A Common Fixed Point Theorem Using in Fuzzy Metric Space Using Implicit Relation

1. International Research Research and Development (IJTSRD) International Open Access Journal mmon Fixed Point Theorem Using in Fuzzy Metric Space Using Implicit Relation Rashmi Rani Assistant Professor, A. S. College for Women, Khanna,Ludhiana, Punjab, India International Journal of Trend in Scientific Scientific (IJTSRD) International Open Access Journal ISSN No: 2456 ISSN No: 2456 - 6470 | www.ijtsrd.com | Volume 6470 | www.ijtsrd.com | Volume - 2 | Issue – 5 A Common Fixed Point Theorem Using i Metric Space Using Implicit Relation n Fuzzy Assistant Professor Khanna ABSTRACT In this paper, we prove a common fixed point theorem in fuzzy metric space by combining the ideas of point wise R- weak commutativity and reciprocal continuity of mappings satisfying contractive conditions with an implicit relation. Keywords: Implicit relation, Common fixed point, R weakly commuting mappings. 1. INTRODUCTION In 1965, Zadeh [11] introduced the concept of Fuzzy set as a new way to represent vagueness in our everyday life. However, when the uncertainty is due to fuzziness rather than randomness, as sometimes in the measurement of an ordinary length, it seems that the concept of a fuzzy metric space is more suitable. We can divide them into following two groups: The first group involves those results in which a fuzzy metric on a set X is treated as a map where represents the totality of all fuzzy points of a set and satisfy some axioms which are analogous to the ordinary metric axioms. Thus, in such an approach numerical distances are set up between fuzzy objects. On the other hand in second group, we keep those results in which the distance between objects is fuzzy and the objects themselves may or may not be fuzzy. Kramosil et al. (1975)[3] have introduced the concept of fuzzy metric spaces in different ways [1 paper, we prove a common fixed point theorem in fuzzy metric space by combining the ideas of wise R- weak commutativity and reciprocal continuity of mappings satisfying contractive conditions with an implicit relation. 2. Preliminaries: The concept of triangular norms ( originally introduced by Menger in study of statistical metric spaces. Definition 2.1 (Schweizer & Sklar, 1985)[9] binary operation *: [0,1]×[0,1] t-norm if * satisfies the following conditions: I.* is commutative and associative; II.* is continuous; III.a * 1 = a for all [0,1] a IV.a * bc * d whenever , , , [0,1] a b c d  . Examples of t-norms are: a*b and a*b = max{a+b-1,0}. Kramosil et al. (1975)[3] introduced the concept of fuzzy metric spaces as follows: Definition 2.2 (Kramosil & Michalek, 1975)[3] tuple ( , ,*) X M is said to be a fuzzy metric space if is an arbitrary set, *is a continuous fuzzy sets on X2×[0, ∞) satisfying the following conditions for all , , x y z X  I.M(x, y, 0) = 0; II.M(x, y, t) = 1 for all t > 0 III.M(x, y, t) = M(y, x, t); IV.M(x, y, t) * M(y, z, s) M V.M(x, y, .) : [0, ∞)→ [0, 1] Then ( , ,*) X M is called a fuzzy metric space on The function M(x, y, t) denote the degree of nearness between x and y w.r.t. t respectively. respectively. In this paper, we prove a common fixed point theorem in fuzzy metric space by combining the ideas of point weak commutativity and reciprocal continuity of mappings satisfying contractive conditions with an The concept of triangular norms (t-norms) is originally introduced by Menger in study of statistical Definition 2.1 (Schweizer & Sklar, 1985)[9] A : [0,1]×[0,1]  [0,1] is continuous satisfies the following conditions: * is commutative and associative; Implicit relation, Common fixed point, R- [11] introduced the concept of Fuzzy set as a new way to represent vagueness in our everyday life. However, when the uncertainty is due to fuzziness rather than randomness, as sometimes in the measurement of an ordinary length, it seems that a fuzzy