Some Fixed Point Results in Dislocated Probability Menger Metric Spaces

Transcription

1 Bol. Soc. Paran. Ma. 3s. v : c SPM ISSN on line ISSN in press SPM: doi:10.569/bspm.v35i.540 Some Fixed Poin Resuls in Dislocaed Probabiliy Menger Meric Spaces M. Shams, H. Shayanpour and F. Ehsanzadeh absrac: In his work, we shall give some new resuls abou generalized common fixed poin heorems for wo mappings f : X X and T : X k X, where X is dislocaed probabiliy quasi Menger meric space briefly, DP qm-space or dislocaed probabiliy Menger meric space briefly, DP M-Space. Our resul exends and generalizes many well known resuls. Key Words: Dislocaed probabilisic Menger meric space, Dislocaed probabilisic quasi Menger meric space, Coincidence and common fixed poins, Weakly compaible mappings, Occasionally weakly compaible mappings. Conens 1 Inroducion 69 Coincidence and common fixed poins in DP q M-space and DPM-space Inroducion An ineresing and imporan generalizaion of he noion of meric space was inroduced by, Karl Menger [13] in 194 under he name of saisical meric space, which is now called probabilisic meric space PM-space. The idea of K. Menger was o use disribuion funcions insead of non-negaive real numbers as values of he meric. The noion of PM-space corresponds o siuaions when we do no know exacly he disance beween wo poins, bu we know probabiliies of possible values of his disance. In fac he sudy of such spaces received an impeus wih he pioneering works of Schweizer and Sklar [17,18]. In an ineresing paper [8], Hicks observed ha fixed poin heorems for cerain conracion mappings on a Menger space endowed wih a riangular -norm may be obained from corresponding resuls in meric spaces. In 1989, Ken and Richardson [11] inroduced he class of probabilisic quasi-meric spaces briefly, PQM-spaces and proved common fixed poin heorems. The sudy of fixed poins of mappings in probabilisic quasi meric spaces is in nascen sage. The projec is parially suppored by he Cener of Excellence for Mahemaics, Universiy of Shahrekord. 000 Mahemaics Subjec Classificaion: Primary 47H10, 54H5; Secondary 54E70, 47H09, 55M0. 69 Typese by B S P M syle. c Soc. Paran. de Ma.

2 70 M. Shams, H. Shayanpour and F. Ehsanzadeh Before we proceed we mus sae some definiions, known facs, and, echnical resuls o be used in he sequel. The conceps used are ha of [6,17]. Definiion 1.1. A disribuion funcion is a funcion F : [, ] [0,1], ha is non-decreasing and lef coninuous on R, moreover, F = 0 and F = 1. The se of all he disribuion funcions d.f. is denoed by, and he se of hose disribuion funcions such ha F0 = 0 is denoed by +. In paricular for every x 0 0, ε x0 is he d.f. defined by { 1 if x > x 0, ε x0 = 0 if x x 0. The space + is parially ordered by he usual poinwise ordering of funcions, he maximal elemen for + in his order is ε 0. Definiion 1.. A probabilisic meric space abbreviaed, P M-space is an ordered pair X,F, where X is a non-empy se and F : X X + Fp,q is denoed by F p,q saisfies he following condiions: PM1 F p,q = 1 for all > 0, iff p = q, PM F p,q = F q,p, PM3 If F p,q = 1 and F q,r s = 1, hen F p,r +s = 1, for every p,q,r X and,s 0. Definiion 1.3. A mapping τ : [0,1] [0,1] [0,1] is called a riangular norm abbreviaed, -norm if he following condiions are saisfied: i τa,b = τb,a, ii τa,τb,c = ττa,b,c, iii τa,b τc,d whenevera candb d, iv τa,1 = a, for every a,b,c,d [0,1]. Definiion 1.4. A Menger space is a riple X,F,τ, where X,F is PM-space and τ is a -norm such ha for all p,q,r X and for all,s 0, F p,r +s τf p,q,f q,r s. Definiion 1.5. A dislocaed probabilisic quasi Menger space abbreviaed, DP q M- space is a riple X,F,τ, where X is a non empy se, τ is a -norm and F : X X + Fp,q is denoed by F p,q saisfies he following condiions: i F p,q = 1andF q,p = 1 p = q,

