A fixed point theorem in probabilistic metric spaces with a convex structure Tatjana»Zikić-Do»senović Faculty oftechnology, University ofnovi Sad Bule

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1 A fixed point theorem in probabilistic metric spaces with a convex structure Tatjana»Zikić-Do»senović Faculty oftechnology, University ofnovi Sad Bulevar Cara Lazara 1, Novi Sad, Serbia tatjanad@tehnol.ns.ac.yu Abstract The inequality F fx;fy qs) F x;y s) s 0), where q 2 0; 1), is generalized for multivalued mappings in many directions. Using Hausdorff distance S.B. Nadler in [7] introduced a generalization of Banach contraction principle in metric spaces. In [3] the definition of probabilistic Nadler q-contraction is given. Using some results given in [12] a fixed point theorem on spaces with a convex structure is obtained. Some fixed point theorems in such spaces are proved in [1, 2]. Keywords: multivalued mappings, coincidence point, probabilistic metric space, Menger space, triangular norm, Menger space with a convex structure. 1 Introduction K. Menger introduced in the notion of a probabilistic metric space as a natural generalization of the notion of a metric space M; d) inwhich the distance dp; q) p; q 2 M) between p and q is replaced by a distribution function F p;q 2 +. F p;q x) can be interpreted as the probability that the distance between p and q is less than x. Since then the theory of probabilistic metric spaces has been developed in many directions[9]). Since Sehgal and Bharucha-Reid proved [10]) a fixed point theorem in Menger spaces S; F; T M ) many fixed point theorem for singlevalued and multivalued mappings on Menger spaces S; F; T) are obtained. Further development of the fixed point theory in a more general Menger space S; F; T) was connected with investigations of the structure of the t-norm T. This problem was very soon in some sence completely solved. V. Radu proved that if f : S! S is a probabilistic q-contraction, where S; F; T) is a complete Menger space and T is a continuous t-norm, f has a fixed point if and only if T is of the H-type [8]. S.B. Nadler proved in [7] the generalization of the Banach contraction principle for multivalied mappings f : X! CBX), where X; d) isametric space, by introducing the condition Dfx; fy)» qdx; y); where D is the Hausdorff metric and q 2 0; 1). Probabilistic version of Nadler's q-contraction is given in [3]. Definition 1 Let S; F) be a probabilistic metric space, M a nonempty subset of S and f : M! 2 S, where 2 S is the family of all nonempty subsets of S. The mapping f is said to be a probabilistic Nadler's q- contraction, where q 2 0; 1) if the following condition is satisfied: For every u; v 2 M, every x 2 fu and every ffi > 0 there exists y 2 fv such that for every >0 F x;y ) F u;v ffi ): q If f is a singlevalued mapping, then the notion of a probabilistic Nadler's q-contraction coincides with the notion of a probabilistic q-contraction by Sehgal and Bharucha-Reid [10], since the function F u;v ) is leftcontinuous. 2 Preinaries Let + be the set of all distribution functions F such that F 0) = 0 F is a nondecreasing, left continuous

2 mapping from R into [0; 1] such that sup F x) =1): x2r The ordered pair S; F) is said to be a probabilistic metric space if S is a nonempty set and F : S S! + Fp; q) written by F p;q for every p; q) 2 S S) satisfies the following conditions: 1. F u;v x) = 1 for every x>0 ) u = v u; v 2 S): 2. F u;v = F v;u for every u; v 2 S: 3. F u;v x) = 1 and F v;w y) =1) F u;w x + y) = 1 for u; v; w 2 S and x; y 2 R + : A Menger space see [9]) is a ordered triple S; F;T); where S; F) is a probabilistic metric space, T is a triangular norm abbreviated t -norm) and the following inequality holds F u;v x + y) T F u;w x);f w;v y)) for every u; v; w 2 S and every x>0;y >0: Recall that a mapping T : [0; 1] [0; 1]! [0; 1] is called a triangular norm a t-norm) if the following conditions are satisfied: T a; 1) = a for every a 2 [0; 1] ; T a; b) =T b; a) for every a; b 2 [0; 1]; a b; c d ) T a; c) T b; d) a; b; c; d 2 [0; 1]); T a; T b; c)) = T T a; b);c) a; b; c 2 [0; 1]): Example 1 The following are the four basic t-norms : i) The minimum t-norm, T M, is defined by T M x; y) = minx; y); ii) The product t-norm, T P, is defined by T P x; y) =x y; iii) The Lukasiewicz t-norm T L is defined by T L x; y) = maxx + y 1; 0); iv) The weakest t-norm, the drastic product T D ; is defined by T D x; y) =ρ minx; y) if maxx; y) =1; 0 otherwise. As regards the pointwise ordering, we have the inequalities