We prove a fixed fuzzy point theorem for fuzzy contraction
mappings (in the S. Heilpern’s sense) over a complete metric
space, and as a consequence we obtain a fixed point theorem in
the context of intuitionistic fuzzy sets

Estruch, Vicente D.

Vidal, Anna

OpenstarTs

Rend. Istit. Mat. Univ. TriesteSuppl. 2 Vol. XXXII, 39–45 (2001)A Note on Fixed Fuzzy Pointsfor Fuzzy MappingsVicente D. Estruch and Anna Vidal (∗)Summary. - We prove a fixed fuzzy point theorem for fuzzy contrac-tion mappings (in the S. Heilpern’s sense) over a complete metricspace, and as a consequence we obtain a fixed point theorem inthe context of intuitionistic fuzzy sets.1. IntroductionAfter the introduction of the concept of a fuzzy set by Zadeh, severalresearches were conducted on the generalizations of the concept of afuzzy set. The idea of intuitionistic fuzzy set is due to Atanassov [1],[2], [3] and recently Çoker [4] has defined the concept of intuition-istic fuzzy topological space which generalizes the concept of fuzzytopological space introduced by Chang [5]. Heilpern [6] introducedthe concept of a fuzzy mapping and proved a fixed point theoremfor fuzzy contraction mappings which is a generalization of the fixedpoint theorem for multivalued mappings of Nadler [7]. In this paperwe give a fixed fuzzy point theorem for fuzzy contraction mappingsover a complete metric space, which is a generalization of the givenby S. Heilpern for fixed points. Then, we introduce the concept(∗) Authors’ addresses: Vicente D. Estruch, Escuela Politécnica Superior deGandia, 46730-Grau de Gandia, Valencia, Spain, e-mail: vdestruc@mat.upv.esAnna Vidal, Escuela Politécnica Superior de Gandia, 46730-Grau de Gandia, Va-lencia, Spain, e-mail: avidal@mat.upv.esAMS Subject Classification: 54A40Keywords: Fixed point, fuzzy mapping, intuitionistic fuzzy set.Partially supported by the grants from UPV ”Incentivo a la Investigación / 99”and from Generalitat Valenciana GV00-122-140 V.D. ESTRUCH and A. VIDALof intuitionistic fuzzy mapping and give an intuitionistic version ofHeilpern’s mentioned theorem.2. PreliminariesLet X be a nonempty set and I = [0, 1]. A fuzzy set of X is anelement of IX . For A,B ∈ IX we denote A ⊂ B iff A(x) ≤ B(x),∀x ∈ X.Definition 2.1. (Atanassov [3]) An intuitionistic fuzzy set (i-fuzzyset, for short) A of X is an object having the form A =< A1, A2 >where A1, A2 ∈ IX and A1(x) + A2(x) ≤ 1, ∀x ∈ X.We denote IFS(X) the family of all i-fuzzy sets of X.Remark 2.2. 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 ordinarysubset {x} of X. For α ∈]0, 1] the fuzzy point [8] xα of X is the fuzzyset of X given by xα(x) = α and xα(z) = 0 if z 6= x. Now we givethe following definition.Definition 2.3. Let xα be a fuzzy point of X. We will say < xα, 1−xα > is an i-fuzzy point of X and it will be denoted by [xα]. Inparticular [x] =< {x}, 1 − {x} > will be called an i-point of X.Definition 2.4. (Atanassov [3]) Let A,B ∈ IFS(X). Then A ⊂ Biff A1 ⊂ B1 and B2 ⊂ A2.Remark 2.5. Notice [xα] ⊂ A iff xα ⊂ A1.Let (X, d) be a metric linear space. Recall the α − level Aα ofA ∈ IX is defined by Aα = {x ∈ X : A(x) ≥ α} for each α ∈]0, 1],and A0 = cl({x ∈ X : A(x) > 0}) where cl(B) is the closure of B.Heilpern [6] called fuzzy mapping a mapping from the set X into afamily W (X) ⊂ IX defined as follows: A ∈ W (X) iff Aα is compactand convex in X for each α ∈ [0, 1] and sup{A(x) : x ∈ X} = 1. Inthis context we give the following definitions.A NOTE ON FIXED FUZZY POINTS etc. 41Definition 2.6. Let A,B ∈ W (X), α ∈ [0, 1]. Definepα(A,B) = inf{d(x, y) : x ∈ Aα, y ∈ Bα},Dα = H(Aα, Bα),D(A,B) = supα Dα(A,B),where His the Hausdorff distance.For x ∈ X we write pα(x,B) instead of pα({x}, B).Definition 2.7. Let X be a metric space and α ∈ [0, 1]. Considerthe following family Wα(X):Wα(X) = {A ∈ IX : Aα is nonempty and compact}.