3 Fixed poin resuls in dislocaed probabiliy Menger meric spaces 71 ii F p,r +s τf p,q,f q,r s, for every p,q,r X and,s 0. Example 1.6. Le X = R, define τa,b = a b and 1 F x,y =, x y + x + y e for all x,y X X, 0,. Then X,F,τ is a DP q M-space. Definiion 1.7. A dislocaed probabilisic Menger space abbreviaed, DP M-space is a DP q M-space such ha for all p,q X, F p,q = F q,p. Le X,F,τ be a DPqM-space and F p,q = min{f p,q,f q,p } p,q X and [0, ], hen i is easy o see ha, X,F,τ is a DPM-space. In addiion, if X,F,τ is a DPM-space, hen F = F. Definiion 1.8. Le X,F,τ be a DP q M-space or DPM-space, T : X k X and f : X X be mappings. A poin z X is said o be a coincidence poin of f and T if Tz,z,...,z = fz. The se of all coincidence poins of he mappings f and T denoed by CT,f. A poin z X is said o be a common fixed poin of f and T if Tz,z,...,z = fz = z. Definiion 1.9. The mappings T : X k X and f : X X, where X,F,τ is a DP q M-space, are said o be weakly compaible wc if he mappings commuing a heir coincidence poins, i.e. Tfz,fz,...,fz = ftz,z,...,z, for all z CT,f. Very recenly many auhors proved some new fixed poin heorems in dislocaed quasi meric spaces, see [7,1,16]. In 1996, Jungck [9] inroduced he noion of weakly compaible mappings which is more general han compaibiliy and proved fixed poin heorems in absence of coninuiy of he involved mappings. In recen years, many mahemaicians esablished a number of common fixed poin heorems saisfying conracive ype condiions and involving condiions on commuaiviy, compleeness and suiable conainmen of ranges of he mappings. Al-Thagafi and Shahzad [] inroduced he noion of occasionally weakly compaible mappings in meric space, which is more general han weakly compaible mappings. Recenly, Jungck and Rhoades [10] exensively sudied he noion of occasionally weakly compaible mappings in semi-meric spaces. The noions of improving commuaiviy of self mappings have been exended o PM-spaces by many auhors. For example, Singh and Jain [19] exended he noion of weak compaibiliy and Chauhan e al. [4] exended he noion of occasionally weak compaibiliy o P M-spaces. The fixed poin heorems for occasionally weakly compaible mappings in differen seings invesigaed by many researchers e.g. [1,3,15,14]. Definiion The mappings T : X k X and f : X X, where X,F,τ is a DP q M-space, are said o be occasionally weakly compaible owc if he mappings commuing a leas one coincidence poin, whenever CT,f φ.

4 7 M. Shams, H. Shayanpour and F. Ehsanzadeh Definiion Le X, F, τ be a DP qm-space. A lef righ open ball abbreviaed, L-open R-open ball wih cener x and radius r 0 < r < 1 in X is he se B L x,r, = {y X : F x,y > 1 r} B R x,r, = {y X : F y,x > 1 r}, for all 0,1. Moreover, a open ball wih cener x and radius r 0 < r < 1 in X is he se Bx,r, = {y X : F x,y > 1 r}, for all 0,1. Definiion 1.1. A sequence x n in a DP q M-space X,F,τ is said o be biconvergen o a poin x X if and only if lim n F x n,x = 1 for all > 0, in his case we say ha limi of he sequence x n is x. A sequence x n is said o be lef righ Cauchy sequence if and only if lim F x n,x n+p = 1 n lim n F x n+p,x n = 1 for all > 0, p N. Also, a sequence x n is said o be bi-cauchy if and only if lim n F x n,x n+p = 1 for all > 0, p N. The concep of lef righ Cauchy sequence is inspired from ha of G-Cauchy sequence i belongs o Grabiec [5]. Definiion A DP q M-space X,F,τ is said o be lef or righ complee if and only if every lef or righ Cauchy sequence in X, is bi-convergen. Also, a DP q M-space is said o be bi-complee if and only if every bi-cauchy sequence in X, is bi-convergen. Clearly a sequence x n in a DP q M-space X,F,τ is bi-cauchy sequence if and only if sequence x n is a Cauchy sequence in he DPM-space X,F,τ. Also, a DP q M-space X,F,τ is bi-complee if and only if he DPM-space X,F,τ is complee. Proposiion 1.1. The limi of a bi-convergen sequence in a DP q M-space X,F,τ is unique. Proof: Le x n be a sequence in X and suppose ha u and v are wo limis of x n. By he hypohesis we have F u,v τf u,x n /,F x n,v / for all n. Now aking he limi as n, so we have F u,v τ1,1 = 1. Hence u = v, he resul follows. Proposiion 1.. Le X,F,τ be a DP q M-space or DPM-space and x n be a sequence in X. If sequence x n bi-converges or converges o x X, hen F x,x = 1 for all > 0. Proof: By using a similar argumen as in he proof of he above proposiion, he resul follows. Proposiion 1.3. Le X,F,τ be a DP q M-space or DPM-space. If f,g : X X is wo mappings such ha fz = gz and F fgz,gfz = 1 or F fgz,gfz = 1 for some z X and [0,, hen F ffz,ffz = 1.

