In this paper, we prove a fixed fuzzy point theorem for
fuzzy mappings over a complete metric space

B. E. Rhoades

D. Turkoglu

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Mathematical Communications 10(2005), 115-121 115A fixed fuzzy point for fuzzy mapping in completemetric spaces∗D.Turkoglu† and B.E.Rhoades‡Abstract. In this paper, we prove a fixed fuzzy point theorem forfuzzy mappings over a complete metric space.Key words: fixed point, fuzzy mapping, intuitionistic fuzzy setAMS subject classifications: 54H25, 54A40Received December 15, 2004 Accepted September 17, 20051. IntroductionAfter the introduction of the concept of a fuzzy set by Zadeh [11], several researcheswere conducted on the generalizations of the concept of a fuzzy set. The ideaof an intuitionistic fuzzy set is due to Atanassov [1] , [2] , [3] and C¸oker [5] hasdefined the concept of fuzzy topological spaces induced by Chang [4] . Heilpern[7], introduced the concept of fuzzy mapping and proved a fixed point theorem forfuzzy contraction mappings which is a generalization of the fixed point theoremfor multivalued mappings of Nadler [8] . Estruch and Vidal [6] give a fixed pointtheorem for fuzzy contraction mappings over a complete metric spaces which is ageneralization of the given Heilpern’s fixed point theorem. In this paper we give afixed point theorem for a fuzzy contraction mapping over a complete metric spacewhich is a generalization of the fixed point theorem given by Estruch and Vidal.They give a fixed point theorem under the condition Dα(F (x), F (y)) ≤ qd(x, y) foreach x, y ∈ X.We give a fixed point theorem under the condition Dα(F (x), F (y)) ≤K(M(x, y)) and some examples, where Dα, F, q,X,K and M are following defined.One of these example has been an extended version of their main theorem.2. PreliminariesLet X be a nonempty set and I = [0, 1]. A fuzzy set of X is an element of IX . ForA,B ∈ IX we denote A ⊆ B if and only if A(x) ≤ B(x) for each x ∈ X.∗Supported by a grant from the Gazi University, Turkey†Department of Mathematics, Faculty of Science and Arts, University of Gazi, Besevler, 06530-Ankara, Turkey, e-mail: dturkoglu@gazi.edu.tr‡Department of Mathematics, Indiana University, Bloomington, Indiana 47405 - 7106, USA,e-mail: rhoades@indiana.edu116 D.Turkoglu and B.E.RhoadesDefinition 1 [3]. An intuitionistic fuzzy set (i−fuzzy set) A of X is an objecthaving the form A =〈A1, A2〉where A1, A2 ∈ IX and A1(x) +A2(x) ≤ 1 for eachx ∈ X.We denote by IFS(X) the family of all i−fuzzy sets of X .Remark 1. If A ∈ IX , then A is identified with the i−fuzzy set 〈A, 1−A〉denoted by [A] .For x ∈ X we write {x} the characteristic function of the ordinary subset {x}of X . For α ∈ (0, 1] the fuzzy point [9] xα of X is the fuzzy set of X given byxα(x) = α and α = x. Now we give the following definition.Definition 2 [6]. Let xα be a fuzzy point of X. We will say that 〈xα, 1− xα〉is an i−fuzzy point of X and it will be denoted by [xα]. In particular [x] =〈{x}, 1− {x}〉 will be called an i−point of X.Definition 3 [3]. Let A,B ∈ IFS(X). Then A ⊂ B if and only if A1 ⊂ B1and B2 ⊂ A2.Remark 2. Notice [xα] ⊂ A if and only if xα ⊂ A1.Let (X, d) be a metric linear space (i. e., a complex or real vector space). Theα− level set of A, denoted by Aα, is defined byAα = {x : A(x) ≥ α} if α ∈ (0, 1],A0 = {x : A(x) > 0}where B denotes the closure of the (non fuzzy)set B. Heilpern [7] called a fuzzymapping a mapping from the set of X into a family W (X) ⊂ IX defined as follows:A ∈ W (X) if and only if Aα is compact and convex in X for each α ∈ [0, 1] andsup {Ax : x ∈ X} = 1. In this context we give the following definitions.Definition 4 [7]. Let A,B ∈W (X), α ∈ [0, 1]. Definepα(A,B) = inf {d(x, y) : x ∈ Aα, y ∈ Bα} ,Dα(A,B) = H(Aα, Bα),D(A,B) = supαDα(A,B),where H is the Hausdorff distance.For x ∈ X we write pα(x,B) instead of pα({x}, B).Definition 5 [6]. Let X be a metric space and α ∈ [0, 1]. Consider the followingfamily Wα(X) :Wα(X) = {A ∈ IX : Aα is nonempty, compact and convex}Now we define the family Φα(X) of i−fuzzy sets of X as follows:Φα(X) = {A ∈ IFS(X) : A1 ∈ Wα(X)}.Clearly, for α ∈ I, W (X) ⊂ Φα(X) in the sense of Remark 1.We need the following lemmas. Let (X, d) be metric space.Lemma 1. [7] Let x ∈ X and A ∈W (X). Then xα ⊂ A if pα(x,A) = 0.Lemma 2. [7] pα(x,A) ≤ d(x, y) + pα(y,A) for x, y ∈ X, A ∈ W (X).Lemma 3. [7] If xα ⊂ A, then pα(x,B) ≤ Dα(A,B), for each A,B ∈ W (X).A fixed fuzzy point for fuzzy mapping 1173. Fixed fuzzy point theoremIn mathematical programming, problems are expressed as optimizing some goalfunction given certain constraints and there are real-life problems that considermultiple objectives. Generally, it is very difficult to get a feasible solution thatbrings us to the optimum of all objective functions. A possible method of resolutionthat is quite useful is the one using Fuzzy Sets. The idea is to relax the pretensesof optimization by means of a subjective gradation which can be modelled intofuzzy membership functions µi. If ∩µi the objective will be to search x such thatmaxF = F (x). If maxF = 1, then there exists x such that F (x) = 1, but ifmaxF = α, α ∈ (0, 1), the solution of the multiobjective optimization is a fuzzypoint xα and F (x) = α˙.In a more general sense than the one given by Heilpern, a mapping F : X → IXis a fuzzy mapping over X ([10]) and (F (x))(x) is the fixed degree of x for F . Inthis context we give the following definition.Definition 6 [6]. Let xα be a fuzzy point of X. We will say that xα is a fixedfuzzy point of the fuzzy mapping F over X if xα ⊂ F (x) (i. e., the fixed degree ofx is at least α). In particular, and according to [7], if {x} ⊂ F (x) , we say that xis a fixed point of F .Theorem 1. Let α ∈ (0, 1] and (X, d) be a complete metric space. Let F be acontinuous fuzzy mapping from X into Wα(X) satisfying the following condition:There exists K : [0,∞) → [0,∞), K(0) = 0,K(t) < t for all t ∈ (0,∞) and Kis non-decreasing such thatDα(F (x), F (y)) ≤ K(M(x, y)) (1)for all x, y ∈ X, whereM(x, y) = max{d(x, y), pα(x, F (x)), pα(y, F (y)), pα(x, F (y)), pα(y, F (x))}.Then there exists x ∈ X such that xα is a fixed fuzzy point of F if and only if thereexists x0, x1 ∈ X such that x1 ∈ (Fx0)α with∞∑n=1Kn(d(x0, x1)) <∞. In