INTERNATIONAL RESEARCH JOURNAL OF MULTIDISCIPLINARY STUDIES

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1 WEAK COMPATIBILE AND RECIPROCALLY CONTINUOUS MAPS IN 2 NON-ARCHIMEDEAN MENGER PM-SPACE Jaya Kushwah School of Studies i Mathematics, Vikram Uiversity, Ujjai (M.P.) ABSTRACT. I this paper, we itroduce weak-compatible maps ad reciprocally cotiuous maps i weak o-archimedea PM-spaces ad establish a commo fixed poit theorem for such maps. Our result geeralizes several fixed poit theorems i the sese that all maps ivolved i the theorem ca be discotiuous eve at the commo fixed poit. Keywords ad Phrases. No-Archimedea Meger probabilistic metric space, Commo fixed poits, Compatible maps, Weakly Compatible, reciprocally cotiuous maps. AMS Subject Classificatio (2000). Primary 47H10, Secodary 54H Itroductio. I this sectio, we give a short survey of the study of fidig weaker forms of commutativity to have a commo fixed poit. I fact, this problem seems to be of vital iterest ad was iitiated by Jugck [9] with thei troductio of the cocept of commutig maps. I 1982, Sessa [16] itroduced the otio of weakly commutativity as a geeralizatio of commutativity ad this was a turig poit i the developmet of Fixed Poit Theory ad its applicatios i various braches of mathematical scieces. To be precise, Sessa [16] defied the cocept of weakly commutig by callig self maps maps A ad B of a metric space(x, d) a weakly commutig pair if ad oly if d (ABx, BAx ) d( Ax, Bx ), for all x Є X.Further to this other authers gave some commo fixed poit theorems for weakly commutig maps [1, 4, 7]. Note that commutig maps are weakly commutig, but the coverse is ot true. I 1986, Jugck [9] itroduced the ew otio of compatibility of maps as a geeralizatio of weak commutativity.thereafter, a flood of commo fixed poit theorems was produced by usig the improved otio of compatibility of maps. Later o, Jugck [9] itroduced the cocept of compatible maps of type (A) or of type (α), Pathak et al.[3,4] itroduced the compatible maps of type (B) or of type (ß), type (C) ad type (P) i metric spaces ad usig these cocepts, several researchers ad mathematicias have proved commo fixed poit theorems. Recetly, Cho et.al, [2] itroduced the otio of compatible maps of type (A) i o-archimedea Meger PM-spaces ad proved some iterestig results. I this directio, a weaker otio of compatible maps, called semi-compatible maps, was itroduced i fuzzy metric spaces by Sigh et. al. [15]. I particular, they proved that the cocept of semicompatible maps is equivalet to the cocept of compatible maps ad compatible maps of type (a) ad of type (ß) uder some coditios o the maps. I this paper, attempts have bee made to itroduce weak-compatible ad reciprocally cotiuous maps i weak o-archimedea Meger PM-spaces Here, we also preset the cocepts of compatible maps of type (A - 1) ad (A-2) Afterwards, Jai et. al. [7] proved the fixed poit theorem usig the cocept of weak compatible maps i Meger space. 2. Prelimiaries. 1 st August, 2017 Page No: 1