metric space is more suitable. We can divide them into following two groups: The first group involves those results in which a fuzzy is treated as a map where X represents the totality of all fuzzy points of a set and e axioms which are analogous to the ordinary metric axioms. Thus, in such an approach numerical distances are set up between fuzzy objects. On the other hand in second group, we keep those results in which the distance between objects is fuzzy cts themselves may or may not be fuzzy. Kramosil et al. (1975)[3] have introduced the concept of fuzzy metric spaces in different ways [1-10]. In this paper, we prove a common fixed point theorem in fuzzy metric space by combining the ideas of point weak commutativity and reciprocal continuity of mappings satisfying contractive conditions with an [0,1]; whenever a c and bd for all a*b = min{a, b}, a*b = ab Kramosil et al. (1975)[3] introduced the concept of spaces as follows: Definition 2.2 (Kramosil & Michalek, 1975)[3] A 3- is said to be a fuzzy metric space if X is a continuous t-norm, and M is ∞) satisfying the following and and s, t > 0, 0 if and only if x = y; M(x, z, t + s); 1] is left continuous. is called a fuzzy metric space on X. ) denote the degree of nearness @ IJTSRD | Available Online @ www.ijtsrd.com www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 291

2. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Remark 2.3(Kramosil & Michalek, 1975)[3] In fuzzy metric space ( , X M decreasing for all , x y X  . Definition 2.4 (Kramosil & Michalek, 1975)[3] Let ( , ,*) X M be a fuzzy metric space. Then a sequence {xn} in X is said to be A.convergent to a point xX if, for all t > 0, a.limn→∞M(xn, x, t) = 1. b. B.Cauchy sequence if, for all t > 0 and p > 0, a.limn→∞M(xn+p, xn, t) = 1. Definition 2.5 (Kramosil & Michalek, 1975) [3] A fuzzy metric space ( , ,*) X M and only if every Cauchy sequence in X is convergent. Definition 2.6 (Vasuki, 1999)[10] Apair of self mappings (A , S) of a fuzzy metric space ( , said to be commuting if M( ASx , SAx , t ) = 1 for all xX. Definition 2.7 (Vasuki, 1999)[10] Apair of self mappings (A , S) of a fuzzy metric space ( , said to be weakly commuting if M(ASx , SAx , t) ≥ M(Ax, Sx, t) for all xX and t > 0. Definition 2.8 (Jungck & Rhoades, 2006)[2] Apair of self mappings (A, S) of a fuzzy metric space ( , ,*) X M is said to be compatible if limn→∞M(ASxn, SAxn, t) = 1 for all t > 0, whenever {xn} is a sequence in X such that limn→∞Axn = limn→∞ Sxn = u for some uX. Definition 2.9(Jungck & Rhoades, 2006)[2] Let ( , ,*) X M be a fuzzy metric space. A and S be self maps on X. A point xX is called a coincidence point of A and S iff Ax = Sx. In this case, w = Ax = Sx is called a point of coincidence of A and S. Definition 2.10 (Jungck & Rhoades, 2006)[2] Apair of self mappings (A, S) of a fuzzy metric space ( , ,*) X M is said to be weakly compatible if they commute at the coincidence points i.e., if Au = Su for some u X , then ASu = SAu. It is easy to see that two compatible maps are weakly compatible but converse is not true. Definition 2.11 (Vasuki, 1999)[10] Apair of self mappings (A, S) of a fuzzy metric space ( , said to be pointwise R-weakly commuting if given x X, there exist R   , , , M ASx SAx t M Ax Sx    Clearly, every pair of weakly commuting mappings is pointwise R-weakly commuting with R = 1. Definition 2.12 (Pant, 1999)[7] Two mappings A and S of a fuzzy metric space ( , reciprocally continuous if whenever {un} is a , n n Au z Su z   for some zX. If A and S are both continuous, then they are obviously reciprocally continuous but converse is not true. Lemma 2.1 (Kramosil & Michalek, 1975)[3] Let {un} is a sequence in a fuzzy metric space ( , If there exists a constant h ( , , ) ( , , ), n n n n M u u ht M u u t    Then {un} is a Cauchy sequence in X. 3. Main Result: Let denote the class of those functions     : 0,1 0,1   such that is continuous and ( ,1,1, , ) . x x x x   There are examples of  : 1.  