Abstract In this paper,   we proved a common fixed point theorem ψ-weakly commuting maps in L-Fuzzy Metric Spaces for integral type inequality. 2010   Mathematics Subject Classification: 47H10, 54H25. Key   words and phrases: L-fuzzy metric space,  ψ-weakly commuting maps, Common fixed point  theorem

Choudhary, Pushpraj

Ramakant Bhardwaj, Rasik M.Patel,

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  106 Common Fixed Point Theorem for ψ-weakly commuting maps in L-Fuzzy Metric Spaces for integral type  Rasik M.Patel, Ramakant Bhardwaj*, Pushpraj Choudhary** The Research Scholar of Sai Nath University, Ranchi (Jharkhand) S.P.B.Patel Engineering College, Mehsana * TIT group of Institutes Bhopal (M.P.), India **Govt. Science Model College, Jabalpur Email: rasik_maths@yahoo.com, drrkbhardwaj100@gmail.com  Abstract In this paper, we proved a common fixed point theorem ψ-weakly commuting maps in L-Fuzzy Metric Spaces for integral type inequality. 2010 Mathematics Subject Classification: 47H10, 54H25. Key words and phrases: L-fuzzy metric space,  ψ-weakly commuting maps, Common fixed point  theorem,  1. Introduction In 1922, Let (X, d) be a complete metric space, c∈ (0, 1) and : X→X be a mapping such that for each x, y ∈ X, d ( ) ≤ c d(x, y) Then f has a unique fixed point a ∈ X, such that for each x ∈ X,  by S. Banach [23].As a generalization of fuzzy sets introduced by L.A.Zadeh [14], K. Atanassov [13] introduced the idea of intuitionistic fuzzy set. Fixed point and common fixed point properties for mappings defined on fuzzy metric spaces by [5], [6], [8], [15], [16], Intuitionistic fuzzy metric spaces by [7], [21]. A. George and P.Veeramani [2] modified the concept of fuzzy metric space introduced by I. Kramosil and J. Michalek [10] and defined a Hausdorff topology on this fuzzy metric space by [12]. Most of the properties which provide the existence of fixed points and common fixed points are of linear contractive type conditions. L-fuzzy metric spaces have been studied by many authors [11], [24]. H. Adibi et al.[9] introduced the concept of compatible mappings and proved common fixed point theorems for four mappings satisfying some conditions in L-fuzzy metric spaces. In the sequel, we shall adopt the usual terminology, notation and conventions of L-fuzzy metric spaces introduced by R. Saadati et al. [19] which are a generalization of fuzzy metric spaces and intuitionistic fuzzy metric spaces [20]. R. Saadati, S.Sedghi and H. Zhou [22] by a common fixed point theorem ψ-weakly commuting maps in L-Fuzzy Metric Spaces 2. Preliminaries Definition 2.1 [1]: Let (X, d) be a complete metric space, c∈ (0, 1) and f: X→X be a mapping such that for each x, y ∈ X,  where : [0,+∞) →[0,+∞) is a Lebesgue integrable mapping which is summable on each compact subset of [0,+∞) , non negative, and such that for each  > o, then f has a unique fixed point  such that for each , . B.E.Rhoades [4], extending the result of Branciari by replacing the above condition by the following    Definition 2.2[11] Let L = (L, ) be a complete lattice, and U a nonempty set called a universe.  An L-fuzzy set A on U is defined as a mapping A : U → L. For each u in U, A(u) represents the degree (in L) to which u satisfies A. Lemma 2.1[8]. Consider the set  and the operation  defined by: Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  107   = {( , ) : ( , ) ∈  and  +  ≤ 1}, ( , )  ( , )   ≤  and  ≥ , for every  ( , ), ( , ) ∈ . Then ( , ) is a complete lattice. Classically, a triangular norm T on ([0, 1], ) is defined as an increasing, commutative, associative mapping T: satisfying T(1, x) = x, for all x  [0, 1]. These definitions can be straightforwardly extended to any lattice L = (L, ) Define first  = inf L and  = sup L. Definition 2.3[19]. A triangular norm (t-norm) on L is a mapping T:  → L satisfying the following conditions: (i) ( x  L)(T (x, ) = x); (boundary condition) (ii) ( (x, y)  )(T (x, y) = T (y, x)); (commutativity) (iii) ( (x, y, z)  )(T (x, T (y, z)) = T (T (x, y), z)); (associativity) (iv) ( (x, , y, )  )(x  _ and y  T(x, y)  T ( , ));(monotonicity) A t-norm   on  is said to be continuous if for any x, y  X and any sequences  which converge to x and y we have  Definition 2.4 [7].A t-norm   on  is called t-represent able if and only if there exist a t-norm T and a t-co norm S on [0, 1] such that, for all ( , ), ( , ) ∈ .  Definition 2.5[19]. A negation on L is any decreasing mapping N: L → L satisfying N ( ) =  and N ( ) =  If N (N(x)) = x, for all x  L, then N is called an involutive negation.  