2 Defiitio 1. A distributio fuctio is a fctio,f: [, ] [0,1] which is left cotiuous o R, o decreasig ad F( ) = 0, F(+ ) = 1.If X is o empty set, F: X X Δ is called a probabilistic distace o X ad F(x,y) is usually deoted by F xy. Defiitio 2. Let X be a o-empty set ad D be the set of all left-cotiuous distributio fuctios. A ordered pair (X, F) is called a o-archimedea probabilistic metric space (shortly a N.A. PM-space) if F is a mappig from X X X ito D satisfyig the followig coditios (the distributio fuctio F(u,v,w) is deoted by F u,v,w for all u, v,w X) : (PM-1 ) F u,v,w (x) = 1, for all x > 0, if ad oly if at least two of the three poits are equal; (PM-2) F u,v,w = F u,w,v = F w,v,u ; (PM-3) F u,v,w (0) = 0 ; PM-4) If F u,v,s (x 1 ) = 1, F u,s,w (x 2 ) = 1 ad F s,v,w (x 3 ) = 1 the F u,v,w (max{x 1, x 2, x 3 }) = 1 for all u, v, w, s X ad x 1, x 2, x 3 0. Defiitio 3. A t-orm is a fuctio Δ : [0,1] [0,1] [0,1] [0,1] which is associative, commutative, o-decreasig i each coordiate ad Δ(a,1,1) = a for every a [0,1]. Defiitio 4. A 2 N.A. Meger PM-space is a ordered triple (X, F,Δ), where (X, F) is a o-archimedea PM-space ad Δ is a t-orm satisfyig the followig coditio: (PM-5) F u,v,w (max{x,y,z}) Δ (F u,v,s (x), F u,s,w (y)), F s,v,w (y) For all u,v,w,s X ad x,y,z 0 If the triagular iequality (PM-5) is replaced by the followig (WNA) Fu,v,w (x) = max{fu,v,s (x), Fu,s,w (x/2) * Fu,s,w (x) Fs,v,w (x/2) * Fs,v,w (x), Fu,v,s, (x/2)}, for all x, y, z X ad t > 0, the the triple (X, F, *) is called a weak o-archimedea Meger probabilistic metric space (shortly Meger WNAPM-space). Obviously every Meger NAPM-space is itself a Meger WNA-space (see Vetro [21] for the same cocept i fuzzy metric spaces). Remark 1. Coditio (WNA) does ot imply that Fu,v,w (x) is odecreasig ad thus a Meger WNAPM- space is ot ecessarily a Meger PM-space. If F u,v,w(x) is odecreasig, the a Meger WNA-space is a Meger PM-space. Remark 2. Recall that a Meger space is also a fuzzy metric space, for more details see Hadzic [17]. Example 1. Let X = [0, + ), a * b = ab for every a, b [0, 1]. Defie Fx,y(t) by: Fx,y (0) = 0, Fx,x (t) = 1 for all t > 0, Fx, y (t) = t for x y ad 0 < t 1, Fx,y (t) = t/2 for x y ad 1 < t 2, Fx,y (t) = 1 for x y ad t > 2. The (X, F, *) is a Meger WNAPM-space, but it is ot a PM-space. We recall that the cocept of eighborhood i Meger PM-spaces was itroduced by Schweizer ad Sklar [13] as follows; If x X, ε > 0 ad (0, 1), the a (ε, )-eighborhood of x, Ux (ε, ) is defied by Ux ( ε, ) = {y X : Fx,y ( ε) > 1 - }. If the t-orm * is cotiuous ad strictly icreasig, the (X, F, *) is a Hausdorff space i the topology iduced by the family {Ux (ε, ) : x X, ε > 0, (0, 1)} of eighborhoods. Let Ω = g{ such that : [0, 1] [0, + ) is cotiuous, strictly decreasig, g(1) = 0 ad g(0) < + }. 1 st August, 2017 Page No: 2