5 Fixed poin resuls in dislocaed probabiliy Menger meric spaces 73 Proof: Since F fgz,gfz = 1, so by he hypohesis we have fgz = gfz. Therefore F ffz,ffz = F fgz,fgz = F fgz,gfz = 1. The following resuls are immediae. Lemma Le X,F,τ be a DP q M-space or DPM-space, T : X k X and f : X X be occasionally weakly compaible mappings. If f and T have a unique poin of coincidence hen f and T are weakly compaible. Thus, if mappings f and T have a unique poin of coincidence, hen he pair f, T are weakly compaible if and only if hey are occasionally weakly compaible. Lemma Le X,F,τ be a DP q M-space or DPM-space, T : X k X and f : X X be occasionally weakly compaible mappings. If pair f,t has a unique poin of coincidence, hen i has a unique common fixed poin. The above lemma is also valid for weakly compaible mappings. The nex example shows ha if he poin of coincidence is no unique, hen occasionally weakly compaible mappings are more general han weakly compaible mappings. Example Take X = [0,1], fx = x, Tx 1,,x k = k x 1x x k. I is obvious ha {0, 1 } Cf,T, ft0 = Tf0 bu ft 1 Tf 1 and so f and T are occasionally weakly compaible bu no weakly compaible. Noe ha 0 and 1 4 are wo poin of coincidence and 0 is he unique common fixed poin. Definiion The mapping f is said o be coincidenally idempoen ci wih respec o T, if and only if f is idempoen a he coincidence poins of f and T, i.e. ffz = fz, for all z CT,f. Definiion The mapping f is said o be occasionally coincidenally idempoen oci wih respec o T, if and only if f is idempoen a leas one coincidence poin, whenever CT, f φ. Clearly if f and T are coincidenally idempoen hen hey are oci. However, he Example 1.16 shows ha he converse is no necessarily rue. Definiion A funcion φ : [0,1] k = [0,1] [0,1] [0,1] [0,1] is said o be a Φ k -funcion if i saisfies he following condiions: i φ is an increasing funcion, i.e, x 1 y 1,x y,,x k y k implies ii φ,,,, for all [0,, iii φ is coninuous in all variables. φx 1,x,,x k φy 1,y,,y k, In his paper, we esablish some coincidence poin heorems for cerain maps and common fixed poin heorems for weakly compaible maps in DPM-space DP q M-space under sric conracive condiions. Our resuls generalize many known resuls in DPM-space DP q M-space.

6 74 M. Shams, H. Shayanpour and F. Ehsanzadeh. Coincidence and common fixed poins in DP q M-space and DPM-space In his secion we prove some coincidence and common fixed poin resuls for wo mappings f : X X and T : X k X in a DP q M-space or DPM-space. Now, we sae our firs main heorem. Theorem.1. Le X,F,τ be a DPM-space, k be an ineger, f : X X and T : X k X be mappings, such ha fx is complee and TX k fx. If F Tx1,x,...,x k,tx,x 3,...,x k+1 q φ F fx1,fx,f fx,fx 3,...,F fxk,fx k+1, where x 1,x,...,x k+1 are arbirary elemens in X, 0 < q < 1, [0, and φ is Φ k -funcion. Then he sequence y n defined by y n = fx n+k = Tx n,x n+1,...,x n+k 1.1 for arbirary elemens x 1,x,...,x k in X, converges o a poin of coincidence of f and T. Proof: If α n = F yn,y n+1 q, hen by he hypohesis, i is easy o see ha α n+k φα n,α n+1,...,α n+k 1. Clearly, if α 1 = α = = α k = 1, hen he sequence y n n=1 is consan and so i is a Cauchy sequence in fx. Oherwise, by inducion on n, we will prove ha K θ n α n K +θ n. { where θ = 1 q and K = min θ1+ } α 1 1 α 1, θ 1+ α 1 α,..., θk 1+ αk 1. We α k can see ha. is rue for n = 1,,...,k by he definiion of K. Assume ha. is rue for n, n+1 o n+k 1. Then by he hypohesis, we have α n+k φα n,α n+1,...,α n+k 1 φ K θ n K θ n+1 K θ, n+k 1 K +θ n K +θ n+1,..., K +θ n+k 1 φ K θ, n+k 1 K θ n+k 1 K θ,..., n+k 1 K +θ n+k 1 K +θ n+k 1 K +θ n+k 1 K θ n+k 1 K +θ n+k 1 K θ n+k K +θ n+k.

