Dubey, Ravi

Patel, Shailendra Kumar

Shukla, Manoj

English

Scitech Research Journals

 Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218   Volume 3, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                           216|   SCITECH                                                                               Volume 3, Issue 3 RESEARCH ORGANISATION|                                         May 6, 2015| Journal of Progressive Research in Mathematics www.scitecresearch.com Results in Fuzzy Probabilistic Metric Space Manoj Shukla1, Ravi Dubey2 and Shailendra Kumar Patel3 1 Govt. Model Science College (autonomous), Jabalpur(M.P.) India. 2 Dr C.V. Raman University, Bilaspur (C.G) India. 3 BIT, Raipur (C.G) India.  Abstract In this paper we prove fixed point theorem in the setting of fuzzy probabilistic metric space using weak commuting mappings and also prove result for rational expression.  AMS Classification: 47H10, 54H25. Key words: Fuzzy Probabilistic Metric Space (FPMr Space); weak commuting mappings Common Fixed Points.  Introduction and Preliminaries Menger [2]  in 1942 introduced the notation of the probabilistic metric space. The probabilistic generalization of metric space appears to be well adopted for the investigation of physical quantities and physiological thresholds. Cho et al. [1] introduced the notation of semi compatible maps in a- topological space. According to them a pair of self-maps (S,T) to be semi compatible if condition (i) Sy = Ty  STy = TSy; (ii) the sequence {xn} in X and x X, {Sxn}  x,  {Txn}  x then STxn  = Tx as n  , hold. We define semi compatible self-maps in probabilistic metric space by (ii) only. Popa in [3]  used the family  of implicit function to find the fixed points of two pairs of semi compatible maps in a d complete topological space, where   be the family of real continuous function  : (R+)4  R satisfying the properties (Gh)  for every u 0, v0 with (u,v,u,v) 0 or (u,v,v,u) 0 we have u  v. (Gu)  (u,u,1,1)  0 implies that  u  1 The main object of this paper we prove fixed point  theorem in the setting of fuzzy probabilistic metric space using weak compatibility also prove result for rational expression.  Definition 4.2.1 A fuzzy probabilistic metric space (FPM space) is an ordered pair (X,Fα) consisting of a nonempty set X and a mapping Fα from XxX into the collections of all fuzzy distribution functions FαR α.  [0,1]. For x, y  X we denote the distribution function Fα (x,y) by Fα(x,y) and Fα(x,y) (u) is the value of Fα(x,y)   at u in R. The functions Fα(x,y)   for all α.  [0,1] assumed to satisfy the following conditions: (a)  Fα(x,y) (u) = 1  u > 0 iff x = y, (b)  Fα(x,y)   (0) = 0  x , y in X, (c)  Fα(x,y)   = Fα(y,x)    x , y in X, (d)  If Fα(x,y)    (u) = 1 and   Fα(y,z)   (v) = 1 then Fα(x,z) (u+v) = 1   Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218   Volume 3, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                           217|              x , y ,z in X  and   u, v  > 0 Definition 4.2.2 A  commutative,  associative  and  non-decreasing  mapping t: [0,1]  [0,1] [0,1] is a t-norm if and only if t(a,1)=a for all a[0,1] , t(0,0)=0 and t(c,d)  t(a,b) for c  a,  d  b   . Definition 4.2.3 A Fuzzy Menger space is a triplet (X,Fα,t), where (X,Fα) is a FPM-space,  t is a t-norm and the  generalized triangle inequality Fα(x,z) (u+v)  t (Fα(x,z) (u), Fα(y,z) (v)) holds for all x, y, z in X u, v > 0 and α.  [0,1] The concept of neighborhoods in Fuzzy Menger space is introduced as Definition 4.2.4 Let (X,Fα,t) be a Fuzzy Menger space. If  xX,  > 0 and  (0,1), then (,)  - neighborhood of x, called Ux  (,), is defined by  Ux (,) = {yX: Fα(x,y)()>(1-)} An (,)-topology in X is  the topology  induced by  the family {Ux(,): xX, >0, α.  [0,1]and  (0,1)} of neighborhood. Remark: If t is continuous, then Fuzzy Menger space (X,Fα,t) is a Housdroff space in (,)-topology. Let (X,Fα,t) be a complete Fuzzy Menger space and AX. Then A is called a bounded set if  lim inf Fα(x,y) (u) = 1          u    x,yA Definition 4.2.5 A sequence {xn} in (X,Fα,t) is said to be convergent to a point x in X if for every >0and  >0, there exists an integer N=N(,) such that xn Ux(,) for all n  N  or equivalently Fα (xn, x; ) > 1- for all n  N and α[0,1]. Definition 4.2.6  A sequence {xn}  in (X,Fα, t)  is said to be  cauchy sequence  if for every  > 0 and   > 0, there exists an integer N=N(,) such that Fα(xn,xm; ) > 1-  n, m  N for all α[0,1]. Definition 4.2.7 A Fuzzy Menger space (X,Fα,t) with the continuous t-norm is said to be complete if every Cauchy sequence  in X converges to a point in X for all α[0,1]. Definition 4.2.8 Let (X,Fα,t) be a Fuzzy Menger space. Two mappings  f, g :XX are said to be weakly comptable if  they commute at coincidence point for all α[0,1]. Lemma 4.2.1 Let {xn}  be a sequence in a Fuzzy Menger space  (X,Fα,t), where t  is continuous and t(p,p)  p for all p[0,1], if there exists a constant k(0,1) such that for all p > 0 and nN  Fα (xn,xn+1; kp)  Fα(xn-1,xn; p), for all α[0,1] then {xn} is cauchy sequence. Lemma 4.2.2 If (X,d) is a metric space, then  the  metric d  induces, a mapping Fα: XxXL defined by Fα (p, q) = Hα(x- d(p, q)), p, q  R for all α[0,1]. Further if  t: [0,1]  [0,1] [0,1] is defined by t(a,b) = min{a,b}, then (X,Fα,t) is a Fuzzy Menger space. It is complete if (X,d) is complete.  Main results  Theorem (1.1): Let (X,Fα,t) be a complete Fuzzy Menger space, where t is continuous and t(p,p)  p for all p and α in [0,1].Let S and T be continuous mappings of X , then S and T have a common  fixed point in X if there exists continuous mappings A of X into S (X)  T (X) which commute weakly with S and T and             (1.1)    , , , , , , , , , , , , , , , ,F Ax Ay kp F Ty Ay p F Sx Ax p F Sx Ty p F Ax Ty p F Sx Ay p                                                                                                                                                                                                                                                                                                                                                                     for all x,y X and p > 0. Then S, T and A have a unique common fixed point. Proof: We define a sequence {xn} such that   Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218   Volume 3, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                           218|  Ax2n  = Sx2n-1  and A,x2n-1 =  Tx2n n = 1, 2,-- We shall prove that {Axn}is a Cauchy sequence.           2 1 2 1 2 2 2 2 12 2 1 2 2 1 2 2 1, , , , , , , , ,, min , , , , ,n n n n n nn n n n n nF Tx Ax p F Sx Ax p F Sx Tx pF Ax Ax kp F Ax Tx p F Sx Ax p                        2 2 1 2 1 2 2 1 22 2 1 2 2 2 1 2 1, , , , , , , , ,, , min , , , , ,n n n n n nn n n n n nF Ax Ax p F Ax Ax p F Ax Ax pF Ax Ax kp F Ax Ax p F Ax Ax p                2 2 1 2 1 2 2 1 2min ( , , ), ( , , ), ( , , ) ,1,1n n n n n nF Ax Ax p F Ax Ax p F Ax Ax p    2 1 2 2 2 1 2 2 1min ( , , ) , ( , ), ( , , ) ,1,1n n n n n np p pF Ax Ax F Ax Ax F Ax Axk k k                     Therefore   2 2 1, ,n nF Ax Ax kp    2 1 2, ,n npF Ax Axk     By induction   2 2 1 2 2 1, , , ,k m m kpF Ax Ax kp F Ax Axk        For every k and m in N, Further if 2m +1 > 2k, then   2 2 1 2 1 2 0 2 1 2 2, , , , ........... , , ---- (8.2.1b)k m k m m k kp pF Ax Ax kp F Ax Ax F Ax Axk k                                                                                                                                                If 2k > 2m+1, then  2 2 1 2 1 2 2 (2 1) 0 2 1, , , , ........... , , ----- (1.1 c)k m k m k m mp pF Ax Ax kp F