COMMON FIXED POINT THEOREM USING CONTROL FUNCTION AND PROPERTY (CLR G ) IN FUZZY METRIC SPACES

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1 Iteratioal Joural of Physics ad Mathematical Scieces ISSN: (Olie) A Ope Access, Olie Iteratioal Joural Available at Vol 4 (2) April-Jue, pp 68-73/Asati et al COMMON FIXED POINT THEOREM USING CONTROL FUNCTION AND PROPERTY (CLR G ) IN FUZZY METRIC SPACES *Alok Asati 1, ADSigh 1 ad Madhuri Asati 2 1 Departmet of Mathematics, GovtMVM, Bhopal (Idia) 2 Departmet of Applied Sciece, SKSITS, Idore (Idia) * Author for Correspodece ABSTRACT The aim of this paper is to prove a commo fixed poit theorem via commo limit rage property i fuzzy metric space satisfyig cotrol fuctio We geeralized the result of Kumar ad Fisher (2010) Mathematical Subject Classificatio: Primary 47H10, Secodary 54H25 Keywords: Commo Fixed Poits, Fuzzy Metric Space, Weak Compatible Map, Commo Limit Rage Property, Fixed Poit, Cotrol Fuctio INTRODUCTION The cocept of fuzzy sets was itroduced by Zadeh i 1965, as a ew way to represet the vagueess i everyday life I mathematical programmig problems are expressed as optimizig some goal fuctio give certai costraits, ad there are real life problems that cosider multiple objectives Geerally it is very difficult to get a feasible solutio that brigs us to the optimum of all objective fuctios A possible method of resolutio that is quite useful is the oe usig fuzzy sets It was developed extesively by may authors ad used i various fields To use this cocept i topology ad aalysis several researchers have bee defied fuzzy metric space i various ways (Aamri ad Moutawakil, 2002; Abbas et al, 2009; Balasubramaiam et al, 2002; Chauha ad Kumar, 2013; Chauha et al, 2012; Cho, 1997; Imdad et al, 2012; Kumar ad Chauha, 2013; Schweizer et al, 1983; Sharma, 2002) George ad Veeramai (1994) modified the cocept of fuzzy metric space itroduced by Kramosil ad Michalek (1975) i order to get the Hausdorff topology Jugck (1996) itroduced the otio of compatible maps for a pair of self mappig Mishra et al, (1994) exteded the otio of compatible mappigs to fuzzy metric spaces ad proved commo fixed poit theorem i preset of cotiuity of at least oe of the mappigs, completeess of the uderlyig space ad cotaimet of the rage amogst ivolved mappigs Sigh et al, (2005) weakeed the otio of compatibility by usig the otio of weak compatible mappig i fuzzy metric space ad show that every pair of compatible mappigs is weak compatible but reverse is ot true The study of commo fixed poit of o-compatible mappigs is also of great iterest due to Pat (1998) Situavarat et al, (2011) itroduced a ew cocept of property (CLR g ) Recetly Chauha et al, (2013) utilized the otio of commo limit rage property to prove uified fixed poit theorems for weakly compatible mappig i fuzzy metric spaces We prove commo fixed poit theorem for weakly compatible mappig with commo limit rage property 2 Prelimiaries Defiitio 21 [18] A biary operatio :[0,1] [0,1] [0,1] is called a cotiuous t orm if [0,1], is a abelia topological mooid with uit 1 such that ab c d wheever a c adb d a, b, c, d [0,1] Example of t-orms are ab ab ad ab mi{a,b} Copyright 2014 Cetre for Ifo Bio Techology (CIBTech) 68

