Common Fixed Point Theorems in Non-Archimedean. Menger PM-Spaces

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1 Iteratioal Mathematical Forum, Vol. 6, 0, o. 40, Commo Fixed Poit Theorems i No-Archimedea Meer PM-Spaces M. Alamir Kha Departmet of Mathematics, Eritrea Istitute of Techoloy Asmara, Eritrea (N. E. Africa alam3333@mail.com Abstract. The aim of this paper is to prove a related commo fixed poit theorem for four mappis i two complete o-archimedea Meer PM-spaces which exteds ad eeralizes the result of Fisher [, ], Jai et al. [4], Nesic [5] ad Popa [6]. Mathematics Subject Classificatio: 47H0, 54H5 Keywords: No-Archimedea Meer PM-space, two complete o-archimedea Meer PM-spaces ad fixed poits. Itroductio Fisher [,], Jai et al. [4] proved some related fixed poit theorems o two ad three complete metric spaces. Later o, Sedhi ad Shobe [7] itroduced this cocept i two M- fuzzy metric spaces ad proved a related fixed poit theorem i this space. Motivated by the work of above metioed authors we prove a related commo fixed poit theorem i two complete o-archimedea Meer PM-space.. Prelimiary defiitios, otatios ad results Defiitio.. Let X be ay o-empty set ad D be the set of all left cotiuous distributio fuctios. A ordered pair (X, F is said to be o-archimedea probabilistic metric space (briefly N. A. PM-space if F is a mappi from X X ito D satisfyi the followi coditios where the value of F at ( x,y X X is represeted by Fx,y or F(x,y for all x,y X such that i F(x, y ; t = for all t > 0 if ad oly if x= y ii F(x, y ; t = F( y, x ; t

2 994 M. Alamir Kha iii F( x, y ; 0 = 0 If F(x, y; t == F(y, z; t =, the F(x, z; max{t, t } = iv Defiitio.. A t-orm is a fuctio Δ :[0,] [0,] [0,] which is associative, commutative, o decreasi i each coordiate ad Δ (a, = a for all a [0,] Defiitio.3. A o-.archimedea Meer PM-space is a ordered triplet (X, F, Δ, where Δ is a t-orm ad (X, F, is a N.A. PM-space satisfyi the followi coditio; F(x,z; max{t,t } Δ(F(x, y;t,f(y,z;t for all x,y,z X,t,t 0. For details of topoloical prelimiaries o o-archimedea Meer PM-spaces, we refer to Cho, Ha ad S.S. Cha [8]. Defiitio.4. A N. A. Meer PM-space (X, F, Δ is said to be of type (C if there exists a Ω such that (F(x,z;t (F(x,y;t (F(y,z;t for all x, y,z X,t 0 where Ω= { / : [0,] [0, is cotiuous, strictly decreasi ( = 0 ad (0 < }. Defiitio.5. A N. A. Meer PM-space (X, F, Δ is said to be of type exists a Ω such that ( Δ(t,t (t (t t,t [0,] Remark. i If N. A. Meer PM-space is of type (D the (X, F, Δ is of type (C. (D if there ii If ( X, F, Δ is N. A. Meer PM-space is ad ( r, s max( r s, the (X, F, Δ is of type (D for Ω ad ( t = t. Throuh out this paper let ( X, F, Δ be a complete N.A. Meer PM-space with a cotiuous strictly icreasi t-orm Δ. Let :0, [ [ 0, ( φ Δ Δ =, φ be a fuctio satisfyi the coditio ( Φ ; Φ is semi upper cotiuous from riht ad φ ( t < t for t > 0. Defiitio.6. A sequece { x } i N. A. Meer PM-space ( X, F, Δ coveres to x if ad oly if for each ε > 0, λ > 0 there exists M ( ε, λ such that Fx ( (, x; ε < ( λ, > M. Defiitio.7. A sequece { x } i N. A. Meer PM-space is Cauchy sequece if ad oly if for each ε > 0, λ > 0 there exists a iteer M ( ε, λ such that Fx ( (, x p; ε < ( λ, M adp.

