IJRRAS 19 (3) Jue 2014 www.arpapress.com/volumes/vol19issue3/ijrras_19_3_05.pdf COMMON FIXED POINT THEOREM FOR FINITE NUMBER OF WEAKLY COMPATIBLE MAPPINGS IN QUASI-GAUGE SPACE Arihat Jai 1, V. K. Gupta 2 & Dhasigh Bamiya 3 1 Departmet of Applied Mathematics, Shri Guru Sadipai Istitute of Techology ad Sciece, Ujjai (M.P.) 456 550 Idia 2 Departmet of Mathematics, Govt. Madhav Sciece College, Ujjai (M.P.) 4560Idia 3 Departmet of Mathematics, Govt. P. G. College, Khargoe E-mail: arihat2412@gmail.com, dr_vkg61@yahoo.com, dsbamiya@gmail.com ABSTRACT The preset paper deals with the commo fixed poit theorem for fiite umber of weak compatible mappigs i Quasi- gauge space. By poitig out the fact that the cotiuity of ay mappig for the existece of the fixed poit is ot ecessary, we improve the result of Rao ad Murthy [6]. Keywords: Commo fixed poit, weakly compatible mappigs, Quasi-gauge space. 2010 AMS Subject Classificatio : 54H25, 47H10. 1. INTRODUCTION The cocept of quasi-gauge space is due to Reilly [7] i the year 1973. Afterwards, Atoy et. al. [2] gave a geeralizatio of a commo fixed poit theorem of Fisher [3] for quasi-gauge spaces. Pathak et. al. [5] proved fixed poit theorems for compatible mappigs of type (P). Rao ad Murthy [6] exteded results o commo fixed poits of self maps by replacig the domai complete metric space with Quasi-gauge space. But i both theorems cotiuity of ay mappig was the ecessary coditio for the existece of the fixed poit. We improve results of Rao ad Murthy [6] ad show that the cotiuity of ay mappig for the existece of the fixed poit is ot required. Defiitio 1.1. A Quasi-pseudo-metric o a set X is a o-egative real valued fuctio p o X X such that (i) p(x,x) = 0 for all x X. (ii) p(x,z) p(x,y) + p(y,z) for all x,y,z X. Defiitio 1.2. A Quasi-gauge structure for a topological space (X,T) is a family P of quasi-pseudo-metrics o X such that T has as a subbase the family {B(x,p, ) : x X, p P, > 0} where B(x,p, ) is the set {y X: p(x,y) < }. If a topological space has a Quasi-gauge structure, it is called a quasigauge space. Defiitio 1.3. A sequece {x } i a Quasi-gauge space (X,P) is said to be P-Cauchy, if for each >0 ad p P there is a iteger k such that p(x m, x ) < for all m, k. Defiitio 1.4. A Quasi-gauge space (X, P) is sequetially complete if ad oly if every P-Cauchy sequece i X is coverget i X. We ow propose the followig characterizatio. Let (X,P) be a Quasi-gauge space. X is a T 0 space iff p(x,y) = p(x,y) = 0 for all p i P implies x = y. Atoy [1] itroduced the cocept of weak compatibility for a pair of mappigs o Quasi-gauge Space. Defiitio 1.5. Let (X, P) be a Quasi-gauge space. The self maps f ad g are said to be (f,g) weak compatible if lim gfx = fz for some z X wheever x is sequece i X such that lim fx = lim gx = z ad lim fgx = lim ffx = fz. 212