Ax Ax F Ax Axk k                                                                                                                                             By simple induction with (1.1 b) and (1.1c) we have     0, , , , .n n p p npF Ax Ax kp F Ax Axk      For n = 2k, p = 2m+1 or n = 2k+1, p = 2m +1   0 1 1, , , , . , , . (1.1 d)2n n p pn np pF Ax Ax kp F Ax Ax F Ax Axk k             If n = 2k, p = 2m or n = 2k+1, p= 2m For every positive integer p and n in N, by nothing that  0 , , 1p npF Ax Ax as nk     Thus {Axn} is a Cauchy sequence. Since the space X is complete there exists z  X, such that  Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218   Volume 3, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                           219|  2 1 2lim lim limn n nn n nAx Sx Tx z      Science A is weakly commute with S and T ,so  Az = Sz = Tz and  Therefore           , , , , , , , , , , , ,, , inf, ,n n n nnnF Tz Az p F Sx Ax p F Sx Tz p F Ax Tz pF Ax Az kpF Sx Az p          On taking limit, we get                     ( , , ) inf , , , , , , , , , , , , , ,( , , ) inf , , , , , , , , , , , , , ,.F z Az kp F Tz Az p F z z p F z Tz p F z Tz p F z Az pF z Az kp F Az Az p F z z p F z Az p F z A p F z Az pAz z            Thus z is common fixed point of A, S and T.  For uniqueness, let v  z be another common fixed point of S, T and A.  By (1.1 a) we write                        , , , , , , , , , , , ,, , inf, ,, , inf , , , , , , , , , , , , , ,F Tv Av p F Sz Az p F Sz Tv p F Az Tv pF Az Av kpF Sz Av pF z v kp F v v p F z z p F z v p F z v p F z v p               Therefore by lemma iii, we write .z v .  This completes the proof of Theorem (1.1)  Now we prove a result for rational expression. Theorem (1.2): Let (X,Fα,t) be a complete Fuzzy Menger space, where t is continuous and t(p,p)  p for all p and α in [0,1].Let S and T be continuous mappings of X in X, then S and T have a common fixed point in X if there exists continuous mappings A of X into S (X)  T (X) which commute weakly with S and T and             (1.2a), ,, , , , , , , , , ,, ,, , min, , , ,, ,, , ) , ,F Sx Ty pF Ty Ay p F Sx Ax p F Sx Ty pF Ax Ty pF Ax Ay kpF Ty Ay p F Sx Ax pF Sx Ax p F Ty Ay p                                                                                                                                                                                                                                                                                                                                                                for all x,y  X, t > 0,and 0 < q < 1. Then S, T and A have a unique common fixed point. Proof: We define a sequence xn such that  A,x2n=S,x2n-1  and Ax2n-1  = T,x2n n = 1, 2,-- We shall prove that {Axn} is a Cauchy sequence. By (1.2a), we have             2 1 2 1 2 2 2 2 12 2 1 2 1 2 1 2 22 2 12 2 1 2 2 2 1 2 1, , , , , , , , ,, , , , , ,, , min , ,, , , , , ,n n n n n nn n n n n nn nn n n n n nF Tx Ax p F Sx Ax p F Sx Tx pF Sx Tx p F Tx Ax p F Sx Ax pF Ax Ax kpF Ax Tx p F Sx Ax p F Tx Ax p                      Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218   Volume 3, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                           220|              2 2 1 2 1 2 2 1 22 2 1 2 1 2 2 2 1 2 1 22 2 2 1 2 2 2 1, , , , , , , , ,, , min , , , , , ,, ,, , , , , ,n n n n n nn n n n n n n nn n n n n nF Ax Ax p F Ax Ax p F Ax Ax pF Ax Ax kp F Ax Ax p F Ax Ax p F Ax Ax pF Ax Ax p F Ax Ax p F Ax Ax p                     2 2 1 2 1 2 2 1 2min ( , , ), ( , , ), ( , , ), 1,1,1n n n n n nF Ax Ax p F Ax Ax p F Ax Ax p      2 1 2 2 2 1)min ( , , ), ( , , )n n n np pF Ax Ax F Ax Axk k       Therefore   2 2 1, ,n nF Ax Ax kp    2 1 2, ,n npF Ax Axk     By induction   2 2 1 2 2 1, , , ,k m m kpF Ax Ax kp F Ax Axk        For every k and m in N, Further