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 2.2.We will use the following lemmas which are adequate modifica-tions of the ones given in [6] for the family W (X), when (X, d) is ametric space.Lemma 2.8. Let x∈X and A ∈ Wα(X). Then xα⊂A if pα(x,A)=0.Lemma 2.9. pα(x,A) ≤ d(x, y)+pα(y,A), for x, y ∈ X, A ∈ Wα(X).Lemma 2.10. If xα ⊂ A, then pα(x,B) ≤ Dα(A,B), for each A,B ∈Wα(X)3. Fixed fuzzy point theoremIn mathematical programming, problems are expressed as optimizingsome goal function given certain constraints and there are real-lifeproblems that consider multiple objectives. Generally, it is very dif-ficult to get a feasible solution that carries us to the optimum of allthe objective functions. A possible method of resolution that is quiteuseful is one using Fuzzy Sets. The idea is to relax the pretenses of42 V.D. ESTRUCH and A. VIDALoptimization by means of a subjective gradation which can be mod-elled into fuzzy membership functions µi . If F = ∩µi the objectivewill be to search x such that max F = F (x). If maxF = 1, thenthere exists x such that F (x) = 1, but if maxF = α,α ∈]0, 1[ thesolution of the multiobjective optimization is a fuzzy point xα andF (x) = α.In a more general sense than the one given by Heilpern, a map-ping F : X −→ IX is a fuzzy mapping over X ([9]) and (F (x))(x)is the fixed degree of x for F . In this context we give the followingdefinition.Definition 3.1. Let xα be a fuzzy point of X. We will say that xαis a fixed fuzzy point of the fuzzy mapping F over X if xα ⊂ F (x)(i.e., the fixed degree of x is at least α). In particular, and accordingto [6] , if {x} ⊂ F (x) we say that x is a fixed point of F .The next proposition is a generalization of [6] Theorem 3.1.Theorem 3.2. Let α ∈ ]0, 1] and let (X, d) be a complete metricspace. Let F be a fuzzy mapping from X into Wα(X) satisfying thefollowing condition: there exists q ∈ ]0, 1[ such thatDα(F (x), F (y)) ≤ qd(x, y), for each x, y ∈ X.Then there exists x ∈ X such that xα is a fixed fuzzy point of F .In particular if α = 1, then x is a fixed point of F .Proof. Let x0 ∈ X. Since (F (x0))α 6= ∅, then there exists x1 ∈ Xsuch that x1 ∈ (F (x0))α. there exists x2 ∈ (F (x0))α. Since (F (x1))αis a nonempty compact subset of X, then there exists x2 ∈ (F (x1))αsuch thatd(x1, x2) = d(x1, (F (x1))α) ≤ Dα(F (x0), F (x1))by Lemma 2.10. By induction we construct a sequence (xn) in Xsuch that xn ∈ (F (xn−1))α and d(xn, xn+1) ≤ Dα(F (xn−1), F (xn)).In a similar way to the proof of [6] Theorem 3.1., it is provedthat (xn) is a Cauchy sequence. Suppose (xn) converges to x ∈ X.Now, by Lemmas 2.9, 2.10 we haveA NOTE ON FIXED FUZZY POINTS etc. 43pα(x, F (x)) ≤ d(x, xn)+pα(xn, F (x))≤d(x, xn)+Dα(F (xn−1), F (x))≤ d(x, xn)+qd(xn−1, x)In consequence pα(x, F (x)) = 0 and by Lemma 2.8, xα ⊂ F (x).Remark 3.3. Theorem 3.1 of [6] states the existence of a fixed pointof the fuzzy mapping F : X −→ W (X) whenever D(F (x), F (y)) ≤qd(x, y) for each x, y ∈ X, being