7 Fixed poin resuls in dislocaed probabiliy Menger meric spaces 75 Thus inducive proof of. is complee. Now for p N and [0,, we have F yn+1,y n+p+1 τ F yn+1,y n+,τ F yn+,y n+3, τ,τ F yn+p 1,y n+p p 1,F yn+p,y n+p+1 p 1 K n,τ K n+1, τ K + n K + n+1 K p 1n+p, τ,τ K + p 1n+p K p 1n+p 1 K + p 1n+p 1 τ1,τ1,τ,τ1,1 = 1, asn. Hence y n is a Cauchy sequence in fx and so here exiss v in fx such ha lim n y n = v. Le v = fu for some u X. Then we have F Tu,u,...,u,fu = lim F Tu,u,...,u,y n n = lim F Tu,u,...,u,Tx n n,x n+1,...,x n+k 1 lim τ F Tu,u,...,u,Tu,u,...,xn, n τ F Tu,u,...,xn,Tu,u,...,x n,x n+1, τ,τ F Tu,u,xn,...,x n+k 3,Tu,x n,...,x n+k k 1, F Tu,xn,...,x n+k,tx n,x n+1,...,x n+k 1 k 1 lim τ φ F fu,fu,f fu,fu,...,f fu,fxn, n τ φ F fu,fu,...,f fu,fxn,f fxn,fx n+1, τ,τ φ F fu,fu,f fxn,fx n+1,...,f fxn+k 3,fx n+k, φ F fu,fxn,f fxn,fx n+1,...,f fxn+k,fx n+k 1 = 1, hence, F Tu,u,...,u,fu = 1. Therefore Cf,T φ and v is a poin of coincidence of f and T, as required. Corollary.. Wih he same hypoheses of he Theorem.1, if T and f are occasionally weakly compaible mappings, 0 < q < 1 and lim F x,y = 1 for all

8 76 M. Shams, H. Shayanpour and F. Ehsanzadeh x,y X. Then he sequence y n defined by.1 converges o a unique common fixed poin of f and T. Proof: By he Theorem.1, we see ha he sequence y n converges o v = fu which is a poin of coincidence of f and T. Then by he hypohesis, we have F fu,fu q = F Tu,u,...,u,Tu,u,...,u q φ F fu,fu,f fu,fu,...,f fu,fu F fu,fu = F Tu,u,...,u,Tu,u,...,u φ F fu,fu q,f fu,fu q,...,f fu,fu q F fu,fu q... F fu,fu q n 1. As n we ge F fu,fu q = 1. Suppose here exiss v X such ha fu = Tu,u,...,u = v for some u in Cf,T. Similarly, we can show ha F fu,fu q = 1. Therefore, we have F v,v q =F Tu,u,u,...,u,Tu,u,u,...,u q τ F Tu,u,u,...,u,Tu,u,u,...,u q τ F Tu,u,u,...,u,Tu,u,u,...,u,u q 3, τ,τ F Tu,u,u,...,u,Tu,u,u,...,u q k, F Tu,u,u,...,u,Tu,u,u,...,u q k τ φ F fu,fu,f fu,fu,...,f fu,fu, τ φ F fu,fu 3,F fu,fu 3,...,F fu,fu 3,F fu,fu 3, τ,τ φ F fu,fu k,f fu,fu k,...,f fu,fu k, φ F fu,fu k,f fu,fu k,...,f fu,fu k τ F fu,fu,τ F fu,fu 3,τ, τ F fu,fu k,f fu,fu k,