3 Commo fixed poit theorems 995 Example ([3]. Let X be ay set with at least two elemets. If we defie 0,t F( x, x; t = for all x X, t > 0 ad F( x, y; t = whe x, y X, x y, the,, t> (,, with Δ a,b = mi a,b or a.b X F Δ is N. A. Meer PM-space ( ( ( Lemma.. If a fuctio φ:[0, [0, satisfies the coditio ( Φ the we et i For all t 0, lim φ (t = 0, where φ (t is the th iteratio of φ(t. ii If {t } is a o decreasi sequece of real umbers ad t φ(t, =,, the lim t = 0. I particular, if t φ(t, for each t 0, the t = 0. Lemma.. Let { y } be a sequece i o-archimedea Meer PM-space ( X, F, Δ with the coditio lim t ( x, y ; t = for all x, y Є X. If there exists a umber ( 0, that ( F ( y, y ; qt ( F ( y, y ; t for all t > 0 ad = 0,,, the { } a Cauchy sequece i X. Proof. For t > 0 ad q ( 0,, we have ( F ( y, y 3; qt ( F ( y, y ; t F y 0, y; t or q ( F ( y, y 3; t F y 0, y; t q t By iductio, we ca have ( F ( y, y ; t F y, y ; q Thus for ay positive iteer p, q such y is ( (, ;, ; t p, ; t F y y t F y y F y y... F y p, y p ; t q q q ( (, ;, ; t t F y y t F y y... F y, y ; p pq pq. Therefore, we have, lim F y, y p ; t... = 0 ( ( ( ( ( (

4 996 M. Alamir Kha Thus { y } is a Cauchy sequece. Theorem. Let (,, X F Δ ad ( Y, F, Δ be two complete N. A. Meer PM-spaces. If A, B be two mappis of X to Y ad T, S be two mappis of Y to X satisfyi the followi coditios; (, ; ( (, ; F SAx TBx k t F x x t for all x, x i ( (, ; ( (, ; ii ( X ad some k > F ATy BSy k t F y y t for all y, y X ad some k > If at least A, B, T or S be cotiuous mappi, the there exists a uique poit z X ad w Y such that SAz = TBw ad ATw = BSw = w. Moreover, Sw = Tw = z, Az = Bz = w Proof. Let x 0 X be a arbitrary poit. We defie Ax = y, Sy = x, Bx = y, Ty = x. So by iductio for =,,3,..., 0 we have Ax3 = y3 = Sy3 = x3, Bx3 = y3, Ty3 = x3 Now, we prove that { x } ad { } Let r ( t = ( F( x, x ; t Now, for 3 we et, y are Cauchy sequeces i X ad Y respectively. ( = ( (, ; = ( F ( x 3, x 3 ; t = r3 ( t r k t F x x k t Ad for 3, we et r k t = F x, x ; k t = F Sy, Ty ; k t ( ( ( ( 3, 3 ; ( ( 3, 3 ; 3 ( ( ( ( = F SAx TBx k t F x x t = r t Also for 3, we et r k t = F x, x ; k t = F Sy, Ty ; k t ( ( ( ( 3, 3 ; ( ( 3, 3 ; 3 ( ( ( ( = F SAx TBx k t F x x t = r t

5 Commo fixed poit theorems 997 Hece for every N, we have r ( k t = r ( t i.e., ( F ( x, x ; t ( F ( x, x ; t Thus by lemma (., { x } is a Cauchy sequece ad by the completeess of X, { x } coveres to z i X. i.e., ( x lim = z. ( Now, let s ( k t = F ( y, y ; k t = F ( Ax, Bx ; k t For 3 we et, 3 3 ( 3 3 ( 3 3 ( ( = F ( ATy, BSy ; k t (, ; ( ( = (, ; = F ( Ax, Bx ; k t s k t F y y k t ( 3 3 F y y t = s t ( 3 3 ( ( = F ( ATy, BSy ; k t (, ; Ad for 3 we et, ( F y y t = s t ( = (, ; = F ( Ax, Bx ; k t s k t F y y k t ( 3 3 ( 3 3 ( ( = F ( ATy, BSy ; k t (, ; Hece for every, we have s ( t i.e., ( F y y t = s t. ( (, ; (, ; F y y k t F y y t. Aai by lemma (., { y } is Cauchy sequece i Y ad by the completeess of Y, { y } coveres to w i Y, i.e., lim ( y = w Let A be cotiuous, hece lim y = lim Ax = A lim x = Az = w. Now, we prove that SAz = z. By (i, we have ( F ( SAz, TBx 3 ; kt ( F ( z, x 3 ; t O taki, we et F ( SAz z k t ( ( F ( z z t, ;, ; = 0, which implies that