3 Defiitio 5. A 2 Meger WNAPM-space (X, F, Δ) is said to be of type (C) g if there exists a g Ω such that g(f x,y,z (t)) g(f x, y, a (t)) + g(f x, a, z (t)) + g(f a, y, z (t)) for all x, y, z, a X ad t 0, where Ω = {g g : [0,1] (0, ) is cotiuous, strictly decreasig, g(1) = 0 ad g(0) < }. Defiitio 6. A 2 Meger WNAPM-space (X, F, * ) is said to be type (D) g if there exists a g Ω such that g(δ(t 1 * t 2 * t 3 ) g(t 1 ) + g(t 2 ) + g(t 3 ) for all t 1, t 2, t 3 [0,1]. Remark 3. (1) If a weak 2 WNA Meger PM-space (X, F, *) is of type (D) g the (X, F, Δ) is of type (C) g.o the other had if (X, F, *) is a 2 WNAPM-space such that a * b max[a+ b 1,0] for all a,b [0,1], the (X, F, *) is of type (D) g For g Ω defied by g(t) = 1- t, t 0 Throughout this paper, eve whe ot specified (X, F, *) will be a complete 2 Meger WNAPM-space of type (D) g with a cotiuous strictly icreasig t-orm Δ. Let : [0, + ) [0, ) be a fuctio satisfied the coditio (Φ) : (Φ) is upper-semicotiuous from the right ad (t) < t for all t > 0. Lemma 1. If a fuctio : [0,+ ) [0,+ ) satisfies the coditio (Φ), the we have (1) For all t 0, lim (t) = 0, (t) is -th iteratio of (t). (2) If {t } is a o-decreasig sequece of real umbers ad t +1 (t ), = 1, 2, the lim t = 0. I particular, if t (t) for all t > 0, the t = 0. Defiitio 7. Let A, S : X X be mappigs. A ad S are said to be compatible if g(f (t)) = 0 for all t > 0, a X, wheever {x ASx, SAx, a } is a sequece i X such that Ax = Sx = z for some z i X. The otio of reciprocal cotiuity was defied by Pat [19] i ordiary metric space. Now, followig the same lie, we itroduce reciprocally cotiuous maps i 2 Meger WNAPMspaces. Defiitio 8. A pair of self-maps (A,S) of a 2 Meger WNAPM-space (X, F, *) is said to be reciprocally cotiuous if g(f ASx,Az, a (t)) 0 ad g(f SAx,Sz, a (t)) 0 for all t > 0, wheever there exists a sequece {x } i X such that Ax z, Sx z for some z i X as. If A ad S both are cotiuous, the, they are obviously reciprocally cotiuous but the coverse geerally is ot true. Propositio 1. Let A ad S be two self-maps of a 2 Meger WNAPM-space (X, F, *). Assume that (A, S) is compatible ad reciprocally cotiuous, the (A,S) is weakly compatible. 1 st August, 2017 Page No: 3

4 Proof. Let { x } be a sequece i X such that A x z ad S x z sice the pair of maps (A, S) is reciprocally cotiuous, the for all t > 0, we have lim g( F ASx, Az, a (t)) = 0 ad lim g( F SAx, Sz, a (t)) = 0 suppose that (A, S) is compatible ad reciprocally cotiuous, the, for t > 0, lim g(f ASx, SAx, a (t)) = 0, for all x X The we get g(f ASx, Sz, a (t)) g(f ASx, SAx, a (t)) + g(f SAx, Sz, a Ad so lettig +,we obtai lim g(f ASx, Sz, a (t))=0 thus,a ad S are weak compatible. Naturally, we ca defie the cocept of compatible mappigs of type (A-1) ad type (A-1) i 2 Meger WNAPM-space is as follows. Defiitio 9. Two self-maps A ad B of a 2 Meger WNAPM-space (X, F, *) are said to be compatible of type (A-1) if, for all t > 0, lim + g(f AB x,bbx, a (t)) = 0, wheever {x } is a sequece i X such that Ax, Bx z for some z X as +. Defiitio 10. Two self-maps A ad B of a 2 Meger WNAPM-spac ( X, F, *) are said to be compatible of type (A-2) if, for all t > 0, lim + g(f BA x,aax, a (t)) = 0, wheever {x } is a sequece i X such that Ax,Bx z for some z X. I the followig propositio, it is show that the cocept of compatible maps of type (A-1), type (A-2) ad if A ad B compatible maps of type (A) the the pair (A,B) is compatible of type (A-1) as well as type (A-2). Propositio 2. Let A ad B be two self-maps of a 2 Meger WNAPM-space (X, F, *). If (A,B) are compatible of type A