ON COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN INTUITIONISTIC FUZZY METRIC SPACES

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1 IJRRAS 9 (1) October 11 ON COMMON FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN INTUITIONISTIC FUZZY METRIC SPACES Kamal Wadhwa 1 & Hariom Dubey 1 Govt. Narmada Mahavidyalaya, Hoshangabad (M.P.) India Technocrats Institute of technology, Bhopal, (M.P.) India omsatyada@gmail.com ABSTRACT The aim of this paper is to obtain some common fixed point theorems in an intuitionistic fuzzy metric space for two pairs of occasionally weakly semi compatible hybrid mappings. Keywords: Intuitionistic fuzzy metric space, occasionally weakly semi-compatible pairs. Mathematics Subject Classification: 47H1, 54H5 1. INTRODUCTION As a generalization of fuzzy sets introduced by Zadeh [16], Atanassov [4] introduced the concept of intuitionistic fuzzy sets. Recently, using the idea of intuitionistic fuzzy sets, Park [9] introduced the notion of intuitionistic fuzzy metric spaces with the help of continuous t-norm and continuous t-conorms as a generalization of fuzzy metric spaces due to George and Veeramani [5]. Jungck and Rhoades [6] gave more generalized concept weak compatibility then compatibility.al-thagafi and Shahzad [3] weakened the concept of weakly compatible maps by giving the concept of occasionally weakly compatible maps.more recently Abbas and Rhoades [1] extended the definition of o.w.c. maps to the setting of set-valued maps. Recently, many authors have studied fixed point theory in intuitionistic fuzzy metric spaces ( [1],[6],[7],[11],[13]).In this paper we obtain common fixed point theorems for hybrid pairs of single and multivalued occasionally weakly semi-compatible mappings. We begin by briefly recalling some definitions and notions from fixed point theory literature that we will use in the sequel. Definition.1 [] - A binary operation :,1,1 conditions: (i) is commutative and associative; (ii) is continuous; (iii) a 1 a a,1 ; for all (iv) a b c d whenever a cand b d for all,,,,1,1 is a continuous t-norms if satisfying a b c d. Definition. A binary operation,1,1,1 is continuous t -conorm if is satisfying the following conditions: (i) is commutative and associative; (ii) is continuous; (iii) a = a for all a,1 ; (iv) a b c d whenever a c and b d for all a, b, c, d,1. Definition.3 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 -conorm and, following conditions: (i) M x y t N x y t,,,, 1 for all x, y X and t ; M N are fuzzy sets on X, satisfying the 81

2 IJRRAS 9 (1) October 11 (ii) M x, y, for all x, y X ; (iii) M x, y, t 1for all x, y X and (iv) M x, y, t M y, x, t for all x, y X (v) M x, y, t M y, z, s M x, z, t s (vi) for all x, y X,,,. :,,1 (vii) lim M x, y, t 1 for all x, y X t ; t if and only if x y and t ; for all x, y, z X and s, t ; M x y is left continuous; and t ; (viii) N x, y, 1 for all x, y X ; (ix) N x, y, t for all x, y X and (x) N x, y, t N y, x, t for all x, y X (xi) N x, y, t N y, z, s N x, z, t s (xii) for all, x, y X, N x, y,. :,,1 (xiii) lim N x, y, t for all x, y X ; t ; t if and only if x y and t ; for all x, y, z X and s, t ; is right continuous; M, N is called an intuitionistic fuzzy metric on X. The functions M x, y, t and,, degree of nearness and the degree of non-nearness between x and y with respect to t, respectively. N x y t denote the Remark An intuitionistic fuzzy metric spaces with continuous t -norm and Continuous t -conorm defined by a a a,1 1 a 1 a 1 a,1 x, y X, M x, y, is non-, a and for all a, Then for all decreasing and, N x, y, is non-increasing. Definition:.4 Let ( X, M, N,, ) be an intuitionistic fuzzy metric space and f : X X, F : X B( X ).The hybrid pairs, F) is said to be occasionally weakly compatible(o.w.c.) iff there exists some point x X such that fx Fx and ffx Ffx. Definition.5: Let ( X, M, N,, ) be an intuitionistic fuzzy metric space and f : X X, F : X B( X ).The hybrid pairs, F) is said to be occasionally weakly semi- compatible(o.w.s.c.) iff there exists some point x X such that fx Fx and f x Ffx. It is clear that owc hybrid pair is owsc pair, but not the converse in view of the following example. Example:.6 Let [,1] X and Define fx 1 x and Then f ( ) F( ), ff( ) [,1] Ff ( ) [, ] and f ( ) Ff ( ). Thus the hybrid pair, F ) is owsc, but not owc. 1 Fx [, ]. Definition.7 [13] Let CB( X ) be the set of all nonempty bounded and closed subsets of X,we define the functions; M ( a, B, t) max M( a, b, t) : b B If a B then from above definition M ( a, B, t) 1 8

