In this paper, we give a fixed point theorem for fuzzy contraction mappings in quasi-pseudo-metric spaces which is a generalization of the corresponding one for metric spaces given by S. Heilpern

Gregori, Valentín

Pastor, José

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Rend. Istit. Mat. Univ. TriesteSuppl. Vol. XXX, 103{109 (1999)A Fixed Point Theorem for FuzzyContraction MappingsValentn Gregori and Jose Pastor ()Summary. - In this paper, we give a xed point theorem for fuzzycontraction mappings in quasi-pseudo-metric spaces which is ageneralization of the corresponding one for metric spaces givenby S. Heilpern.1. IntroductionS. Heilpern [1] introduced the concept of a fuzzy mapping, i.e., map-ping from an arbitrary set to a certain subfamily of fuzzy sets in ametric linear space X. He proved a xed point theorem for fuzzycontraction mappings which is a generalization of the xed pointtheorem for multivalued mappings of Nadler [3] arising from the set-representation of fuzzy sets [4]. In this paper we extend the result ofHeilpern to quasi- pseudo-metric spaces which are left K-sequentiallycomplete.() Authors` addresses: Valentn Gregori, Departamento de Matematica Apli-cada, Universidad Politecnica de Valencia, Escola Universitaria de Gandia Car-retera Nazaret-Oliva S.N. 46730-Grau de Gandia, Valencia, Spain, e-mail: vgre-gori@mat.upv.esJose Pastor, Departamento de Matematica Aplicada, Universidad Politecnica deValencia, Escola Universitaria de Gandia Carretera Nazaret-Oliva S.N. 46730-Grau de Gandia, Valencia, Spain, e-mail: jpastogi@mat.upv.esAMS Classication: 54A40.Keywords: Fuzzy mapping, left K-Cauchy sequence, quasi-pseudo-metric.While working on this paper the rst-listed author has been partially supportedby a grant from Ministerio de Educacion y Ciencia DGES PB 95-0737.104 VALENTIN GREGORI and JOSE PASTOR2. PreliminariesThe set of positive integers is denoted by N. Recall that a nonnega-tive real valued function d dened on a nonempty set X is said to bea quasi-pseudo-metric provided it satises the following properties:for every x; y; z 2 X,d(x; z)  d(x; y) + d(y; z)d(x; x) = 0.The set B"(x) = fy 2 X : d(x; y) < "g is the d-ball with centre xand radius " > 0. The topology T (d), having as a base the familyof all d-balls B"(x) with x 2 X and " > 0, is the topology on Xinduced by d. (X; d) is called a quasi-pseudo-metric space, if d isa quasi-pseudo-metric on X and we will suppose it is endowed withthe topology T (d), in the following.If d is a quasi-pseudo-metric onX, then d 1, dened by d 1(x; y) =d(y; x) whenever x; y 2 X, is also a quasi-pseudo-metric on X. Wewill denote B 1" the d 1-ball with centre x and radius " > 0. Onlyif confusion is possible we write d-closed or d 1-closed, for example,to distinguish the topological concept in (X; d) or (X; d 1). We willdenote min(d; d 1) by d ^ d 1. We will make use of the followingnotion, which has been studied by various authors under dierentnames (see e.g. [2], [5]).A sequence (xn) in a quasi-pseudo-metric space (X; d) is calledleft K-Cauchy if for each " > 0 there is k 2 N such that d(xr; xs) <" for all r; s 2 N with k  r  s. A quasi-pseudo-metric space(X; d) is said to be leftK-sequentially complete if each leftK-Cauchysequence in X converges (with respect to the topology T (d)).Let x be a point in X and A a nonempty subset of X. We denethe distance d(x;A) from x to A byd(x;A) = inffd(x; a) : a 2 Ag:Thus