On Common Fixed Points in Menger Probabilistic Metric Spaces

Int. J. Contemp. Math. Sciences, Vol. 2, 2007, no. 8, 383-391 On Common Fixed Points in Menger Probabilistic Metric Spaces Servet Kutukcu Department of Mathematics, Faculty of Science and Arts Ondokuz Mayis University, Kurupelit, 55139 Samsun, Turkey Cemil Yildiz Department of Mathematics, Faculty of Science and Arts Gazi University, Teknikokullar, 06500 Ankara, Turkey Adnan Tuna Department of Mathematics, Faculty of Science and Arts Gazi University, Teknikokullar, 06500 Ankara, Turkey Abstract In this paper, we prove a common fixed point theorem for self maps satisfying a new contraction type condition in Menger probabilistic metric spaces. Mathematics Subject Classification: 54H25, 54E70 Keywords: Fixed point, contraction map, Menger probabilistic metric space. 1 Introduction There have been a number of generalizations of metric space. One such generalization is Menger space introduced in 1942 by Menger [6] who was use distribution functions instead of nonnegative real numbers as values of the metric. Schweizer and Sklar [9] studied this concept and gave some fundamental results on this space. The important development of fixed point theory in Menger spaces was due to Sehgal and Bharucha-Reid [11]. The study of

384 S. Kutukcu, C. Yildiz and A. Tuna common fixed points of maps satisfying some contractive type condition has been at the centre of vigorous research activity. It is observed by many authors that contraction condition in metric space may be translated into probabilistic metric space endoved with minimum norm. Milovanovic-Arandelovic [7] and Vasuki [13] proved some fixed point theorems for a sequence of selfmaps satisfying a contraction type condition in Menger spaces. In this paper, we prove a common fixed point theorem in Menger probabilistic metric spaces as a generalization of the results given by Milovanovic- Arandelovic [7] and Vasuki [13]. 2 Preliminaries In this section, we recall some definitions and known results in Menger probabilistic metric space. For more details, we refer the readers to [1-9,12]. Definition 2.1 A triangular norm (shorty t-norm) is a binary operation on the unit interval [0, 1] such that for all a, b, c, d [0, 1] the following conditions are satisfied: (a) a 1=a; (b) a b = b a; (c) a b c d whenever a c and b d; (d) a (b c) =(a b) c. Examples of t-norms are a b = max {a + b 1, 0} and a b = min {a, b}. Definition 2.2 A distribution function is a function F :[, ] [0, 1] which is left continuous on R, non-decreasing and F ( ) =0, F ( ) =1. We will denote by Δ the family of all distribution functions on [, ]. H is a special element of Δ defined by H(t) = { 0 if t 0, 1 if t>0.. If X is a nonempty set, F : X X Δ is called a probabilistic distance on X and F (x, y) is usually detoned by F xy. Definition 2.3 ([6]) (see also [1-3,9]) The ordered pair (X, F) is called a probabilistic semimetric space (shortly PSM-space) if X is a nonempty set and F is a probabilistic distance satisfying the following conditions: for all x, y, z X and t, s > 0,