the they are weakly compatible. Proof. To prove let { x } be a sequece i X such that Ax, Bx z for some z i X, as + ad let the pair (A, B) be compatible of type (A). we have lim g(f ABx, BBx, a (t)) = 0 ad lim g(f BAx, AAx, a (t)) = 0 g(f AB x, BAx (t)) g(f AB x,bbx (t)) + g(f BB x, BAx, a lettig +, we have g(f AB x, BAx (t)) g = 0 for all t > 0 Thus ABx =BAx ad we get (A, B) is weakly compatible.usig similar argumets as above, the reader ca easily prove the followig result. Propositio 3. Let A ad B be two self-maps of a 2 Meger WNAPM-space. If the pair (A,B) is weakly-compatible ad reciprocally cotiuous ad {x } is a sequece i X such that Ax, Bx z for some z X as +, the ABz = BAz. Proof. Suppose {x } is a sequece i X defied by x = z, = 1, 2,... ad Az = Bz. The we have, have (3) 1 st August, 2017 Page No: 4

5 Ax, Bx Bz as Sice (A, B) is weakly compatible, by triagle iequality g(f TBx,BBx, a (t)) g(f TBx,Bz, a (t)) + g(f Bz,BBx, a (t)) Sice BAx Az ad AAx Az as the g(f TBx,Bz, a (t)) = 0 ad g(f Bz,BBx, a (t)) = 0 g(f TBx,BBx, a (t)) = 0 g(f TBz,BBz, a (t)) = 0 i.e. BAz = AAz. (1) Similarly, we ca have ABz = BBz. (2) Hece, by (1) ad (2), we have ABz = BAz = AAz = BBz Before provig our mai theorem, we eed the followig lemma. Lemma 2. Let A, B, L, M, S ad T be self-maps of a complete 2 Meger WNAPM-space (X, F, *) of type (D) g, satisfyig (i) L(X) ST(X), M(X) AB(X); (ii) for all x, y X ad t > 0, g(f Lx,My (t)) (max{g(f ABx,Sty g(f Lx,ABx g(f My,Sty ½ [g(f ABx,My (t)) + g(f Lx,Sty (t))]}), where the fuctio : [0,+ ) [0,+ ) satisfies the coditio (Φ) proof: Let x 0 X The the sequece {y } i X, defied by Lx 2 = STx 2+1 = y 2 ad ABx 2+1 = Mx 2+2 = y 2+1 for = 0, 1, 2,..., such that g(f y (t)) = 0 for all t > 0 is a Cauchy sequece i X. If is ot a Cauchy sequece,y +1 i X, there exist ε > 0, t > 0 ad two sequeces { m i },{ i } of positive iteger such that (a) m i > i + 1 ad i as i (b) F ymi, y i, a (t 0)) < 1 ε ad F ymi - 1, y i, a (t 0)) 1- ε0, i = 1,2,3 Thus, we have g (1- ε 0 ) < g( F ymi, y i, a (t 0)) g( F ymi, y mi - 1, a (t 0 )) + g( F ymi - 1, y i, a (t 0)) g( F ymi, y mi - 1, a (t 0 )) + g(1- ε0 ) Ad lettig i +,we get g(f ymi, y i, a (t 0)) = g(1- ε0 ) (5) o the other had, we have g(1- ε0 ) < g(f ymi, y i, a (t 0)) g( F ymi, y i + 1, a (t 0 ))+ g( F y i + 1, y i, a (t 0)) (6) Let us assume that both m ad are eve, By cotractive coditio (ii),we get g( F ymi, y i + 1, a (t 0 )) = g( F Lxmi, Mx i + 1, a (t 0 )) that is φ(max{g(f ABx,STy g(f Lx,ABx g(f My,STy ½ [g(f ABx,My (t)) + g(f Lx,STy (t))]}), 1 st August, 2017 Page No: 5

6 g( F ymi, y i + 1, a (t 0 )) φ(max{g(f ymi - 1, y (t i, a 0)), g(f ymi - 1, y (t mi, a 0)), g(f yi, y i + 1, a (t 0 )),½ F y mi - 1, y i + 1, a (t 0 )) +g(f yi, y mi, a (t 0))]}), puttig this values i (6), usig (5) ad lettig i +, we get g(1- ε0 ) φ(max{g(1- ε0 ), 0, 0 g(1- ε0)}) = φ{g(1- ε0 ) } < g(1- ε0 ) a cotradictio. Hece {y }is a Cauchy sequece i X. 3 Mai Result. Theorem 3.1. Let A,B,L,M,S ad T be self-maps of a complete 2 Meger WNAPM-space (X, F, *) of type (D) g, satisfyig (3.1.1) L(X) ST(X), M(X) AB(X); (3.1.2) for all x, y X ad t > 0, g(f Lx,My (t)) (max{g(f