3 IJRRAS 9 (1) October 11 N ( B, y, t) min N( b, y, t); b B Definition.8 [11] Let CB( X ) be the set of all nonempty bounded and closed subsets of X,we define the functions; ( A, B, t) inf M( a, b, t) : a A, b B. 3. MAIN RESULTS Theorem1: Let, ( X, M, N, be a intuitionistic fuzzy metric space and f, g : X X, F, G : X CB( X ) be mappings satisfying (i) the pairs, F ) and ( gg, ) are o.w.s.c. min M x, gy, t ), M x, Fx (,, ), t ), M ( gy, Gy, t M fx Gy qt ) (ii). ( t) ( t) and N fx, gy, t ), N x (,, ), Fx, t ), N ( gy, Gy, t N fx Gy qt ) ( t) ( t) x y X, where q 1, : R R for all, and such that ( t) for each. is a Lebesgue-integrable mapping which is summable, nonnegative Proof: Since the pairs, F ) and ( gg, ) are occasionally weakly semi-compatible,there exist u, v X fu Fu, Ffu, gv Gv and gv.from (ii),we have Suppose fu M u, gv, qt) M u, Gv, qt) and g v Ggv. min M u, gv, t ), M u, Fu, t ), M ( gv, Gv, t ) ( t) ( t) ( t) min M u, gv, t),1,1 () t M u, gv, t) () t...(i) such that max N u, gv, t ), N u (,, ) (,, ), Fu, t ), N ( gv, Gv, t N fu gv qt N fu Gv qt ) ( t) ( t) ( t) max Nu, gv, t),, () t 83

4 IJRRAS 9 (1) October 11 (I) and (II) implies that fu gv Nu, gv, t) () t.(ii) Suppose fu.i.e. gv.since fu Fu and M u, gv, qt) M u, Gv, qt) Ffu,from (ii),we have min M u, gv, t), M u, Ffu, t), M ( t) ( t) ( t) M u, fu, qt) min M u, fu, t), M u, Ffu, t), M ( t) ( t) min ( M, fu, t ),1,1 M u, fu, qt) ( t) ( t) M u, fu, qt) M u, fu, t) ( t) ( t).(iii) max N u, fu, t), N u, Ffu, t), N N u, fu, qt) ( t) ( t) max N u, fu, t), N u, Ffu, t), N N u, fu, qt) ( t) ( t) N u, fu, qt) max N u, fu, t ),, ( t) ( t) N u, fu, qt) N u, fu, t) ( t) ( t)...(iv) fu (II) and (IV) implies that Thus fu Ffu.Hence fu is a common fixed point of f and F.Similarly we can show that gv is common fixed point of g and G since fu gv,it follows that fu is common fixed point of f, g, F and G.Uniqueness of common fixed point follows easily from (ii). Theorem : Let ( X, M, N,, be a fuzzy metric space and f, g : X X, F, G : X CB( X ) be mappings satisfying (i) the pairs, F ) and ( gg, ) are occasionally weakly semi-compatible, (ii) ( Fx, Gy, qt) inf M x, gy, t), M x, Fx, t), M ( gy, Gy, t), M x, Gy, t), M ( gy, Fx, t) ( t) ( t) 84