particular,if α = 1, then x is a fixed point of F .Proof. If there exists x ∈ X such that xα is a fixed fuzzy point of F , thenxα ⊂ F (x) and d(x, x) = 0, 0 = K(0) = K2(0) = ... = Kn(0) = ... and∞∑n=1Kn(d(x, x)) = 0. Let x0 ∈ X and suppose that there exists x1 ∈ (F (x0))αsuch that∞∑n=1Kn(d(x0, x1)) <∞. Since (F (x1))α is a nonempty compact subset ofX , then there exists x2 ∈ (F (x1))α such thatd(x1, x2) = pα(x1, F (x1)) ≤ Dα(F (x0), F (x1))by Lemma 3. By induction we construct a sequence {xn} in X such that xn ∈(F (xn−1))α and d(xn, xn+1) ≤ Dα(F (xn), F (xn−1)).118 D.Turkoglu and B.E.RhoadesNoting that K is nondecreasing and using inequality (1), we haved(xn, xn+1) ≤ Dα(F (xn), F (xn−1))≤ K(max{d(xn−1, xn), pα(xn−1, F (xn−1)), pα(xn, F (xn)),pα(xn−1, F (xn)), pα(xn, F (xn−1))})which implies thatd(xn, xn+1) ≤ K(max{d(xn−1, xn), d(xn, xn+1)}). (2)Suppose d(xn, xn+1) > d(xn−1, xn) for some n. Then from (2) and K(t) < t forall t ∈ (0,∞), we haved(xn, xn+1) ≤ K(d(xn, xn+1)) < d(xn, xn+1)which is a contradiction. Therefore, we haved(xn, xn+1) ≤ Dα(F (xn−1), F (xn))≤ K(d(xn−1, xn))= K(pα(xn−1, F (xn−1)))≤ K(Dα(F (xn−1), F (xn−2)))≤ K(K(d(xn−1, xn−2)))...≤ Kn(d(x0, x1)).Hence we obtaind(xn, xn+m) ≤ d(xn, xn+1) + . . .+ d(xn+m−1, xn+m)≤ Kn(d(x0, x1)) + . . .+Kn+m−1(d(x0, x1))=n+m−1∑k=nKk(d(x0, x1)).Since∞∑n=1Kn(d(x0, x1)) <∞, it follows that there exists r such that d(xn, xn+m) <r. Therefore the sequence {xn} is a Cauchy sequence in X . Suppose {xn} convergesto x ∈ X . Now by Lemmas 1 and 2 we havepα(x, F (x)) ≤ d(x, xn) + pα(xn, F (x))≤ d(x, xn) +Dα(F (xn−1), F (x))≤ d(x, xn) +K(d(xn−1, x)).consequently, pα(x, F (x)) = 0 and by Lemma 1, xα ⊂ F (x). ✷Remark 3. K needs only be defined on the range d. If one replaces the metricwith an equivalent metric with d(x, y) < 1, then clearly Theorem 4 holds for K :A fixed fuzzy point for fuzzy mapping 119[0, 1)→ [0,∞). To apply Theorem 4, one needs a nondecreasing function K and xin X with∞∑n=1Kn(pα(x, F (x))) <∞.The following examples satisfy these conditions and therefore illustrate the gen-erality of Theorem 4. Let (X, d) be a complete metric space.Example 1. Suppose 0 <q <1. Let K(t) = qt for t≥0. Then Dα(F (x), F (y)) ≤K(M(x, y)) = qM(x, y) and Kn(pα(x, F (x))) = qnpα(x, F (x)) for any x in X. Itis known that there exist x ∈ X such that xα ⊂ F (x).Remark 4. Example 1 is a generalization of Theorem 3.2. of [6].Remark 5. Theorem 3.1. of [7]. A fixed point of the fuzzy mapping F : X →W (X) exists whenever D(F (x), F (y)) ≤ qd(x, y) for each x, y ∈ X, being q ∈ (0, 1).Now, in Example 1, it is clear that the condition D(F (x), F (y)) ≤ qd(x, y) can beweakened to D1(F (x), F (y)) ≤ qd(x, y).Example 2. Suppose that F satisfies Dα(F (x), F (y)) ≤ φ(M(x, y))d(x, y) forall x, y in X, where φ : [0,∞)→ [0, 1) and φ is nondecreasing . Then K(t) = tφ(t)is nondecreasing and K : [0,∞) → [0,∞). It follows by induction that Kn(t) ≤t[φ(t)]n, since φ(t) < 1 and∞∑n=1Kn(t) <∞.Example 3. Consider K(M(x, y)) =M(x, y)φ(M(x, y)) for all x, y in X, whereφ : [0,∞)→ [0,∞), φ(t) ≤ t for t ≤ 1. If t < 1, it follows that Kn(t) ≤ t[φ(t)]n. IfK is nondecreasing, then