COMMON FIXED POINT THEOREMS IN MENGER SPACE FOR SIX SELF MAPS USING AN IMPLICIT RELATION

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1 BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p), ISSN (o) Vol. 4(2014), hp:// JOURNALS / BULLETIN Formerly BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o), ISSN X (p) COMMON FIXED POINT THEOREMS IN MENGER SPACE FOR SIX SELF MAPS USING AN IMPLICIT RELATION I.H.Nagaraja Rao 1, S. Rajesh 2, and G.Venkaa Rao 3 Absrac. The aim of his paper is o prove, mainly, a common fixed poin heorem for six self mappings of a Menger space using wo weakly compaible pairs saisfying an implici relaion. This generalizes several known resuls including hose of Kohli e.al [2] and Sasry e.al [7]. 1. Inroducion The pursui of fixed poin heorems in Menger space is an acive area of research in he presen days. Menger [4] inroduced he concep of probabilisic Menger space. Singh e.al [10] inroduced he noion of weakly commuing mappings on Menger spaces. Kohli e. al [2] esablished a common fixed poin heorem for six self mappings using poinwise R-weakly commuing mappings wih a conracive ype implici relaion. This generalizes he resuls of Kumar and Pan [3]. Sasry e. al [7] made some modificaions o he resuls of Kohli e. al [2]. In his paper, we furher generalized he resuls of [2] and [7]. As usual R sands for he se of all real numbers, R + sands for he se of all non-negaive real numbers, Q sands for he se of raional numbers and N sands for he se of naural numbers. 2. Preliminaries We ake he sandard definiions given in Schweizer and Sklar [8] Mahemaics Subjec Classificaion. 47H10; 54H25. Key words and phrases. Menger space, weakly compaible mappings, poinwise R-weakly commuing mappings, common fixed poin. 89

2 90 I.H.NAGARAJA RAO, S. RAJESH, AND G.VENKATA RAO We hereunder give he following definiions and he resul required in subsequence secion. Definiion 2.1. ([10]) Self mappings f and g of a probabilisic meric space (X, F ) are said o be weakly commuing if and only if (iff) F fgx,gfx () F fx,gx () for each x X and > 0. Definiion 2.2. ([1]) Self mappings f and g of a probabilisic meric space (X, F ) are said o be poinwise R-weakly commuing if given z in X, here exiss R > 0(depending on x) such ha F fgx,gfx () F fx,gx ( R ) for > 0. Noe 2.1. Weakly commuing mappings are poinwise R-weakly commuing wih R = 1. Definiion 2.3. ([3]) Self mappings f and g of a probabilisic meric space (X, F ) are said o be reciprocally coninuous if fgx n fz and gfx n gz, whenever {x n } is a sequence such ha fx n, gx n z for some z in X. Noe 2.2. Every pair of coninuous mappings is reciprocally coninuous. Definiion 2.4. Self mappings f and g of a probabilisic meric space (X, F ) are said o be weakly compaible iff fx = gx for some x X implies fgx = gfx. Definiion 2.5. ([5]) Self mappings f and g of a probabilisic meric space (X, F ) are said o be weakly compaible if F fgxn,gfx n () 1 for all > 0 whenever {x n } is a sequence in X such ha fx n, gx n z for some z X. Noe 2.3. Compaible implies weakly compaible bu he converse is no rue. We, hereunder give a pair of self mappings on a Menger space ha are weakly compaible bu no compaible, R-weakly commuing and weakly commuing. Example 2.1. Le X = [0, λ] (λ 2), a b = min{a, b} for all a, b [0, 1] and F x,y () = + x y for all x, y X and for all > 0. Then (X, F, ) is a complee Menger space. Define self mappings f and g on X by { x if 0 x < λ f(x) = 2, λ if λ 2 x λ, { λ x if 0 x < λ g(x) = 2, λ if λ 2 x λ. Claim 1: {f, g} is weakly compaible. For x [0, λ 2 ), fx < λ 2 < gx. Hence, fx gx, for every x [0, λ 2 ). For every x [ λ 2, λ], fx = λ = gx and fg(x) = f(λ) = λ = g(λ) = gf(λ).