6 998 M. Alamir Kha Sw = SAz = z. Now, we prove that Bz = w. ( 3, ; ( ( 3, ; Usi (ii we have ( i.e., ( F ATy BSw k t F y w t ( ( ( F w, BSw; k t F w, w; kt = 0 ( ( ( F ATw, BSw; k t F w, w; t = 0. Therefore, BSw = Bz = w. Aai by (ii ( Therefore, ATw = BSw = w. ( ( F ( z z t Now, we prove that TBz = z. From (i F ( SAw TBw k t, ;, ; = 0, i.e., TBz = Tw = z. Hece SAz = TBz = z. Now, we have Sw = SAz = z ad Tw = TBz = z. Therefore, Az = Bz = w ad Sw = Tw = z. For uiqueess, let be aother commo fixed poit of mappis A ad B, (, ; ( ( ( (, ;, ; F z z k t F SAz TSz k t F z z t, a cotradictio. the, ( Therefore, z z = is the uique commo fixed poit of self maps A ad B. Let w be aother commo fixed poit of S ad T, (, ; ( ( ( (, ;, ; F zw kt F ATw BSw kt F ww t, a cotradictio. The, ( Therefore, w w = is uique commo fixed poit of self maps S ad T. Example. Let X = [ 0, ], Y = [,]. If, :[,] [ 0,] T ( y if y is ratioal = 0 if y is irratioal ad S ( y Moreover, if A, B :[ 0,] [,] defied ( S T defied if y is ratioal = if y is irratioal 3 if x is ratioal B x = if x is irratioal A x = ad ( The is N. A. Meer PM-space. Also it is easy to see

7 Commo fixed poit theorems 999 that A= B= ad T = S=. Hece SA= TB= ad AT = BS = Moreover, the coditio (i ad (ii of our theorem are also satisfied as (i If x is a ratioal umber the, ( ( ( ( 0= F,, k t F,, t = 0 If x is irratioal umber the, ( F ( kt ( F ( t Similarly, coditio (ii is also satisfied. Corollary. Let (,, X F Δ ad ( Y, F, 0=,,,, = 0. Δ be two complete N. A. Meer PM-spaces respectively. If A be mappi of X to Y ad B mappi of Y to X satisfyi the followi coditios; (, ; ( (, ; (i ( F BAx BAx k t F x x t for all (, ; ( (, ; (ii ( x, x F ABy ABY k t F y y t for all X ad some k y, y X ad some k If at least A or B be cotiuous mappi, the there exist a uique poit Ad such that BAz = z ad ABw = w. Moreover, Bw = z, Az = w. Ackowledmet. This paper is dedicated to my dearest fried Dr. Sumitra who helped me a lot at every stae of my life. Refereces [] B. Fisher, Fixed poits o two metric spaces, Glasik Mat. 6(36(98, [] B. Fisher, Fixed poits o two metric spaces, Math. Sem. Ntes, Kobe Uiv.,0(98, 7-6. [3] M. Alamir Kha ad Sumitra, A commo fixed poit theorem i o-archimedea Meer PM-Spaces, Novi Sad Joural of Mathematics, 39( (009, [4] R. K. Jai, H.K. Sahu, Brai Fisher, Related fixed poit theorems for three metric spaces, Novi Sad J. Math., 6((996, -7.

8 000 M. Alamir Kha [5] S. C. Nesic, A fixed poit theorem i two metric spaces, Bull. Math. Soc. Sci. Math. Roumaie (N.S 44 (9(00, [6] V. Popa, Fixed poits o two complete metric spaces, Zb. Rad. Prirod, Mat. Fak, (N. S Ser. Mat. ( (99, [7] Shaba Sedhi ad Nabi Shobe, A commo fixed poit theorem i two M-fuzzy metric spaces, Commu. Korea Math. Soc. (4(007, [8] Y. J. Cho ad K. S. Ha ad S. S. Cha, Commo fixed poit theorems for compatible mappis of type (A i o-archimedea Meer PM-space, Math Japoica (46(( , CMP zbl Received: December, 00