International Journal of Mathematical Archive-5(10), 2014, 217-224 Available online through www.ijma.info ISSN 2229 5046 COMMON FIXED POINT OF WEAKLY COMPATIBLE MAPS ON INTUITIONISTIC FUZZY METRIC SPACES FOR INTEGRAL INEQUALITY Rasik M. Patel Ramakant Bhardwaj* *The Research Scholar of Sai Nath University, Ranchi (Jharkh), India. Truba Institute of Engineering & Information Technology, Bhopal (M.P), India. (Received On: 23-08-14; Revised & Accepted On: 17-10-14) ABSTRACT In this paper, we introduce the concept of - chainable intuitionistic fuzzy metric space akin to the notion of - chainable fuzzy metric space, introduced by Cho, Jungck [6] prove a common fixed point theorem for weakly compatible mappings for integral type inequality. 2010 Mathematics Subject Classification: 47H10, 54H25. Key words phrases: - chainable intuitionistic fuzzy metric space, weakly Compatible mappings, Common fixed point. 1. INTRODUCTION Let (X, d) be a complete metric space, c (0, 1) f: 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 [13] in 1922. K. Atanassov [11] introduced the notion of intuitionistic fuzzy sets by generalizing the notion of fuzzy set by treating memberships a fuzzy logical value rather than a single truth value. The application of Intuitionistic fuzzy sets instead of fuzzy sets means the introduction of another degree of freedom into a set description. By employing intuitionistic fuzzy sets in databases we can express a hesitation concerning examined objects. In 2006, Alaca et. al [3],[4] using the idea of intuitionistic fuzzy sets, defined the notion of intuitionistic fuzzy metric space with the help of continuous t-norm continuous t co-norms. Cho, Jungck [6], Sanjay Kumar, M. Alamgir Khan Sumitra [13] introduced the notion of - chainable fuzzy metric space proved common fixed point theorems for four weakly compatible mappings. D. Turkoglu, C. Alaca, Y. J. Cho C. Yildiz [8] J.S. Park [10], Gregori [9] Common fixed point theorems in intuitionistic fuzzy metric spaces. D. Coker [7] Zadeh [14] an introduction to intuitionistic fuzzy topological spaces. 2. PRELIMINARIES Definition 2.1 [1]: Let (X, d) be a complete metric space, c (0, 1) 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, such that for each > o, then f has a unique fixed point such that for each,. Corresponding Author: Rasik M. Patel The Research Scholar of Sai Nath University, Ranchi (Jharkh), India. International Journal of Mathematical Archive- 5(10), Oct. 2014 217
B.E.Rhoades [5], extending the result of Branciari by replacing the above condition by the following Definition 2.2[2]: A binary operation *: [0, 1] [0, 1] [0, 1] is a continuous t-norm if it satisfies the following conditions: (1) * is associative commutative, (2) * is continuous, (3) a * 1 = a for all a [0, 1], (4) a * b c * d whenever a c b d for all a, b, c, d [0, 1], Two typical examples of continuous t-norm are a * b = ab a * b = min (a, b). Definition 2.3[2]: A binary operation : [0, 1] [0, 1] [0, 1] is a continuous t-co norm if it satisfies the following conditions: (1) is associative commutative, (2) is continuous, (3) a 1 = a for all a [0, 1], (4) a b c d whenever a c b d for all a, b, c, d [0, 1], Two typical examples of continuous t-norm are a b = ab a b = min (a, b). Definition 2.4[1]: A 5-tuple (X, M, N, ) is said to be an intuitionistic fuzzy metric space if X is an arbitrary set, is a continuous t-norm, is a continuous t-co norm M, N are fuzzy sets on satisfying the following conditions: (i) (ii) (iii) (iv) (v) (vi) (vii) (ix) (x) (xi) (xii) (xiii) 2014, IJMA. All Rights Reserved 218
Then (M, N) is called an intuitionistic fuzzy metric space on X. The functions M(x, y, t) N(x, y, t) denote the degree of nearness the degree of non-nearness between x y w.r.t. t respectively. Remark 2.1: Every fuzzy metric space is an intuitionistic fuzzy metric space of the form such that t-norm t-co norm are associated as Remark 2.2: In intuitionistic fuzzy metric space is non-decreasing is non-increasing for all Alaca, Turkoglu Yildiz [3], [4], [8] introduced the following notions: Definition 2.5: Let (X, M, N, ) be an intuitionistic fuzzy metric space. Then (a) a sequence in X is called Cauchy-sequence if, for all t > 0 P > 0, (b) a sequence in X is said to be convergent to a point if, for all Definition 2.6: A pair of self-mappings (f, g) of an intuitionistic fuzzy metric space (X, M, N, ) is said to be compatible if & for every t > 0, whenever { } is a sequence in X such that for some. Definition 2.7: An intuitionistic fuzzy metric space (X, M, N, sequence in X is convergent. ) is said to be complete if