In this paper we work out two fixed point theorems for fuzzy mappings on Hilbert spaces. The proofs rely on the paralellogram law in Hilbert spaces

B. S. Choudhury

P. N. Dutta

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Mathematical Communications 7(2002), 91-96 91Fixed points of fuzzy mappings in Hilbert spacesP.N.Dutta∗ and B. S. Choudhury†Abstract. In this paper we work out two fixed point theorems forfuzzy mappings on Hilbert spaces. The proofs rely on the paralellogramlaw in Hilbert spaces.Key words: fuzzy mapping, Hilbert space, fixed point, approximatequantityAMS subject classifications: 54H25Received February 6, 2002 Accepted July 4, 20021. IntroductionHeilpern [2] introduced the concept of fuzzy mappings as a mapping from an ar-bitrary set to one subfamily of fuzzy sets in a metric linear space and proved afixed point theorem for fuzzy mappings. Various authors extended and generalisedHeilpern’s result [1], [3], [4], [5], [6] and [7]. In the present paper, we prove fixedpoint theorems of fuzzy mappings as introduced by Heilpern applied to Hilbertspaces.2. PreliminariesIn the following discussions we mainly follow the definitions and notations due toHeilpern [2].Let H be a Hilbert space and F (H) be collection of all fuzzy sets in H . LetA ∈ F (H) and α ∈ [0, 1]. The α-level set of A, denoted by Aα is defined asAα = {x : A(x) ≥ α} if α ∈ (0, 1]A0 = {x : A(x) > 0},where B stands for the closure of a set B.Definition 1. A fuzzy subset A of H is said to be an approximate quantityiff its α-level set is a nonfuzzy compact convex subset of H for each α ∈ [0, 1] andsupx∈H A(x) = 1.∗Department of Mathematics, Hooghly Mohsin College, Chinsurah, West Bengal, India, e-mail:pndut1@cal3.vsnl.net.in†Department of Mathematics, B.E. College (D.U.), Howrah 711 103, West Bengal, India, e-mail:bsc@math.becs.ac.in92 P.N.Dutta and B. S. ChoudhuryFrom the collection F (H), the subcollection of all approximate quantities isdenoted by W (H).Definition 2. Let A,B ∈W (H) and α ∈ [0, 1], then(i) Pα(A,B) = infx∈Aα,y∈Bα‖x− y‖(ii) Dα(A,B) = dist (Aα, Bα), where “dist” denotes the Housdorff metric betweenAα and Bα(iii) D(A,B) = supαDα(A,B) and(iv) P (A,B) = supαPα(A,B).It is to be noted that for any ‘α’, Pα is a nondecreasing as well as continuousfunction.Definition 3. Let A,B ∈ W (H). An approximate quantity A is said to bemore accurate than B, denoted by A ⊂ B, iff A(x) < B(x) for each x ∈ H. Therelation ⊂ induces a partial ordering on W (H),Definition 4. A mapping F from the set H onto W (H) is said to be a fuzzymapping. Any x ∈ H is called a fixed point of a mapping F : H →W (H) if{x} ⊂ Fxwhere {x} is the fuzzy set with a membership function equal to the chracteristicfunction