T D <T L <T P <T M : The ffl; ) topology in S is introduced by the family of neighbourhoods U = fu v ffl; )g v;ffl; )2S R+ 0;1), where U v ffl; ) =fu; F u;v ffl) > 1 g: If a t-norm T is such that sup T x; x) =1; then S is x<1 in the ffl; ) topology a metrizable topological space. Each t-norm T can be extended see [6]) by associativity) in a unique way to an n-ary operation taking for x 1 ;:::;x n ) 2 [0; 1] n the values T 0 i=1x i =1; T n i=1x i = T T n 1 i=1 x i;x n ): We can extend T to a countable infinitary operation taking for any sequence x n ) n2n from [0; 1] the values T 1 i=1 x i = n!1 Tn i=1 x i: Limit of right side exists since the sequence T n x i=1 i) n2n is nonincreasing and bounded from below. In the fixed point theoryitisofinterest to investigate the classes of t-norms T and sequences x n ) n2n from the interval [0; 1] such that n!1 x n =1,and n!1 T1 i=nx i = T 1 i=1x n+i =1: In [5] itisproved that for the Dombi, Aczél-Alsina and Sugeno-Weber family of t-norms exists the sequence x n ) n2n from 0; 1] such that the last condition is satisfied. Definition 2 : Let S; F; T) be a Menger space, ; 6= M ρ S; f : M! M i A : M! 2 M the family of nonempty subsets of M). The mapping A is f- strongly demicompact if for every sequence fx n g n2n from M, such that n!1 F fxn;ynffl) =1, for some sequence fy n g n2n ;y n 2 Ax n ;n2 N and every ffl > 0, there exists a convergent subsequence ffx n k g k2n. A mapping A : M! 2 M is weakly commuting with f : M! M if for every x 2 M fax) ρ Afx): In [12] the following theorem is proved. Theorem 1 : Let S; F; T) be a complete Menger space, sup a<1 T a; a) = 1, A is a nonempty and closed subset of S, f : A! A acontinuous mapping, L; L 1 : A! 2 f c A) closed, multivalued mappings such

3 that the following condition is satisfied: space S; F; T M ) with the same function W. For every u; v 2 A, x 2 Lu and ffi > 0, there exists y 2 L 1 v such that F x; y ) F fu;fv ffi ); for all >0; where q 2 0; 1): q If L and L 1 are weakly commuting with f and i) or ii) are satisfied, then there exists x 2 A such that fx 2 Lx L 1 x where i) L or L 1 is f-strongly demicompact. ii) There exists x 0 ;x 1 2 A, fx 1 2 Lx 0 and μ 2 0; 1) such that t-norm T satisfies the following condition 3 Main results n!1 T1 i=nf fx0;fx 1 1 μ i )=1: W. Takahashi introduced in [11] the notion of a metric space with a convex structure. This class of metric spaces includes normed linear spaces and metric spaces of the hiperbolic type. Let us recall that a metric space S; d) has a convex structure in the sense of Takahashi, if there exists a mapping W : S S [0; 1]! S such that for every u; x; y; ffi) 2 S S S [0; 1] It is well-known that every probabilistic normed space is a probabilistic metric space with a convex structure W x; y; ffi) =ffix +1 ffi)y x; y 2 S) since for every >0andffi 2 [0; 1] we have F u;w x;y;ffi) = F u ffix 1 ffiy) 2) = F ffiu+1 ffiu) ffix 1 ffi)y 2) = F ffiu x)+1 ffi)u y) 2) T F ffiu x) ;F 1 ffi)u y) )) = T F u x ffi );F u y 1 ffi ))): In this paper we shall suppose that a convex structure W on a Menger space S; F; T) satisfies the condition F W x;z;ffi);w y;z;ffi) ffi) F x;y ) 1) for every x; y; z) 2 S S S, every > 0 and ffi 2 0; 1). For the next theorem we shall need the following definitions. Definition 3 Let S; F; T) be Menger space with a convex structure W. A nonempty subset M of S is called W -starshaped if there exists x 0 2 M such that the set fw x; x 0 ; ): x 2 M; 2 0; 1)g ρm. The point x 0 is said to be the star-centre ofm. du; W x; y; ffi))» ffidu; x)+1 ffi)du; y): This definition can be generalized in Menger spaces S; F; T). The notion of convex structure in probabilistic metric spaces, as well as Definition 3 and Definition 4 belong to O. Had»zić [4]. A mapping W : S S [0; 1]! S is said to be a convex structure on S if for every x; y) 2 S S Clearly, every convex set is a starshaped set and the inverse is not true. Definition 4 Let S; F; T) be a Menger space with a convex structure W and M a nonempty subset of S. A mapping f : M! S is said to be W; x 0 )-convex if for each x; ) 2 M [0; 1] W x; y; 0) = y; Wx; y; 1) = x, and for every ffi 2 0; 1), u 2 S, >0 F u;w x;y;ffi) 2) T F u;x ffi );F u; y 1 ffi )): It is easy to see that every metric space S; d) with a convex structure W can be considered as a Menger W fx; x 0 ; )=fw x; x 0 ; )): Lemma 1 Let S; F; T) be a Menger space, M a nonempty subset of S which is W -starshaped with the star-centre inx 0, f : M! S is the mapping which is W; x 0 )-convex. Then the fm) is the W; x 0 )-convex.