3 COMMON FIXED POINT THEOREMS 91 Therefore, {f, g} is weakly compaible. Claim 2: {f, g} is no compaible. Take x n = { λ 2 1 n }. fx n = { λ 2 1 n } λ 2 as n and gx n = {λ λ n } λ 2 fg(x n ) = f( λ ) = λ and gf(x n) = g( λ ) = λ 2. F fgxn,gfx n () = F λ, λ () < 1 as n. 2 + λ 2 Hence, {f, g} is no compaible. as n. Claim 3: {f, g} is no weakly commuing. Take x = 3λ 8. fx = 3λ 8 and gx = λ 3λ 8 = 5λ 8. fg(x) = f( 5λ 8 ) = λ and gf(x) = g( 3λ 8 ) = 5λ 8. Since 3λ 8 > λ 4, follows ha F fgx,gfx() < F fx,gx (). Therefore, {f, g} is no weakly commuing. Claim 4: {f, g} is no R-weakly commuing. Take x [ 3λ 8, λ 2 ). fx = x and gx = λ x. fg(x) = f(λ x) = λ and gf(x) = g(x) = λ x. Le R > 0. F fgx,gfx () = F λ,λ x () = F fx,gx ( R ) = F x,λ x( R ) = +x and R R +(λ 2x) = +R(λ 2x). Now, F fgx,gfx () F fx,gx ( R ) x R(λ 2x) R x (λ 2x). Since, Sup{(λ 2x) : x [ 3λ 8, λ 2 )} = +, i follows ha such R does no exis. Therefore, {f, g} is no R-weakly commuing. (Observe ha he pair {f, g} is poinwise R-weakly commuing, since for any x [ 3λ 8, λ 2 ), we can selec R x x (λ 2x) ). Definiion 2.6. ([6]) A funcion ϕ : (R + ) 4 R is said o be an implici relaion if (i.) ϕ is coninuous, (ii.) ϕ is Monoonic increasing in he firs argumen and (iii.) ϕ saisfies he following condiions: (a) for x, y 0, ϕ(x, y, x, y) 0 or ϕ(x, y, y, x) 0 implies x y, (b) ϕ(x, x, 1, 1) 0 implies x 1. Example 2.2. Define ϕ : (R + ) 4 R by ϕ(x 1, x 2, x 3, x 4 ) = ax 1 +bx 2 +cx 3 +dx 4 wih a + b + c + d = 0,a + b > 0, a + c > 0 and a + d > 0. Clearly, ϕ is an implici relaion. In paricular, (i.) ϕ(x 1, x 2, x 3, x 4 ) = 6x 1 3x 2 2x 3 x 4, (ii.) ϕ(x 1, x 2, x 3, x 4 ) = 5x 1 3x 3 2x 4

4 92 I.H.NAGARAJA RAO, S. RAJESH, AND G.VENKATA RAO are implici relaions. Noaion: Le Φ be he class of all implici relaions. Lemma 2.1. ([9]) Le {x n }(n = 0, 1, 2,... ) be a sequence in a Menger space (X, F, ). If here is a k (0, 1) such ha F xn,x n+1 (k) F xn 1,x n () for all > 0 and n N, hen {x n } is a Cauchy sequence in X. Kohli e.al [2] proved he following: 3. Main heorem Theorem 3.1. ([2]) Le (X, F, T ) be a complee Menger space, where T denoes a coninuous -norm. Le f, g, h, k, p and q be self maps of X. Furher, le {p, hk} and {q, f g} be poinwise R-weakly commuing mappings, saisfying: (3.1.1) p(x) fg(x), q(x) hk(x); (3.1.2) ϕ(f px,qy (α), F hkx,fgy (), F px,hkx (), F qy,fgy (α)) 0, (3.1.3) ϕ(f px,qy (α), F hkx,fgy (), F px,hkx (α), F qy,fgy ()) 0, for all x, y X & > 0 and for some ϕ Φ & α (0, 1); (3.1.4) k commues wih p & h and g commues wih q & f; (3.1.5) one of he mappings in he compaible pair {p, hk} or {q, fg} is coninuous. Then f, g, h, k, p and q have a unique common fixed poin in X. The conceps of compaibiliy and he reciprocal coninuiy are used in obaining his resul. Sasry e.al [7] made he modificaion of replacing poinwise R-weakly commuing by weakly compaible and deduced he resul using he conceps compaibiliy and reciprocal coninuiy. Now, we modify and generalize heir resuls and esablish he following: Theorem 3.2. Le (X, F, T ) be a Menger space, where T denoes a coninuous -norm and f, g, h, k, p and q be self maps of X. Furher, le {p, hk} and {q, fg} be weakly compaible mappings, saisfying: (3.2.1) p(x) fg(x), q(x) hk(x); (3.2.2) ϕ(f px,qy (α), F hkx,fgy (), F px,hkx (), F qy,fgy (α)) 0, (3.2.3) ϕ(f px,qy (α), F hkx,fgy (), F px,hkx (α), F qy,fgy ()) 0, for all x, y X & > 0 and for some ϕ Φ & α (0, 1); (3.2.4) fg = gf and eiher qg = gq or qf = fq ; (3.2.5) hk = kh and eiher pk = kp or hp = ph ; (3.2.6) one of p(x), q(x), hk(x), fg(x) is a complee subspace of X. Then f, g, h, k, p and q have a unique common fixed poin in X say z. Also z is he unique common fixed poin h, k & p as well as f, g & q.