2 Iteratioal Joural of Physics ad Mathematical Scieces ISSN: (Olie) A Ope Access, Olie Iteratioal Joural Available at Vol 4 (2) April-Jue, pp 68-73/Asati et al Defiitio 22 [9] A Triplet ( X, M, ) is called a fuzzy metric space if X is a arbitrary set, is a 2 cotiuous t-orm ad M is a fuzzy set o X [0, ) satisfyig the followig coditios: for all x, y, z X ad ts, 0, [1] M ( xy,,0) 1, [2] M( x, y, t) M( y, x, t), [3] M( x, y, t) 1For all t 0 if ad oly if x y, [4] M( x, y,) :[0, ) [0,1] is cotiuous, [5] M(x, y, t) (y, z,s) M(x, z, t s) [6] lim M ( x, y, t) 1 t Note that M (x, y,t) ca be cosidered as the degree of earess betwee x ad y with respect to twe idetify x ywith M( x, y, t) 1for all t 0 t Defiitio 23[9] Let (X, d) be a metric space Defie ab mi{a, b} ad M ( x, y, t) t d( x, y) for all x, y X ad all t 0The ( X, M, ) is a fuzzy metric space It is called the fuzzy metric space iduced b the metric d Defiitio 24 [10] Let (X, M, ) be a fuzzy metric space The (a) A sequece {x } i X is said to coverges to x i X if for each >0 ad each t > 0, there exists 0 N such that M (x, x, t) > 1- for all 0 (b) A sequece {x } i X is said to be Cauchy if for each > 0 ad each t > 0, there exists 0 N such that M (x, x m, t) > 1 for all, m 0 (c) A fuzzy metric space i which every Cauchy sequece is coverget is said to be complete Defiitio 25[17] Two maps F ad G, from a fuzzy metric space (X, M, ) ito itself are said to be R- weakly commutig if there exists a positive real umber R such that for each x X M( FGx, GFx, Rt) M( Fx, Gx, t) for all t 0 Defiitio 26 [16] Let F ad G be maps from a fuzzy metric space ( X, M, ) ito itself The maps F ad G are said to compatible if lim M ( FGx, GFx, t) 1 For all t 0wheever {x } is a sequece i X such that lim Fx lim Gx z for some z X Defiitio 27[20] Two maps F ad G, from a fuzzy metric space (X, M, ) ito itself are said to be weak-compatible if they commute at their coicidece poits, ie Fx Gx FGx GFx Remark 28[20] Let (F, G) be a pair of self-maps of a fuzzy metric space (X, M, )The (F, G) is R- weakly commutig implies that (F, G) is compatible, which implies that (F, G) is weak compatible But the coverse is ot true Copyright 2014 Cetre for Ifo Bio Techology (CIBTech) 69

3 Iteratioal Joural of Physics ad Mathematical Scieces ISSN: (Olie) A Ope Access, Olie Iteratioal Joural Available at Vol 4 (2) April-Jue, pp 68-73/Asati et al Defiitio 29 [6] A pair (A, S) of self-mappigs of a fuzzy metric space (X, M, ) is said to satisfy the commo limit rage property with respect to mappig S (briefly, (CLR S ) property),if there exists a sequece {x } i X such that lim Ax lim Sx z where z S( X ) Defiitio 210[6] Two pairs (A, S) ad (B, T) of self-mappigs of a fuzzy metric space (X, M, ) is said to satisfy the commo limit rage property with respect to mappig S ad T (briefly, (CLR ST ) property), if there exists a sequece {x },{y } i X such that lim Ax lim Sx lim By lim Sy z Where z S( X ) T( X ) Lemma 211 [20] Let ( X, M, ) be a fuzzy metric space The for all x, yx, M( x, y, ) is a odecreasig fuctio Lemma 212 [8] Let ( X,M, ) be a fuzzy metric space If there exists