of the crist set {x}.We shall use the following lemma due to Heilpern [2].Lemma 1. Let x ∈ H, A ∈ W (H), then {x} ⊂ A if and only if Pα(x,A) = 0for each α ∈ [0, 1].Lemma 2. Pα(x,A) ≤ ‖x− y‖+ Pα(y,A) for any x, y ∈ H.Lemma 3. If {x0} ⊂ A, then Pα(x0, B) ≤ Dα(A,B) for each B ∈W (H).3. Main resultsIn this section we prove common fixed point theorems for a pair of fuzzy mappings.Theorem 1. Let H be a Hilbert space, F and G are fuzzy mappings from Hinto W (H) satisfyingD2(Fx,Gy) ≤ a‖x− y‖2 + bP 2α(x, Fx) + cP 2α(y,Gy)+e2{P 2α(x,Gy) + P2α(y, Fx)}(1)for all x, y in H and for all α ∈ [0, 1] and a, b, c, e are nonnegative numbers satisfyinga+ b+ c+ 2e < 1. (2)Then there exists a point z in H such that{z} ⊂ Fz ∩Gz.Fixed points of fuzzy mappings in Hilbert spaces 93Proof. Let x0 ∈ H . We construct the sequence {{xn}} as follows.{x1} ⊂ Fx0, {x2} ⊂ Gx1, . . . , {x2n+1} ⊂ Fx2n, {x2n+2} ⊂ Gx2n+1and‖xi − xi+1‖ ≤ D(Fxi−1, Gxi), i = 1, 2, . . .Now,‖x2n − x2n+1‖2 ≤ D2(Fx2n, Gx2n−1)≤ a‖x2n − x2n−1‖2 + bP 2α(x2n, Fx2n) + cP 2α(x2n−1, Gx2n−1)+e2{P 2α(x2n, Gx2n−1) + P 2α(x2n−1, Fx2n)}≤ a‖x2n − x2n−1‖2 + b‖x2n − x2n+1‖2 + c‖x2n−1 − x2n‖2+e2{‖x2n − x2n‖2 + ‖x2n−1 − x2n+1‖2}≤ a‖x2n − x2n−1‖2 + b‖x2n − x2n+1‖2 + c‖x2n−1 − x2n‖2+e2{‖(x2n−1 − x2n) + (x2n − x2n+1)‖2}≤ a‖x2n − x2n−1‖2 + b‖x2n − x2n+1‖2 + c‖x2n−1 − x2n‖2+e{‖x2n−1 − x2n‖2 + ‖x2n − x2n+1‖2}which gives‖x2n − x2n+1‖2 ≤ k1‖x2n − x2n−1‖2where0 < k1 =a+ c+ e1− b− e < 1.Again,‖x2n−1 − x2n‖2 ≤ D2(Fx2n−2, Gx2n−1)≤ a‖x2n−2 − x2n−1‖2 + bP 2α(x2n−2, Fx2n−2) + cP 2α(x2n−1, Gx2n−1)+e2{P 2α(x2n−2, Gx2n−1) + P 2α(x2n−1, Fx2n−2)}≤ a‖x2n−2 − x2n−1‖2 + b‖x2n−2 − x2n−1‖2 + c‖x2n−1 − x2n‖2+e2{‖x2n−2 − x2n‖2 + ‖x2n−1 − x2n−1‖2}≤ a‖x2n−2 − x2n−1‖2 + b‖x2n−2 − x2n−1‖2 + c‖x2n−1 − x2n‖2+e{‖x2n−2 − x2n−1‖2 + ‖x2n−1 − x2n‖2}which gives‖x2n−1 − x2n‖2 ≤ k2‖x2n−2 − x2n−1‖2where0 < k2 =a+ b+ e1− c− e < 1.Choosing k = max{k1, k2}, it follows that‖xn+1 − xn‖2 ≤ k‖xn − xn−1‖2where 0 < k < 1.94 P.N.Dutta and B. S. ChoudhuryHence {xn} is a Cauchy sequence in H and therefore it converges to a limit inH . We assumelimn→∞ xn = z.Again, using Lemma 3 and for all α ∈ [0, 1]P 2α(x2n+2, F z) ≤ D2α(Gx2n+1, F z)≤ D2(Gx2n+1, F z)≤ a‖x2n+1 − z‖2 + bP 2α(x2n+1, Gx2n+1) + cP 2α(z, Fz)+e2{P 2α(z,Gx2n+1) + P 2α(x2n+1, F z)}≤ a‖x2n+1 − z‖2 + b‖x2n+1 − x2n+2‖2 + cP 2α(z, Fz)+e2{P 2α(z, x2n+2) + P 2α(x2n+1, F z)}Making n→∞ and using the fact that Pα is continuous,P 2α(z, Fz) ≤(c+e2)P 2α(z, Fz).As {c + (e/2)} < 1, it follows that P 2α(z, Fz) = 0, hence by Lemma1, {z} ⊂ Fz.Similarly, {z} ⊂ Gz. Hence, {z} ⊂ Fz ∩Gz. ✷Theorem 2. Let H be a Hilbert space and F and G fuzzy mappings from Hinto W (H) satisfyingD2(Fx,Gy) ≤ qmax{‖x− y‖2, P 2α(x, Fx), P 2α(y,Gy),1/2{P 2α(x,Gy) + P 2α(y, Fx))}(3)for