Common Fixed Points in Menger Spaces 385 (FM-1) F xy (t) =H(t) x = y; (FM-2) F xy = F yx. If, in addition, the following inequality takes place: (FM-3) F xz (t) =1,F zy (s) =1 F xy (t + s) =1, then (X, F) is called a probabilistic metric space. The ordered triple (X, F, ) is called Menger probabilistic metric space (shortly Menger space) if (X, F) is a PM-space, is a t-norm and the following condition is also satisfies: for all x, y, z X and t, s > 0, (FM-4) F xy (t + s) F xz (t) F zy (s). For every PSM-space (X, F), we can consider the sets of the form U ε,λ = {(x, y) X X : F xy (ε) > 1 λ}. The family {U ε,λ } ε>0,λ (0,1) generates a semi-uniformity denoted by U F and a topology τ F called the F-topology or the strong topology. Namely, A τ F iff x A ε >0 and λ (0, 1) such that U ε,λ (x) A. U F is also generated by the family {V δ } δ>0 where V δ := U δ,δ ([9]). In [10] it is proved that if sup t<1 (t t) = 1, then U F is a uniformity, called F-uniformity, which is metrizable. The F-topology is generated by the F-uniformity and is determined by the F-convergence: x n x F xnx(t) 1, t >0. Proposition 2.4 ([11]) Let (X, d) be a metric space. Then the metric d induces a distribution function F defined by F xy (t) =H(t d(x, y)) for all x, y X and t>0. If t-norm is a b = min {a, b} for all a, b [0, 1] then (X, F, ) is a Menger space. Further, Radu [8] proved that it is complete if (X, d) is complete. Definition 2.5 ([9]) A sequence {x n } in a Menger space (X, F, ) is called converge to a point x in X (written as x n x) if for every ε>0 and λ (0, 1), there is an integer n 0 = n 0 (ε, λ) such that F xnx(ε) > 1 λ for all n n 0. The sequence called Cauchy if for every ε>0 and λ (0, 1), there is an integer n 0 = n 0 (ε, λ) such that F xnx m (ε) > 1 λ for all n, m n 0. A Menger space (X, F, ) is said to be complete if every Cauchy sequence in it converges to a point of it. Lemma 2.6 ([12]) Let {x n } be a sequence in a Menger space (X, F, ) with continuous t-norm and t t t. If there exists a constant k (0, 1) such that F xnxn+1 (kt) F xn 1 x n (t) for all t>0 and n =1, 2..., then {x n } is a Cauchy sequence in X.

386 S. Kutukcu, C. Yildiz and A. Tuna Lemma 2.7 ([12]) Let (X, F, ) be a Menger space. If there exists k (0, 1) such that F xy (kt) F xy (t) for all x, y X and t>0, then x = y. 3 Main Results Theorem 3.1 Let (T n ),n=1, 2,... be a sequence of mappings of a complete Menger space (X, F, ) into itself with t t t for all t [0, 1], and S : X X be a continuous mapping such that T n (X) S(X) and S is commuting with each T n. If there exists a constant k (0, 1) such that for any two mappings T i and T j [1+aFSxSy r (kp)] F T r i xt j y (kp) a[f SxT r i x (kp) F SyT r j y (kp) F SyT r i x (kp) F SxT r j x (2kp)] { (1) } + min FSxT r i x (p),fr SyT j y (p),fr SxSy (p),fr 1 SxT j y (2p)F SyT i x(p), SxT j y (2p)F SxT i x(p) holds for all a ( 1, 0],r 1,p>0 and x, y X, then there exists a unique common fixed point for all T n and S. by Proof. Let x 0 be an arbitrary point of X and {x n } be a sequence defined Sx n = T n x n 1,n=1, 2,... Then for each p>0 and 0 <k<1, we have [1 + af r Sx 0 Sx 1 (kp)] F r (kp) = [1 + af r Sx 0 Sx 1 (kp)] F r T 1 x 0 T 2 x 1 (kp) a[fsx r 0 T 1 x 0 (kp) FSx r 1 T 2 x 1 (kp) FSx r 1 T 1 x 0 (kp) FSx r 0 T 2 x 1 (2kp)] + min{fsx r 0 T 1 x 0 (p),fsx r 1 T 2 x 1 (p), Sx 0 T 2 x 1 (2p)F Sx1 T 1 x 0 (p), Sx 0 T 2 x 1 (2p)F Sx0 T 1 x 0 (p)} = a[fsx r 0 Sx 1 (kp) FSx r 1 Sx 2 (kp) FSx r 1 Sx 1 (kp) FSx r 0 Sx 1 (kp) FSx r 1 Sx 2 (kp)] + min{fsx r 0 Sx 1 (p),fsx r 1 Sx 2 (p), (2p)F Sx1 Sx 1 (p), (2p)F Sx0 Sx 1 (p)}. Thus it follows that FSx r 1 Sx 2 (kp) min{fsx r 0 Sx 1 (p),fsx r 1 Sx 2 (p), (2p)F Sx1 Sx 1 (p), (2p)F Sx0 Sx 1 (p)}.