q ∈]0, 1[. Now, in the light of theabove theorem, it is clear that the condition D(F (x), F (y)) ≤ qd(x, y)can be weakened to D1(F (x), F (y)) ≤ qd(x, y).The next example illustrates when the above theorem has certainadvantages if compared with Heilpern’s (Theorem 3.1 of [6]).Example 3.4. Take a, b, c ∈] − ∞,+∞[, such that a < b < c. LetX = {a, b, c} and d : X × X −→ [0,+∞[ the Euclidean metric. Letα ∈]0, 0.5[ and suppose F : X −→ IX defined byF (a)(x) =1 if x=a2α if x=bα/2 if x=cF (b)(x) =1 if x=aα if x=bα/2 if x=cF (c)(x) =1 if x=aα if x=b0 if x=cThenF (a)1 = F (b)1 = F (c)1 = {a}, F (a)α = F (b)α = F (c)α = {a, b},F (a)α2= F (b)α2= {a, b, c} and F (c)α2= {a, b} .ConsequentlyD1(F (x), F (y)) = H(F (x)1, F (y)1) = 0,Dα(F (x), F (y)) = H(F (x)α, F (y)α) = 0, ∀x, y ∈ X .44 V.D. ESTRUCH and A. VIDALBy Theorem 3.2 there exists a fixed fuzzy point x1 (a fixed point)and a fixed fuzzy point xα of the fuzzy mapping F . We can seeby the definition of F that a is a fixed point and bα is a fuzzypoint. Nevertheless Heilpern’s theorem is not useful in this exam-ple because Dα2(F (a), F (c)) = H({a, b, c} , {a, b}) ≥ d(a, c) and thenD(F (a), F (c)) = supr H(F (a)r, F (c)r) ≥ d(a, c).4. Fixed i-fuzzy pointDefinition 4.1. An i-fuzzy mapping over X is a mapping F fromX into IFS(X). We will say that [xα] is a fixed i-fuzzy point of Fif [xα] ⊂ F (x). In particular we will say that [x] is a fixed i-point ofF if [x] ⊂ F (x).Definition 4.2. Let (X, d) be a metric space and 0 < α ≤ 1. ForA,B ∈ Φα(X) we define D∗α(A,B) = max{Dα(A1, B1),Dα(1 −A2, 1 − B2)}. Clearly, D∗α is a pseudometric on Φα(X).Remark 4.3. Let A,B ∈ Wα(X). If we consider [A], [B] ∈ Φα(X)thenD∗α([A], [B]) = max{Dα(A,B),Dα(1−(1−A), 1−(1−B)) = Dα(A,B)Now, as a consequence of Theorem 3.2 we have the followingcorollary.Corollary 4.4. Let (X, d) be a complete metric space and let Fbe an i-fuzzy mapping from X into Φα(X), α ∈]0, 1], satisfying thefollowing condition: There exists q ∈]0, 1[ such thatD∗α(F (x), F (y)) ≤ qd(x, y), for each x, y ∈ XThen there exists x ∈ X such that [xα] is a fixed i-fuzzy point ofF . In particular if α = 1, then [x] is a fixed i-point of F .AcknowledgmentThe authors are grateful to the referee for his valuable suggestions.A NOTE ON FIXED FUZZY POINTS etc. 45References[1] K. Atanassov, Intuitionistic fuzzy sets, (Sofia) (V. Sgurev, ed.), 1984,(June 1983 Central Sci. and techn. Library, Bulg. Academy of Sciences).[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,Preprint IM-MFAIS-1-88, 1988.[4] D. Çoker, An introduction to intuitionistic fuzzy topological spaces,Fuzzy Sets and Systems 88 (1997), 81–89.[5] C.L. Chang, Fuzzy topological spaces, J. Math.Anal. Appl. 24 (1986),182–190.[6] S. Heilpern, Fuzzy mappings and fixed point theorem, J. Math. Anal.Appl. 83 (1981), 566–569.[7] S.B. Nadler, Multivalued contraction mapping, Pacific. J. Math. 30(1969), 475–488.[8] P-M. Pu and Y-M. Liu, Fuzzy Topology. I. Neighborhood Structureof a Fuzzy Point and Moore-Smith Convergence., J. Math. Anal. Appl.76 (1980), 571–599.[9] Chang Shih-sen, Fixed degree for fuzzy mappings and a generalizationof Ky Fan’s theorem, Fuzzy Sets and Systems 24 (1987), 103–112.Received January 18, 2000.

A Note on Fixed Fuzzy Points for Fuzzy Mappings

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A Note on Fixed Fuzzy Points for Fuzzy Mappings

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