k (0,1) such that for all x, yx, M( x, y, kt) M(x, y,t), t > 0, the x y Lemma 213[4] The oly t-orm satisfyig r r r for all r [0,1] is the miimum t-orm, that is ab mi{a, b} for all ab, [0,1] 3 Mai Result Theorem 31 Suppose that (X, M, ) be a fuzzy metric space with ab mi{a,b} Let ( PJ, ) ad ( QLis, ) weak compatible pairs of self maps of X the followig coditios satisfies [i] P( X ) L( X ), the pair ( PJ, ) satisfies the property ( CLR ), [ii] Q( X ) J( X ), the pair ( QLsatisfies, ) the property ( CLR T ), [iii] There exists k (0,1) such that for every x, y X ad t 0 M (Px,Qy,kt) mi M ( Jx, Ly, t),m(px,jx,t),m(qy,ly,t), M(Px,Ly,t) Where :[0,1] [0,1] is a cotiuous fuctio such that () t t, for each 0t 1 [iv]oe of the subset P( X ), Q( X ), J( X ) or LX ( ) is a closed subset of X The pair ( PJ, ) ad ( QLare, ) weak compatible The P, Q, J ad L have a uique commo fixed poit i X Proof Let Q( X ) J( X ) ad the pair ( QLsatisfy, ) the property CLRT the there exists a sequece {x } i X such that Qxad Lx coverges to Lx for some x i X as Sice Q( X ) J( X ) so that there exists a sequece { y } i X such that Qx Jy, hece Jy Lx as We show that lim Py Lx Suppose that lim Py z Step 1 We put x y ad y x i equatio (iii) M (Py,Qx,kt) mi M ( Jy, Lx, t),m(py,jy,t),m(qx,lx,t), M(Py,Lx,t) M (Py,Qx,kt) mi M ( Jy, Lx, t),m(py,jy,t),m(qx,lx,t), M(Py,Lx,t) Takig limit M ( z,lx,kt) mi M ( Lx,Lx, t),m( z,lx,t),m( z,lx,t),m(lx,lx,t) Copyright 2014 Cetre for Ifo Bio Techology (CIBTech) 70 S

4 Iteratioal Joural of Physics ad Mathematical Scieces ISSN: (Olie) A Ope Access, Olie Iteratioal Joural Available at Vol 4 (2) April-Jue, pp 68-73/Asati et al M ( z,lx,kt) mi 1,M( z,lx,t),m( z,lx,t),1 M( z,lx,kt) M( z,lx,t) M( z,lx,t) Therefore usig lemma 212 z Lx We have Qx, Lx, Pyad Jy coverges to z We show that Qx z Step2 We put x y ad y xi (iii) M (Py,Qx,kt) mi M ( Jy, Lx, t),m(py,jy,t),m(qx,lx,t),m(py,lx,t) Takig limit M ( z,qx,kt) mi M (z,z, t),m(z,z,t),m(qx,z,t),m(z,z,t) M( z,qx, kt) mi 1,1, M(z,Qx, t),1 M( z,qx, kt) [M( z,qx, t)] M(z,Qx, t) Therefore usig lemma 212 z Qx Therefore z Qx Lx Sice the pair ( Q,L) is weak compatible it follows that Qz Lz also sice Q( X ) J( X ) there exists some y i x such that Qy Jy( z) Step 3 We show that Jy Py( z) We put x yad y x i (iii) M (Py,Qx,kt) mi M( Jy, Lx, t),m(py,jy,t),m(qx,lx,t), M(Py,Lx,t) Takig limit M (Py, z,kt) mi M (z,z, t),m(py, z,t),m(z,z,t),m(py,z,t) M (Py, z,kt) mi 1,M(Py, z,t),1,m(py,z,t) M(Py, z,kt) M(Py, z,t) M(Py, z,t) Py z By lemma 212 which implies that Py Jy z But the pair ( PJ, ) is weak compatible We have Pz Jz Step 4 We claim Pz z We put x zad y x i equatio (iii) M (Pz,Qx,kt) mi M ( Jz, Lx, t),m(pz,jz,t),m(qx,lx,t), M(Pz,Lx,t) Takig limit M (Pz, z,kt) mi M( Pz,z, t),m(pz, Pz,t),M(z,z,t),M(Pz, z,t) M (Pz, z,kt) mi M( Pz, z, t),1,1,m(pz,z,t) M(Pz, z,kt) M ( Pz, z, t) M ( Pz, z, t) Copyright 2014 Cetre for Ifo Bio Techology (CIBTech) 71