9 Fixed poin resuls in dislocaed probabiliy Menger meric spaces 77 =τ F Tu,u,...,u,Tu,u,...,u,τ,τ F Tu,u,...,u,Tu,u,...,u τ k, F Tu,u,...,u,Tu,u,...,u k τ F fu,fu q,τ F fu,fu 3 q, τ,τ F fu,fu k q,f fu,fu k q F Tu,u,...,u,Tu,u,...,u 3,. Repeaing he above process n imes we ge F v,v q τ F fu,fu n+1 q n,τ F fu,fu n+ q n, τ,τ F fu,fu n+k 1 q n,f fu,fu n+k 1 q n. Taking he limi as n we ge F v,v q 1 and so v = v i.e. v is he unique poin of coincidence of f and T. Finally, by he Lemma 1.15, v is a unique common fixed poin of f and T. Noe ha he condiion lim F x,y = 1 ensures he uniqueness of he poin of coincidence. However in he nex resul we will remove he condiion lim F x,y = 1 and also increase he range of q. Corollary.3. Wih he same hypoheses of he Theorem.1, if one of he following wo condiions are saisfied: i f is oci wih respec o T and he pair f,t is weakly compaible, ii f is coincidenally idempoen wih respec o T and he pair f,t is owc. Then f and T have a common fixed poin. Proof: By he Theorem.1, i follows ha CT,f φ. Now suppose ha i is saisfied. So here will exis z Cf,T such ha ffz = fz and also ftz,z,...,z = Tfz,fz,...,fz. Thus we have fz=ffz=ftz,z,...,z = Tfz,fz,...,fz, i.e., fz is a common fixed poin of f and T. The proof follows on he same lines in he oher case also. Theorem.4. Le X,F,τ be a DP q M-space, k be an ineger, f : X X and T : X k X be mappings, such ha fx is R-complee L-complee and TX k fx. If F Tx1,x,...,x k,tx k+1,x 1,x,...,x k 1 q φ F fx1,fx k+1,f fx,fx 1,...,F fxk,fx k 1,

10 78 M. Shams, H. Shayanpour and F. Ehsanzadeh where x 1,x,...,x k+1 are arbirary elemens in X, 0 < q < 1, [0, and φ is Φ k -funcion. Then he sequence y n defined by.1 converges o a poin of coincidence of f and T. Proof: By modifying he proof of he Theorem.1, we can show ha Cf,T φ. Corollary.5. Wih he same hypoheses of he Theorem.4, if T and f are weakly compaible mappings, 0 < q < 1 and lim F x,y = 1 for all x,y X. Then he sequence y n defined by.1 converges o a unique common fixed poin of f and T. Proof: By modifying he proof of he Corollary., we can show ha he sequence y n defined by.1 converges o a unique common fixed poin. Proof of he following corollary follows on he same lines as ha of he Corollary.3. Corollary.6. Wih he same hypoheses of he Theorem.4, if one of he following wo condiions are saisfied: i f is oci wih respec o T and he pair f,t is weakly compaible, ii f is coincidenally idempoen wih respec o T and he pair f,t is owc. Then f and T have a common fixed. Taking k = 1 in he previous resuls, we ge he following. Theorem.7. Le X,F,τ be a DP q M-space, f : X X and T : X X be mappings, such ha fx is R-complee L-complee and TX fx. If F Tx,Ty q φ F fx,fy,.3 where x,y are arbirary elemens in X, 0 < q < 1, [0, and φ is Φ1 -funcion. Then f and T have a coincidence poin, i.e., Cf,T φ. Moreover, if f and T are weakly compaible and lim F x,y = 1 for all x,y X, or one of he following wo condiions are saisfied: i f is oci wih respec o T and he pair f,t is weakly compaible, ii f is coincidenally idempoen wih respec o T and he pair f,t is owc. Then he pair f,t have a unique common fixed poin. The above heorem is also valid for complee fx in DPM-space. If we ake f o be he ideniy mapping in he above corollaries, we ge he following

11 Fixed poin resuls in dislocaed probabiliy Menger meric spaces 79 Corollary.8. Le X,F,τ be a R-complee L-complee DP q M-space, T : X X be a mapping, such ha F Tx,Ty F x,y, where x,y X, 0 < q < 1 and [0,. Then T has a fixed poin. In wha follows, we presen some illusraive examples which demonsrae he validiy of he hypoheses and degree of uiliy of our resuls proved in his paper. Example.9. Le X = [0,] and d : X X X be given by dx,y = x y +x+y and define F x,y : [, ] [0,1] by F x,y = +dx,y, for every x,y X. Clearly, d is dislocaed meric on X and X,F,τ is DPMspace wih τa,b = ab for all a,b [0,. Le k be an ineger and T : X k X and f : X X be defined by Tx 1,x,,x k = x 1 +x + +x k k and fx = x. Then we have F Tx1,x,,x k,tx,x 3,,x k+1 q q = q+ 1 k x 1 x k k x 1 +x + +x k +x k+1 q q+ 1 k x 1 x + 1 k x k x k x k x k k x k + 1 q = 1 k x k k x 1 x + 1 k x k x k x k x k k x k + 1 k x k+1 { } min,, + x 1 x +x 1 +x + x k x k+1 +x k +x k+1 = φ F fx1,fx,f fx,fx 3,...,F fxk,fx k+1. Therefore, f and T saisfy condiions of he Theorem.1 wih φ 1,, k = min{ 1,, k } and q = 1. So we see ha CT,f = {0}, f and T commue a 0. Finally 0 is he unique common fixed poin of f and T. Example.10. Le X = R and d : X X X be given by and define F x,y : [, ] [0,1] by dx,y = x y + x + y F x,y = e 1 1 dx, y,