Common Fixed Points in Menger Spaces 387 { Since F Sx0 Sx 2 (2p) min {F Sx0 Sx 1 (p),f Sx1 Sx 2 (p)}, we have (2p) min F r 1 Sx 0 Sx 1 (p), (p) },so (2p)F Sx0 Sx 1 (p) min{fsx r 0 Sx 1 (p), (p) F Sx0 Sx 1 (p)}. Therefore we have FSx r 1 Sx 2 (kp) min{fsx r 0 Sx 1 (p),fsx r 1 Sx 2 (p), Sx 0 Sx 1 (p), (p), (p)f Sx0 Sx 1 (p)} = min { FSx r 0 Sx 1 (p),fsx r 1 Sx 2 (p), (p)f Sx0 Sx 1 (p) }. Since min {a r,b r } ab r 1 and min {a r,b r } min {ab r 1,a r,b r } for all a, b [0, 1], r 1, we have F r (kp) min { F r Sx 0 Sx 1 (p),f r (p) }, that is F Sx1 Sx 2 (kp) min {F Sx0 Sx 1 (p),f Sx1 Sx 2 (p)} (2) Suppose that p>0 is such that F Sx0 Sx 1 (p) >F Sx1 Sx 2 (p) =r. Put q = sup{p : F Sx1 Sx 2 (p) =r}. Then there exists p 1 >qsuch that kp 1 >q. So we have r = F Sx1 Sx 2 (kp 1 ) <F Sx1 Sx 2 (p 1 ). From (3.2) follows F Sx1 Sx 2 (kp 1 ) min {F Sx0 Sx 1 (p 1 ),F Sx1 Sx 2 (p 1 )}. Hence F Sx1 Sx 2 (kp 1 ) F Sx0 Sx 1 (p 1 ). Now we have F Sx0 Sx 1 (p) <F Sx0 Sx 1 (p 1 ) F Sx1 Sx 2 (kp 1 ) <F Sx1 Sx 2 (p) which is a contradiction with F Sx0 Sx 1 (p) >F Sx1 Sx 2 (p). So (3.2) implies that F Sx1 Sx 2 (kp) F Sx0 Sx 1 (p) for all p>0 and 0 <k<1. By induction F SxnSx n+1 (kp) F Sxn 1 Sx n (p),n=1, 2,... Thus, by Lemma 2.5, {Sx n } is a Cauchy sequence in X. Since X is complete, there exists some u X such that Sx n u. Since Sx n = T n x n 1, {T n x n 1 } also converges to u. Since S commutes with each T n, using (3.1), we have [1 + af r SSx n 1 Su (kp)] F r SSx nt (kp) = [1 + af r SSx n 1 Su (kp)] F r ST nx n 1 T (kp) = [1+aFSSx r n 1 Su (kp)] F T r nsx n 1 T (kp) a[fssx r n 1 SSx n (kp) FSuT r (kp) FSuSSx r n (kp) FSSx r n 1 T (2kp)] + min{fssx r n 1 SSx n (p),fsut r (p),fr SSx n 1 Su (p), SSx n 1 T (2p)F SuSSx n (p), SSx n 1 T (2p)F SSx n 1 SSx n (p)}. Using the continuity of S and taking limits on both sides, we have [1 + a] FSuT r (kp) af SuT r (kp) + min{f r SuSu (p),fr SuT (p),fr SuSu (p), SuT (2p)F SuSu(p), SuT (2p)F SuSu(p)}