Theorem 4 can be applied.Example 4. K(M(x, y)) = M(x, y)φ(M(x, y)) for all x, y in X, where φ :[0,∞) → [0,∞), φ(qM(x, y)) ≤ qφ(M(x, y)) for q ∈ (0, 1). If φ(t) < 1, thenKn(t) ≤ K(t)[φ(t)]n for all n ≥ 2.Example 5. Assume that K is nondecreasing, K is convex on [0, 1) andK(M(x, y)) < M(x, y) for all x, y in X. If t < 1, K(t) < t for all 0 < t < 1,then K(t) = qt for some 0 < q < 1. It can be shown that Kn(t) ≤ qnt for all n and∞∑n=1Kn(t) <∞.Example 6. Let X = [0, 1] and let d : X ×X → IR+ be the Euclidean metric.Let α ∈ (0, 12)and suppose F : X → IX defined byF (0)(x) =1 , x = 0α, x ∈ (0, 12 ]α2 , x ∈ (12 , 1]F (1)(x) =1 , x = 02α, x ∈ (0, 12 ]α2 , x ∈ (12 , 1]and for z ∈ (0, 1)F (z)(x) =1 , x = 0α, x ∈ (0, 12 ]0, x ∈ (12 , 1].Then F (0)1 = F (z)1 = F (1)1 = {0}, F (0)α = F (z)α = F (1)α =[0, 12], and120 D.Turkoglu and B.E.RhoadesF (0)α2= F (1)α2= [0, 1], F (z)α2=[0, 12]. ConsequentlyD1(F (x), F (y)) = H(F (x)1, F (y)1) = 0, ∀x, y ∈ X,Dα(F (x), F (y)) = H(F (x)α, F (y)α) = 0, ∀x, y ∈ X,Dα2(F (x), F (y)) = H(F (x)α2, F (y)α2) = 0, ∀x, y ∈ {0, 1} and ∀x, y ∈ (0, 1),Dα2(F (x), F (y)) = H(F (x)α2, F (y)α2) =12, ∀x ∈ {0, 1} and ∀y ∈ (0, 1).Let φ : [0,∞)→ [0,∞), φ(t) = t21+t . Thus φ(t) = t t1+t < t. Let K(t) = t31+t = tφ(t).If t ≤ 1, it follows that Kn(t) ≤ t[φ(t)]n = t2n+1(1+t)n and∞∑n=1Kn(t) < ∞. ThereforeDα(F (x), F (y)) = 0 ≤ K(d(x, y)) for each x, y ∈ X. By Theorem 4, there existsa fixed fuzzy point (a fixed point ) of the fuzzy mapping F . We can see by thedefinition of F that 0 is a fixed point. Nevertheless, Heilpern’s theorem is not usefulin this example because Dα2(F (0), F (13 )) = H(F (0)α2 , F (13 )α2 ) =12 >13 = d(0,13 )and then D(F (0), F (13 )) = supkH(F (0)k, F (13 )k) > d(0,13 ).Acknowledgements. We are thankful to the referees for their valuable com-ments in modifying the first version of this paper.References[1] K.Atanassov, Intuitionistic fuzzy sets, in: VII ITKRs Session, (V. Sgurev,Ed.), Bulgarian Academy of Sciences, Sofia, 1984.[2] K.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20(1986),87-96.[3] K.Atanassov, Review and new results on intuitionistic fuzzy sets, PreprintIM-MFAIS-1-88, 1988.[4] C. L.Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24(1986), 182-190.[5] D. C¸oker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Setsand Systems 88(1997), 81-89.[6] V.D.Estruch, A.Vidal, A note on fixed fuzzy points for fuzzy mappings,Rend Istit. Univ. Trieste 32(2001), 39-45.[7] S.Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal. Appl.83(1981), 566-569.[8] S.B.Nadler, Multivalued contraction mapping, Pacific J. Math. 30(1969),475-488.[9] P.M.Pu, Y.M. Liu, Fuzzy topology I: Neighborhood structure of a fuzzy pointand Moore-Smith convergence, J. Math. Anal. Appl. 76(1980), 571-599.A fixed fuzzy point for fuzzy mapping 121[10] C. S. Sen, Fixed degree for fuzzy mappings and a generalization of Ky Fan’stheorem, Fuzzy Sets and Systems 24(1987), 103-112.[11] L.A. Zadeh, Fuzzy sets, Information and control 8(1965), 338-353.

A fixed fuzzy point for fuzzy mapping in complete metric spaces

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