International Journal of Algebra, Vol. 5, 0, no. 8, 855-865 Weak Comatibility in Menger Saces Abhishek Sharma*, Arihant Jain** and Sanjay Choudhari* *Deartment of Mathematics, Govt. Narmada P.G. College Hoshangabad (M.P.) 46 00, India **Deartment of Alied Mathematics, Shri Guru Sandiani Institute of Technology and Science, Ujjain (M.P.) 456 650, India arihant4@gmail.com Abstract In this aer, we rove a common fixed oint theorem for weakly comatible mas in Menger sace which generalizes the result of Kutukcu [6] without aeal to continuity on this sace. We also cited an examle in suort of our result. Mathematics Subject Classification: Primary 47H0, Secondary 54H5 Keywords : Menger sace, t-norm, common fixed oint, comatible mas, weak comatible mas f tye (A), uzzy metric sace.. Introduction There have been a number of generalizations of metric sace. One such generalization is Menger sace initiated by Menger [7]. It is a robabilistic generalization in which we assign to any two oints x and y, a distribution function x,y. Schweizer and Sklar [9] studied this concet and gave some fundamental results on this sace. Sehgal and Bharucha-Reid [] obtained a generalization of Banach Contraction Princile on a comlete Menger sace which is a milestone in develoing fixed oint theory in Menger sace.
856 A. Sharma, A. Jain and S. Choudhari Recently, Jungck and Rhoades [5] termed a air of self mas to be coincidentally commuting or euivalently weakly comatible if they commute at their coincidence oints. Sessa [0] initiated the tradition of imroving commutativity in fixed oint theorems by introducing the notion of weak commuting mas in metric saces. Jungck [4] soon enlarged this concet to comatible mas. The notion of comatible maing in a Menger sace has been introduced by Mishra [8]. Using the concet of comatible maings of tye (A), Jain et. al. [, ] roved some interesting fixed oint theorems in Menger sace. Afterwards, Jain et. al. [3] roved the fixed oint theorem using the concet of weak comatible mas in Menger sace. Using the concet of semi-comatible mas, Singh et. al. [] roved a common fixed oint theorem in Menger sace. The aim of this aer is to rove a common fixed oint theorem for weakly comatible mas in Menger sace which generalizes the result of Kutukcu [6] without aeal to continuity on this sace. We also cited an examle in suort of our result.. Preliminaries or terminologies, notations and roerties of robabilistic metric saces, refer to [9] and []. Definition.. [3] A maing : R R + is called a distribution if it is nondecreasing left continuous with inf{ (t) : t R} = 0 and su{ (t) : t R} =. We shall denote by L the set of all distribution functions while H will always denote the secific distribution function defined by H(t) = 0, t 0; H(t) =, t > 0 Definition.. [6] A triangular norm * (shortly t-norm) is a binary oeration on the unit interval [0, ] such that for all a, b, c, d [0, ] the following conditions are satisfied : (t-) a * = a; (t-) a * b = b * a; (t-3) a * b c * d whenever a c and b d; (t-4) a * (b * c) = (a * b) * c. Examles of t-norms are a * b = max {a + b, 0} and a * b = min {a, b}. Definition.3. [9] A robabilistic metric sace (PM-sace) is an ordered air (X,) consisting of a non emty set X and a function : X X L, where L is
Weak comatibility in Menger saces 857 the collection of all distribution functions and the value of at (u, v) X X is reresented by u,v. The function u,v assumed to satisfy the following conditions : (PM-) u,v (x) =, for all x > 0, if and only if u = v ; (PM-) u,v (0) = 0 ; (PM-3) u,v = v,u ; (PM-4) If u,v (x) = and v,w (y) = then u,w (x + y) =, for all u, v, w X and x, y > 0. A Menger sace is a trilet (X,, *) where (X, ) is a PM-sace and * is a t-norm such that the ineuality (PM-5) u,w (x + y) u,v (x) * v,w (y), for all u, v, w X and x, y 0. Proosition.. [] If (X,d) is a metric sace then the metric d induces a maing X X L defined by, (x) = H(x d(, )), for all, X and x > 0. urther, if the t-norm * is a * b = min{a, b} for all a, b [0, ], then (X,, *) is a Menger sace. It is comlete if (X, d) is comlete. The sace (X,, *) so obtained is called the induced Menger sace. Definition.4. [8] A seuence {x n } in a Menger sace X is said to be convergent and converges to a oint x in X if and only if for each ε > 0 and λ > 0, there is an integer M(ε, λ) such that xn,x (ε) > - λ, for all n M(ε, λ). urther the seuence {x n } is said to be Cauchy seuence if for ε > 0 and λ > 0, there is an integer M(ε, λ) such that xn,x m (ε) > - λ, for all m, n M(ε, λ). A Menger sace is said to be comlete if every Cauchy seuence in X converges to a oint in X. Definition.5. [8] Self mas S and T of a Menger sace (X,, *) are said to be comatible if STxn,TSx n (t) for all t > 0, whenever {x n } is a seuence in X such that Sx n, Tx n u, for some u in X, as n. Definition.6. [3] Self mas S and T of a Menger sace (X,, *) are said to be weakly comatible (or coincidentally commuting) if they commute at their coincidence oints, i.e. if S = T for some X then ST = TS. Proosition.. [3] Self maings A and S of a Menger sace (X,, *) are comatible then they are weakly comatible. Remark.. [3] The concet of weak comatibility is more general than that of comatibility.
