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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 2, No. 1, 2001pp. 39 - 49

Common �xed point theorems for acountable family of fuzzy mappings
Anna Vidal�Abstract. In this paper we prove �xed point theorems forcountable families of fuzzy mappings satisfying contractive-typeconditions and a rational inequality in left K-sequentially com-plete quasi-pseudo-metric spaces. These results generalize thecorresponding ones obtained by other authors.2000 AMS Classi�cation: 54A40.Keywords: Fuzzy mapping, left K-Cauchy sequence, quasi-pseudo-metric.1. IntroductionHeilpern [4] introduced the concept of fuzzy mappings and proved a �xedpoint theorem for fuzzy contraction mappings which is a fuzzy analogue ofNadler's [7] �xed point theorem for multivalued mappings. Bose and Sahani [1]extended Heilpern's �xed point theorem to a pair of fuzzy contraction mappings.Park and Jeong [8] proved the existence of common �xed points for pairs offuzzy mappings satisfying contractive-type conditions and rational inequality incomplete metric spaces. In [2] the authors extended the theorems of [8] to leftK-sequentially complete quasi-pseudo-metric spaces and in [3] they obtained�xed point theorems for fuzzy mappings in Smyth-sequentially complete quasi-metric spaces. This study was motivated by the e�ciency of quasi-pseudo-metric spaces as tools to formulate and solve problems in theoretical computerscience. In this paper we generalize the theorems of [2] and present a partialgeneralization for theorem 3.1 of [1] to countable families of fuzzy mappings inleft K-sequentially complete quasi-pseudo-metric spaces.�While working on this paper the author has been partially supported by the grants fromUPV "Incentivo a la Investigaci�on / 99" and from Generalitat Valenciana GV00-122-1



40 Anna Vidal2. PreliminariesRecall that (X;d) is a quasi-pseudo-metric space, and d is called a quasi-pseudo-metric if d is a non-negative real valued function on X � X, whichsatis�es d(x;x) = 0 and d(x;z) � d(x;y) + d(y;z) for every x;y;z 2 X. If dis a quasi-pseudo-metric on X, then the function d�1 : X � X ! R, de�nedby d�1(x;y) = d(y;x) for all x;y 2 X, is also a quasi-pseudo-metric on X.Only if confusion is possible, we write d-closed or d�1-closed, for example, todistinguish the topological concept in (X;d) or (X;d�1).We will make use of the following notion, which has been studied underdi�erent names by various authors (see e.g. [5], [9]).De�nition 2.1. 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 allr;s 2 N with k � r � s. (X;d) is said to be left K-sequentially complete if eachleft K-Cauchy sequence in X converges (with respect to the topology T (d)).A fuzzy set in X is an element of IX where I = [0;1]. The r-level set ofA, denoted by Ar, is de�ned by Ar = fx 2 X : A(x) � rg if r 2 (0;1], andA0 = cl fx 2 X : A(x) > 0g. For x 2 X we denote by fxg the characteristicfunction of the ordinary subset fxg of X. If A; B 2 IX, as usual in fuzzytheory, we denote A � B when A(x) � B(x), for each x 2 X.Let (X;d) be a quasi-pseudo-metric space. We consider the families of [2]W 0(X) = fA 2 IX : A1 is nonempty and d-closedgW�(X) = fA 2 W 0(X) : A1 is d�1-countably compactgand the following concepts for A, B 2 W 0(X):� p(A;B) = inf fd(x;y) : x 2 A1;y 2 B1g = d(A1;B1),� �(A;B) = supfd(x;y) : x 2 A0;y 2 B0g and� D(A;B) = supfH(Ar;Br) : r 2 Ig,where H(Ar;Br) is the Hausdor� distance deduced from the quasi-pseudo-metric d.We will use the following lemmas for a quasi-pseudo-metric space (X;d):Lemma 2.2. Let x 2 X and A 2 W 0(X). Then fxg � A if and only ifp(x;A) = 0:Lemma 2.3. p(x;A) � d(x;y) + p(y;A), for any x;y 2 X; A 2 W 0(X).Lemma 2.4. If fx0g � A then p(x0;B) � D(A;B) for each A;B 2 W 0(X).Lemma 2.5. Suppose K 6= ? is countably compact in the quasi-pseudo-metricspace (X;d�1). If z 2 X, then there exists k0 2 K such that d(z;K) = d(z;k0).3. Fixed point theoremsFirst we generalize the theorems of [2] to countable families of fuzzy map-pings. From now on (X;d) will be a quasi-pseudo-metric space.