5 IJRRAS 9 (1) October 11 (iii) for all, ( Fx, Gy, qt) max N x, gy, t), N x, Fx, t), N ( gy, Gy, t), N x, Gy, t), N ( gy, Fx, t) ( t) ( t) x y X, where q 1, : R R nonnegative and such that ( t) for each. is a Lebesgue-integrable mapping which is summable, Proof: Since the pairs, F ) and ( gg, ) are occasionally weakly semi-compatible,there exist u, v X fu Fu, Ffu, gv Gv and gv.from (ii),we have Suppose fu M u, gu, qt) ( Fu, Gv, qt) g v Ggv. such that inf M u, gv, t), M u, Fu, t), M, M u, Gv, t), M ( gv, Fu, t) M u, gu, qt) ( t) ( t) ( t) inf M u, gv, t),1,1, M u, gv, t), M ( gv, fu, t) ( t) ( t) M u, gv, qt) M u, gv, t) and ( t) ( t).(i) Nu, gv, t) ( Fu, Gv, qt) max Nu, gv, t), N u, Fu, t), N, N u, Gv, t), N ( gv, Fu, t) Nu, gv, t) ( t) ( t) ( t) max Nu, gv, t),,, Nu, gv, t), N ( gv, fu, t) ( t) ( t) Nu, gv, qt) Nu, gv, t) ( t) ( t)..(ii) (I) and (II) implies that fu gv Suppose M u, gv, t) fu.i.e. gv.since fu Fu and ( Ffu, Gv, qt) Ffu,from (ii),we have inf M u, gv, t), M u, Ffu, t), M, M u, Gv, t), M ( gv, Ffu, t) ( t) ( t) ( t) M u, gv, t) inf M u, gv, t),1,1, M u, gv, t), M ( gv, f u, t) ( t) ( t) 85

6 IJRRAS 9 (1) October 11 M u, gv, t) M gv t And (,, ) ( t) ( t) (III) N u, gv, t) ( Ffu, Gv, qt) max N u, gv, t), N u, Ffu, t), N, N u, Gv, t), N ( gv, Ffu, t) N u, gv, t) ( t) ( t) ( t) max N u, gv, t),,, N u, gv, t), N ( gv, f u, t) ( t) ( t) N u, gv, t) N u, gv, t) ( t) ( t) (IV) (II) and (IV) implies that fu Thus fu Ffu.Hence fu is a common fixed point of f and F.Similarly we can show that gv is common fixed point of g and G since fu gv,it follows that fu is common fixed point of f, g, F and G.Uniqueness of common fixed point follows easily from (ii). 4. REFERENCES: [1]. Abbas,M.,Rhoades, B.E, Common Fixed Point Theorems for Hybrid pairs of Occasionally Weakly Compatible Mappings Satisfying Generalized Contractive Condition of Intregal type.fixed Point Theory and Applications, 7,Article ID 5411,9 pages. []. C. Alaca, D. Turkoglu, and C. Yildiz, Fixed point in intuitionistic fuzzy metric spaces,chaos, Solitons and Fractals 9 (6), 173{178. [3]. Al-Thagafi, M.A, Shahzad, N., Generalized I-nonexpansive self maps and invariant approximations.acta.math.sinica. 4(5)(8), [4]. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems (1986), 87{96. [5]. A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy sets andsystems 64 (1994), 395{399. [6]. G. Jungck and B. E. Rhoades, Fixed point for set valued functions without continuity,ind. J. Pure & Appl. Math. 9 (1998), no. 3, 7{38. [7]. I. Kubiaczyk and S. Sharma, Common fixed point in fuzzy metric space, J. Fuzzy Math.(to appear). [8]. S. Kutukcu, A common fixed point theorem for a sequence of self maps in intuitionisticfuzzy metric spaces, Commun. Korean Math. Soc. 1 (6), no. 4, 679{687. [9]. J. H. Park, Intuitionistic fuzzy metric spaces, Chaos, Solitons and Fractals (4),139{146. [1]. B. Schweizer and A. Sklar, Statistical spaces, Paci c J. Math. 1(196),313{334. [11]. S. Sedghi Common fixed point theorems for multivalued contractions International Mathematical Forum,, 7, no. 31, [1]. S. Sharma, On fuzzy metric space, Southeast Asian Bull. Math. 6 (), no. 1, 145{157. [13]. Sushil Sharma, Servet Kutuku, and R.S.Rathore,commun. Common fixed point for multivalued mappings in intuitionistic fuzzy metric spaces Korean Math.Soc.(7), No.3,pp [14]. D. Turkoglu, C. Alaca, and C. Yildiz, Compatible maps and compatible maps of type ( ) and ( ) in intuitionistic fuzzy metric spaces, Demonstratio Math. 39 (6), no. 3,671{684). [15]. D. Turkoglu, C. Alaca, Y. J. Cho, and C. Yildiz, Common xed point theorems inintuitionistic fuzzy metric spaces, J. Appl. Math. & Computing (6), no. 1-,411{44. [16]. L. A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965), 338{353). 86