858 A. Sharma, A. Jain and S. Choudhari Lemma.. [3] Let {x n } be a seuence in a Menger sace (X,, *) with continuous t-norm * and t * t t. If there exists a constant k (0, ) such that xn,x n+ (kt) xn-,x n (t) for all t > 0 and n =,,..., then {x n } is a Cauchy seuence in X. Lemma.. [3] Let (X,, *) be a Menger sace. If there exists a constant k (0, ) such that x,y (kt) x,y (t) for all x, y X and t > 0, then x = y. 3. Main Result Theorem 3.. Let A, B, S, T, L and M be self mas on a comlete Menger sace (X,, *) with t * t t for all t [0, ], satisfying () L(X) ST(X), M(X) AB(X); () there exists a constant k (0, ) such that Lx,My (kt)* [ ABx,Lx (kt). STy,My [ ABx,Lx (t)+ ABx, STy (t)]. ABx, My (kt) for all x, y X and t > 0 where 0 <, < such that + = ; (3) (a) L and AB have a oint of coincidence, (b) M and ST have a oint of coincidence; (4) AB = BA, ST = TS, LB = BL, MT = TM; (5) the airs (L, AB) and (M, ST) are weakly comatible. Then A, B, S, T, L and M have a uniue common fixed oint. Proof. Let x 0 X. rom condition () x, x X such that Lx 0 = STx = y 0 and Mx = ABx = y. Inductively, we can construct seuences {x n } and {y n } in X such that Lx n = STx n+ = y n and Mx n+ = ABx n+ = y n+ for n = 0,,,.... Ste. By taking x = x n and y = x n+ in (), we have Lx,Mx n n+ (kt) * [ ABxn,Lx n (kt). STxn+,Mx n+ [ ABxn,Lx n (t) + ABxn, STx n+ (t)]. ABxn, Mx n+ (kt) n n+ (kt) * [ y n-,y n (kt). yn,y n+ n n+ [ yn,y n- (t) + yn-, y n (t)]. yn-, y n+ (kt) (kt) [ y n-,y n (kt) * yn,y n+
Weak comatibility in Menger saces 859 n n+ Hence, we have n n+ Similarly, we also have n+ n+ ( + ) yn,y n- (t). yn-, y n+ (kt). y n-,y n+ (kt) yn-,y n (t). yn-, y n+ (kt). (kt) y n-,y n (t). (kt) y n,y n+ (t). In general, for all n even or odd, we have n n+ (kt) y n-,y n (t) for k (0, ) and t > 0. Thus, by lemma., {y n } is a Cauchy seuence in X. Since (X,, *) is comlete, it converges to a oint z in X. Also its subseuences converges as follows : {Mx n+ } z and {STx n+ } z, {Lx n } z and {ABx n } z. Ste. Since M(X) AB(X), then there exists a oint u X such that z = ABu. Put x = u and y = x n+ in (), we get Lu,Mx n+ (kt)* [ ABu,Lu (kt). STxn+,M x n+ z,lu [ ABu,Lu (t)+ ABu, STxn+ (t)]. ABu, M xn+ (kt). (kt)* [ z,lu (kt). z,z [ z,lu (t)+ z, z (t)]. z, z (kt) z,lu Noting that (kt)* [ z,lu [ z,lu (t)+]. z,lu(kt) and using (t-3) in definition., we get z,lu (kt) z,lu (t) + z,lu (kt) +
860 A. Sharma, A. Jain and S. Choudhari z,lu (kt) = for k (0, ) and t > 0. Thus, we have z = Lu. Therefore, z = Lu = ABu. Thus u is a coincidence oint of L and AB. Since (L, AB) is weakly comatible, then LABu = ABLu. Thus, Lz = ABz. Ste 3. By taking x = z and y = x n+ in (), we get Lz,Mx n+ (kt)* [ ABz,Lz (kt). STxn+,M x n+ [ ABz,Lz (t)+ ABz, STxn+ (t)]. ABz, M xn+ (kt). z,lz (kt)* [ Lz,Lz (kt). z,z [ Lz,Lz (t)+ z, Lz (t)]. z, Lz (kt) z,lz (kt) [ + z,lz (t)]. z,lz. z,lz (kt) + z,lz (t) + z,lz (kt) z,lz (kt) = for k (0, ) and t > 0. Thus, we have z = Lz. Therefore, z = Lz = ABz. Ste 4. By taking x = Bz and y = x n+ in (), we get LBz,Mx n+ (kt)* [ ABBz,LBz (kt). STxn+,M x n+ [ ABBz,LBz (t)+ ABBz, STxn+ (t)]. ABBz, M xn+ (kt). Since AB = BA and BL = LB, we have LBz = BLz = Bz and ABBz = BABz = Bz. z,bz (kt)* [ Bz,Bz (kt). z,z [ Bz,Bz (t)+ z, Bz (t)]. z, Bz (kt)
