P is approximately compact. Theorem 4.9. Let B and C be nonempty subsets of an intuitionistic fuzzy metricspace (X, MM,N , T ). If B is approximately compact and C is compact, then K {b B : c C, MM,N (b, c, t) MM,N (B, c, t)} is compact.Proof. Let yn be a sequence in K and for every n N choose cn in C so that yn minimizes the distance from B to cn . Since C is compact, cn cn0 c0 C. Hence,for every ε 0, there exists n0 such that for all n0 n0 , MM,N (cn0 , c0 , t) L (Ns (ε), ε), and therefore, for all n0 n0 ,MM,N (B, c0 , t) L MM,N (yn0 , c0 , t) L T 2 (MM,N (B, c0 , t 2ε), MM,N (cn0 , c0 , ε), MM,N (cn0 , c0 , ε)) L T 2 (MM,N (B, c0 , t 2ε), (Ns (ε), ε), (Ns (ε), ε)).Since ε 0 was arbitrary, then MM,N (B, c0 , t) limn0 MM,N (yn0 , c0 , t).Therefore, yn0 converges in distance to c0 , so it converges sub-sequentially. Acknowledgements. It is a pleasure to thank Professor Glad Deschrijver forhelpful suggestions about the subject of this paper.References[1] M. Amini and R. Saadati, Topics in fuzzy metric space, Journal of Fuzzy Math., 4 (2003),765-768.[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87–96.[3] C. Cornelis, G. Deschrijver and E. E. Kerre, Classification of intuitionistic fuzzy implicators:an algebraic approach, In H. J. Caulfield, S. Chen, H. Chen, R. Duro, V. Honaver, E. E.Kerre, M. Lu, M. G. Romay, T. K. Shih, D. Ventura, P. P. Wang and Y. Yang, editors,Proceedings of the 6th Joint Conference on Information Sciences, (2002), 105-108.[4] C. Cornelis, G. Deschrijver and E. E. Kerre, Intuitionistic fuzzy connectives revisited, Proceedings of the 9th International Conference on Information Processing and Management ofUncertainty in Knowledge-Based Systems, (2002), 1839-1844.[5] G. Deschrijver, C. Cornelis and E. E. Kerre, On the representation of intuitionistic fuzzytnorms and t-conorms, IEEE Transactions on Fuzzy Systems, 12 (2004), 45–61.[6] G. Deschrijver and E. E. Kerre, On the relationship between some extensions of fuzzy settheory, Fuzzy Sets and Systems, 23 (2003), 227-235.[7] M. S. Elnaschie, On the uncertainty of Cantorian geometry and two-slit expriment, Chaos,Soliton and Fractals, 9 (1998), 517–529.[8] M. S. Elnaschie, On a fuzzy Kahler-like manifold which is consistent with two-slit expriment,Int. Journal of Nonlinear Science and Numerical Simulation, 6 (2005), 95–98.[9] M. S. Elnaschie, A review of E infinity theory and the mass spectrum of high energy particlephysics, Chaos, Soliton and Fractals, 19 (2004), 209–236.www.SID.ir

Archive of SID64S. B. Hosseini, D. O‘Regan and R. Saadati[10] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and System,64 (1994), 395–399.[11] A. George and P. Veeramani, On some results of analysis for fuzzy metric spaces, Fuzzy Setsand Systems, 90 (1997), 365–368.[12] S. B. Hosseini , J. H. Park and R. Saadati , Intuitionistic fuzzy invariant metric spaces, Int.Journal of Pure Appl. Math. Sci., 2(2005).[13] C. M. Hu , C-structure of FTS. V. fuzzy metric spaces, Journal of Fuzzy Math., 3(1995)711–721.[14] P. C. Kainen , Replacing points by compacta in neural network approximation, Journal ofFranklin Inst., 341 (2004), 391–399.[15] E. Kreyszig, Introductory functional analysis with applications, John Wiley and Sons, NewYork, 1978.[16] R. Lowen, Fuzzy set theory, Kluwer Academic Publishers, Dordrecht, 1996.[17] J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals, 22 (2004) 10391046.[18] R. Saadati and S. M. Vaezpour, Some results on fuzzy Banach spaces, Journal of Appl. Math.Comput., 17 (2005), 475–484.[19] R. Saadati and J. H. Park, On the intuitionistic fuzzy topological spaces, Chaos, Solitons andFractals, 27 (2006), 331–344.[20] B. Schweizer and A. Sklar, Statistical metric spaces, Pacific Journal of Math., 10 (1960),314–334.[21] Y. Tanaka, Y. Mizno and T. Kado, Chaotic dynamics in Friedmann equation, chaos, solitonand fractals, 24 (2005), 407–422.[22] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338–353.S. B. Hosseini, Islamic Azad University-Nour Branch, Nour, IranDonal O’Regan, Department of Mathematics, National University of Ireland, Galway, IrelandE-mail address: donal.oregan@nuigalway.ieReza Saadati Department of Mathematics, Islamic Azad University-Ayatollah AmolyBranch, Amol, Iran and Institute for Studies in Applied Mathematics 1, Fajr 4, Amol46176-54553, IranE-mail address: rsaadati@eml.cc Corresponding authorwww.SID.ir

Iranian Journal of Fuzzy Systems Vol. 4, No. 1, (2007) pp. 53-64 53 SOME RESULTS ON INTUITIONISTIC FUZZY SPACES S. B. HOSSEINI, D. O'REGAN AND R. SAADATI Abstract. In this paper we define intuitionistic fuzzy metric and normed . of fuzzy metric (normed) spaces introduced by George and Veeramani [10, 11] and