1 1 2 3 4 5 ( , , , , ) min x x x x x     3. 3 1 2 3 4 5 1 2 3 4 5 ( , , , , ) x x x x x x x x x x   Now we prove our main results. , M(x, y, .) is non- ,*) is X M ,*) > t R 0 such that     for all t > 0. , will be called , n Az SAu such X M ASu sequence ,*)   Sz that , n is said to be complete if is X M ,*) . X M ,*) is X M ,*) such that (0,1) n = 1, 2, 3, … 1 1 5  x x x x x , , , , ; 1 2 3 4 5      x x x  x  x  x   1 1 2 3 4 2 5 2. ; ( , x x x x x , , , )  2 1 2 3 4 5 x x x 1 4 5 3 @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 292

3. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Theorem 3.1 Let f and g be conditionally reciprocally continuous self-mappings of a complete fuzzy metric space ( , ,*) X M satisfying the conditions: (3.1) ( ) ( ) f X g X  ; (3.2) for any , x y X  , 0 t  and (0,1) k  such that:  ( , , ) min ( , , ), ( , ,2 ), M fx fy kt M gx gy t M gx fy t M fx gx t M fx gy t M fy gy t  If f and g are either compatible or g- compatible or f- compatiblethen f and g have a unique common fixed point. Proof. Let x0 be any point in X. Then as ( ) f X g X  ( ) ( ) n n f x g x  . Also, define a sequence {yn} in X as 1 ( ) ( n n n y f x g x   Now, we show that {yn} is a Cauchy sequence in X. For proving this, by (3.2), we have ( , , ) ( , , ) n n n n M y y kt M fx fx kt M gx gx t M gx fx t M fx gx t M fx gx t M fx gx t         , ), ( , , ), ( ( , , ),1, ( , , ) n n n n M y y t M y y t M y y t M y y t     Then, by lemma 2.2, {yn} is a Cauchy sequence in X. As, X is complete, there exist a point z in X such that lim n n  n n   reciprocally continuous and lim ( ) lim ( ) n n n n   lim ( ) lim ( ) ( ) n n n n   n  sn , there exist zn in X such that fsn= gzn. Thus, lim ( ) lim ( ) lim ( ) n n n n n n    ( , , ), ( , ,2 ), ( , , ) min ( , , ), ( , , ) n n n n M fs gz t M fz gz t n M u u t M u fz t M u u t M u fz kt M u u t M fz u t              ( , , ), ( , , ), ( , , ) ; , there exist a sequence of points {xn} such that ( ) 1 ) . (3.3)    1 1      ( ( , , , ), , ), ( ( , ,2 ), , ) ( , , ),   n n 1 n n 1 n n  min , n n 1 n 1 n 1    M y M y y t M y ( ( , y t M y , ), , ), ( , y t y ,2 ), t M y y ( , , ), t     n 1 n n 1 n 1 n n 1 min , ( , , )  n n y t M y n 1 n y t M y y       M y ( , , , ), t    n 1 n n 1 n n n 1  min   1 1   min ( , , ), ( , , ) n 1 n n n 1 M y y ( , , ) kt M y ( , y t , )   n n 1 n 1 n     . Since f and g be conditionally . Therefore, by (3.3), we have y z lim y lim ( f x ) lim ( n  g x ) z  n n n 1   , there exist a sequence {sn} satisfying f x g x z      such that lim and lim . Since, , for each f s g s u say fg s ( ) fu gf s ( ) gu f X ( ) g X ( ) n n  n    . By using (3.2), we get f s g s g z u       M gs gz t M gs fz t M fs gs t ( , , ), n n n n n n  M fs fz kt n n        ( , , ), ( , , ), ( , ( ,2 ), , , ) ( , , ), n  ( ,lim n , ) min n  n 1, M u ( ,lim n fz t , ),1, n  fz u t   min 1, M (lim n fz t , , ) n M u ( ,lim n u  , ) n  . Hence, lim ( )      this gives, lim ( (3.4) f z ) f s lim ( ) n  g s lim ( n  g z ) lim ( n  f z ) u n n n n n   n n   , that is, lim n  M fg s ( ), n gf s ( ), n t 1 Suppose that f and g are compatible mappings. Then  , this gives, fu = gu. Also, fgu = ffu = fgu = gfu. Using (3.2), we get lim n  fg s ( ) lim n  gf s ( ) n n @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 293

4. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470          M gu gfu t M gu ffu t M fu gu t M fu gfu t M ffu gfu t ( ( , , , ), , ), ( ( , , ,2 ), , ) ( , , ),  M fu ffu kt ( , , ) min    M fu ffu t M fu ffu t M fu fu t M fu ffu t M ffu ffu t ( ( , , , ), , ), ( ( , , ), , ) ( , , ),  min ,  M fu ffu t ( , , ) That is fu = ffu. Hence, fu = ffu= gfu and f u is a common fixed point of f and g. Now, Suppose that f and g are g-compatible mappings. Then    , that is, lim n  M ff s ( ), n gf s ( ), n t 1   . Using (3.2), we get lim n  M fu ffs kt   ff s ( ) lim n  gf s ( ) gu n n    ( , , ) min M gu gfs t M gu ffs ( , , ), ( , ,2 ), t M fu gu t M fu gfs t M ffs gfs t ( , , ), ( , , ), ( , , ) n n n n n n n M fu gu kt     ( , , ) min M gu gu t M gu gu ( , , ), ( , ,2 ), t M fu gu t M fu gu t M gu gu t ( , , ), ( , , ), ( , , ) M fu gu kt This gives, fu = gu. Also, fgu = ffu = fgu = gfu. Using (3.2), we get   min ( , M fu ffu t M fu ffu t M fu fu t M fu ffu t M ffu ffu t  ( , , ) M fu gu t ( , , )   M fu ffu kt ( , , ) min M gu gfu t M gu ffu t M fu gu t M fu gfu t M ffu gfu t ( , , ), ( , ,2 ), ( , , ), ( , , ), ( , , )  , ), ( , , ), ( , , ), ( , , ), ( , , )  M fu ffu t ( , , ) That is fu = ffu. Hence, fu = ffu= gfu and f u is a common fixed point of f and g. Finally, Suppose that f and g are f-compatible mappings. Then    , that is, lim n  M fg z ( ), n gg z ( ), n t 1    . Also, lim . lim n  Therefore, lim fg z ( ) lim n  gg z ( ) gf s ( ) lim n  gg z ( ) gu n n n n  n (   fg z ( ) lim n  gg z ) gu . n n  n Using (3.2), we get       M gu ggz t M gu fgz M fu gu t M fu ggz t M fgz ggz t ( ( , , , ), ( , ,2 ), , ), t n n  M fu fgz kt ( , , ) min n , ), ( , ( , , ) n n n   n       M gu gu t M gu gu t M fu gu t M fu gu t M gu gu t M fu gu t This gives, fu = gu. Also, fgu = ffu = fgu = gfu. Using (3.2), we get ( , ( , , ) min ( , M fu gu t M fu gfu t M ffu gfu t       That is fu = ffu. Hence, fu = ffu= gfu and f u is a common fixed point of f and g. Uniqueness of the common fixed point theorem follows easily in each of the three cases. ( ( , , , ), , ), ( ( , ,2 ), , ),  M fu gu kt ( , , ) min , ( , , )  M fu gu kt ( , , ) ( , , )         M gu gfu t M gu ffu t , ), , ), ( , ,2 ), , ),  M fu ffu kt ( , ( , , ) M fu ffu t M fu ffu t M fu fu t M fu ffu t M ffu ffu t M fu ffu t ( ( , , , ), , ), ( ( , ,2 ), , ) ( , , ), M fu ffu kt ( , , ) min , M fu ffu kt ( , , ) ( , , ) @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 294

5. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Theorem 3.2 Let f and g be non-compatible self-mappings of a fuzzy metric space ( , conditions: (3.4) ( ) ( ) f X g X  ; (3.5) for all (0,1) k  such that:  ( , , ) min ( , , ), ( , M fx fy kt M gx gy t M gx fy t M fx gx t M fx gy t M fy gy t  ( , , ) ( , , ) M fx ffx t M gx ggx t  whenevergx Suppose f and g be conditionally reciprocally continuous If f and g are either g- compatible or f- compatible then f and g have fixed point. Proof: Since f and g are non-compatible maps, there exists a sequence {xn} in X such that f xn→ z and gxn→ z for some z in X as n→∞but either lim satisfying the X M ,*)  ,2 ), ( , , ), ( , , ), ( , , ) ;  t  .  ggx 0 (3.6) for all , x y and X  or the limit does not exist. M fgx gfx t ( , , ) 1 n n  n   , there exist a Also, since f and g be conditionally reciprocally continuous and lim ( f x ) lim ( n  and lim g x ) z n n  n fg y     sequence {yn} satisfying lim ( such that lim . Since, f y ) lim ( n  g y ) u say ( ) ( ) fu gf y ( ) gu n n n n    g y n n n    .  , for each yn , there exist zn in X such that fyn= gzn. Thus, lim ( f y ) lim ( n  ) lim ( ) n  g z u f X ( ) g X ( ) n n n  n By using (3.5), we get       M gy gz t M gy M fy gz t M fz gz t ( ( , , , ), , ), ( ( , fz ,2 ), , ) t M fy gy t ( , , ), n n n n n n  M fy ( , fz kt , ) min n n , n n n n   n           ( , , ), M u u t M u ( ,lim n fz ,2 ), t M u u t ( , , ), n   M u ( ,lim n fz kt , ) min n ( , , ), M u u t M (lim n  fz u t , , )  n    min 1, M u ( ,lim n , ) n fz t f , ),1,1, z t M (lim n  fz u t , , ) n n   M u ( ,lim n ) u   . Therefore, we have lim (     . this gives, lim ( f z f y ) lim ( n  g y ) lim ( n  lim n g z ) lim ( n  ( n f z ) u n n n n n   n n    , that is, M ff y ), gf y ( ), t 1 Now, Suppose that f and g are g-compatible mappings. Then n    . Using (3.5), we get lim n  M fu ffy kt   ff y ( ) lim n  gf y ( )  gu n n   ( , , ) min M gu gfy t M gu ffy ( , , ), ( , ,2 ), t M fu gu t