ABx,Sty g(f Lx,ABx g(f My,Sty ½ [g(f ABx,My (t)) + g(f Lx,Sty (t))]}), where the fuctio : [0,+1) [0,+1) satisfies the coditio (Φ). (3.1.3) AB = BA, ST = TS, LB = BL, MT = TM; (3.1.4) the pair (M,ST) is compatible. If the pair (L,AB) is weakly-compatible ad reciprocally cotiuous, the A, B, L, M, S ad T have a uique commo fixed poit. Proof. Let x 0 X From coditio (3.1) x 1, x 2 X such that Lx 1 = STx 2 = y 1 ad Mx 0 = ABx 1 = y 0. Iductively, we ca costruct sequeces {x } ad {y } i X such that (3.1.5) Lx 2 = STx 2+1 = y 2 ad Mx 2+1 = ABx 2+2 = y 2+1 for = 0, 1, 2,.... We prove that g(f y (t)) = 0 for all t > 0.,y +1 From (3.1.4) ad (3.1.5), we have g(f y2, y 2+1 (t)) = g(f Lx 2, Mx 2+1 (t)) (max{g(f ABx2, STx 2+1 g(f ABx 2, Lx 2 g(f STx2+1, Mx 2+1 ½(g(F ABx2, Mx 2+1 (t)) + g(f STx 2+1, Lx 2 = (max{g(f y2-1,y 2 g(f y 2-1, y 2 g(f y 2, y 2+1 ½(g(F y2-1 (t)) + g(1))}), y 2+1 (max{g(f y2-1, y 2 g(f y 2, y 2+1 ½(g(F y2-1, y 2 (t)) + g(f y 2, y 2+1 (t))}). If g(f (t)) g(f (t)) for all t > 0, the by (3.1.4) y 2-1, y 2 y 2, y 2+1 g(f y 2, y 2+1 (t)) g(f y 2, y 2+1 (t))), o applyig Lemma 2, we have g(f (t)) = 0 for all t > 0. y 2, y st August, 2017 Page No: 6

7 Similarly, we have g(f (t)) = 0 for all t > 0. y 2+1, y 2+2 Thus, we have g(f (t)) = 0 for all t > 0. y, y +1 O the other had, if g(f (t)) g(f y2-1 the by (3.1.4), we have,y 2 y 2,y 2+1 g(f (t)) (g(f y2 (t))) for all t > 0.,y 2+1 y 2-1,y 2 Similarly, g(f (t)) (g(f y2+1 (t))) for all t > 0.,y 2+2 y 2,y 2+1 Thus, we have g(f (t)) (g (F y (t))) for all t > 0 ad = 1, 2, 3,.,y +1 y -1,y Therefore, by Lemma 2, g(f (t)) = 0 for all t > 0, which implies that {y y } is a Cauchy sequece i X by, y +1 Lemma 1. Sice (X, F, ) is complete, the sequece {y } coverges to a poit z X. Also its subsequeces coverges as follows : (3.7) {Mx 2+1 } z ad {STx 2+1 } z, (3.8) {Lx 2 } z ad {ABx 2 } z. Now sice the pair of maps (L, AB)is reciprocally cotiuous, therefore, we have g(f (t)) 0 ad g(f LABx2 (t)) 0 as +., Lz ABLx 2, ABz As (L, AB) is weakly compatible, so by Propositio 3, we have L(AB)x 2 ABz. That is ABz = Lz Puttig x = ABx 2 ad y = x 2+1 for t > 0 i (3.5), we get g(f LABx2,Mx 2+1 (t)) (max{g(f ABABx 2,STx 2+1 g(f ABABx 2, LABx 2 g(f STx2+1, Mx 2+1 ½(g(F ABABx2, Mx 2+1 (t)) + g(f STx 2+1, LABx 2. Lettig, we get g(f ABz,z (t)) (max{g(f ABz,z g(f ABz, ABz g(f z, z ½(g(F ABz, z (t)) + g(f z, ABz. i.e. g(f ABz,z (t)) (g(f ABz,z (t))) which implies that g(f ABz,z (t)) = 0 by Lemma 2 ad so we have ABz =z Puttig x = z ad y = x 2+1 for t > 0 i (3.5), we get g(f (t)) (max{g(f g(f Lz,Mx2+1 ABz,STx 2+1 ABz, Lz g(f STx2+1, Mx 2+1 ½(g(F ABz, Mx2+1 (t)) + g(f STx 2+1, Lz. Lettig, we get g(f Lz,z (t)) (max{g(f z,z g(f z, Lz g(f z, z ½(g(F z, z (t)) + g(f z, Lz 1 st August, 2017 Page No: 7