Weak comatibility in Menger saces 86 z,bz (kt) [ + z,bz (t)]. z,bz (kt) z,bz (kt) [ + z,bz (t)] z,bz (kt) [ + z,bz z,bz (kt) for k (0, ) and t > 0. Thus, we have z = Bz. Since z = ABz, so z = Az. Therefore, z = Az = Bz = Lz. Ste 5. Since L(X) ST(X), there exists v X such that z = Lu = STv. = By taking x = x n and y = v in (), we get Lx n,mv (kt)*[ ABxn,Lx n (kt). STv,Mv [ ABxn,Lx n (t)+ ABxn, STv(t)]. z,mv ABxn, Mv(kt). (kt)*[ z,z (kt). z,mv [ z,z (t)+ z, z (t)]. z, Mv (kt) Noting that z,mv (kt)* z,mv (kt) [ + ]. z, Mv (kt). z,mv(kt) and using (t-3) in definition., we get z,mv (kt) z,mv (kt) z,mv (kt). Thus, by lemma., we have z = Mv and so z = Mv = STv. Thus, v is a coincidence oint of M and ST. Since (M, ST) is weakly comatible, we have MSTv = STMv. Thus, STz = Mz. Ste 6. By taking x = x n and y = z in (), we get Lx n,mz (kt)*[ ABxn,Lx n (kt). STz,Mz [ ABxn,Lx n (t)+ ABxn, STz(t)].
86 A. Sharma, A. Jain and S. Choudhari z,mz ABxn, Mz(kt). (kt)*[ z,z (kt). Mz,Mz [ z,z (t)+ z, Mz (t)]. z, Mz (kt) z,mz (kt) [ + z, Mz. z, Mz (kt) z, Mz (kt) [ + z, Mz z, Mz (kt) for k (0, ) and t > 0. Thus, we have = z = Mz and so z = Az = Bz = Lz = Mz = STz. Ste 7. By taking x = x n and y = Tz in (), we get Lx n,mtz (kt)*[ ABxn,Lx n (kt). STTz,MTz [ ABxn,Lx n (t)+ ABxn, STTz(t)]. ABxn, MTz(kt). Since MT = TM and ST = TS, we have MTz = TMz = Tz and ST(Tz) = T(STz) = Tz. z,tz (kt)*[ z,z (kt). Tz,Tz [ z,z (t)+ z, Tz (t)]. z, Tz (kt) z, Tz (kt) = for k (0, ) and t > 0. Thus, we have z = Tz. Since z = STz, so Sz = z. Therefore, z = Az = Bz = Lz = Mz = Sz = Tz, i.e. z is a common fixed oint of six mas. Ste 8. or uniueness, let z (z z) be another common fixed oint of the given six self mas. Then z = Az = Bz = Lz = Mz = Sz = Tz. By taking x = z and y = z in (), we get Lz,Mz (kt)* [ ABz,Lz (kt). STz,M z [ ABz,Lz (t)+ ABz, STz (t)]. ABz, Mz (kt)
Weak comatibility in Menger saces 863 which imlies that z,z (kt)* [ z, z (kt). z, z [ z, z (t)+ z, z (t)]. z, z (kt) Thus, we have z = z. [ + z, z (t)]. z, z (kt) z, z (kt) [ + z, z z, z (kt) [ + z, z. z, z (kt) Hence, z is a uniue common fixed oint of six self mas A, B, S, T, L and M. This comletes the roof. =. Remark 3.. If we take B = T = I, the identity ma on X in theorem 3., then the condition (4) is satisfied trivially and we get Corollary 3.. Let A, S, L and M be self mas on a comlete Menger sace (X,, *) with t * t t for all t [0, ], satisfying (6) L(X) S(X), M(X) A(X); (7) there exists a constant k (0, ) such that Lx,My (kt)* [ Ax,Lx (kt). Sy,My [ Ax,Lx (t)+ Ax, Sy (t)]. Ax, My (kt) for all x, y X and t > 0 where 0 <, < such that + = ; (8) (a) L and A have a oint of coincidence, (b) M and S have a oint of coincidence; (9) the airs (L, A) and (M, S) is weakly comatible. Then A, S, L and M have a uniue common fixed oint. Remark 3.. Theorem 3. is a generalization of the result of Kutukcu [6] in the sense that condition of comatibility of the air of self mas has been restricted to weak comatible self mas and none of the maings of the weak comatible airs are needed to be continuous. The following examle illustrates Corollary 3.. Examle 3.. Let X = [0, 30) with the metric d defined by d(x, y) = x y and for each t [0, ] define