only if every Cauchy Definition 2.8[3]: A pair of self mappings the coincidence points i.e. of a metric space is said to be weakly compatible if they commute at then Definition 2.9[3]: Let (X, M, N, ) be an intuitionistic fuzzy metric space. be self maps on X. A point x in X is called a coincidence point of iff Definition 2.10[3]: A pair of self mappings they commute at the coincidence points i.e. intuitionistic fuzzy metric space is said to be weakly compatible if then Lemma 2.1[1]: Let (X, M, N, ) be an intuitionistic fuzzy metric space if for a number such that then Definition 2.11: Let (X, M, N, ) be intuitionistic fuzzy metric space. A finite sequence is called - chain from x to y if there exists a positive number > 0 such that for all t > 0 & i = 1,.. n. An intuitionistic fuzzy metric space (X, M, N, ) is called - chainable if for any, there exists a - chain from x to y. 3. MAIN RESULTS Theorem 3.1: Let A, B, S T be self maps of a complete - chainable intuitionistic fuzzy metric spaces (X, M, N, ) with continuous t-norm * continuous t-co norm defined by for all satisfying the following condition: (3.1) A(X) T(X) B(X) S(X), (3.2) A S are continuous, 2014, IJMA. All Rights Reserved 219
(3.3) The pairs (A, S) (B, T) are weakly compatible, (3.4) There exist such that.then A, B, S T have a unique common fixed point in X. Proof: Since A(X) T(X), therefore for any, there exists a point such that for the point, we can choose a point such that as B(X) S(X). Inductively, we can Find a sequence in x as follows for n = 0, 1, 2.. By theorem of Alaca et al. [4], we can conclude that is Cauchy sequence in X. Since X is complete, therefore sequence in X converges to z for some z in X also the sequences also converges to z. Since X is - chainable, there exists - chain from to, that is,there exists a finite sequence such that for all t > 0 i = 1,2.l. Thus we have for m > n Therefore, is a Cauchy sequence in X hence there exists in X such that From (3.2), as limit n. By uniqueness of limits, we have since pair (A, S) is weakly compatible, therefore, so From (3.2), we have Also, form Continuity of S, we have, from (3.4), we get 2014, IJMA. All Rights Reserved 220
Proceeding limit as n, we have From lemma (2.1), we get hence. Since A(X) T(X), there exists in X such that From (3.4), we have Letting as n, we have 2014, IJMA. All Rights Reserved 221
By Lemma (2.1), we have therefore, we have Since (B, T) is weakly compatible, Therefore hence From (3.4) Letting as n, we have Which implies that. Therefore, Hence A, B, S T have common fixed point z in X. Uniqueness: Let w be another common fixed point of A, B, S T. Then from (3.4), 2014, IJMA. All Rights Reserved 222
By lemma (2.1), z = w. Hence A, B, S T have unique common fixed point z in X. 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. B.Schweizer A. Sklar, Probabilistic Metric Spaces, North Holl Amsterdam,10(1960),313-334. 3. Alaca, D. Turkoglu C. Yildiz: Fixed points in intuitionistic fuzzy metric spaces, Chaos, Solitons & Fractals, 29(5) (2006), 1073 1078. 4. Alaca, D. Turkoglu C. Yildiz: Common fixed points of compatible maps in intuitionistic fuzzy metric spaces, Southeast Asian Bulletin of Mathematics, 32(2008), 21 33. 5. B.E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics Mathematical Sciences, 63, (2003), 4007-4013. 6. Cho. Jungck Minimization theorems for fixed point theorems in fuzzy metric spaces applications, Fuzzy Sets Systems 61 (1994) 199-207. 7. D. Coker: An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets System, 88(1997), 81 89. 8. D. Turkoglu, C. Alaca, Y. J. Cho C. Yildiz: Common fixed point theorems in intuitionistic fuzzy metric spaces, J. Appl. Math. & Computing, 22(2006), 411 424. 9. Gregori, S. Ramaguera P. Veeramani: A note on intuitionistic fuzzy metric Spaces, chaos, solitons & Fractals, 28(2006), 902-905. 2014, IJMA. All Rights Reserved 223
10. J.S. Park, Y.C. Kwun, J.H. Park: A fixed point theorem in the intuitionistic fuzzy metric spaces, Far East J. Math. Sci., 16(2005), 137 149. 11. K.Atanassov: Intuitionistic fuzzy sets, Fuzzy Sets System, 20(1986), 87-96. 12. S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrals, Fund. Math.3,(1922)133-181 (French). 13. Sanjay Kumar, M. Alamgir Khan Sumitra, - chainable intuitionistic fuzzy metric space, International Mathematical Forum, Vol. 6, 37(2011),1837-1843 14. Zadeh, Fuzzy sets, Infor. Control, 8 (1965), 338-353. Source of support: MPCOST Bhopal, India. Conflict of interest: None Declared [Copy right 2014. This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited.] 2014, IJMA. All Rights Reserved 224