8 i.e. g(f Lz,z (t)) (g(f Lz,z (t))) which implies that g(f Lz,z (t)) = 0 by Lemma 2 ad so we have Lz = z. Therefore, ABz = Lz = z. Puttig x = Bz ad y = x 2+1 for t > 0 i (3.1.4), we get g(f (t)) (max{g(f g(f LBz,Mx2+1 ABBz,STx 2+1 ABBz, LBz g(f STx2+1, Mx 2+1 ½(g(F ABBz, Mx2+1 (t)) + g(f STx 2+1, LBz. As BL = LB, AB = BA, so we have L(Bz) = B(Lz) = Bz ad AB(Bz) = B(ABz) = Bz. Lettig, we get g(f Bz, z, a (t)) (max{g(f Bz,z g(f Bz, Bz g(f z,z ½(g(F Bz, z (t)) + g(f z, Bz i.e g(f Bz, z, a (t)) (g(f Bz, z (t))) which implies that g(f Bz, z, a (t)) = 0 by Lemma 2 ad so we have Bz = z. Also, ABz = z ad so Az = z. Therefore, Az = Bz = Lz = z. (3.9) As L(X) ST(X), there exists w X such that z = Lz = STw. Puttig x = x 2 ad y = w for t > 0 i (3.5), we get g(f (t)) (max{g(f g(f g(f Lx2,Mw ABx 2,STw ABx 2, Lx 2 STv, Mw ½(g(F ABx2, Mw, a (t)) + g(f STw, Lx 2, a. Lettig ad usig equatio (3.8), we get g(f z,mw (t)) (max{g(f z, z, a g(f z, z, a g(f z, Mw, a ½(g(F z, Mw, a (t)) + g(f z, z, a i.e. g(f z,mw (t)) (g(f z,mw (t))) which implies that g(f z,mw (t)) = 0 by Lemma 2 ad so we have z = Mw. Hece, STw = z = Mw. As (M, ST) is weakly compatible, we have STMw = MSTw. Thus, STz = Mz. Puttig x = x 2, y = z for t > 0 i (3.5), we get g(f Lx2,Mz (t)) (max{g(f ABx 2,STz g(f ABx 2, Lx 2 g(f STz, Mz ½(g(F ABx2, Mz (t)) + g(f STz, Lx 2. Lettig ad usig equatio (3.8) ad Step 5, we get g(f z,mz (t)) (max{g(f z,mz g(f z, z, a g(f Mz, Mz, a ½(g(F z, Mz, a (t)) + g(f Mz, z, a 1 st August, 2017 Page No: 8

9 i.e. g(f z,mz (t)) (g(f z,mz (t))) which implies that g(f z,mz (t)) = 0 by Lemma 2 ad so we have z = Mz. Puttig x = x 2 ad y = Tz for t > 0 i (3.1.4), we get g(f (t)) (max{g(f g(f Lx2,MTz ABx 2,STTz ABx 2, Lx 2 g(f STTz, MTz ½(g(F ABx2, MTz (t)) + g(f STTz, Lx 2. As MT = TM ad ST = TS, we have MTz = TMz = Tz ad ST(Tz) = T(STz) = Tz. Lettig we get g(f z,tz (t)) (max{g(f z,tz g(f z,z g(f Tz,Tz ½(g(F z,tz (t)) + g(f Tz,z i.e g(f z,tz (t)) (g(f z,tz (t))), which implies that g(f z,tz (t)) = 0 by Lemma 2 ad so we have z = Tz. Now STz = Tz = z implies Sz = z. Hece Sz = Tz = Mz = z. (3.10) Combiig (3.9) ad (3.10), we get Az = Bz = Lz = Mz = Tz = Sz = z. Hece, the six self maps have a commo fixed poit z. (Uiqueess) Let u be aother commo fixed poit of A, B, S, T, L ad M; the Au = Bu = Su = Tu = Lu = Mu = u. Puttig x = z ad y = u for t > 0 i (3.5), we get g(f Lz,Mu (t)) (max{g(f ABz,STu g(f ABz, Lz, a g(f STu, Mu, a ½(g(F ABz, Mu, a (t)) + g(f STu, Lz, a. Lettig, we get g(f z, u, a (t)) (max{g(f z, u, a g(f z, z, a g(f u, u, a ½(g(F z, u, a (t)) + g(f u, z, a = (g(f z, u, a (t))), which implies that g(f z,u (t)) = 0 by Lemma 2 ad so we have z = u. Therefore, z is a uique commo fixed poit of A, B, S, T, L ad M. This completes the proof. Remark 3.1. If we take B = T = I, the idetity map o X i theorem 1, the the coditio (b) is satisfied trivially ad we get Corollary 3.1. Let A, S, L, M : X X be mappigs satisfyig the coditio : (a) L(X) S(X), M(X) A(X); (b) Either A or L is cotiuous; (c) (L, A) is reciprocally cotiuous ad weakly compatible. (d) g(f Lx,My (t)) (max{g(f Ax, Sy, a g(f Ax, Lx, a g(f Sy, My, a 1 st August, 2017 Page No: 9