5 COMMON FIXED POINT THEOREMS 93 Proof: Le x 0 X. By (3.2.1) we consruc sequences {x n } and {y n } in X such ha px 2n = fgx 2n+1 = y 2n (say) and qx 2n+1 = hkx 2n+2 = y 2n+1 (say), for n = 0, 1, 2,... By puing x = x 2n (n 1) and y = x 2n+1 in (3.2.2), we ge ha ϕ(f px2n,qx 2n+1 (α), F hkx2n,fgx 2n+1 (), F px2n,hkx 2n (), F qx2n+1,fgx 2n+1 (α)) 0 i.e, ϕ(f y2n,y 2n+1 (α), F y2n 1,y 2n (), F y2n,y 2n 1 (), F y2n+1,y 2n (α)) 0. So, by he propery of ϕ, F y2n,y 2n+1 (α) F y2n 1,y 2n (). By puing x = x 2n+2 and y = x 2n+1 in (3.2.3), we ge ha ϕ(f px2n+2,qx 2n+1 (α), F hkx2n+2,fgx 2n+1 (), F px2n+2,hkx 2n+2 (α), F qx2n+1,fgx 2n+1 ()) 0 i.e, ϕ(f y2n+2,y 2n+1 (α), F y2n+1,y 2n (), F y2n+2,y 2n+1 (α), F y2n+1,y 2n ()) 0 Thus for all n N, F y2n+1,y 2n+2 (α) F y2n,y 2n+1 (). F yn,y n+1 (α) F yn 1,y n (). By Lemma(2.12), {y n } is a Cauchy sequence in X. {y 2n } and {y 2n±1 } are Cauchy sequences in X. Case I: Suppose p(x) or fg(x) is a complee subspace of X. Since {y n } p(x)( fg(x)), here is a z X such ha y 2n z as n. Since p(x) fg(x), by our supposiion, here is a v X such ha fgv = z. By puing x = x 2n (n 1) and y = v in (3.2.2), we ge ha ϕ(f px2n,qv(α), F hkx2n,fgv(), F px2n,hkx 2n (), F qv,fgv (α)) 0 i.e, ϕ(f y2n,qv(α), F y2n 1,z(), F y2n,y 2n 1 (), F qz,z (α)) 0. Since ϕ is coninuous, leing n, we ge ha ϕ(f z,qv (α), F z,z (), F z,z (), F qz,z (α)) 0. By he propery of ϕ, follows ha F z,qv (α) F z,z () = 1. z = qv. Thus fgv = qv = z. Since {q, fg} is weakly compaible, q(fg)v = fg(q)v. i.e, qz = fgz. By puing x = x 2n (n 1) and y = z in (3.2.2), we ge ha ϕ(f px2n,qz(α), F hkx2n,fgz(), F px2n,hkx 2n (), F qz,fgz (α)) 0 i.e, ϕ(f y2n,qz(α), F y2n 1,qz(), F y2n,y 2n 1 (), F qz,qz (α)) 0.