M fu gfy t M ffy gfy t ( , , ), ( , , ), ( , , ) n n n n n n n M fu gu kt M fu gu kt this gives, fu = gu. Also, fgu = ffu = fgu = gfu. If fu ( , , ) ( , , ) M fu ffu t M gu ggu t  point of f and g. Finally, Suppose that f and g are f-compatible mappings. Then     ( ( , , , , ) ) min M fu gu t M gu gu t M gu gu t M fu gu t M fu gu t M gu gu t ( , , ), ( , ,2 ), ( , , ), ( , , ), ( , , ) ( , , )  , using (3.6), we get ffu  , a contradiction. Hence, fu = ffu= gfu and f u is a common fixed M fu ffu t ( , , )    , that is, lim n  M fg z ( ), n gg z ( ), n t 1    . Also, lim . lim n  Therefore, lim fg z ( ) lim n  gg z ( ) gf y ( ) lim n  gg z ( ) gu n n n n  n   fg z ( ) lim n  gg z ( ) gu . n n  n Using (3.5), we get @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 295

6. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470       M gu ggz t M gu fgz M fu gu t M fu ggz t M fgz ggz t ( ( , , , ), ( , ,2 ), , ), t n n  M fu fgz kt ( , , ) min n , ), ( , ( , , ) n n n   n       M gu gu t M gu gu t M fu gu t M fu gu t M gu gu t M fu gu t This gives, fu = gu. Also, fgu = ffu = fgu = gfu. If fu ( , , ) ( , , ) M fu ffu t M gu ggu t   point of f and g. Theorem 3.3 Let f and g be non-compatible self-mappings of a fuzzy metric space ( , conditions: (3.7) ( ) ( ) f X g X  ; (3.8) for all (0,1) k  such that:  ( , , ) min ( , , ), ( , ,2 ), M fx fy kt M gx gy t M gx fy t M fx gx t M fx gy t M fy gy t        Whenever fx f x  for all , x y X  and 0 t  . Suppose f and g be conditionally reciprocally continuous If f and g are either g- compatible or f- compatible then f and g common fixed point. Proof: Since f and g are non-compatible maps, there exists a sequence {xn} in X such That f xn→ z and gxn→ z for some z in X as n→∞but either lim ( ( , , , ), , ), ( ( , ,2 ), , ),  M fu gu kt ( , , ) min , ( , , )  M fu gu kt ( , , ) ( , , )  , using (3.6), we get ffu ,a contradiction. Hence, fu = ffu= gfu and f u is a common fixed M fu ffu t ( , , ) satisfying the X M ,*)  ( , , ), ( , , ), ( , , ) ;      2 M gx gfx t M fx gx t M f x gfx t ( , , ), ( , , ), ( , , ), 2 M fx f x t ( , , ) max (3.9) 2 M fx gfx t M gx f x t ( , , ), ( , , ) 2  or the limit does not exist. M fgx gfx t ( , , ) 1 n n  n   , there exist a Also, since f and g be conditionally reciprocally continuous and lim ( f x ) lim ( n  and lim g x ) z n n  n fg y     sequence {yn} satisfying lim ( such that lim . Since, f y ) lim ( n  g y ) u say ( ) ( ) fu gf y ( ) gu n n n n    g y n n n    .  , for each yn , there exist zn in X such that fyn= gzn. Thus, lim ( f y ) lim ( n  ) lim ( ) n  g z u f X ( ) g X ( ) n n n  n By using (3.8), we get       M gy gz t M gy M fy gz t M fz gz t ( ( , , , ), , ), ( ( , fz ,2 ), , ) t M fy gy t ( , , ), n n n n n n  M fy ( , fz kt , ) min n n , n n n n   n           ( , , ), M u u t M u ( ,lim n fz ,2 ), t M u u t ( , , ), n   M u ( ,lim n fz kt , ) min n ( , , ), M u u t M (lim n  fz u t , , )  n    min 1, M u ( ,lim n , ) n fz t f , ),1,1, z t M (lim n  fz u t , , ) n n   M u ( ,lim n )   u     . This gives, lim ( . Therefore, we have lim ( f z f y ) lim ( n  g y ) lim ( n  lim n  g z )  lim ( n  ( n f z ) u n n n n n   n n   , that is, M ff y ), gf y ( ), t 1 Now, Suppose that f and g are g-compatible mappings. Then n   . Using (3.8), we get lim n  ff y ( ) lim n  gf y ( ) gu n n @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 296

7. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470  , , ) min ( , , ), ( , ,2 ), n n n M fu ffy kt M gu gfy t M gu ffy t M fu gu t M fu gfy t M ffy gfy t n M fu gu kt M gu gu t M gu gu t M fu gu t M fu gu t M gu gu t   This gives, fu = gu. Also, fgu = ffu = fgu = gfu. If fu ( , , ), ( , , ), ( , , ) max ( , , ), ( , , ) M fu gfu t M gu f u t          a contradiction. Hence, fu = ffu= gfu and f u is a common fixed point of f and g. Finally, Suppose that f and g are f-compatible mappings. Then   ( ( , , ), ( , , ), ( , , ) n n n     ( , , ) min ( , , ), ( , ,2 ), ( , , ), ( , , ), ( , , ) M fu gu kt ( , , ) M fu gu t ( , , )  ( , using (3.9), we get           ffu    2 M gu gfu t M fu gu t M f u gfu t , , ),  2 M fu f u t 2 2 M fu ffu t M fu fu t M f u ffu t ( , , ), ( , , ), ( , , ), 2 M fu f u t ( , , ) max 2 M fu ffu t M fu f u t ( , , ), ( , , ) 2 2 M fu f u t ( , , ) M fu f u t ( , , )    , that is, lim n  M fg z ( ), n gg z ( ), n t 1    . Also, lim . lim n  Therefore, lim fg z ( ) lim n  gg z ( ) gf y ( ) lim n  gg z ( ) gu n n n n  n   fg z ( ) lim n  gg z ( ) gu . n n  n Using (3.8), we get       M gu ggz t M gu fgz M fu gu t M fu ggz t M fgz ggz t ( ( , , , ), ( , ,2 ), , ), t n n  M fu fgz kt ( , , ) min n , ), ( , ( , , ) n n n   n       M gu gu t M gu gu M fu gu t M fu gu t M gu gu t M fu gu t This gives, fu = gu. Also, fgu = ffu = fgu = gfu. If fu ( , ( , , ) max ( , M fu gfu t M gu f u t          A contradiction Hence, fu = ffu= gfu and f u is a common fixed point of f and g. If we take θ as θ1, θ2, θ3, then we get the following corollaries: Corollary 4.3 Let f and g be weakly reciprocally continuous non-compatible self-mappings of a fuzzy metric space ( , ,*) X M satisfying the conditions: (4.1) ( ) ( ) f X g X  ;         ( ( , , , ), , ), ( ( , ,2 ), , ), t  M fu gu kt ( , , ) min , ( , , )  M fu gu kt ( , , ) ( , , )  , using (3.9), we get           ffu ,    2 M gu gfu t M fu gu t M f u gfu t , ), ( , , ), ( , ),  2 M fu f u t 2 , ), ( , , ) 2 M fu ffu t M fu fu t M f u ffu t ( , , ), ( , , ), ( , , ), 2 M fu f u t ( , , ) max 2 M fu ffu t M fu f u t ( , , ), ( , , ) 2 2 M fu f u t ( , , ) M fu f u t ( , , ) M fx fy t ( , , ) M gx gy t ( , , ) ( ) s ds   ( ) s ds  (4.7) ; 0 0    2 M gx gfx t M fx gx t M f x gfx t M fx gfx t M gx f x t ( ( , , , ), , ), ( ( , , , ), ( , , ), 2 M fx f x t ( , , ) min ( ) s ds  ( ) s ds  2 , ) (4.8) 0 0 @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 297

8. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 t  , where      2 0 Whenever fx f x for all , x y ,  : ¡ ¡ is a Lebesgue integrable mapping which is X    . ( ) s ds 0 0 summable nonnegative and such that for each 0 If f and g are R-weakly commuting of type (Ag) or R-weakly commuting of type (Af) or R-weakly commuting of type (P) then f and g have a unique common fixed point. Corollary 4.4 Let f and g be weakly reciprocally continuous non-compatible self-mappings of a fuzzy metric space ( , ,*) X M satisfying the conditions: (4.1) ( ) ( ) f X g X  ;    2 ( , , ) ( , , ) 0 0   Whenever fx f x  for all , x y X  , 0 t  , where :     If f and g are R-weakly commuting of type (Ag) or R-weakly commuting of type (Af) or R-weakly commuting of type (P) then f and g have a unique common fixed point. Corollary 4.5 Let f and g be weakly reciprocally continuous non-compatible self-mappings of a fuzzy metric space ( , ,*) X M satisfying the conditions: (4.1) ( ) ( ) f X g X  ;     whenever fx f x  for all , x y X  , 0 t  , where :     If f and g are R-weakly commuting of type (Ag) or R-weakly commuting of type (Af) or R-weakly commuting of type (P) then f and g have a unique common fixed point. Let denote the class of those functions    : 0,1 0,1   ( ,1, ,1) . x x x   There are examples of  : 1.   1 1 2 3 4 1 2 3 4 ( , , , ) min , , , x x x x x x x x   ; ( , , , ) x x x x x x x x   ;  3 1 2 3 4 1 2 3 4 ( , , , ) min , x x x x x x x x . Now we prove our main results. Theorem 4.5 Let f and g be weakly reciprocally continuous non-compatible self-mappings of a fuzzy metric space ( , ,*) X M satisfying the conditions: (4.1) ( ) ( ) f X g X  ;   M fx fy t ( , , ) M gx gy t ( , , ) ( ) s ds   ( ) s ds  (4.7) ; 0 0  2 2     M gx gfx t M gx gfx t ( , , ) ( , , ) M fx gx t ( , , ) M f x gfx t ( , , ) M fx gfx t ( , , ) M gx f x t ( , , ) M fx f x t ( ) s ds   ( ) s ds  2   , ) 2  M gx gfx t M fx gfx t ( , , ) M gx f x t ( , (4.8)  2 ¡ ¡ is a Lebesgue integrable mapping which is   . ( ) s ds 0 0 summable nonnegative and such that for each 0 M fx fy t ( , , ) M gx gy t ( , , ) ( ) s ds   ( ) s ds  (4.7) ; 0 0 2 2 2 3 M fx f x t ( , , ) M gx gfx t M fx gx t M f x gfx t M fx gfx t M gx f x t ( , , ). ( , , ). ( , , ). ( , , ). ( , , ) ( ) s ds   ( ) s ds  (4.8) 0 0  2 ¡ ¡ is a Lebesgue integrable mapping which is   . ( ) s ds 0 0 summable nonnegative and such that for each 0  5 such that is continuous and 2. 