10 ½(g(F Ax, My, a (t)) + g(f Sy, Lx, a for all t > 0, where a fuctio : [0,+ ) [0,+ ) satisfies the coditio (Φ). The A, S, L ad M have a uique commo fixed poit i X. Remark 3.2. I view of remark 3.1, corollary 3.1 is a geeralizatio of the result of Cho et. al. [2] i the sese that coditio of compatibility of the pairs of self maps i a 2 weak o-archimedea Meger PM-space has bee restricted to weak compatible i a 2 weak o-archimedea Meger PM-space ad oly oe of the mappigs of the compatible pair is eeded to be reciprocally cotiuous. Refereces. 1. Chag, S.S., Cho, Y.J., Kag, S.M. ad Fa, J.X., Commo fixed poit theorems for multivalued mappigs i Meger PM-space, Math. Japo. 40 (2), (1994), Cho, Y.J., Ha, K.S. ad Chag, S.S., Commo fixed poit theorems for compatible mappigs of type (A) i o-archimedea Meger PM-spaces, Math. Japoica 48 (1), (1997), H. K. Pathak, M. S. Kha, Compatible mappigs of type (B) ad commo fixed poit theorems of Gregus type, Czechoslovak Mathematical Joural 45 (1995) H. K. Pathak, Y. J. Cho, S. S. Chag, S. M. Kag, Compatible mappigs of type (P) ad fixed poit theorems i metric spaces ad probabilistic metric spaces, Novi Sad Joural of Mathematics 26 (1996) Istrătescu, V.I., Fixed poit theorems for some classes of cotractio mappigs o oarchimedea probabilistic metric space, Publ. Math. (Debrece) 25 (1978), Istrătescu, V.I. ad Crivat, N., O some classes of oarchimedea Meger spaces, Semiar de spatii Metrices probabiliste, Uiv. Timisoara Nr. 12 (1974). 7. Jai, A. ad Sigh, B., Commo fixed poit theorem i Meger space through compatible maps of type (A), Chh. J. Sci. Tech. 2 (2005), Jai, A. ad Sigh, B., Commo fixed poit theorem i Meger Spaces, The Aligarh Bull. of Math. 25 (1), (2006), Jugck, G., Murthy, P.P. ad Cho,Y.J., Compatible maps of type (A) ad commo fixed poits, Math. Japo. 38 (1996), Kutukcu, S. ad Sharma, S., A Commo Fixed Poit Theorem i No-Archimedea Meger PM-Spaces, Demostratio Mathematica Vol. XLII (4), (2009), Meger, K., Statistical metrics, Proc. Nat. Acad. Sci. USA. 28(1942), Mishra, S.N., Commo fixed poits of compatible mappigs i PM-spaces, Math. Japo. 36(2), (1991), Schweizer, B. ad Sklar, A., Statistical metric spaces, Pacific J. Math. 10 (1960), Sehgal, V.M. ad Bharucha-Reid, A.T., Fixed poits of cotractio maps o probabilistic metric spaces, Math. System Theory 6(1972), Sessa, S., O a weak commutativity coditio i fixed poit cosideratio, Publ. Ist. Math. Beograd 32(46), (1982), Sigh, A., Bhatt, S. ad Chaukiyal, S., A Uique Commo Fixed Poit Theorem for Four Maps i No-Archimedea Meger PM-spaces, It. Joural of Math. Aalysis, Vol. 5, 2011, o. 15, Sigh, Bijedra, Jai, Arihat ad Agarwal, Pallavi, Commo fixed poit of coicidetally commutig mappigs i o-archimedea Meger PM-spaces, Italia Joural of Pure ad Applied Mathematics 25(2009), O. Hadzic, A ote o I. Istratescu s xed poit theoremi o- Archimedea Meger spaces, Bulleti Mathematique de la Societedes Scieces Mathematiques de Roumaie 24 (1980) st August, 2017 Page No: 10

11 18. O. Hadzic, Z. Ovci, Fixed poit theorems i fuzzy metric ad probabilistic metric spaces, Novi Sad Joural of Mathematics 24(1994) Pat, B.D. ad Sigh, S.L., Commo fixed poits of weakly commutig mappigs o oarchimedea Meger spaces, Vikram Math. J., 6 (1986), Schweizer, B. ad Sklar, A., Statistical metric spaces, Pacific J. Math. 10 (1960), C. Vetro, Some xed poit theorems for occasioally weakly compatible mappigs i probabilistic semi-metric spaces, Iteratioal Joural of Moder Mathematics 4 (2009) st August, 2017 Page No: 11