all x, y in H and for all α ∈ [0, 1] and q ∈ (0, 1/2). Then there exists a point zin H such that {z} ⊂ Fz ∩Gz.Proof. Let x0 ∈ H , we construct the sequence {xn} as in Theorem 1 and corre-spondingly‖x2n − x2n+1‖2 ≤ D2(Fx2n, Gx2n−1)≤ qmax [‖x2n − x2n−1‖2, P 2α(x2n, Fx2n), P 2α(x2n−1, Gx2n−1),12{P 2α(x2n, Gx2n−1) + P2α(x2n−1, Fx2n)}]≤ qmax [‖x2n − x2n−1‖2, ‖x2n − x2n+1‖2, ‖x2n − x2n−1‖2,12‖x2n−1 − x2n+1‖2]≤ qmax[‖x2n − x2n−1‖2, 12‖x2n−1 − x2n+1‖2]≤ qmax [‖x2n − x2n−1‖2, ‖x2n − x2n−1‖2 + ‖x2n − x2n+1‖2]≤ qmax [‖x2n − x2n−1‖2 + ‖x2n − x2n+1‖2]which yields‖x2n − x2n+1‖2 ≤ qq − 1‖x2n − x2n−1‖2. (4)Fixed points of fuzzy mappings in Hilbert spaces 95Again,‖x2n − x2n−1‖2 ≤ D2(Fx2n−2, Gx2n−1)≤ qmax [‖x2n−2 − x2n−1‖2, P 2α(x2n−2, Fx2n−2), P 2α(x2n−1, Gx2n−1),12{P 2α(x2n−2, Gx2n−1) + P2α(x2n−1, Fx2n−2)}]≤ qmax [‖x2n−2 − x2n−1‖2, ‖x2n−2 − x2n−1‖2, ‖x2n−1 − x2n‖2,12‖x2n−2 − x2n‖2]≤ qmax[‖x2n−2 − x2n−1‖2, 12‖x2n−2 − x2n‖2]≤ qmax [‖x2n−2 − x2n−1‖2, ‖x2n−2 − x2n−1‖2 + ‖x2n−1 − x2n‖2]≤ qmax [‖x2n−2 − x2n−1‖2 + ‖x2n−1 − x2n‖2]which yields‖x2n − x2n−1‖2 ≤ q1− q ‖x2n−2 − x2n−1‖2 (5)From (4) and (5) it follows that‖xn+1 − xn‖2 ≤ k1‖xn − xn−1‖2where0 < k1 =q1− q < 1.Hence, {xn} is a Cauchy sequence in H and therefore it converges to a limit in H .We assumelimn→∞ xn = z.Again, using Lemma 3,P 2α(x2n+2, F z) ≤ D2α(Gx2n+1Fz)≤ D2(Gx2n+1, F z)≤ qmax [‖z − x2n+1‖2, P 2α(z, Fz), P 2α(x2n+1, Gx2n+1),12{P 2α(z,Gx2n+1) + P2α(x2n+1, F z)}]≤ qmax [‖z − x2n+1‖2, P 2α(z, Fz), ‖x2n+1 − x2n+2‖,12{‖z − x2n+2‖2 + P 2α(x2n+1, F z)}].Making n→∞ and using the fact that Pα is continuous,P 2α(z, Fz) ≤ qmax{P 2α(z, Fz),12P 2α(z, Fz)}≤ qP 2α(z, Fz)As q ∈ (0, 1/2), it follows that P 2α(z, Fz) = 0. Hence, by Lemma1 {z} ⊂ Fz.Similarly, {z} ⊂ Gz. Hence, {z} ⊂ Fz ∩Gz. ✷Acknowledgement. This work is supported partly by University Grants Com-mission, Govt. of India under Minor Research Project No. F-PSW-19/97(ERO)and partly by Dept. of Atomic Energy, Govt. of India under NBHM project No.48/2/2000 - R&D-II/1317. The supports are gratefully acknowledged.96 P.N.Dutta and B. S. ChoudhuryReferences[1] R.K.Bose, D. Sahani, Fuzzy mappings and fixed point theorems, Fuzzy Setsand Systems 21(1987), 53–58.[2] S.Heilpern, Fuzzy mappings and fixed point theorems, J. Math. Anal. Appl.83(1981), 566–569.[3] B. S. Lee, S. J. Cho, A fixed point theorem for contractive - type fuzzy map-pings, Fuzzy Sets and Systems 61(1994), 309–312.[4] B. S. Lee, M.K.Kang, A generalisation of Som and Mukherjee’s fixed pointtheorem, Indian Journal of Mathematics 41(1999), 205–209.[5] B.E.Rhoades, Fixed points of some fuzzy mappings, Soochow Journal ofMathematics 22(1996), 111–115.[6] S. L. Singh, R. 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