4 Proof: Let u 2 fm). Then there exists x 2 M so that u = fx). Let us prove that for every 2 [0; 1], As u = fx) it follows z = W u; x 0 ; ) 2 fm): z = W fx); x 0 ; ): Since f is W; x 0 )-convex it follows z = fw x; x 0 ; )): From the condition we have that the set M is W - starshaped, and it follows that W x; x 0 ; ) ρ M, i.e. z 2 fm). Lemma 2 Let S; F; T) be a Menger space, T a t-norm such that sup T x; x) = 1, M a nonempty x<1 subset of S, f : M! S a continuous mapping, L : M! CS) where CS) is the family of all nonempty and closed subset of S) and the following inequality is satisfied: For every u; v 2 M, every x 2 Lu and every ffi > 0, there exists y 2 Lv such that F x;y ) F fu;fv ffi ) 2) q for every >0. Then the mapping L is closed. Proof: Let x n ) n2n be a sequence from M such that n!1 x n = x and let y n 2 Lx n, for every n 2 N such that n!1 y n = y. Let us prove thaty 2 Lx. Since Lx is closed we shall prove that y 2 Lx. Let >0and 2 0; 1) be given. It remains to be proved that there exists b 2 Lx such that b 2 U y ; ), i.e. F b;y ) > 1 : Using condition 2, where u = x n, v = x and ffi = 3 from y 4 n 2 Lx n it follows that there exists b n 2 Lx such that So, F y n;bn) F fx n;fx 4q ): F y;b n) T F y;y n 2 );F yn;bn 2 )) T F y;y n 2 );F fxn;fx 4q )): From the continuity of the mapping f n!1 fx n = fx, i.e. it follows n!1 F fxn;fx) =1. We also have n!1 F y;yn) =1, for every >0: From the condition sup T x; x) =1, x<1 it follows that for a given 2 0; 1) there exists 2 0; 1) such that T ; ) > 1, so there exists n 0 ) such that for every n n 0 ) Hence, i.e. b n0 2 U y ; ) Lx: F y;y n 2 ) > i F fxn;fx 4q ) > : F y;b n) T ; ) > 1 ; Theorem 2 Let S; F; T) be a complete Menger space with a convex structure W and a continuous t- norm T and M is nonempty, closed and W -starshaped subset of S with the star-centre x 0. Let f : M! M be a continuous, W; x 0 )-convex mapping and L : M! 2 f c M) such that LM) is compact set and the following condition is satisfied: For every u; v 2 M, x 2 Lu and ffi > 0 there exists y 2 Lv such that F x;y ) F fu;fv ffi); for every >0: 3) If L is weakly comuting with f, then there exists x 2 M such that fx 2 Lx. Proof: Let ) n2n be a sequence from the interval 0; 1) such that n!1 = 1. For every n 2 N, and every x 2 M let [ L n x = W Lx; x 0 ; ) i.e. L n x = W z; x 0 ; ); n 2 N; x 2 M: z2lx From the Lemma 1 it follows that fm) is W; x 0 )- convex. We shall prove that L n x ρ fm), i.e. that for every z 2 L n x it follows z 2 fm). Since z 2 L n x = W Lx; x 0 ; ), there exists u 2 Lx such that z = W u; x 0 ; ). Since Lx ρ fm) it follows that u 2 fm), so W u; x 0 ; ) ρ fm) i.e. z 2 fm): It means that L n x ρ fm): From the condition 1) it follows that the mapping W

5 is continuous in respect to the first variable, and since Lx is closed it follows that Lx is compact as a subset of LM)) such that W Lx; x 0 ; ) is closed for every n 2 N. It follows that L n x is closed for every n 2 N and x 2 M. Next, we shall prove that for every u; v 2 M and every x 2 L n u and ffi>0 there exists y 2 L n v such that F x;y ) F fu;fv ffi ); >0: Let u; v 2 M, ffi > 0andx 2 L n u = W Lu; x 0 ; ). Then, there exists z 2 Lu such thatx = W z; x 0 ; ) and from 3) there exists y 0 2 Lv such that F z;y 0) F fu;fv ffi ): Let y = W y 0 ;x 0 ; ) 2 L n v: Then F x;y ) = F W z;x0;kn);wy 0 ;x 0;kn) ) F z;y 0 ffl ) F fu;fv