864 A. Sharma, A. Jain and S. Choudhari t t+ x y,if t> 0 (t) = x,y 0, if t = 0 for all x, y X. Clearly (X,, *) is Menger sace where * defined by t * t t. Define A, S, L and M : X X by 0, if x = 0; A(X) =, if 0 < x < 5; x 9, if 5 < x < 30 0, if x = 0; S(X) = 6, if 0 < x < 5; x 6, if 5 < x < 30 0, if x = 0; L(X) = and 0, if x = 0; M(X) =. 6, if 0< x < 30 9, if 0 < x < 30 Then A, S, L and M satisfy all the conditions of Corollary 3. with k (0, ) and have a uniue common fixed oint 0 X. It may be noted that in this examle that the maings L and S commute at coincidence oint 0 X. So L and A are weakly comatible mas. Similarly, M and S are weakly comatible mas. To see the airs (L, A) and (M, S) are not comatible. Let us consider a seuence {x n } defined as x = 5 +, n then x n n 5 as n. n Then lim n Lx n = 6, n lim Ax t n = 6 but lim (t) =. n LAx n,alx n t+ 6 Thus, the air (L, A) is not comatible. Also, lim n Mx n = 9, n lim Sx n = 9 but t lim (t) =. n MSx n,smxn t+ 9 6 So the (M, S) is not comatible. All the maings involved in this examle are discontinuous even at the common fixed oint x = 0. REERENCES. A. Jain and B. Singh, Common fixed oint theorem in Menger sace through comatible mas of tye (A), Chh. J. Sci. Tech. (005), -.. A. Jain and B. Singh, A fixed oint theorem in Menger sace through comatible mas of tye (A), V.J.M.S. 5(), (005), 555-568.
Weak comatibility in Menger saces 865 3. A. Jain and B. Singh, Common fixed oint theorem in Menger Saces, The Aligarh Bull. of Math. 5 (), (006), 3-3. 4. G. Jungck, Comatible maings and common fixed oints, Internat. J. Math. and Math. Sci. 9(4), (986), 77-779. 5. G. Jungck and B.E. Rhoades, ixed oints for set valued functions without continuity, Indian J. Pure Al. Math. 9(998), 7-38. 6. K. Kutukcu, A fixed oint theorem in Menger saces, International Mathematical orum (3), (006), 543-554. 7. K. Menger, Statistical metrics, Proc. Nat. Acad. Sci. USA. 8(94), 535-537. 8. S.N. Mishra, Common fixed oints of comatible maings in PM-saces, Math. Jaon. 36(), (99), 83-89. 9. B. Schweizer and A. Sklar, Statistical metric saces, Pacific J. Math. 0 (960), 33-334. 0. S. Sessa, On a weak commutativity condition of maings in fixed oint consideration, Publ. Inst. Math. Beograd 3(46), (98), 46-53.. V.M. Sehgal and A.T. Bharucha-Reid, ixed oints of contraction mas on robabilistic metric saces, Math. System Theory 6(97), 97-0.. B. Singh, A. Jain and B. Lodha, On common fixed oint theorems for semicomatible maings in Menger sace, Commentationes Mathematicae 50(), (00), 7-39. 3. B. Singh and S. Jain, A fixed oint theorem in Menger sace through weak comatibility, J. Math. Anal. Al., 30 (005), 439-448. Received: ebruary, 0