Common �xed point theorems 41De�nition 3.1. F is said to be a fuzzy mapping if F is a mapping from theset X into W 0(X). We say that z 2 X is a �xed point of F if z 2 F(z)1, i.e.,fzg � F(z).Theorem 3.2. Let (X;d) be a left K-sequentially complete space and letfFi : X ! W�(X)g1i=1 be a countable family of fuzzy mappings. If there existsa constant h, 0 � h < 1, such that for each x;y 2 X,D(Fi(x);Fi+1(y)) � hmaxf (d ^ d�1)(x;y);p(x;Fi(x));p(y;Fi+1(y));p(x;Fi+1(y))+p(y;Fi(x))2 g; i = 1;2;3; : : :D(Fi(x);F1(y)) � hmaxf (d ^ d�1)(x;y);p(x;Fi(x));p(y;F1(y));p(x;F1(y))+p(y;Fi(x))2 g; i = 2;3;4; : : : ;then there exists z 2 X such that fzg � Fi(z), i = 1;2;3; ldotsProof. Assume � = ph. Let x01 2 X and suppose x11 2 (F1(x01))1. By Lemma2.5 there exists x12 2 (F2(x11))1 such that d(x11;x12) = d(x11;(F2(x11))1) since(F2(x11))1 is d�1-countably compact. We haved(x11;x12) = d(x11;(F2(x11))1) � D1(x11;F2(x11)) � D(F1(x01);F2(x11))Again, we can �nd x21 2 X such that x21 2 (F1(x12))1 and d(x12;x21) �D(F2(x11);F1(x12)). Continuing in this manner we produce a sequence�x11;x12;x21;x22;x23;x31;x32;x33;x34; : : : ;xn1;xn2; : : : ;xn(n+1); : : :	in X such thatxn1 2 (F1(x(n�1)n))1; d(x(n�1)n;xn1) � D(Fn(x(n�1)(n�1));F1(x(n�1)n));xn2 2 (F2(xn1))1; d(xn1;xn2) � D(F1(x(n�1)n);F2(xn1));n = 1;2; : : : andxni 2 (Fi(xn(i�1)))1; d(xn(i�1);xni) � D(F(i�1)(xn(i�2));Fi(xn(i�1)));i = 3;4; : : : ;(n + 1), n = 2;3; : : :We will prove that (xrs) is a left- K-Cauchy sequence. Firstlyd(x11;x12) � D(F1(x01);F2(x11))< � maxf(d ^ d�1)(x01;x11);p(x01;F1(x01));p(x11;F2(x11));p(x01;F2(x11)) + p(x11;F1(x01))2 g� � maxf(d ^ d�1)(x01;x11);d(x01;x11);d(x11;x12);d(x01;x12) + d(x11;x11)2 g� � maxfd(x01;x11);d(x11;x12); d(x01;x11) + d(x11;x12)2 g= � maxfd(x01;x11);d(x11;x12)g



42 Anna VidalIf d(x11;x12) > d(x01;x11), then d(x11;x12) < �d(x11;x12), a contradiction.Thus, d(x11;x12) � d(x01;x11), and d(x11;x12) < � d(x01;x11): Similarlyd(x12;x21) � D(F2(x11);F1(x12))< � maxf(d ^ d�1)(x11;x12);p(x11;F2(x11));p(x12;F1(x12));p(x11;F1(x12)) + p(x12;F2(x11))2 g� � maxfd(x11;x12);d(x12;x21)gand d(x12;x21) < �d(x11;x12) < �2d(x01;x11);d(x21;x22) � D(F1(x12);F2(x21))< � maxfd(x12;x21);d(x21;x22)gand d(x21;x22) < �d(x12;x21) < �3d(x01;x11);d(x22;x23) � D(F2(x21);F3(x22))< � maxfd(x21;x22);d(x22;x23)gand so d(x22;x23) < �d(x21;x22) < �4d(x01;x11).Let y0 = x01. Now, we rename the constructed sequence (xrs) as follows:y1 = x11;y2 = x12;y3 = x21;y4 = x22; : : :and so, we obtain the sequence (yn) of points of X such thatyn = xij 2 (Fj(yn�1))1 for n = (i+1)i2 + j � 1where i = 1;2; : : : , j = 1; : : : ; i + 1. By the above relations, one can verifythat d(yn;yn+1) < �d(yn�1;yn) < �n d(y0;y1) n = 1;2; ::: and for m > n itis easy to see that d(yn;ym) � �n1��d(y0;y1). Then, from [6], (yn) is a leftK-Cauchy sequence in X, so there exists z 2 X such that d(z;yn) ! 0 (andd(z;xi(i+1)) ! 0, d(z;xii) ! 0; as i ! 1).Next, we show by induction that p(z;Fj(z)) = 0, j = 1;2;3; :::By lemmas2.3, 2.4 we have:p(z;F1(z)) � d(z;x12) + p(x12;F1(z))� d(z;x12) + D(F2(x11);F1(z)):Similarly p(z;F1(z)) � d(z;x23) + p(x23;F1(z))� d(z;x23) + D(F3(x22);F1(z))p(z;F1(z)) � d(z;x34) + p(x34;F1(z))� d(z;x34) + D(F4(x33);F1(z))and in general, for i = 1;2;3; :::(3.1) p(z;F1(z)) � d(z;xi(i+1)) + D(Fi+1(xii);F1(z))