388 S. Kutukcu, C. Yildiz and A. Tuna and so FSuT r (kp) min{fsusu(p),f r SuT r (p),fsusu(p),f r r 1 SuT (2p)F SuSu(p), SuT (2p)F SuSu(p)} min { FSuT r (p),fr 1 SuT (2p)}. Since F SuT(2p) min {F SuSu (p),f SuT(p)} = F SuT(p), we have SuT (2p) (p) and hence SuT FSuT r (kp) min { FSuT r (p), SuT (p)} min { F } SuT r (p),fsut r (p) FSuT r (p) So F SuT(kp) F SuT(p) for each p>0 and 0 <k<1. Hence Su = T α u for any fixed integer α. Moreover [1 + af r Sx n 1 Su (kp)] F r Sx nt (kp) = [1 + af r Sx n 1 Su (kp)] F r T nx n 1 T (kp) a[fsx r n 1 Sx n (kp) FSuT r (kp) F r SuSx n (kp) FSx r n 1 T (2kp)] + min{fsx r n 1 Sx n (p),fsut r (p),fsx r n 1 Su(p), Sx n 1 T (2p)F SuSx n (p), Sx n 1 T (2p)F Sx n 1 Sx n (p)} and letting n, we have [1 + afusu r (kp)] F ut r (kp) a[f uu r (kp) F SuT r (kp) F r usu (kp) F ut r (2kp)] + min{fuu(p),f r SuSu(p),F r ut r (p), ut (2p)F ut (p), ut (2p)F uu(p)} a[fsusu r (kp) F usu r (kp) F ut r (kp)] + min { FuT r (p),fr 1 ut (p)f ut (p), ut (p)} a[fut r (kp) FuSu(kp)] r + min { FuT r (p),fr 1 ut (p)}. Thus, it follows that FuT r (kp) F r ut (p). So F ut(kp) F ut(p) for each p>0 and 0 <k<1. Hence u = T α u = Su for any fixed integer α. Thus u is a common fixed point of S and T n for n =1, 2,... For uniquenesses, let v be another common fixed point of S and T n for n =1, 2,... Then, using (3.1), we have [1 + afuv r (kp)] F uv r (kp) = [1 + af SuSv r (kp)] F T r i ut j v (kp) a[fsuu(kp) r FSvv(kp) r FSvu(kp) r FSuT r j v(2kp)]

Common Fixed Points in Menger Spaces 389 + min{fsuu r (p),fr Svv (p),fr SuSv (p),fr 1 Suv (2p)F Svu(p), Suv (2p)F Suu(p)} = a[fuu(kp) r Fvv(kp) r Fvu(kp) r Fuv(kp)] r + min{fuu(p),f r vv(p),f r uv(p),f r uv r 1 (2p)F vu (p), Fuv r 1 (2p)F uu(p)}. It follows that F r uv(kp) F r uv(p). So F uv (kp) F uv (p) for each p>0 and 0 <k<1. Hence, by Lemma 2.6, u = v. This completes the proof. If we take a = 0 and r = 2 in the main Theorem, we have the following: Corollary 3.2 ([7]) Let (T n ),n =1, 2,... be a sequence of mappings of a complete Menger space (X, F, ) into itself with t t t for all t [0, 1], and S : X X be a continuous mapping such that T n (X) S(X) and S is commuting with each T n. If there exists a constant k (0, 1) such that for any two mappings T i and T j FT 2 i xt j y(kp) min{fsxt 2 i x(p),fsyt 2 j y(p),fsxsy(p),f 2 SxTj y(2p)f SyTi x(p), F SxTj y(2p)f SxTi x(p)} holds for all p>0 and x, y X, then there exists a unique common fixed point for all T n and S. If we take a = 0 and S = I X (the identity map on X) in the main Theorem, we have the following: Corollary 3.3 Let (T n ),n =1, 2,... be a sequence of mappings of a complete Menger space (X, F, ) into itself with t t t for all t [0, 1]. If there exists a constant k (0, 1) such that for any two mappings T i and T j FT r i xt j y (kp) min{f xt r i x (p),fr yt j y (p),fr xy (p),fr 1 xt j y (2p)F yt i x(p), (3) xt j y (2p)F xt i x(p)} holds for all r 1,p > 0 and x, y X, then for any x 0 X the sequence (x n )=(T n x n 1 )(n =1, 2,...) converges and its limit is the unique common fixed point for all T n. Proof. It is easy to verify from Theorem 3.1.