d(x;A) = 0 i x 2 clA, the closure of A in X.Now let A and B be nonempty subsets of X. We dene thedistance d(A;B) from A to B byd(A;B) = inffd(x; y) : x 2 A; y 2 BgA FIXED POINT THEOREM FOR FUZZY etc. 105and clearly d(A;B) 6= d(B;A) in general. Now, we dene the Haus-dor separation of A from B bydH(A;B) = supfd(a;B) : a 2 Ag:Thus we have dH(A;B)  0 with dH(A;B) = 0 i A  clB. Inaddition, the triangle inequalitydH(A;C)  dH(A;B) + dH(B;C)holds for all nonempty subsets A; B and C of X. In general, howeverdH(A;B) 6= dH(B;A).We dene the Hausdor distance, deduced from the quasi-pseudo-metric d, between nonempty subsets A and B of X byH(A;B) = maxfdH(A;B); dH(B;A)g:This is now symmetric in A and B. Consequently, H(A;B)  0with H(A;B) = 0 i clA = clB, H(A;B) = H(B;A) and H(A;C) H(A;B) + H(B;C) for any nonempty subsets A; B and C of X.When d is a metric on X, clearly H is the usual Hausdor distance.Remark 2.1. Given a quasi-pseudometric d : XX  ! R+ let  =maxfd; d 1g. Then  is obviously a pseudometric on X. Moreover,it is easy to notice that the Hausdor distance H(A;B) = Hd(A;B)determined by d and the Hausdor distance H(A;B) determined bythe pseudometric  coincide.A fuzzy set on X is an element of IX where I = [0; 1]. The  level set A of a fuzzy set A on X is dened asA = fx 2 X : A(x)  g for each  2]0; 1],A0 = cl(fx 2 X : A(x) > 0g).For x 2 X we denote by fxg the characteristic function of theordinary subset fxg of X.Definition 2.2. Let (X; d) be a quasi-pseudo-metric space. We de-ne the family of fuzzy sets on X, W (X), as follows:W (X) = fA 2 IX : A1 is nonempty d closed and d 1 compactg.106 VALENTIN GREGORI and JOSE PASTORFor a metric linear space (X; d), in [1] it is dened the familyW (X) of fuzzy sets on X, as follows, A 2 W (X) i A is compactand convex in X for each  2 [0; 1] and supx2X A(x) = 1. Clearly,A is closed for  2 [0; 1] and it is easy to verify that A1 is nonempty.Then, in a metric linear space (X; d) we have the following inclusions:W (X) W (X)  IX .For working with a similar notation to [1] we introduce the nextdenition.Definition 2.3. . Let (X; d) be a quasi-pseudo-metric space and letA; B 2W (X);  2 [0; 1]. Then we dene:p(A;B) = inffd(x; y) : x 2 A; y 2 Bg = d(A; B)D(A;B) = H(A; B)where H is the Hausdor distance deduced from the quasi-pseudo-metric d on X;D(A;B) = supD(A;B)Notice that p is non-decreasing function of , and then p1(A;B) =d(A1; B1).The following denition is more general than the one given in [1].Definition 2.4. . Let X and Y be an arbitrary set and a quasi-pseudo-metric space, respectively. F is said to be a fuzzy mapping ifF is a mapping from the set X into W (Y ).Definition 2.5. . Let A; B 2 IX . As usual in fuzzy theory, wedenote A  B when A(x)  B(x), for each x 2 X. We say x is axed point of the mapping F : X  ! IX , if fxg  F (x).We will use the following three lemmas, whose proofs we omit,given for a quasi-pseudo-metric space (X; d). They were given inmetric version (the rst one modied) by Heilpern [1], for the familyW (X).Lemma 2.6. . Let x 2 X and A 2 W (X). Then fxg  A if andonly if p1(x;A) = 0.A FIXED POINT THEOREM FOR FUZZY etc. 107Lemma 2.7. . p(x;A)  d(x; y) + p(y;A), for any x; y 2 X; A 2W (X).Lemma 2.8. . If fx0g  A then p(x0; B)  D(A;B) for eachA;B 2W (X).We will need the following lemma.Lemma 2.9. . Suppose K 6= ; is compact in the quasi-pseudo-metricspace (X; d 1). If z 2 X, then there exists k0 2 K such thatd(z;K) = d(z; k0).Proof. Let