Common �xed point theorems 43ButD(Fi+1(xii);F1(z)) � h maxf(d ^ d�1)(xii;z);p(xii;Fi+1(xii));p(z;F1(z));p(xii;F1(z)) + p(z;Fi+1(xii))2 g� h maxf(d ^ d�1)(xii;z);d(xii;xi(i+1));d(z;xi(i+1)) + D(Fi+1(xii);F1(z));d(xii;xi(i+1)) + D(Fi+1(xii);F1(z)) + d(z;xi(i+1)))2 g:(3.2)In the sequel, the expression (2) will be denoted by hmaxfCg. Now, thereare four cases:Case I: If maxfCg = (d ^ d�1)(xii;z), then the inequality (3.1) becomesp(z;F1(z)) � d(z;xi(i+1))) + h(d ^ d�1)(xii;z)� d(z;xi(i+1)) + hd(z;xii) ! 0; as i ! 1:The other three cases II-IV coincide with the corresponding ones in [8], andp(z;F1(z)) = 0 in all them. Thus, p(z;F1(z)) = 0.Suppose p(z;Fj(z)) = 0. Then, by lemma 2.2 fzg � Fj(z) and by lemma 2.4we havep(z;Fj+1(z)) � D(Fj(z);Fj+1(z))� hmax(d ^ d�1)(z;z);p(z;Fj(z));p(z;Fj+1(z));p(z;Fj+1(z)) + p(z;Fj(z))2 g= hp(z;Fj+1(z))Thus (1 � h)p(z;Fj+1(z)) � 0;and therefore p(z;Fj+1(z)) = 0. Hence, bylemma 2.2 it follows that fzg � Fj(z), for each j 2 N. �Theorem 3.3. Let (X;d) be a left K-sequentially complete space and letfFi : X ! W�(X)g1i=1 be a countable family of fuzzy mappings. If there existsa constant h 2]0;1[, such that for each x;y 2 XD(Fi(x);Fi+1(y)) � k [p(x;Fi(x)) � p(y;Fi+1(y))]1=2; i = 1;2;3; :::D(Fi(x);F1(y)) � k [p(x;Fi(x)) � p(y;F1(y))]1=2; i = 2;3;4; :::;then there exists z 2 X such that fzg � Fi(z), i = 1;2;3; :::Proof. Let x01 2 X: Let (xrs) be the sequence in the proof of theorem 3.2.Now, d(x11;x12) � D(F1(x01);F2(x11)) � 1phD(F1(x01);F2(x11))� hph[p(x01;F1(x01)) � p(x11;F2(x11))]1=2� h1=2[d(x01;x11) � d(x11;x12)]1=2