Definition 2.6[19]. The 3-tuple (X, M, ) is said to be an L-fuzzy metric space if X is an arbitrary (non-empty) set,  is a continuous t–norm on L and M is an L-fuzzy set on  satisfying the following conditions for every x, y, z in X and t, s in : (a) M(x, y, t)  ; (b) M(x, y, t) =  for all t > 0 if and only if x = y; (c) M(x, y, t) = M(y, x, t); (d) (M(x, y, t), M(y, z, s))   M(x, z, t + s); (e) M(x, y, ·) : → L is continuous. Let (X, M, ) be an L-fuzzy metric space. For t  , we define the open ball B(x, r, t) with center x  X and radius r  L  { , }, as B(x, r, t) = {y ∈ X :M(x, y, t)  N(r)}. A subset A ⊆ X is called open if for each x  A, there exist t > 0 and r  L { , } such that B(x, r, t) ⊆ A. Let  denote the family of all open subsets of X. Then  is called the topology induced by the L-fuzzy metric M. Example 2.1 [21]. Let (X, d) be a metric space. Denote (a, b) = ( , min (   , 1)) for all  a = ( , ) and b = ( , ) in  and let M and N be fuzzy sets on  ×(0,∞) be defined as follows: (x, y, t) = (M(x, y, t), N(x, y, t)) = Then (X, , ) is an intuitionistic fuzzy metric space. Example 2.2 [19]. Let X = N. Define (a, b) = (max (0, −1), ) for all a = ( , ) and b = ( , ) in , and let M (x, y, t) on  (0,∞) be defined as follows:  for all x, y  X and t  0. Then (X, M, ) is an L-fuzzy metric space. Let (X, M, ) be an L-fuzzy metric space. For t  , we define the open ball B(x, r, t) with center x  X and radius r  L  { , }, as B(x, r, t) = {y ∈ X :M(x, y, t)  N(r)}. A subset A ⊆ X is called open if for Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  108 each x  A, there exist t > 0 and r  L  { , } such that B(x, r, t) ⊆ A. Let  denote the family of all open subsets of X. Then  is called the topology induced by the L-fuzzy metric M. Lemma 2.2 [9]. Let (X, M, ) be an L-fuzzy metric space. Then M(x, y, t) is non decreasing with respect to t, for all x, y in X. Definition 2.7[19]. A sequence in an L-fuzzy metric space (X, M, ) is called a Cauchy sequence, if for each ε ∈  L  { } and t  0, there exists   N such that for all m  n   (n  m ),  M ( , , t)  N(ε). The sequence  is said to be convergent to x  X in the L-fuzzy metric space (X, M, ) if  whenever  for every t  0. A L-fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent. Definition 2.8 [22] Let (X, M, ) be an L-fuzzy metric space. M is said to be continuous on (0, ) if  whenever a sequence { } in (0 , ) converges to a point (x, y, t) (0 , ) i.e.,  Lemma 2.3 [22] Let (X, M, ) be an L-fuzzy metric space. Then M is continuous function on (0, ). Definition 2.9[22] Let  be maps from an L-fuzzy metric space (X, M, ) into itself. The maps  are said to be weakly commuting if  for each  in X &  Definition 2.10[22].Let  be maps from an L-fuzzy metric space (X, M, ) into itself. The maps  are said to be ψ-weakly commuting if there exists a positive real function  such that  for each  in X and  Example 2.3[22]. Let X = R. Let (a, b) = ( , min (   , 1)) for all a = ( , ) and b = ( , ) in  and let M and N be fuzzy sets on  ×(0,∞) be defined as follows: (x, y, t) = , for all  Then (X, , ) is an intuitionistic fuzzy metric space. Define      Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  109  . Therefore, for   weakly commuting. But  not weakly commuting since the exponential function is strictly increasing. 3. Main Results  Theorem 3.1. Let (X, M, ) be a left L-fuzzy metric space and let  and  be  weakly commuting self-mappings of X satisfying the following conditions: (3.1.1)  (3.1.2) either  or  is continuous;  (3.1.3)   where  is a continuous function such that   for each  for every x, y in X .Then  and  have a unique common fixed point in X. Proof. Let  be an arbitrary point in X. By (3.1.1), there exists  such that  In general choose  such that  Then for t > 0,      Thus, is an  increasing sequence in L and therefore, tends to a limit  we claim that  For if   when  in the above inequality we get  a contradiction. Hence  i.e.  If we define (2.9)    then  Now, we prove that  is a Cauchy sequence in  suppose that  is not a Cauchy sequence in  For convenience, Let  for n =1,2,3……Then there is an   such that for each integer k, there exists integers  with  such that  (2.10)                      for k = 1, 2, 3….. We may assume that Example.2.3                       by choosing  to be the smallest number exceeding  for which (2.10) holds. Using (2.9), we have (3.1)  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  110 Hence,  for every  as  We know that                                                Thus, as  in the above inequality we have  which is a contradiction. Thus,  is a Cauchy and by the completeness of X,  converges to z in X. Also  converges to z in X. Let us suppose that the mapping A is continuous. Then and Further we have since  and  be  weakly commuting   On letting  in the above inequality we get  by lemma (2.3). We now prove that  Suppose   then  By (3.1.3)   Letting  in the above inequality we get  a contradiction. Therefore,  Since  we can find  in X such that . Now,  Letting  in the above inequality we get  Since  this implies that  also for any   which again implies that  thus  is a common fixed point of A and B. Now, to prove uniqueness suppose is another common fixed point of A and B. Then there exists  such that  and     Which is contradiction. Therefore, . i.e,  is a unique common fixed point A and B. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.5, No.7, 2015  111 Example 2.4[22].Consider example 2.1 in which   Define  and   on X. It is evident that  A is continuous   and B is discontinuous. Define   by  ), then  )  for  and  for all  in X,  and  be  weakly commuting. Thus all the conditions of last theorem are satisfied and 1 is a common fixed point of A and B.  Acknowledgement: One of the Author (Dr. Ramakant Bhardwaj) is thankful to MPCOST Bhopal for the project No.2556.  References: [1] A.Branciari. A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int.J.Math.Sci. 29(2002), no.9, 531 - 536.  [2] A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets Syst, 64 (1994), 395–399. [3] B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl. 301 (2005) 439-448. [4] B.E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences, 63, (2003), 4007 - 4013. [5] D. Mihet ,¸ A Banach contraction theorem in fuzzy metric spaces, Fuzzy Sets Syst, 144(2004) 431–439. [6] E. Pap, O. Hadzic and R. Mesiar, A fixed point theorem in probabilistic metric spaces and an application, J. Math. Anal. Appl. 202 (1996) 433–449. [7] G. Deschrijver, C. Cornelis, and E.E. Kerre, On the representation of intuitionistic fuzzy t-norms and t-co norms, IEEE Transactions on Fuzzy Sys. 12 (2004), 45-61. [8] G. Deschrijver, E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets Syst. 33 (2003) 227-235. [9] H. Adibi, Y.J. Cho, D. O’Regan, R. Saadati, Common fixed point theorems in L- fuzzy metric spaces, Appl. Math. Compu. 182 (2006), 820-828. [10] I. Kramosil and J. Michalek, Fuzzy metric and statistical metric space, Kybernetica, 11(1975) 326-334. [11] J. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145-174. [12] J. Rodriguez L o´pez and S. Romaguera, The Hausdorff fuzzy metric on compact sets, Fuzzy Sets Syst, 147 (2004) 273–283. [13] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and Systems, 20(1986), 87-96.  [14] L. A. Zadeh, Fuzzy sets Inform and Control 8 (1965), 338-353. [15] M. A. Erceg, Metric spaces in fuzzy set theory, J. Math. Anal. Appl. 69 (1979), 205-230. [16] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets Syst. 12 (1984), 215-229. [17] R. Saadati and J. H. Park, Intuitionistic fuzzy Euclidean normed spaces, Commun. Math. Anal. 1 (2006), 1-6. [18] R. Saadati, A. Razani and H.Adibi, A common fixed point theorem in L-fuzzy metric spaces and its generalization to L-fuzzy metric spaces, Chaos, Solitons and Fractals 35 (2008), 176-180. [19] R. Saadati, A. Razani, and H. Adibi, A Common fixed point theorem in L-fuzzy metric spaces Chaos, Solitons and Fractals, 33(2007), 358-363. [20] R. Saadati and J.H. Park, On the Intuitionistic Fuzzy Topological Spaces, Chaos, Solitons and Fractals 27 (2006), 331–344. [21] R. Saadati and J.H. Park, Intuitionistic fuzzy Euclidean normed spaces, Commun. Math. Anal., 1(2) (2006), 86-90. [22] R. Saadati, S.Sedghi and H. Zhou, A common fixed point theorem  ψ-weakly commuting maps in L-Fuzzy Metric Spaces, Int. J. Fuzzy systems, 5(1) (2008), 47-53. [23]. S. Banach, Sur les operations dans les ensembles abstraits et leur application aux questions integrals’, Fund. Math.3,(1922)133-181 (French). [24] V. Gregori, A. Sapena, On fixed point theorems in fuzzy metric spaces, Fuzzy Sets Syst. 125 (2002), 245-253. [25] Y. J. Cho, H. K. Pathak, S. M. Kang and J. S. Jung, Common fixed points of compatible maps of type (β) on fuzzy metric  spaces, Fuzzy Sets Syst, 93 (1998), 99–111. [26] Z. K. Deng, Fuzzy pseduo-metric spaces, J. Math. Anal. Appl. 86 (1982), 74-95. The IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   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Common Fixed Point Theorem for ψ-weakly commuting maps in L-Fuzzy Metric Spaces for integral type

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Common Fixed Point Theorem for ψ-weakly commuting maps in L-Fuzzy Metric Spaces for integral type

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