A COMMON FIXED POINT THEOREM IN FUZZY METRIC SPACE USING SEMI-COMPATIBLE MAPPINGS
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1 Volume 2 No. 8 August 2014 Joural of Global Research i Mathematical Archives RESEARCH PAPER Available olie at A COMMON FIXED POINT THEOREM IN FUZZY METRIC SPACE USING SEMI-COMPATIBLE MAPPINGS Dr. V.K.Agarwal 1 Dr. K. Wadhwa 2 Abhilasha Bhigare 3 Vedprakash Bhardwaj 4 1 Govt. girls P.G. college Hoshagabad 2 Govt. Narmada college Hoshagabad 3 Corporate Istitute of Sciece & Techology Bhopal 4 Govt. Narmada College Hoshagabad. Abstract: I this paper the cocept of semi-compatibility ad weak compatibility i Fuzzy metric space has bee applied to prove a commo fixed poit theorem. We geeralize the result of Arihat Jai et.al. [11] usig ratioal iequality. Keyword: Fuzzy metric space commo fixed poit compatible maps semi-compatible maps ad weak compatible maps. 1. INTRODUCTION The cocept of fuzzy sets was itroduced by Zadeh [22] i 1965.Sice the this cocept was used i topology ad aalysis. May authors have extesively developed the theory of fuzzy sets ad its applicatio. Especially Deg[4]Erceg[5]Kaleva ad Seikkala[12] Kramosil ad Michalek[13] have itroduced the cocept of fuzzy metric space i differet ways.george ad Veeramai[8] have modified the cocept of fuzzy metric space itroduced by Kramosil ad Michalek[13] ad have defied the Housdroff topology o fuzzy metric spaces. It has bee see that the study of Kramosil ad Michalek[13] of fuzzy metric space covered almost all the poits i the way for developig this theory to the field of fixed poit theorem i particular for the study of cotractive type maps. They have also show that every metric iduces a fuzzy metric. Sigh [21] proved various fixed poit theorems usig the cocepts of semi-compatibility compatibility ad implicit relatios i Fuzzy metric space. Kumar ad Pat [15] have give a commo fixed poit theorem for two pairs of compatible mappig satisfyig expasio type coditio i probabilistic Meger space. Recetly Arihat Jai et.al.[11] improved the result of Kumar ad Pat [15] by droppig the coditio of cotiuity of the mappig ad usig semi ad weak compatibility of the mappig i place of compatibility. I this paper we are geeralizig the result of Arihat Jai et.al.[11] usig ratioal iequality. 2. PRELIMINARIES Defiitio 2.1.[18] A biary operatio : [01] [01] [01] is called a cotiuous t-orm if [01] is a abelia topological mooid with uit 1 such that a b c d wheever a c ad b d for all abcd [01]. Example of t-orm are a b=ab ad a a=mi{ab}. JGRMA 2014 All Rights Reserved 1
2 Dr. V. K. Agrawal et al. Joural of Global Research i Mathematical Archives 2(8) August Defiitio 2.2.[13] The 3-tuple XM * is said to be a Fuzzy metric space if X is a arbitrary set* is a cotiuous t-orm ad M is a Fuzzy set i 2 X [0 ) satisfyig the followig coditios: for all x y z X ad st 0. (FM-1) M x y 0 0 (FM-2) M x y t 1 for all t 0 if ad oly if x y (FM-3) M x y t M y x t (FM-4) M x y t * M y z s M x z t s (FM-5) M x y. :[0 ) [01] is left cotiuous. Where M x y t ca be cosidered as the degree of earess betwee x ad y with respect tot. We idetify x y with M x y t 1 for all t 0. Defiitio 2.3. [9] Let XM * be a fuzzy metric space: (1) A sequece x i X is said to be coverget to a poit x X (deoted by lim x x)if lim M x x t 1 for all 0 (2) A sequece t. x i X is said to be a Cauchy sequece lim M x x t 1for all t 0 ad 0 p (3) A Fuzzy metric space i which every Cauchy sequece is coverget is said to be complete. p. Let XM * be a fuzzy metric space with followig coditio: (FM-6) lim M x y t 1 for all x y X. t Defiitio 2.4. [19] A fuctio M is cotiuous i Fuzzy metric space if ad oly if wheever x x y y the lim M x y t M x y t for all 0 t t. Defiitio 2.5. [14] Let A ad B be mappigs from a Fuzzy metric space XM * ito itself. The mappigs A ad B are said to be weakly compatible if they commute at their coicidece poits i.e. Ax Bx implies ABx BAx. Defiitio 2.6. [21] Suppose A ad S be two maps from a Fuzzy metric space XM * ito itself. The they are said to be semi-compatible if lim ASx Sx wheever x is a sequece such that lim Ax lim Sx x X. JGRMA 2014 All Rights Reserved 2