ffi ): Let us prove that L m is a f-strongly demicompact. Suppose that x n ) n2n is a sequence in M such that for every >0 n!1 F fxn;yn) =1 for some sequence y n ) n2n y n 2 L m x n. Since L n M) =W LM); x 0 ; ), n 2 N is relatively compact, from y n 2 L m x n it follows that y n ) n2n has a convergent subsequence y n k) k2n and let y nk = z. k!1 Then wehave F fx nk ;z ) T F fx nk ;yn k 2 );F ynk ;z 2 )) i.e. k!1 fx nk = z, which means that the mapping L m is f-strongly demicompact. We are going to prove that the mapping L n is weakly comuting with f, i.e. that for every x 2 M fl n x) ρ L n fx)=w Lfx); x 0 ; ): Let u 2 L n x = W Lx; x 0 ; ). Then there exists z 2 Lx such that u = W z; x 0 ; ), so fu) = fw z; x 0 ; )) = W fz; x 0 ; ) ρ W flx); x 0 ; ): From the condition that the mapping L is weakly commuting with f it follows W flx); x 0 ; ) = W Lfx); x 0 ; ), i.e. fu) 2 L n fx). Hence, all the conditions of the Theorem 1 are satisfied and according to it, for every n 2 N, there exists x n 2 M such that Since L n x n = fx n 2 L n x n : S z2lxn W z; x 0 ; ) there exists z n 2 Lx n such that fx n = W z n ;x 0 ; ). Then, for every n 2 N F fx n;zn) = F z n;w zn;x 0;kn)) T F z n;zn );F z n;x 2k 0 n 21 ) )) = T 1; F z n;x 0 21 ) )) = F z n;x 0 21 ) ): Since n!1 n!1 F zn;x 0 21 kn) = 1, it follows that ) = 1 because z 21 kn) n 2 LM) which is probabilistically bounded. Then we have n!1 F fxn;zn) = 1. Since z n 2 Lx n and LM) is compact it follows that there exists convergent subsequence z n k) k2n of the sequence z n ) n2n. From n!1 fx nk = z it follows n!1 z nk = z. It remains to be proved that fz 2 Lz. From n!1 z nk = z and from the continuity of the mapping f it follows follows that n!1 fz nk = fz. Since z n k 2 Lx nk it fz n k 2 flx nk) ρ Lfx n k): From the Lemma 2 it follows that the mapping L is closed i.e. Acknowledgement fz 2 Lz: Research supported by MNTRRS References [1] Shih-sen Chang, Yeol Je Cho, Shin Min Kang, Probabilistic metric spaces and nonlinear operator theory, Sichuan University Press, 1994.

6 [2] O. Had»zić, A fixed point theorem in Menger spaces, Publ. Inst. Math. Beograd T ), [3] O. Had»zić, Fixed point theorems for multivalued mappings in probabilistic metric spaces, Mat. Vesnik 3 16)31), 1979), [4] O. Had»zić, Fixed point theory in probabilistic metric spaces, Serbian academy of science and arts branch in Novi Sad, University of Novi Sad, Institute of Mathematics, [5] O. Had»zić, E. Pap, Fixed Point Theory in Probabilistic Metric Spaces, Theory in Probabilistic Metric Spaces, Kluwer Academic Publishers, Dordrecht, [6] E.P. Klement, R. Mesiar, E. Pap, Triangular Norms, Kluwer Academic Publishers, Trends in Logic 8, Dordrecht 2000a). [7] S.B. Nadler, Multivalued contraction mappings, Pacific J. Math ), [8] V. Radu, On the contraction principle in Menger spaces, Seminarul the theoria probabilitatilor siaplicatii, University of Timisoara, Faculty of Natural Science, [9] B. Schweizer, A. Sklar, Probabilistic metric spaces, Elsevier North-Holland, New York, [10] V.M. Sehgal, A.T. Baharucha-Reid Fixed points of contraction mappings on probabilistic metric spaces, Math. Syst. Theory ), [11] W. Takahashi, A convexity in metric spaces and nonexpansive mappings I, Kodai Math. rep ), [12] T.»Zikić, Multivalued probabilistic q-contraction, Journal of Electrical Engineering, Vol. 53 No.12/s, 2002),