44 Anna VidalSo, d(x11;x12) � hd(x01;x11). Similarlyd(x12;x21) � 1phD(F2(x11);F1(x12))� h1=2[d(x11;x12) � d(x12;x21)]1=2and d(x12;x21) � hd(x11;x12) < h2d(x01;x11);d(x21;x22) � 1phD(F1(x12);F2(x21))� h1=2[d(x12;x21) � d(x21;x22)]1=2and d(x21;x22) � hd(x12;x21) � h3d(x01;x11);d(x22;x23) � 1phD(F2(x21);F3(x22))� h1=2 [d(x21;x22) � d(x22;x23)]1=2and d(x22;x23) � hd(x21;x22) � h4d(x01;x11).Let y0 = x01. Now, we rename the constructed sequence (xrs) as theorem3.2. By the above relations one can verify that d(yn;yn+1) � hd(yn�1;yn) �hn d(y0;y1); n = 1;2; :::and from [6], (yn) is a left K-Cauchy sequence in X.Then, there exists z 2 X such that d(z;yn) ! 0.Next we will show by induction that p(z;Fj(z)) = 0, j = 1;2;3; ::: By lemmas2.3 and 2.4 it follows that for i = 1;2;3; :::p(z;F1(z)) � d(z;xi(i+1)) + p(xi(i+1);F1(z))� d(z;xi(i+1)) + D(Fi+1(xii);F1(z))� d(z;xi(i+1)) + h[d(xii;xi(i+1)) � p(z;F1(z))]1=2 ! 0 as i ! 1:Then, p(z;F1(z)) = 0: Now, suppose p(z;Fj(z)) = 0. Then, by lemmas 2.2and 2.4 we havep(z;Fj+1(z)) � D(Fj(z);Fj+1(z))� h[p(z;Fj(z)) � p(z;Fj+1(z))]1=2 = 0:It follows that p(z;Fj+1(z)) = 0 and fzg � Fj(z), for each j 2 N. �Since D(A;B) � �(A;B), 8A;B 2 W 0(X), then we deduce the followingcorollary.Corollary 3.4. Let (X;d) be a left K-sequentially complete space and letfFi : X ! W�(X)g1i=1 be a countable family of fuzzy mappings. If there existsa constant h 2]0;1[, such that for each x;y 2 X�(Fi(x);Fi+1(y)) � k[p(x;Fi(x)) � p(y;Fi+1(y))]1=2; i = 1;2;3; : : :�(Fi(x);F1(y)) � k[p(x;Fi(x)) � p(y;F1(y))]1=2; i = 2;3;4; : : : ;then there exists z 2 X such that fzg � Fi(z), i = 1;2;3; : : :



Common �xed point theorems 45Theorem 3.5. Let (X;d) be a left K-sequentially complete space and letfFi : X ! W�(X)g1i=1 be a countable family of fuzzy mappings. If there existconstants h, k > 0, with h + k < 1, such that for each x;y 2 XD(Fi(x);Fi+1(y)) � hp(y;Fi+1(y))[1+p(x;Fi(x))]1+d(x;y) + kd(x;y); i = 1;2;3; : : :D(Fi(x);F1(y)) � hp(y;F1(y))[1+p(x;Fi(x))]1+d(x;y) + kd(x;y); i = 2;3;4; : : :D(F1(x);Fi(y)) � hp(y;Fi(y))[1+p(x;F1(x))]1+d(x;y) + kd(x;y); i = 3;4; : : : ;then there exists z 2 X such that fzg � Fi(z), i = 1;2;3; : : :.Proof. Let x01 2 X: Let (xrs) be the sequence in the proof of theorem 3.2.Now, d(x11;x12) � D(F1(x01);F2(x11)) and using one of the two boundaryconditions for D, it is proved thatd(x11;x12) � k1 � hd(x01;x11) and d(x12;x21) � k1 � hd(x11;x12):Similarly we haved(x21;x31) � k1 � hd(x12;x21);d(x32;x31) � k1 � hd(x21;x31); : : : :Let y0 = x01. Now, we rename the constructed sequence (xrs) as theorem3.2 and we can see thatd(yn;yn+1) � k1 � h d(yn�1;yn) � � k1 � h�n d(y0;y1):Furthermore, taking t = k1�h, for m > n the following relation is satis�edd(yn;ym) � tn1 � t d(y0;y1):In consequence (yn) is a left K-Cauchy sequence and hence converges to z inX. We will see that p(z;Fj(z)) = 0; j = 1;2;3; ::: First,p(z;F1(z)) � d(z;xi(i+1)) + hd(xii;xi(i+1))[1 + p(z;F1(z))]1 + d(z;xii) + kd(z;xii) ! 0;as i ! 1.Then we have p(z;F1(z)) = 0: Now, suppose p(z;Fj(z)) = 0. Then bylemmas 2.2 and 2.4 we have p(z;Fj+1(z)) � hp(z;Fj+1(z)) and it follows thatp(z;Fj+1(z)) = 0. Hence, by lemma 2.2 it follows that fzg � Fj(z), for eachj 2 N. �We consider the following theorem for complete metric spaces.Theorem 3.6 (Bose and Sahani [1]). Let (X;d) be a complete linear spaceand let F1 and F2 be fuzzy mappings from X to W(X) satisfying the followingcondition: For any x;y in X,D(F1(x);F2(y)) � a1p(x;F1(x)) + a2p(y;F2(y)) + a3p(y;F1(x))+a4p(x;F2(y)) + a5d(x;y)where a1, a2, a3, a4, a5, are non-negative real numbers, a1+a2+a3+a4+a5 < 1and a1 = a2 or a3 = a4. Then there exists z 2 X such that fzg � Fi(z), i = 1;2:



46 Anna VidalWe will present two similar theorems for a countable family of fuzzy mappingsin a quasi-pseudo-metric space (X;d).Theorem 3.7. Let (X;d) be a left K-sequentially complete space and letfFi : X ! W�(X)g1i=1 be a countable family satisfying the following condition:For any x, y 2 X,D(Fi(x);Fi+1(y)) � a1p(x;Fi(x)) + a2p(y;Fi+1(y)) + a3p(y;Fi(x))+a4p(x;Fi+1(y)) + a5(d ^ d�1)(x;y); i = 1;2; :::D(Fi(x);F1(y)) � a1p(x;Fi(x)) + a2p(y;F1(y)) + a3p(y;Fi(x))+a4p(x;F1(y)) + a5(d ^ d�1)(x;y); i = 2;3; :::where a1, a2, a3, a4, a5, are non-negative real numbers and a1+a2+2a4+a5 < 1.Then there exists z 2 X such that fzg � Fi(z), i = 1;2;3; :::Proof. Let x01 2 X: Let (xrs) be the sequence in the proof of theorem 3.2.Nowd(x11;x12) � D(F1(x01);F2(x11))� a1p(x01;F1(x01)) + a2p(x11;F2(x11)) + a3p(x11;F1(x01))+a4p(x01;F2(x11)) + a5(d ^ d�1)(x01;x11)� a1d(x01;x11) + a2d(x11;x12) + a4(d(x01;x11) + d(x11;x12))+a5d(x01;x11);i.e., d(x11;x12) � a1 + a4 + a51 � a2 � a4 d(x01;x11):Let r = a1 + a4 + a51 � a2 � a4 . Then 0 < r < 1 and d(x11;x12) � rd(x01;x11): Againd(x12;x21) � D(F2(x11);F1(x12))� a1d(x11;x12) + a2d(x12;x21) + a4(d(x11;x12) + d(x12;x21))+a5d(x11;x12);i.e., d(x12;x21) � rd(x11;x12) � r2d(x01;x11):Let y0 = x01. Now, we rename the constructed sequence (xrs) as theorem3.2. By the above relations one can verify that d(yn;yn+1) � rn d(y0;y1); n =1;2; :::: and there exists z 2 X such that d(z;yn) ! 0.We will show by induction that p(z;Fj(z)) = 0, j = 1;2;3; ::: By lemmas 2.3and 2.4 it follows that for i = 1;2;3; :::p(z;F1(z)) � d(z;xi(i+1)) + p(xi(i+1);F1(z))� d(z;xi(i+1)) + D(Fi+1(xii);F1(z))