A be a nonempty subset of X. From d(z; x)  d(z; y) +d(y; x) whenever x; y; z 2 X, we conclude, taking the inmum of thelast expresion for z 2 A, thatd(A; x)  d(A; y) + d(y; x) (1)We will see that d(A; x) is a d 1-lower-semicontinous (lsc) func-tion ofX. Let x0 2 X and " > 0. By (1) we have d(A; y)  d(A; x0) d(y; x0) and then for y 2 B 1" (x0) we have d(A; y) > d(A; x0)   "and so d(A; x) is a d 1-lsc function.In particular if A is the one-point set fzg, the function d(z; k) is ad 1-lsc function of k 2 K, and sinceK is d 1-compact then there ex-ists k0 2 K such that d(z; k0) = minfd(z; k) : k 2 Kg, i.e, d(z; k0) =d(z;K):3. Fixed point theoremNow, we prove a xed point theorem for fuzzy contraction mappingsin quasi-pseudo-metric spaces.Theorem 3.1. Let (X; d) be a left K-sequentially complete quasi-pseudo-metric space, and F be a fuzzy mapping from X to W (X)satisfying the following condition: there exists q 2]0; 1[, such thatD(F (x); F (y))  q (d ^ d 1)(x; y) for each x; y 2 X:Then there exists x 2 X such that fxg  F (x).108 VALENTIN GREGORI and JOSE PASTORProof. Let x0 2 X and fx1g  F (x0). By Lemma 2.9 there existsx2 2 X such that fx2g  F (x1) and d(x1; x2)  d(x1; (F(x1))1) since(F (x1))1 is d 1- compact. We haved(x1 ; x2)  d(x1; (F (x1))1)  H(x1; (F (x1))1)  D(F (x0); F (x1)):Continuing in this way we produce a sequence (xn) in X suchthat fxng  F (xn 1) and d(xn; xn+1)  D(F (xn 1); F (xn)) for eachn 2 N. We will prove that (xn) is a left K-Cauchy sequence.d(x1; x2)  D(F (x0); F (x1))  q (d ^ d 1)(x0; x1)  q d(x0; x1)andd(xk; xk+1)  D(F (xk 1); F (xk))  q (d ^ d 1)(xk 1; xk) q d(xk 1; xk)  qk d(x0; x1); for k = 0; 1; 2 : : :For n < m we haved(xn; xm) m n 1Xi=0d(xn+i; xn+i+1) m 1Xi=nqi d(x0; x1)qn1  qd(x0; x1)whenever q 2]0; 1[ and then (xn) is a left K-Cauchy sequence, sinceqn converges to 0 as k ! 1. Then, since X is left K-sequentiallycomplete in X, there exists x 2 X such that limnxn = x.Now, by Lemma 2.7p1(x; F (x))  d(x; xn) + p1(xn; F (x)).Then by Lemma 2.8 (compare with [1]):p1(x; F (x))  d(x; xn) +D1(xn; F (x)) d(x; xn) +D(F (xn 1); F (x)) d(x; xn) + q (d ^ d 1)(xn 1; x) d(x; xn) + q d(x; xn 1):Now, d(x; xn) and d(x; xn 1) converge to 0 as n!1. Hence,by Lemma 2.6 we conclude that fxg  F (x).A FIXED POINT THEOREM FOR FUZZY etc. 109When d is a complete metric on X, we get the following result ofHeilpern [1]Corollary 3.2. Let X be a complete metric linear space and F bea fuzzy mapping from X to W (X) satisfying the following condition:there exists q 2]0; 1[ such thatD(F (x); F (y))  qd(x; y) for each x; y 2 X:Then, there exists x 2 X such that fxg  F (x).Acknowledgement. The authors are grateful to the referee forhis valuable suggestions.References[1] S. Heilpern, Fuzzy mappings and xed point theorem, J. Math. Anal.Appl. 83 (1981), 566{569.[2] J.C. Kelly, Bitopological spaces, Proc. London Math. Soc. 13 (1963),71{89.[3] S. B. Nadler, Multivalued contraction mappings, Pacic J. Math. 30(1969), 475{488.[4] H.T. Nguyen, A note on the extension principle for fuzzy sets, J.Math Anal. Appl. 64 (1978), 369{380.[5] I.L. Reilly, P.V. Subrahmanyam, and M.K. Vamanamurthy,Cauchy sequences in quasi-pseudometric spaces, Mh. Math. 93 (1982),127{140.Received October 24, 1997.

A fixed point theorem for fuzzy contraction mappings

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A fixed point theorem for fuzzy contraction mappings

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