6 94 I.H.NAGARAJA RAO, S. RAJESH, AND G.VENKATA RAO Leing n, we ge ha ϕ(f z,qz (α), F z,qz (), F z,z (), F qz,qz (α)) 0. i.e, ϕ(f z,qz (α), F z,qz (), 1, 1) 0. Since ϕ is non-deceasing in he firs argumen, we ge ha qz = z. Thus z = qz = fgz = gfz (since fg = gf). Suppose qg = gq, hen qgz = gqz = gz. Since fg = gf, we have fg(gz) = gf(gz) = g(fgz) = gz. By puing x = x 2n (n 1) and y = gz in (3.2.2), we ge ha ϕ(f px2n,gv(α), F hkx2n,gz(), F px2n,hkx 2n (), F gz,gz (α)) 0 i.e, ϕ(f y2n,gz(α), F y2n 1,gz(), F y2n,y 2n 1 (), F gz,gz (α)) 0. Leing n, we ge ha ϕ(f z,gz (α), F z,gz (), F z,z (), F gz,gz (α)) 0. ϕ(f z,gz (), F z,gz (), 1, 1) 0 F z,gz () 1 gz = z. Since fgz = z, follows ha fz = z. Thus z = fz = gz = qz. Suppose qf = fq, so qfz = fqz = fz. Since fg = gf, we have fgfz = f(gfz) = fz. By puing x = x 2n (n 1) and y = fz in (3.2.2) as above we ge ha z = fz. Hence z = fz = gz = qz. Since q(x) hk(x), here is a w X such ha z = hkw. By puing x = w and y = x 2n+1 in (3.2.2), we ge ha pw = z. Thus pw = z = hkw. Since {p, hk} is weakly compaible, phkw = hkpw. i.e, pz = hkz. By puing x = z and y = x 2n+1 in (3.2.2), we ge ha pz = z. Thus z = pz = hkz = khz (since hk = kh). Suppose pk = kp, hen pkz = kpz = kz. Since hk = kh, we have hk(kz) = kh(kz) = k(hkz) = kz. By puing x = kz and y = x 2n+1 in (3.2.2), we ge ha kz = z. Since hkz = z, follows ha hz = z. Thus z = hz = kz = pz. Suppose ph = hp, so phz = hpz = hz. Since hk = kh, we have hk(kz) = h(khz) = hz. By puing x = hz and y = x 2n+1 in (3.2.2), we ge ha hz = z. Since hkz = z, follows ha kz = z. Thus z = hz = kz = pz. Hence z = fz = gz = hz = kz = pz = qz. Case II: Suppose q(x) or hk(x) is a complee subspace of X. On similar lines, firs we ge ha z = hz = kz = pz and hen z = fz = gz = qz.

7 COMMON FIXED POINT THEOREMS 95 Thus z = fz = gz = hz = kz = pz = qz. Hence z is a common fixed poin of f, g, h, k, p and q. Uniqueness: if z 1 is also a common fixed poin of f, g, h, k, p and q, hen z 1 = fz 1 = gz 1 = hz 1 = kz 1 = pz 1 = qz 1. By puing x = z and y = z 1 in (3.2.2), we ge ha z 1 = z. Hence z is he unique common fixed poin of f, g, h, k, p and q. We now prove ha z is he unique common fixed poin of h, k & p. Suppose z 1 is also a common fixed poin of h, k & p. By puing x = z 1 and y = z in (3.2.2), we ge ha z 1 = z. Hence z is he unique common fixed poin of h, k & p. So is he case wih f, g & q. This complees he proof of he heorem. Corollary 3.1. ([7] Theorem 3.2 ) Le (X, F, T ) be a complee Menger space, where T denoes a coninuous -norm. Le f, h, p and q be self maps of X. Furher, le {p, h} and {q, f} be weakly compaible mappings, saisfying: (3.3.1) p(x) f(x), q(x) h(x); (3.3.2) ϕ(f px,qy (α), F hx,fy (), F px,hx (), F qy,fy (α)) 0, (3.3.3) ϕ(f px,qy (α), F hx,fy (), F px,hx (α), F qy,fy ()) 0, for all x, y X & > 0 and for some ϕ Φ & α (0, 1); (3.3.4) Suppose {p, h} and {q, f} are compaible pairs; (3.3.5) one of he mappings in he compaible pairs {p, h} or {q, f} is coninuous. Then f, h, p and q have a common fixed poin in X. Proof: This can be deduced from our Theorem by aking f = r, h = s and g = k = I(he ideniy map). + x y Now we give he following example in suppor of our Theorem (3.2). Example 3.1. Le X = Q, a b = min{a, b} for all a, b [0, 1] and F x,y () = for all x, y X and for all > 0. Then (X, F, ) is a Menger space. Define self mappings f, g, h, k, p and q on X by px = qx = l(> 1), { 0 if x 1, h(x) = l if x > 1, { 2 x if x 1, f(x) = l if x > 1, kx = gx = x, for all x X. Define ϕ : (R + ) 4 R by ϕ(x 1, x 2, x 3, x 4 ) = 6x 1 3x 2 2x 3 x 4 hen ϕ is an implici relaion. For x, y 1, ϕ(f l,l (α), F 0,(2 y) (), F l,0 (), F l,(2 y) (α)) > = 0.