2 1 2 3 4 1 2 3 4    3. M fx fy t ( , , ) M gx gy t ( , , ) ( ) s ds   ( ) s ds  (4.7) ; 0 0 @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 298

9. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470    2 2  M fx f x t ( , , ) M gx gfx t M fx gx t M fx gfx t M f x gfx t ( , , ), ( , , ), ( , , ), ( , , )   ( ) s ds   ( ) s ds (4.8) 0 0 t  and for some   where     2 0 whenever fx f x for all , x y , : ¡ ¡ is a Lebesgue integrable X      . ( ) s ds 0 0 mapping which is summable nonnegative and such that for each 0 If f and g are R-weakly commuting of type (Ag) or R-weakly commuting of type (Af) or R-weakly commuting of type (P) then f and g have a unique common fixed point. Proof. Since f and g are non-compatible maps, there exists a sequence {xn} in X such that f xn→ z and gxn→ z for some z in X as n→∞ but either lim ( , , ) 1 n n n  {yn} in X such that f xn= gyn. Thus f xn→ z, gxn→ z and gyn→ z as n→∞. By virtue of this andusing (4.7) we obtain ( , , ) ( , , ) ( ) ( ) s ds s ds       or the limit does not exist. Since  , for each {xn} there exists M fgx gfx t f X ( ) g X ( ) M fx fy t M gx gy t n n n n 0 0   n  which implies that, fyn→ z as n→∞. Therefore, we have fxn→z, gxn→ z, gyn→ z, fyn→ z. Suppose that f and g are R-weakly commuting of type (Ag). Then weak reciprocal continuity of f and g implies that fgxn→ f z or g fxn→ gz. Similarly, fgyn→ fz or gfyn→ gz. Let us first assume that gfyn→ gz. Then R-weak commutativity of type (Ag) of f and g yields   , , , , n n n n M ffy gfy t M fy gy R n t M ffy gz t M z z R  This gives, f f yn→ gz. Using (4.7), we get ( , , ) ( , , ) ( ) ( ) s ds s ds      M z ( ,lim n fy t , ) ( , , ) M z z t  ( ) s ds   ( ) s ds  n  0 0   t          lim n , , , , 1 n M ffy fz kt M gfy gz t n n 0 0   n  This implies that f z = gz. Again, by virtue of R-weak commutativity of type (Ag),   , , , , 1 M ffz gfz t M fz gz R using (4.8), we get  ( ) s ds        a contradiction. Hence f z = f f z = g f z and f z is a common fixed point of f and g. M gz fz kt ( , , ) M gz gz t ( , , )  ( ) s ds   ( ) s ds  0 0   t   . This yields f f z = g f z and f f z = f gz = g f z = ggz. If fz  then by ffz  2 2  M fz f z t ( , , ) M gz gfz t M fz gz t M fz gfz t M f z gfz t ( , , ), ( , , ), ( , , ), ( , , ) ( ) s ds  0 0   2 2 2 2  M fz f z t M fz fz t M fz f z t M f z f z t ( , , ), ( , , ), ( , , ), ( , , )  ( ) s ds  0   2 2  M fz f z t ( , , ),1, M fz f z t ( , , ),1  ( ) s ds  0 2 M fz f z t ( , , )  ( ) s ds  0 @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 299

10. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 Similarly, we can prove, if fgyn→ fz, then again f z is a common fixed point of f and g. Proof is similar if f and g are R-weakly commuting of type (Af) or (P). Uniqueness of the common fixed point theorem follows easily in each of the two cases. If we take as 1 2 3 , ,    then we get the following corollaries: Corollary 4.5 Let f and g be weakly reciprocally continuous non-compatible self-mappings of a fuzzy metric space ( , ,*) X M satisfying the conditions: (4.1) ( ) ( ) f X g X  ;    ( , , ) min ( , , ), ( , ( ) s ds     whenever fx f x  for all , x y X  , 0 t  where :     If f and g are R-weakly commuting of type (Ag) or R-weakly commuting of type (Af) or R-weakly commuting of type (P) then f and g have a unique common fixed point. Corollary 4.5 Let f and g be weakly reciprocally continuous non-compatible self-mappings of a fuzzy metric space ( , ,*) X M