5 Iteratioal Joural of Physics ad Mathematical Scieces ISSN: (Olie) A Ope Access, Olie Iteratioal Joural Available at Vol 4 (2) April-Jue, pp 68-73/Asati et al Therefore by usig lemma 212 we get Pz z Which implies that Pz Qz Jz Lz z Therefore z is the commo fixed poit of P, Q, J, ad L REFERENCES 1 Aamri M ad El Moutawakil D (2002) Some ew commo fixed poit theorems uder strict cotractive coditios Joural of Mathematical Aalysis ad Applicatios Abbas M, Altu I ad Gopal D (2009) Commo fixed poit theorems for o compatible mappigs i fuzzy metric spaces Bulleti of Mathematical Aalysis ad Applicatios 1(2) Balasubramaiam P, Muralisakar S ad Pat RP (2002) Commo fixed poits of four mappigs i a fuzzy metric space The Joural of Fuzzy Mathematics 10(2) Chauha MS, Badshah VH ad Chouha VS (2010) Commo fixed poit of semi compatible Maps i fuzzy metric spaces Kathmadu Uiversity Joural of Sciece, Egieerig ad Techology 6(1) Chauha S ad Kumar S (2013) Coicidece ad fixed poits i fuzzy metrtc spaces usig commo property (EA) Kochi Joural of Mathematics Chouha et al, (2013) Uified fixed poit theorems i fuzzy metric spaces via commo limit rage property, Joural of Iequalities ad Applicatios 1(182) 7 Chauha S, Situavarat W ad Kumam P (2012) Commo fixed poit theorems for weakly compatible mappigs i fuzzy metric spaces usig (JCLR) property Applied Mathematics 3(9) doi:104236/am Cho YJ (1997) Fixed poits i fuzzy metric spaces The Joural of Fuzzy Mathematics 5(4) George A ad Veeramai P (1994) O some result i fuzzy metric space Fuzzy Sets ad Systems Grabiec M (1988) Fixed poits i fuzzy metric spaces Fuzzy Sets ad Systems 27(3) Imdad M, Pat BD ad Chauha S (2012) Fixed poit theorems i Meger spaces usig the (CLRST) property ad applicatios Joural of Noliear Aalysis ad Optimizatio 3(2) Jugck G (1996) Commo fixed poits for ocotiuous oself maps o ometric spaces Far East Joural of Mathematical Scieces 4(2) Kramosil I ad Michalek J (1975) Fuzzy metric ad statistical metric spaces Kyberetika 11(5) Kumar S ad Chauha S (2013) Commo fixed poit theorems usig implicit relatio ad property (EA) i fuzzy metric spaces Aals of Fuzzy Mathematics ad Iformatics 5(1) Kumar S ad Fisher B (2010) A commo fixed poit theorem i fuzzy metric space usig property (EA) ad implicit relatio Thai Joural of Mathematics 8(3) Mishra SN, Sharma N ad Sigh SL (1994) Commo fixed poits of maps o fuzzy metric spaces Iteratioal Joural of Mathematics ad Mathematical Scieces 17(2) Pat RP (1998) Commo fixed poits of four mappigs Bulleti of Calcutta Mathematical Society 90(4) Schweizer B ad Sklar A (1983) Probabilistic Metric Spaces North-Hollad Series i Probability ad Applied Mathematics (North-Hollad, New York) ISBN: Sharma S (2002) Commo fixed poit theorems i fuzzy metric spaces Fuzzy Sets ad Systems 127(3) Copyright 2014 Cetre for Ifo Bio Techology (CIBTech) 72

6 Iteratioal Joural of Physics ad Mathematical Scieces ISSN: (Olie) A Ope Access, Olie Iteratioal Joural Available at Vol 4 (2) April-Jue, pp 68-73/Asati et al 20 Sigh B ad Jai S (2005) Semicompatibility ad fixed poit theorems i fuzzy metric space usig implicit relatio Iteratioal Joural of Mathematics ad Mathematical Scieces Situavarat W ad Kumam P (2011) Commo fixed poit theorems for a pair of weakly compatible mappigs i fuzzy metric spaces Joural of Applied Mathematics Article ID Zadeh LA (1965) Fuzzy sets Iformatio ad Cotrol 8(3) Copyright 2014 Cetre for Ifo Bio Techology (CIBTech) 73