IJRRAS 19 (3) Jue 2014 Jai et al. Commo Fixed Poit Theorem f ad g are said to be weak compatible to each other if (f,g) ad (g,f) are weak compatible. Lemma 1.1. [4] Suppose that : [0, ) [0, ) is o-decreasig ad upper semi-cotiuous from the right. If (t) < t for every t > 0, the lim (t) = 0. Rao ad Murty [6] proved the followig. Theorem 2.1 Let A,B,S ad T be self maps o a left (right) sequetially complete Quasi-gauge T 0 space (X, P) such that (i) (A,S), (B,T) are weakly compatible pairs of maps with T(X) A(X); S(X) B(X); (ii) A ad B are cotiuous; (iii) max{p²(sx,ty),p²(ty,sx)} {p(ax,sx)p(by,ty), p(ax,ty) p(by,sx), p(ax,sx)p(ax,ty), p(by,sx)p(by,ty), p(by,sx)p(ax,sx), p(by,ty) p(ax,ty)}; for all x,y X ad for all p i P, where : [0, )⁶ (0, + ) satisfies the followig : (iv) is o-decreasig ad upper semi-cotiuous i each coordiate variable ad for each t > 0 : (t)=max{ (t,0,2t,0,0,2t), (t,0,0,2t,2t,0), (0,t,0,0,0,0)} < t; the A, B, S ad T have a uique commo fixed poit. Theorem 2.2 Let A, B, S ad T be self maps o a left (right) sequetially complete Quasi-gauge T 0 space (X,P) with coditio (iii) ad (iv) of Theorem 2.1 such that (i) (S, A), (A, S), (B, T) ad (T, B) are weakly compatible pairs of maps with T(X) A(X); S(X) B(X); (ii) Oe of A, B, S ad T is cotiuous; the the same coclusio of Theorem 2.1 holds. We prove Theorem 2.1 ad Theorem 2.2 without assumig that ay fuctio is cotiuous for fiite umber of mappigs. 3. MAIN RESULTS Theorem 3.1. Let A, B, S, T, I, J, L, U, P ad Q be mappigs o left sequetially complete Quasi-gauge T₀ space (X, P) such that (3.1) (P,STJU) ad (Q,ABIL) are weakly compatible pairs of mappigs with ABIL(X) P(X); STJU(X) Q(X); (3.2) max{p²(stjux,abily),p²(abily,stjux)} {p(px,stjux)p(qy,abily), p(px,abily) p(qy,stjux), p(px,stjux)p(px,abily), 213
IJRRAS 19 (3) Jue 2014 Jai et al. Commo Fixed Poit Theorem p(qy,stjux)p(qy,abily), p(qy,stjux)p(px,stjux), p(qy,abily) p(px,abily)}; for all x,y X ad for all p i P,where : [0, )⁶ (0, + ) satisfies the followig: (3.3) is o-decreasig ad upper semi-cotiuous i each coordiate variable ad for each t >0: (t) = max{ø(t,0,2t,0,0,2t), ø(t,0,0,2t,2t,0), ø(0,t,0,0,0,0), ø(0,0,0,0,0,t) ø(0,0,0,0,t,0)} < t. The A, B, S, T, I, J, L, U, P ad Q have a uique commo fixed poit. Proof. Let x 0 be a arbitrary poit i X. Sice (3.1) holds we ca choose x 1, x 2 i X such that Qx 1 = STJUx 0 ad Px 2 = ABILx 1. I geeral we ca choose x 2+1 ad x 2+2 i X such that (3.4) y 2 = Qx 2+1 = STJUx 2 ad y 2+1 = Px 2+2 = ABILx 2+1 ; = 0,1,2... We deote d = p(y,y +1 ) ad e =p(y +1,y ); ow applyig (3.2) we get max{d 2 2+2, e 2 2+2} = max{p²(stjux 2+2, ABILx 2+3 ), p²(abilx 2+3, STJUx 2+2 )} ø{p(px 2+2, STJUx 2+2 )p(qx 2+3, ABILx 2+3 ), p(px 2+2,ABILx 2+3 )p(qx 2+3,STJUx 2+2 ), p(px 2+1,STJUx 2+2 )p(px 2+2,ABILx 2+1 ), p(qx 2+1,STJUx 2+2 )p(qx 2+2,ABILx 2+3,), p(qx 2+3,STJUx 2+2,) p(px 2+2,STJUx 2+1 ) p(qx 2+3, ABILx 2+3 ) p(px 2+2, ABILx 2+3 )}; = {p(y 2+1,y 2+2 ) p(y 2+2,y 2+3 ), p(y 2+1,y 2+3 )p(y 2+2,y 2+2 ), p(y 2+1,y 2+2,) p(y 2+1,y 2+3 ), p(y 2+2,y 2+2 ) p(y 2+1,y 2+3 ), p(y 2+2,y 2+2 ) p(y 2+2,y 2+3 ), p(y 2+2,y 2+2 )p(y 2+1,y 2+2 ), p(y 2+2,y 2+3 )p(y 2+1,y 2+3 )} (3.5) {d 2+1 d 2+2,0, d 2+1 (d 2+1 + d 2+2 ), 0, 0, d 2+2 ( d 2+1 + d 2+2 )}. If d 2+2 > d 2+1 the (3.6) max{d² 2+1, e² 2+2 } { d² 2+2,0, 2d² 2+2,0,0,2d² 2+2 } < d² 2+2, 214