if 2m +1 > 2k, then   2 2 1 2 1 20 2 1 2 2, , , , .................. , , ----- (1.2b)k m k mm k kpF Ax Ax kp F Ax AxkpF Ax Axk                                                                                                                                        If 2k > 2m+1, then  2 2 1 2 1 2 2 (2 1) 0 2 1, , , , ........... , , ....... (1.2c)k m k m k m mp pF Ax Ax kp F Ax Ax F Ax Axk k                                                                                                                                                    By simple induction with (1.2b) and (1.2c) We have     0, , , , .n n p p npF Ax Ax kp F Ax Axk      For n = 2k, p = 2m+1 or n = 2k+1, p = 2m +1  0 1 1, ,, , . , , . (1.2d)2n n ppn nF Ax Ax kpp pF Ax Ax F Ax Axk k             If n = 2k, p = 2m or n = 2k+1, p= 2m For every positive integer p and n in N, by nothing that  0 , , 1p npF Ax Ax as nk     Thus { nAx } is a Cauchy sequence.  Since the space X is complete there exists z  X, such that  Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218   Volume 3, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                           221|  2 1 2lim lim limn n nn n nAx Sx Tx z      It follows that Ax = Sz  = Tz .Therefore              , , , , , , , , ,, , min , , , , , ,, , ,, , , , , ,F TAz AAz p F Sz Az p F Sz TAz pF Az AAz kp F Sz TAz p F TAz AAz p F Sz Az pF Az TAz p F Sz Az p F TAz AAz p              ( , , ) ( , , )F Az AAz kp F Sz TAz p  ( , , ) ( , , )............. ( , , )npF Sz ATz p F Az AAz p F Az AAzk      lim ( , , ) 1nnpSince F Az AAz Az AAzk    Thus z is common fixed point of A, S and T.  For uniqueness, let v(( )(v( ) z( )) be another common fixed point of S, T and A.  By (1.2a), we write               , , , , , , , , ,, , min , , , , , ,, ,, , , , , ,F Tv Av p F Sz Az p F Sz Tv pF Az Av kp F Sz Tv p F Tv Av p F Sz Az pF Az Tv p F Sz Az p F Tv Av p                     , , , , , , , ,F Az Av kp F z v p F z v kp F z v p       Therefore by lemma iii, we write z( )=v( ).  This completes the proof of Theorem (1.2). References [1] Beg, I. and Shahzad, N. “Random approximations and random fixed point theorems” J. Appl. Math. Stochastic Anal. 7 (1994). No 2, 145-150. [2] Beg, I. and Shahzad, N. “Random fixed points of weakly inward operators in conical shells” J. Appl. Math, Stoch. Anal. 8(1995) 261-264. [3] Bharucha –Reid, A.T. Fixed point theorems in probabilistic analysis, Bull. Amer. Math. Soc.82 (1976) 641-657. [4] Choudhary, B.S. and Ray, M. “Convergence of an iteration leading to a solution of a random operator equation” J. Appl. Math. Stochastic Anal. 12 (1999). No 2,161-168. [5] Choudhary, B.S. and Upadhyay, A. “An iteration leading to random solutions and fixed points of operators” Soochow J. Math.25 (1999). No 4,395-400. [6] Choudhary, B.S. “A common unique fixed point theorem for two random operators in Hilbert spaces” I.  J. M.M. S. 32 (2002) 177-182. [7] Dhagat, V.B., Sharma, A. and Bhardwaj, R.K.  “Fixed point theorem for random operators in Hilbert spaces” International Journal of Math. Analysis 2(2008) no. 12, 557-561. [8] Grabiec, M.: Fixed points in fuzzy metric space, Fuzzy Sets and Systems, 27, 385- 389 (1988).  [9] Mishra, S.N., Sharma, N., Singh, S.L.: Common fixed points of maps on fuzzy metric spaces,   Internet. J. Math. & Math Sci., 17, 89-108 (1966).  Journal of Progressive Research in Mathematics(JPRM) ISSN: 2395-0218   Volume 3, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                           222|  [10] Som, T. “Some results on common fixed point in fuzzy Metric spaces” Soochow Journal of Mathematics 4 (2007) 553-561. 

Results in Fuzzy Probabilistic Metric Space

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Results in Fuzzy Probabilistic Metric Space

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