390 S. Kutukcu, C. Yildiz and A. Tuna Corollary 3.4 ([13]) Let (T n ),n =1, 2,... be a sequence of mappings of a complete Menger space (X, F, ) into itself with t t t for all t [0, 1]. If there exists a constant k (0, 1) such that for any two mappings T i and T j FT 2 i xt j y(kp) min{f xy (p)f xti x(p),f xy (p)f ytj y(p),f xti x(p)f yti y(p), (4) F xtj y(2p)f yti x(p)} holds for all p>0 and x, y X, then for any x 0 X the sequence (x n )= (T n x n 1 )(n =1, 2,...) converges and its limit is the unique common fixed point for all T n. Proof. Since min {a 2,b 2,c 2 } min {ab, bc, ca} for all a, b, c 0, we have } min {F 2xTix (p),f2ytjy(p),f2xy (p) min { F xy (p)f xti x(p),f xy (p)f ytj y(p),f xti x(p)f yti y(p) }. From (4), we have F 2 T i xt j y(kp) min { } FxT 2 i x(p),fyt 2 j y(p),fxy(p),f 2 xtj y(2p)f yti x(p). So it follows that if the sequence (T n ) satisfies the condition (4), then it also satisfies our condition (3) of Corollary 3.3 for r =2. References [1] G. Constantin, I. Istratescu, Elements of probabilistic analysis, Ed. Acad. Bucureşti and Kluwer Acad. Publ., 1989. [2] O. Hadzic, Fixed point theory in probabilistic metric spaces, Serbian Academy of Science and Arts, Novi Sad University, 1995. [3] O. Hadzic, E. Pap, Fixed point theory in probabilistic metric spaces, Kluwer Acad. Publ., Dordrecht, 2001. [4] S. Kutukcu, D. Turkoglu, C. Yildiz, A common fixed point theorem of compatible maps of type (α) in fuzzy metric spaces, J. Concr. Appl. Math., in press. [5] S. Kutukcu, A fixed point theorem in Menger spaces, Int. Math. F. 1 (32) (2006), 1543-1554. [6] K. Menger, Statistical metric, Proc. Nat. Acad. Sci. 28 (1942), 535-537. [7] M.M. Milovanovic-Arandelovic, A common fixed points theorem for contraction type mappings on Menger spaces, Filomat 11 (1997), 103-108.

Common Fixed Points in Menger Spaces 391 [8] V. Radu, Lectures on probabilistic analysis, Timişoara University, Surveys Lectures Notes and Monograph Series on Probability Statistics and Applied Mathematics, Vol.2, 1994. [9] B. Schweizer, A. Sklar, Statistical metric spaces, Pasific J. Math. 10 (1960), 313-334. [10] B. Schweizer, A. Sklar, E. Thorp, The metrization of SM-spaces, Pasific J. Math. 10 (1960), 673-675. [11] V. Sehgal, A. Bharucha-Reid, Fixed points of contraction mappings on probabilistic metric spaces, Math. System Theory 6 (1972), 97-102. [12] B. Singh, S. Jain, A fixed point theorem in Menger Space through weak compatibility, J. Math. Anal. Appl. 301 (2005), 439-448. [13] R. Vasuki, A fixed point theorem for a sequence of maps satisfying a new contraction type condition in Menger spaces, Math. Japonica 35 (1990), 1099-1102. Received: August 11, 2006