IJRRAS 19 (3) Jue 2014 Jai et al. Commo Fixed Poit Theorem by (3.3) a cotradictio; hece d 2+2 d 2+1. Similarly, we get (3.7) d 2+1 d 2. By (3.5) ad (3.6) max{d² 2+2, e² 2+2,} { d² 2+1,0, 2d² 2+1,0,0, 2d² 2+1 }. (3.8) (d² 2+1 ) = {p²(y 2+1,y 2+2 )} Similarly, we have max{d² 2+1, e² 2+1 } {d² 2, 0, 0, 2d² 2, 2d² 2, 0}. {p²(y 2,y 2+1 )}. So (3.10) d² = p²(y,y +1 ) {p²(y +1,y )}... ⁿ ¹{p²(y₁,y₂)} ad (3.11) e² = p²(y +1,y ) {p²(y -1,y )}... ⁿ ¹{p²(y 1,y₂)}. Hece by Lemma 1.1 ad from (3.10) ad (3.11), we obtai (3.12) lim d = e =0. Now we prove {y } is a P-Cauchy sequece. To show {y } is P-Cauchy it is eough if we show {y 2 } is P-Cauchy. Suppose {y 2 } is ot a P-Cauchy sequece the there is a ε > 0 such that for each positive iteger 2k there exist positive itegers 2m(k) ad 2(k) such that for some p i P, (3.13) p(y 2(k),y 2m(k) ) > ε for 2m(k) > 2(k) > 2k ad (3.14) p(y 2m(k),y 2(k) ) > ε for 2m(k) > 2(k) > 2k for each positive eve iteger 2k, let 2m(k) be the least positive eve iteger exceedig 2(k) ad satisfyig (3.13); hece p(y 2(k),y 2m(k)-2 ) ε the for each eve iteger 2k, ε < p(y 2 (k),y 2m (k) (3.15) p(y 2(k),y 2m(k)-2 )+ (d 2m(k)-2 + d 2m(k)-1 ) From (3.12) ad (3.15), we obtai lim p(y 2(k),y 2m(k) ) = ε. By the triagle iequality p(y 2(k),y 2m(k) ) p(y 2(K), y 2m(k)-1 )+ d 2m(k)-1 p(y 2(K), y 2m(k)-1 ) p(y 2(k),y 2m(k) )+ e 2m(k)-1. So (3.16) p(y 2(k),y 2m(k) ) - p(y 2(k),y 2m(k)-1 ) max d 2m(k)-1, e 2m(k)-1 }. 215
IJRRAS 19 (3) Jue 2014 Jai et al. Commo Fixed Poit Theorem Similarly By triagle iequality (3.17) p(y 2(K)+1,y 2m(k)-1 ) - p(y 2(K), y 2m(k) ) max{e 2(k) + e 2m(k)-1, d 2(k) + d 2m(k)-1 }. From (3.16) ad (3.17) as k, {p(y 2(K),y 2m(k)-1) } ad p(y 2(k)+1,y 2m(k)-1) } coverge to ε. Similarly if p(y 2m(k), y 2(k) ) > ε, By (3.2), lim p(y 2m(K),y 2(k) ) = lim p(y 2m(k)-1, y 2(k)+1 ) ε < p(y 2(k), y 2m(k) ) p(y 2(k), y 2(k)+1 ) + p(y 2(k)+1, y 2m(k) ) = lim p(y 2m(k)-1,y 2(k) ) = ε as k. d 2(k) + max{p(y 2(k)+1, y 2m(k) ), p(y 2(k), y 2(k)+1 )} = d 2(k) + max{p(abilx 2(k)+1, STJUx 2m(k) ), p(stjux 2m(k), ABIL x 2(k)+1 )} d 2(k) + [ {p(y 2m(k)-1, y 2m(k) )p(y 2(k),y 2(k)+1 ), p(y 2m(k)-1,y 2(k)+1 ) p(y 2(k),y 2m(k) ), p(y 2m(k)-1,y 2m(k) ) p(y 2m(k)-1,y 2(k)+1 ), p(y 2(k),y 2m(k) ) p(y 2(k),y 2(k)+1 ), p(y 2(k),y 2m(k) ) p(y 2m(k)-1,y 2m(k) ), p(y 2(k),y 2(k)+1 ) p(y 2m(k)-1,y 2(k)+1) }] 1/2. Sice is upper semi-cotiuous, as k we get that ε { (0, ε², 0, 0, 0, 0)} 1/2 < ε, which is a cotradictio. Therefore {y } is P-Cauchy sequece i X. Sice X is complete there exists a poit z i X such that lim Px 2 = lim ABILx 2-1 = z lim y = z. ad lim Qx 2+1 = lim STJUx 2-2 = z. Sice STJU(X) Q(X), there exist a poit u X such that z = Qu. The usig (3.2), max{p²(stjux 2, ABILu), p²(abilu,stjux 2 )} {p(px 2,STJUx 2 )p(qu,abilu), p(px 2,ABILu)p(Qu,STJUx 2 ), p(px 2,STJUx 2 ) p(px 2,ABILu), p(qu,stjux 2 ) p(qu,abilu), p(qu,stjux 2 ) p(px 2,STJUx 2 ), p(qu,abilu) p(px 2,ABILu)}. Takig limit as, 216
IJRRAS 19 (3) Jue 2014 Jai et al. Commo Fixed Poit Theorem max{p²(z, ABILu), p²(abilu,z)} {p(z,z)p(z,abilu), p(z,abilu)p(z,z), p(z,z)p(z,abilu), p(z,z) p(z,abilu), p(z,z) p(z,z), p(z,abilu) p(z,abilu)} {0, 0, 0, 0, 0, p(z, ABILu) p(z,abilu)}, < p(z, ABILu) p(z, ABILu) a cotradictio. Thus ABILu = z. Therefore ABILu = z = Qu. Similarly, sice ABIL(X) P(X), there exist a poit v X, such that z = Pv. The usig (3.2), max{p²(stjuv, ABILx 2+1 ), p²(abilx 2+1,STJUv)} {p(pv,stjuv)p(qx 2+1,ABILx 2+1 ), p(pv,abilx 2+1 )p(qx 2+1,STJUv), p(pv,stjuv) p(pv,abilx 2+1 ), p(qx 2+1,STJUv) p(qx 2+1,ABILx 2+1 ), p(qx 2+1,STJUv) p(pv,stjuv), p(qx 2+1,ABILx 2+1 ) p(pv,abilx 2+1 )}. Takig limit as, max{p²(stjuv, z), p²(z,stjuv)} {p(z, STJUv) p(z, z), p(z, z) p(z, STJUv), p(z, STJUv) p(z, z), p(z, STJUv) p(z,z), p(z, STJUv) p(z, STJUv), p(z, z) p(z, z)} {0, 0, 0, 0, p(z, STJUv) p(z,stjuv), 0} < p(z, STJUv, p(z, STJUv) a cotradictio. Thus z = STJUv. Therefore z = STJUv = Pv. Hece, z = Qu = ABILu = Pv = STJUv. Sice the pair of mappigs Q ad ABIL are Weakly Compatible, the QABILu = ABILQu. i.e. Qz = ABILz. Now we show that z is a fixed poit of ABIL. If ABILz z, the by (3.2) max{p²(stjux 2, ABILz), p²(abilz, STJUx 2 )} {p(px 2,STJUx 2 )p(qz,abilz), p(px 2,ABILz)p(Qz,STJUx 2 ), p(px 2,STJUx 2 )p(px 2,ABILz), p(qz, STJUx 2 )p(qz, ABILz), p(qz, STJUx 2 ) p(px 2, STJUx 2 ), p(qz, ABILz) p(px 2, ABILz)}. 217
IJRRAS 19 (3) Jue 2014 Jai et al. Commo Fixed Poit Theorem Takig limit as, max{p²(z, ABILz), p²(abilz,z)} {p(z, z) p(qz, ABILz), p(z, ABILz) p(qz, z), p(z, z) p(z, ABILz), p(qz, z) p(qz, ABILz), p(qz, z) p(z,z), p(qz, ABILz) p(z, ABILz)} {0, 0, 0, 0, 0, p(z, ABILz) p(z, ABILz)}, < p(z, ABILz) p(z, ABILz) a cotradictio. Thus ABILz = z. Therefore ABILz = z = Qz. Similarly we prove that STJUz = z = Pz. Hece Pz = Qz = STJUz = ABILz = z; thus z is a commo fixed poit of A, B, S, T, I, J, L, U, P ad Q. Uiqueess follows trivially. Therefore z is a uique commo fixed poit of A, B, S, T, I, J, L, U, P ad Q. REFERENCES [1] Atoy, J., Studies i fixed poits ad Quasi-Gauges, Ph.D. Thesis, I.I.T., Madras(1991). [2] Atoy, Jessy ad Subrahmayam, P.V., Quasi-Gauges ad Fixed Poits, Uiv. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 24 (1), (1994), 31-42. [3] Fisher, B., Theorems o commo fixed poits, Fud. Math. 113 (1981), 37-43. [4] Matkowski, J.,Fixed poit theorems for mappigs with a cotractive iterate at a poit, Proc. Amer. Math. Soc. 62 (1977), No.2, 344-348. [5] Pathak, H. K., Chag, S. S. ad Cho, Y. J., Fixed poit theorems for compatible mappigs of type (P), Idia J. Math. 36 (1994), No. 2, 151-166. [6] Rao, I. H. N. ad Murty, A. S. R., Commo fixed poit of weakly compatible mappigs i Quasi-gauge spaces, J. Idia Acad. Math. 21(1999), No. 1, 73-87. [7] Reilly, I.L.., Quasi-gauge spaces. J. Lod. Math. Soc. 6 (1973), 481-487. 218