satisfying the conditions: (4.1) ( ) ( ) f X g X  ;     whenever fx f x  for all , x y X  , 0 t  where :     If f and g are R-weakly commuting of type (Ag) or R-weakly commuting of type (Af) or R-weakly commuting of type (P) then f and g have a unique common fixed point. Corollary 4.5 Let f and g be weakly reciprocally continuous non-compatible self-mappings of a fuzzy metric space ( , ,*) X M satisfying the conditions: (4.1 ( ) ( ) f X g X  ;    ( ) s ds     whenever fx f x  for all , x y X  , 0 t  where :     If f and g are R-weakly commuting of type (Ag) or R-weakly commuting of type (Af) or R-weakly commuting of type (P) then f and g have a unique common fixed point. M fx fy t ( , , ) M gx gy t ( , , ) ( ) s ds   ( ) s ds  (4.7) ; 0 0  2 2 M fx f x t M gx gfx t M fx gx t M fx gfx t M f x gfx t , ), ( , , ), ( , , ) ( ) s ds  (4.8) 0 0  2 ¡ ¡ is a Lebesgue integrable mapping which is   . ( ) s ds 0 0 summable nonnegative and such that for each 0 M fx fy t ( , , ) M gx gy t ( , , ) ( ) s ds   ( ) s ds  (4.7) ; 0 0 2 2 M fx f x t ( , , ) M gx gfx t M fx gx t M fx gfx t M f x gfx t ( , , ). ( , , ). ( , , ). ( , , ) ( ) s ds   ( ) s ds  (4.8) 0 0  2 ¡ ¡ is a Lebesgue integrable mapping which is   . ( ) s ds 0 0 summable nonnegative and such that for each 0 M fx fy t ( , , ) M gx gy t ( , , ) ( ) s ds   ( ) s ds  (4.7) ; 0 0  2 2 M fx f x t ( , , ) min M gx gfx t M fx gx t ( , , ). ( , , ), M fx gfx t M f x gfx t ( , , ). ( , , ) ( ) s ds  (4.8) 0 0  2 ¡ ¡ is a Lebesgue integrable mapping which is   . ( ) s ds 0 0 summable nonnegative and such that for each 0 @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Page: 300

11. International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456 International Journal of Trend in Scientific Research and Development (IJTSRD) ISSN: 2456-6470 6.Pant, R. P. (1994). Common fixed points of non commuting mappings, J. Math. An. Appl 436-440. References: 1.Imdad, M. and Ali, J.(2008). Jungck's common fxed point theorem and E.A property, Mathematica Sinica, 24(1), 87-94. Imdad, M. and Ali, J.(2008). Jungck's common (1994). Common fixed points of non J. Math. An. Appl., 188, property, Acta 2.Jungck, G. and Rhoades, B. E.(2006). Fix Theorems for occasionally weakly compatible mappings, Fixed point theory, 7, 286 7.Pant, R. P.(1999). A common fixed point theorem under a new condition, Math., 30 (2), 147-152. E.(2006). Fixed point P.(1999). A common fixed point theorem under a new condition, Indian J. Pure Appl. Theorems for occasionally weakly compatible , 286-296. 3.Kramosil, I. and Michalek, J. (1975). Fuzzy metric and statistical metric spaces, Kybernetica 326-334. 8.Pathak, H. K., Cho, Y. J., and Kang, S. Remarks on R-Weakly commuting mappings and common fixed point theorems, Soc., 34 (2), 247-257. (1975). Fuzzy metric Kybernetica, 11, J., and Kang, S. M. (1997). Weakly commuting mappings and common fixed point theorems, Bull. Korean Math. 4.Kumar, S. and Pant, B. D. (2008), Faculty of sciences and mathematics, University of Nis Serbia, 22(2), 43-52. (2008), Faculty of University of Nis 9.Schweizer, B. and Sklar, A. metric spaces, North Holland Amsterdam North Holland Amsterdam. Schweizer, B. and Sklar, A. (1983). Probabilistic 5.Mishra, S. N., Sharma, N. and Singh, S. Common fixed points of maps on fuzzy metric spaces, Int. J. Math. Math. Sci,17, 253 10.Vasuki, R. (1999). Common fixed points for R weakly commuting maps in fuzzy metric spaces, Indian J. Pure Appl. Math., Indian J. Pure Appl. Math.,30, 419-423. N., Sharma, N. and Singh, S. L.(1994). Common fixed points of maps on fuzzy metric (1999). Common fixed points for R- weakly commuting maps in fuzzy metric spaces, , 253-258. 11.Zadeh, L. (1965). Fuzzy sets, control, 89, 338-353. Fuzzy sets, Information and @ IJTSRD | Available Online @ www.ijtsrd.com @ IJTSRD | Available Online @ www.ijtsrd.com | Volume – 2 | Issue – 5 | Jul-Aug 2018 Aug 2018 Page: 301