Common �xed point theorems 47ButD(Fi+1(xii);F1(z)) � a1p(xii;Fi+1(xii)) + a2p(z;F1(z))+a3p(z;Fi+1(xii)) + a4p(xii;F1(z))+a5(d ^ d�1)(xii;z)� a1d(xii;xi(i+1))+a2 �d(z;xi(i+1)) + D(Fi+1(xii);F1(z))	+a3d(z;xi(i+1))+a4 �d(xii;xi(i+1)) + D(Fi+1(xii);F1(z))	+a5d(z;xii):Thus D(Fi+1(xii);F1(z)) � a1+a41�a2�a4 d(xii;xi(i+1))+ a2+a31�a2�a4 d(z;xi(i+1))+ a51�a2�a4 d(z;xii):So p(z;F1(z)) � d(z;xi(i+1)) + a1 + a41 � a2 � a4 d(xii;xi(i+1))+ a2 + a31 � a2 � a4 d(z;xi(i+1)) + a51 � a2 � a4 d(z;xii) ! 0as i ! 1:Then, p(z;F1(z)) = 0. Now, suppose p(z;Fj(z)) = 0. Then, by lemma 2.2fzg � Fj(z) and by lemma 2.4 we havep(z;Fj+1(z)) � D(Fj(z);Fj+1(z))� a1p(z;Fj(z)) + a2p(z;Fj+1(z))+a3p(z;Fj(z)) + a4p(z;Fj+1(z)) + (d ^ d�1)(z;z)= (a2 + a4)p(z;Fj+1(z)):Thus (1 � a2 � a4)p(z;Fj+1(z)) � 0, and it follows that p(z;Fj+1(z)) = 0. Bylemma 2.2 it follows that fzg � Fj(z), for each j 2 N. �We notice the above theorem is not a generalization of theorem 3.6. Now wepresent a partial generalization of this theorem.Theorem 3.8. Let (X;d) be a left K-sequentially complete space and letfFi : X ! W�(X)g1i=1 be a countable family of fuzzy mappings, satisfying thefollowing condition: For any x, y 2 X,D(Fi(x);Fi+1(y)) � a1p(x;F1(x)) + a2p(y;F2(y)) + a3p(y;F1(x))+a4p(x;F2(y)) + a5(d ^ d�1)(x;y); i = 1;2; :::D(F1(x);Fi(y)) � a1p(x;F1(x)) + a2p(y;Fi(y)) + a3p(y;F1(x))+a4p(x;Fi(y)) + a5(d ^ d�1)(x;y); i = 3;4;where a1, a2, a3, a4, a5, are non-negative real numbers and a1+a2+2a3+a5 < 1,a1 + a2 + 2a4 + a5 < 1. Then there exists z 2 X such that fzg � Fi(z),i = 1;2;3; : : : (compare with 3.6).



48 Anna VidalProof. Let x01 2 X. Let (xrs) be the sequence in the proof of theorem 3.2.Now d(x11;x12) � D(F1(x01);F2(x11))and, as in the proof of the above theorem, we haved(x11;x12) � a1 + a4 + a51 � a2 � a4 d(x01;x11):Again d(x12;x21) � a2 + a3 + a51 � a1 � a3 d(x11;x12):Let r = a1 + a4 + a51 � a2 � a4 , and s = a2 + a3 + a51 � a1 � a3 . Then 0 < r;s < 1. Taket = maxfr;sg < 1. So, we haved(x11;x12) � rd(x01;x11) � td(x01;x11);d(x12;x21) � sd(x11;x12) � td(x11;x12) � t2d(x01;x11):Let y0 = x01. Now, we rename the constructed sequence (xrs) as theorem 3.2.By the above relations one can verify that d(yn;yn+1) � tn d(y0;y1), n = 1;2; : : :Then there exists z 2 X such that d(z;yn) ! 0 and as in the proof of the abovetheorem it can be shown that fzg � Fj(z), for each j 2 N. �References[1] R. K. Bose and D. Sahani, Fuzzy mappings and �xed point theorems, Fuzzy Sets andSystems, 21 (1987), 53{58.[2] V. Gregori and S. Romaguera, Common �xed point theorems for pairs of fuzzy mappings,Indian Journal of Mathematics, 41, N� 1 (1999), 43{54.[3] V. Gregori and S. Romaguera, Fixed point theorems for fuzzy mappings in quasi-metricspaces, Fuzzy Sets and Systems, 115 (2000), 477{483.[4] S. Heilpern, Fuzzy mappings and �xed point theorem, J. Math. Anal. Appl., 83 (1981),566{569.[5] J. C. Kelly, Bitopological spaces, Proc. London Math. Soc., 13 (1963), 71{89.[6] H. P. A. K�unzi, M. Mrsevic, I. L. Reilly, M. K. Vamanamurthy, Convergence, precom-pactness and symmetry in quasi-uniform spaces, Math. Japonica, 38 (1993), 239{253.[7] S. B. Nadler, Multivalued contraction mappings, Paci�c J. Math., 30 (1969), 475{488.[8] J. Y. Park and J. U. Jeong, Fixed point theorems for fuzzy mappings, Fuzzy Sets andSystems, 87 (1997), 111{116.[9] I. L. Reilly, P. V. Subrahmanyam and M. K. Vamanamurthy, Cauchy sequences in quasi-pseudometric spaces, Mh. Math., 93 (1982), 127{140.Received September 2000Revised version January 2001
A. VidalDep. de Matem�atica AplicadaEscuela Polit�ecnica Superior de Gandia



Common �xed point theorems 49Carreyera Nazaret-Oliva s/n46730-Grau de Gandia, (Valencia), SpainE-mail address: avidal@mat.upv.es