3 Dr. V. K. Agrawal et al. Joural of Global Research i Mathematical Archives 2(8) August I [9] Grabiec has give two importat lemmas for cotractio coditio.we have the followig lemmas for expasio type coditio. Lemma 2.1. Let {x } be a sequece i a Fuzzy metric space XM * with (FM-6).If there exists a umber h 1 such that M x x ht M x x t for all t 0 ad The x is Cauchy sequece i X. Lemma 2.2. If for all x y X t 0 ad for a umber h 1 M x y ht M x y t the x y 3. MAIN RESULT I this sectio we preset our mai result. Theorem 3.1: Let (X M *) be a complete Fuzzy metric space where * is cotiuous t-orm ad satisfies x * x x for all x [0 1]. Let A B S ad T be self mappigs of a Fuzzy metric space satisfyig the followig coditios: (3.1.1) A ad B are surjective. (3.1.2) (A S) is semi-compatible ad (B T) is weakly compatible. (3.1.3) u v X ad h>1 Where a b r s 0 with a & b ad r&s caot be simultaeously 0 the A B S ad T have a uique commo fixed poit i X. Proof: Let u 0 X. Sice A ad B are surjective we choose a poit u 1 X such that Au 1 = Tu 0 = v 0 ad for this poit u 1 there exists a poit u 2 i X such that Bu 2 = Su 1 = v 1. Cotiuig i this maer we obtai a sequece {v } i X as follows: Au 2+1 = Tu 2 = v 2 ad Bu 2+2 = Su 2+1 = v 2+1. Usig (3.1.3) we have M(v 2 v 2+1 ht) = M(Au 2+1 Bu 2+2 ht) M(v 2+1 v 2+2 t) Therefore by Lemma 2.1 {v } is a Cauchy sequece. Sice X is complete {v } coverges to some poit z X. Cosequetly the subsequeces {Au 2+1 } {Bu 2 } (Su 2+1 } ad {Tu 2 } also coverges to z. Sice A ad S are semi-compatible so lim ASu 2+1 = Sz. Also lim ASu 2+1 = Az. Sice the limit i Fuzzy metric space is uique we get Az = Sz. Agai by (3.1.3) M(Az Bu 2 ht) JGRMA 2014 All Rights Reserved 3
4 Dr. V. K. Agrawal et al. Joural of Global Research i Mathematical Archives 2(8) August Lettig we get which gives M(Az z ht) M(Az z t) implyig Az = z. Therefore Az = Sz = z. Now suppose z = Bp for some p X the by (3.1.3) M(Au 2+1 Bp ht) Lettig we get M(z z ht) which gives M(z z ht) M(z Tp t). It gives that z = Tp. Therefore Bp = Tp = z. p is a coicidece poit of B ad T. Sice B ad T are weakly compatible ad Bp = Tp = z. Therefore TBp = BTp implies Tz = Bz. Now by (3.1.3) ad usig Tz = Bz M(Au 2+1 Bz ht) Lettig we get M(z Tz ht) which gives M(z Tz ht) M(z Tz t). It gives that z = Tz. Hece Bz = z. Therefore Az = Bz = Tz = Sz = z. Hece z is a commo fixed poit of A B S ad T. Fially for the uiqueess of fixed poit let w X be aother fixed poit. The from (3.1.3) M(Aw Bz ht) M(w z ht) Which gives M(w z ht) M(w z t) w = z. This completes the proof of the theorem. Remark 3.1: We geeralized the result of Arihat Jai et.al.[11] usig ratioal iequality. REFERENCES: [1]. Badard R. Fixed poit theorem for fuzzy umbers Fuzzy Sets ad Systems 13(1984) [2]. Baach S.:Theorie des Operatios Lieaires Moography Mathemateycze WarszawaPolad [3]. Butariu D. : Fixed poit theorems for fuzzy mappigs fuzzy sets ad systems 7 (1982) JGRMA 2014 All Rights Reserved 4
5 Dr. V. K. Agrawal et al. Joural of Global Research i Mathematical Archives 2(8) August [4]. Deg. Z. K.: Fuzzy pseudo metric space J.Math. Aal.Appl. 86(1982) [5]. Erceg M. A.: Metric spaces i fuzzy set theory J. Math. Aal. Appl. 69(1979) [6]. Fag J. X. O fixed poit theorem i fuzzy metric spaces Fuzzy Sets ad Systems 46(1992) [7]. Fag J. X. A ote o fixed poit theorem of Hadzic Fuzzy Sets ad Systems 48(1992) [8]. GeorgeA. Veeramai P. O some results of aalysis for Fuzzy Metric Spaces Fuzzy Sets ad System 90(1997) [9]. Grabiec M.Fixed poit i fuzzy metric space Fuzzy Sets ad Systems 27(1988) [10]. Gregori V. ad Sapea A. O fixed poit theorem i fuzzy metric spaces Fuzzy Sets ad Systems 125(2002) [11] Jai A. Badshah V. H. ad Prasad S. K. Fixed poit theorem i fuzzy metric space for semi-compatible mappigs IJRRAS 12 (3) September [12]. Kaleva O. ad Seikkala S. O fuzzy metric spaces Fuzzy sets ad systems 12(1984) [13]. Kramosil I. ad Michalek J. Fuzzy metric ad Statistical metric spaces Kyberetica 11(1975) [14]. Kumar S. Commo fixed poit theorem for miimal commutativity type mappigs i Fuzzy metric spaces.thai Joural of Mathematics6(2008) [15]. Kumar S. ad Pat B. D. A Commo fixed poit theorem for expasio mappigs i Probabilistic metric spaces Gaita 57(2006) [16]. Pap E.Hadzic O. ad Mesiar R. A fixed poit theorem i Probabilistic metric spaces ad a applicatio J. Math. Aal. Appl. 202(1996) [17]. Razai A. ad Shirdaryadzi M. Some results o fixed poits i fuzzy metric space J. Appl.Math. Comp.20(2006) [18]. Schweizer B. ad Sklar A. Statistical metric spaces Pacific Joural Math. 10(1960) [19]. Sharma S. O fuzzy metric space Southeast Asia Bull. Of Math. 26(2002) [20]. Sharma S. Commo fixed poit i fuzzy metric space Fuzzy sets ad systems. 127(2002) [21]. Sigh B. ad Jai S. Semi-compatibility compatibility ad fixed poit theorems i fuzzy metric space Joural of Chugecheog Math. Soc. 18(1) (2005) [22]. Zadeh L. A. Fuzzy sets iformatio ad cotrol 8(1965) JGRMA 2014 All Rights Reserved 5
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