8 96 I.H.NAGARAJA RAO, S. RAJESH, AND G.VENKATA RAO For x, y > 1, For x 1 and y > 1, For x > 1 and y 1, ϕ(f l,l (α), F l,l (), F l,l (), F l,l (α)) = = 0. ϕ(f l,l (α), F 0,l (), F l,0 (), F l,l (α)) > = 0. ϕ(f l,l (α), F l,(2 y) (), F l,l (), F l,(2 y) (α)) = 0. The oher condiions of he Theorem are rivially saisfied. Clearly l is he unique common fixed poin of f, g, h, k, p and q in X as well as f, g & p and h, k & q. (Observe ha X is no complee.) References [1] Lj. B. Ciric and M. M. Milovanovic - Arandjelovic, Common fixed poin heorem for R-weakly commuing mappings in Menger spaces, J. Indian Acad. Mah. 22(2000), [2] Kohli. J. K, Sachin Vashisha and Durgesh Kumar, A Common fixed poin heorem for six mappings in Probabilisic meric spaces saisfying conracive ype implici relaions, In. J. Mah. Anal., Vol.4 (2010), no.2, [3] S. Kumar and B. D.Pan, Common fixed poin heorem in Probabilisic meric space using implici relaion, Filoma 22:2 (2008), [4] K. Menger, Saisical Merices, Proc. Na. Acad. Sci. U.S.A. 28(1942), [5] S. N. Mishra, Common fixed poin heorem of compaible mappings in probabilisic meric spaces, Mah. Japan, 36(1991), [6] B.D.Pan and Sunny Chauhan, Common fixed poin heorems for semi-compaible mappings using implici relasion, In. Journal of Mah. Analysis, Vol.3, (2009), no.28, [7] K.P.R.Sasry, Ch. Srinivasa Rao, K. Saya Mury and S.S.A.Sasri, Common fixed poin heorem for four mapping in Probabilisic meric spaces using implici relaions, In. J. Conemp. Mah. Sciences, Vol. 6 (2011), no.13, [8] B. Schweizer and A. Sklar, Probabilisic Meric Spaces, Norh Hooland Amsferdam, (1983). [9] S.L.Singh and B.D.Pan, Common fixed poin heorems in probabilisic meric spaces and exension o uniform spaces, Honam. Mah. J., (1984) [10] S.L.Singh, B.D.Pan and R.Talwar, Fixed poins of weakly commuing mappings on Menger spaces, Jnanabha, 23(1993), Received by ediors ; Available online Sr. Prof. & Direcor,, G.V.P. College for Degree and P.G. Courses,, Rushikonda, Visakhapanam-45, India. address: ihnrao@yahoo.com 2 Assisan Professor,, Deparmen of Mahemaics, G.V.P. College for Degree and P.G. Courses,, School of Engineering, Rushikonda,, Visakhapanam-45, India. address: srajeshmahs@yahoo.co.in 3 Associae Professor,, Deparmen of Mahemaics, Sri Prakash College of Engineering,, Tuni, India. address: venkaarao.gunur@yahoo.com