12 80 M. Shams, H. Shayanpour and F. Ehsanzadeh for every x,y X. Clearly, d is R-complee dislocaed quasi meric on X and X,F,τ is R-complee DP q M-space wih τa,b = ab for all a,b [0,. Le T : X X and f : X X be defined by Tx,y = x+y 4 and fx = x. I is obvious f and T are weakly compaible mappings and f is coincidenally idempoen wih respec o T. Also i is easy o see ha f and T saisfy condiions of he Theorem.4 whi φ 1, = 1 and q = 1 4. So we see ha CT,f = {0}. Finally 0 is he unique common fixed poin of f and T. References 1. M. Abbas, B. E. Rhoades, Common fixed poin heorems for occasionally weakly compaible mappings saisfying a generalized conracive condiion, Mah. Commun. 13, 008, M. A. Al-Thagafi, N. Shahzad, Generalized I-nonexpansive selfmaps and invarian approximaions, Aca Mah. Sinica English Series 4, 5 008, A. Bha, H. Chandra, D. R. Sahu, Common fixed poin heorems for occasionally weakly compaible mappings under relaxed condiions, Nonlinear Analysis 73, 1 010, S. Chauhan, S. Kumar, B. D. Pan, Common fixed poin heorems for occasionally weakly compaible mappings in Menger spaces, J. Advanced Research in Pure Mah. 3, 4 011, M. Grabiec, Fixed poins in fuzzy meric spaces, Fuzzy Ses Sys O. Had zić, E. Pap, Fixed poin heory in probabilisic meric spaces, Kluwer Academic Publishers, Dordrech F. He, Common fixed poin of four self maps on dislocaed meric spaces, J. Nonlinear Sci. Appl T. L. Hicks, Fixed poin heory in probabilisic meric spaces, Univ. u Novom Sadu. Zb. Rad. Prirod.-Ma. Fak. Ser. Ma. 13, G. Jungck, Common fixed poins for nonconinuous nonself maps on nonmeric spaces, Far Eas J. Mah. Sci.4, 1996, G. Jungck, B. E. Rhoades, Fixed poin heorems for occasionally weakly compaible mappings, Fixed Poin Theory 7, 006, D. C. Ken, G. D. Richardson, Ordered probabilisic meric spaces, J. Aus. Mah. Soc. Ser. A, C. Klin-eam, C. Suanoom, Dislocaed quasi-b-meric spaces and fixed poin heorems for cyclic conracions, Fixed poin heory Appl K. Menger, Saisical merics, Proc. Na. Acad. of Sci., U.S.A., 8 194, B. D. Pan, S. Chauhan, Fixed poin heorems for occasionally weakly compaible mappings in Menger spaces Ma. Vesnik 64, 4 01, B. D. Pan, S. Chauhan, B. Fisher, Fixed poin heorems for families of occasionally weakly compaible mappings, J. Indian Mah. Soc. N.S. 79, , M. Sarway, M. UR. Rahman, G. Ali, Some fixed poin resuls in dislocaed quasi meric d q-meric spaces, J. Ineq. Appl. 014, B. Schweizer, A. Sklar, Probabilisic meric spaces, Norh-Holland Series in Probabiliy and Applied Mahemaics, Norh-Holland Publishing Co., New York, B. Schweizer, A. Sklar, Saisical meric spaces, Pacific J. Mah , B. Singh, S. Jain, A fixed poin heorem in Menger space hrough weak compaibiliy, J. Mah. Anal. Appl. 301, 005,

13 Fixed poin resuls in dislocaed probabiliy Menger meric spaces 81 M. Shams, H. Shayanpour and F. Ehsanzadeh Faculy of Mahemaical Sciences, Deparmen of Pure Mahemaics, Universiy of Shahrekord, P. O. Box , Shahrekord, Iran. address: address: address: