Some Fixed Point Theorems of Semi Compatible and Occasionally Weakly Compatible Mappings in Menger Space

Transcription

1 America Joural of Alied Mahemaics ad Saisics, 05, Vol. 3, No., 9-33 Available olie a h://ubs.scieub.com/ajams/3//6 Sciece ad Educaio Publishig DOI:0.69/ajams-3--6 Some Fixed Poi Theorems of Semi Comaible ad Occasioally Weakly Comaible Maigs i Meger Sace Y. Rohe Sigh,,*, L. Premila Devi NIT Maiur, Takyela, Imhal, Pi, Maiur, Idia D. M. College of Sciece, Imhal, Pi, Maiur, Idia *Corresodig auhor: ymehor008@yahoo.com Received December 05, 04; Revised February 05, 05; Acceed February, 05 Absrac The oio of semi comaible maigs was iroduced by Cho. Sharma ad Sahu (Semicomaibiliy ad fixed ois, Mah. Jao, 4(), 995, 9-98) ad he oio of occaioally weakly comaible maigs was iroduced by Al-Thagafi M. A., Shahzad N. (Geeralized I-o exasive selfmas ad ivaria aroximaios, Aca. Mah. Siica (Eglish series) 4(5), 008, ). I his aer, we rove a commo fixed oi heorem i Meger sace usig he coce of semi comaible ad occasioally weakly comaible maigs. Some resuls are also give as corollaries. Our resuls geeralise some similar resuls i he lieraure. Keywords: commo fixed oi, comaible mas, meger sace, robabilisic meric sace, semi-comaible, weakly comaible, occasioally weakly comaible Cie This Aricle: Y. Rohe Sigh, ad L. Premila Devi, Some Fixed Poi Theorems of Semi Comaible ad Occasioally Weakly Comaible Maigs i Meger Sace. America Joural of Alied Mahemaics ad Saisics, vol. 3, o. (05): doi: 0.69/ajams Iroducio May researchers have bee geeralisig he oio of meric sace i differe ways ad Meger sace is oe of such geeralisaios iroduced by he grea mahemaicia Karl Meger [] i he year 94 who used disribuio fucios isead of o-egaive real umbers as he value of meric. Schweizer ad Sklar [4] sudied his coce ad gave some fudameal resuls o his sace. I 97, Sehgal ad Bharucha-Reid [] obaied a geeralizaio of Baach Coracive Pricile o a comlee Meger sace which is a milesoe i develoig fixed oi heory i Meger sace. I 98, Sessa [3] imroved he defiiio of commuaiviy i fixed oi heorems by iroducig he oio of weakly commuig mas. The i 986, Jugck [5] iroduced he coce of comaible mas ad his oio of comaible maigs i Meger sace was iroduced by Mishra [6]. Furher his codiio has bee weakeed by iroducig he oio of weakly comaible maigs by Jugck ad Rhoades [8]. Recely, Sigh ad Jai [0] iroduced weakly comaible mas i Meger sace o esablish a commo fixed oi heorem. Al. Thagafi ad Shahzad [3] iroduced he oio of occasioally weakly comaible maigs i meric sace which is more geeral ha weakly comaible maigs. Recely, Jugck ad Rhoades [] exesively sudied he oio occasioally weakly comaible maigs i semi-meric sace ad Chauha e.al. [3] exeded he oio of occassioally weakly comaible maigs o PM-sace. Cho, Sharma ad Sahu [7] iroduced he coce of semi-comaibiliy i a d-comlee oological sace ad usig his coce of semi comaibiliy i Meger sace, Sigh e.al. [9] roved a fixed oi heorem usig imlici relaio. Recely, Rohe ad Chharaji [6] used he coce of semi comaible maigs i coe meric sace o rove some commo fixed oi heorems. I his aer, we rove a commo fixed oi heorem i Meger sace usig he coce of semi comaible ad occasioally weakly comaible maigs. Some resuls are also give as corollaries. Our resuls geeralise some similar resuls [,5,6].. Prelimiaries Defiiio. A riagular orm * (shorly -orm) is a biary oeraio o he ui ierval [0, ] such ha for all a, b, c, d [0, ] he followig codiios are saisfied: i) a * = a; ii) a * b = b * a; ii) a * b c * d wheever a c ad b d; iv) a * (b * c) = (a * b) * c. Examle: a * b = mi {a, b}. Defiiio. A disribuio fucio is a fucio F : [, ] [0,] which is lef coiuous o R, odecreasig ad F( ) = 0, F( ) =. We will deoe he family of all disribuio fucios o [, ] by. H is a secial eleme of defied by

2 30 America Joural of Alied Mahemaics ad Saisics 0if 0 H () = if > 0 If X is a o-emy se, F : X X is called a robabilisic disace o X ad Fxy (, ) is usually deoed by F xy. Defiiio.3 (Schweizer ad Sklar [4]): The ordered air ( X, F ) is called a robabilisic meric sace (shorly PM-sace) if X is a oemy se ad F is a robabilisic disace saisfyig he followig codiios: (i) Fxy () = x = y; (ii) F xy (0) = 0 ; (iii) Fxy = Fyx ; (iv) Fxz () =, Fzy () s = Fxy ( s) =. The ordered rile ( X, F,*) is called Meger sace if ( X, F ) is a PM-sace, * is a -orm ad he followig codiio is also saisfied i.e. (v) Fxy ( s) Fxz () * Fzy () s. Proosiio.4 (Sehgal ad Bharucha-Reid []) Le ( X, d ) be a meric sace. The he meric d iduces a disribuio fucio F defied by Fxy ( ) = H ( dxy (, )) for all xy, X ad > 0. If - orm * is a * b = mi { ab, } for all ab, [0,] he ( X, F,*) is a Meger sace. Furher, ( X, F,*) is a comlee Meger sace if ( X, d ) is comlee. Defiiio.5 (Mishra [6]) Le ( X, F,*) be a Meger sace ad * be a coiuous orm. i) A seuece { x } i X is said o coverge o a oi x i X (wrie as x x ) iff for every ε > 0 ad λ (0,), here exiss a ieger 0 = 0 ( ε, λ ) such ha Fx x( ε ) > - λ for all 0. x i X is said o be Cauchy if for ii) A seuece { } every ε > 0 ad λ (0,), here exiss a ieger 0 = 0 ( ε,λ ) such ha Fxx ( ε ) > -λ for all 0 ad > 0. iii) A Meger sace i which every Cauchy seuece is coverge is said o be comlee. Remark.6 If * is a coiuous - orm, i follows from defiiio.3 (v) ha he limi of seuece i Meger sace is uiuely deermied. Defiiio.7 (Mishra[6]) Two self-mas S ad T of a Meger sace ( X, F,*) are said o be comaible if F () for all > 0, wheever { x } is a STxTSx seuece i X such ha Sx, Tx x for some x i X as. Defiiio.8 (Sigh ad Jai [0]) Two self-mas S ad T of a Meger sace ( X, F,*) are said o be weakly comaible (or coicideally commuig) if hey commue a heir coicide ois i.e. if Ax = Bx for some x X he ABx = BAx. Defiiio.9(Al Thagafi ad Shahzad []) Two selfmas S ad T of a Meger sace ( X, F,*) are said o be occasioally weakly comaible (owc) if ad oly if S ad T commue a heir coicidece oi. Defiiio.0 (Sigh B. ad Jai S.[0]) Two selfmas S ad T of a Meger sace ( X, F,*) are said o be semi comaible if FSTx T ( x) for all x > 0, x is a seuece i X such ha Sx, Tx, for some i X, as. Lemma.(Sigh B. ad Jai S. [0]) Le{ x } be a wheever { } seuece i a Meger sace ( X, F,*) wih coiuous - orm * ad *. If here exiss a cosa k (0,) such ha Fxx ( ) Fx x() for all > 0 ad =,,3,... he { x } is a Cauchy seuece i X. 3. Mai Resuls Theorem 3. Le A, B, S, T, L ad M be self-mas o a comlee Meger sace (X, F, *) wih * for all [0,], saisfyig: i) L(X) ST(X), M(X) AB(X); ii) There exiss a cosa k (0,) such ha FLxMy *[ FABxLx. FSTyMy ] [ FABxLx FABxSTy ]. FABxMy for all x, y X ad > 0 where 0 <, < such ha =; iii) AB = BA, ST = TS, LB = BL, MT = TM; iv) Eiher AB or L is coiuous; v) The air (L, AB) is semi comaible ad (M, ST) is occasioally weakly comaible. The A, B, S, T, L ad M have a uiue commo fixed oi. Proof: Le us choose a arbirary oi x 0 i X he by (i), here exis x, x X such ha Lx 0 = STx = y 0 ad Mx = ABx = y. By iducio we ca cosruc seueces {x } ad {y } i X such ha Lx = STx = y ad Mx = ABx = y for = 0,,, 3 F *[ FABx Lx F ] Lx Mx STx Mx FABx Lx () F ABx STx () F ABx Mx F *[ Fy ] y y y F yy Fy y () F (), y y F y y Fy y F y * y F yy Fy, y F y y Fy y F y y Fy y F y y Hece, Fyy ( ) F Similarly, we also have Fy y ( ) F y y () yy ()

3 America Joural of Alied Mahemaics ad Saisics 3 I geeral, for all eve or odd, Fy y F () y y for k (0, ) ad > 0. Thus, by Lemma., { y } is a Cauchy seuece i X. Sice (X, F, *) is comlee, i coverges o a oi z i X. Also { Lx } z, { ABx } z, { Mx } z ad { STx } z. Firs, le AB be coiuous he, AB(AB) x ABz ad (AB) Lx ABz. Sice (L, AB) is semi comaible, L(AB) x ABz. Agai, by (ii), F AB( AB) x L( AB) x F * L( AB) x Mx FSTx Mx F AB( AB) x L( AB) x () F AB( AB) x () STx FAB( AB) x Mx Leig FzABz * F ABzABz Fzz ( F ABzABz () FzABz () F zabz FzABz () F zabz FzABz FzABz FzABz FzABz =. For k (0, ) ad all > 0. Thus, z = ABz. Now by (ii), F * F AB F LzMx STx Mx F AB () FABzSTx () F ABzMx Leig Noig ha *[ zz ] F F F () Fzz () Fzz ( ) ad usig (iii) i defiiio., F F () F = for k ( 0,) ad all > 0. Thus, z = Lz = ABz. L( Bz) Mx AB( Bz) L( Bz) STx Mx F *[ F F ] F F AB( Bz) L( Bz) () AB( Bz) STx () AB( Bz) Mx Sice AB=BA ad BL=LB, L(Bz)=B(Lz)=Bz ad AB(Bz)=B(ABz)=Bz. Leig, FzBz *[ FBzBz Fzz ] [ FBzBz () FzBz ()] FzBz FzBz [ FzBz ] FzBz [ FzBz ] FzBz FzBz FzBz FzBz FzBz = For k (0, ) ad all > 0. Thus, z = Bz. Sice z = ABz, we also have z = Az. Therefore, z = Az = Bz = Lz. Sice L(X) ST(X), here exiss v X such ha z = Lz = STv. F *[ F ABx Lx F ] STvMv Lx Mv [ FABx () ()] Lx F ABxSTv FABx Mv Leig Noig ha *[ Fzz ] [ Fzz () Fzz ()] * ( ) ( ) ad usig (iii) i defiiio., Thus, by Lemma., z = Mv ad so z = Mv = STv. Sice (M, ST) is occasioally weakly comaible, we have STMv = MSTv. Thus, STz = Mz. F * F LxMz ABxLx F STzMz F ABx () () Lx F ABxSTz FABx Mz Leig as zmz *[ zz MzMz ] [ zz zmz ] zmz [ ] [ F () ] F F F F F () F () F FzMz FzMz FzMz zmz zmz FzMz FzMz FzMz FzMz = Thus, z = Mz ad herefore z = Az = Bz = Lz =Mz = STz. LxM ( Tz) ABxLx ST ( Tz) M ( Tz) F * F F F F ABxLx ABxST ( Tz) () () ABxM ( Tz) Sice MT = TM ad ST = TS, MTz = TMz = Tz ad ST(Tz) =T(STz) = Tz. Leig,

4 3 America Joural of Alied Mahemaics ad Saisics FzTz * Fzz FTzTz [ Fzz FzTz ] FzTz FzTz [ FzTz ] FzTz FzTz FzTz FzTz FzTz = Thus, z = Tz. Sice Tz = STz, we also have z = Sz. Therefore, z = Az = Bz = Lz =Mz = Sz = Tz, ad hece z is he commo fixed oi of A, B, L, M, S ad T. Secodly, le L be coiuous he, LL x Lz ad L(AB) x Lz. Sice (L, AB) is semi comaible, L (AB) x ABz ad ABz =Lz. F ABLx LLx F * LLx Mx FSTx Mx FABLx () LLx ABLx () Mx F ABLxSTx Leig *[ L zz ] [ L ] [ ] [ F () ] F F F F F () F () F = Thus, z = Lz. Hece z = Lz = Mz = S z = Tz. Sice M(X) AB(X), here exiss vϵ X such ha z = Mz = ABv. F * F ABvLv FSTx Mx LvMx FABvLv () () ABvMx F ABvSTx Leig Noig ha FzLv *[ FzLv Fzz ] [ FzLv () Fzz () ] Fzz FzLv * FzLv FzLv FzLv FzLv ( ) ad usig (iii) i defiiio., FzLv = Thus, z = Lv = ABv. Sice (L, AB) is occasioally weakly comaible, Lz = ABz ad usig z = Bz as show above. Hece z = Az = Bz = Sz = Tz = Lz = Mz, ha is, z is he commo fixed oi of he six maigs i his case also. I order o rove he uiueess of fixed oi le w be aoher commo fixed oi of A, B, S, T, L ad M. The by (ii), LzMw*[ AB STwMw] F F F FAB () FABzSTw () FABzMw which imlies ha Fzw [ Fzw] Fzw [ Fzw] Fzw Fzw Fzw() Fzw Fzw = Thus, z = w. This comlees he roof of he heorem. If we ake B = T = I X ( he ideiy ma o X) i he mai heorem, he followig: Corollary 3.: Le A, S, L ad M be self-mas o a comlee Meger sace (X, F, *) wih * for all [0, ], saisfyig: (i) L(X) S(X), M(X) A(X); (ii) There exiss a cosa kϵ (0, ) such ha (iii) (iv) FLxMy *[ FAxLx. FSyMy ] FAxLx () FAxSy () FAxMy for all x, y X ad > 0 where 0 <, < such ha =; eiher A or L is coiuous; he air (L, A) is semicomaible ad (M, S) is occassioally weakly comaible. The A, S, L, ad M have a uiue commo fixed oi. If we ake A = S, L = M ad B = T = I X i he mai Theorem, he followig: Corollary 3.3: Le (X, F, *) be a comlee Meger sace wih * for all [0, ] ad le A ad L be comaible mas o X such ha L(X) A(X). If A is coiuous ad here exiss a cosa k (0, ) such ha FLxLy *[ FAxLx. FAyLy ] [ FAxLx () FAxAy ()] FAxLy for all x, y X ad > 0 where 0 <, < such ha =, he A ad L have a uiue fixed oi. Examle 3.4: Le X = [0, ] wih he meric d defied by d(x, y) = x-y ad defied Fxy () =H (- d (x, y)) for all x, y X, > 0. Clearly (X, F, *) is a comlee Meger sace where -orm * is defied by a*b = mi{a, b} for all a, b [0, ]. Le A, B, S, T, L ad M be mas from X io iself defied as Ax = x, Bx = x, Sx = x, Tx = x, Lx = x, Mx = x for all x X. The

5 America Joural of Alied Mahemaics ad Saisics 33 L( X ) = 0, 0, 6 = ST X ad M ( X ) = 0, 0, = AB ( X ). 5 Clearly AB = BA, ST = TS, LB = BL, MT = TM ad AB, L are coiuous. If we ake k = ad =, we see ha he 3 codiio (ii) of he mai Theorem is also saisfied. Moreover, he mas L ad AB are semi comaible if lim x = 0, where { x } is a seuece i X such ha lim Lx = lim ABx = 0 for 0 X. The mas M ad ST are occasioally weakly comaible a 0. Thus, all codiios of he mai Theorem are saisfied ad 0 is he uiue commo fixed oi of A, B, S, T, L, ad M. Ackowledgeme The firs auhor is suored by UGC, New Delhi vide MRP F. No. 4-/03 (SR). Refereces [] K. Meger, Saisical Meric, Proc. Na. Acad. Sci. 8 (94), [] Sehgal V. M. ad Bharucha-Reid A.T., Fixed ois of coracio mas o robabilisic meric saces, Mah. Sysem Theory 6(97), [3] S. Sessa, O a weak commuaive codiio i fixed oi cosideraio, Publ. Is. Mah. (Beograd) 3 (98) [4] Schweizer B., Sklar A., Probailisic Meric Saces, Elsevier, Norh- Hollad, New York (983). [5] G. Jugck, Comaible maigs ad commo fixed ois, Iera. J. Mah. Mah. Sci. (986) [6] S. N. Mishra, Commo fixed ois of comaible maigs i PM- saces, Mah. Jao. 36(99) [7] Cho. Y. J., Sharma B. K. ad Sahu R. D., Semi-comaibiliy ad fixed ois, Mah. Jao. 4(), 995, [8] Jugck G., Rhoades B.E., Fixed ois for se valued fucios wihou coiuiy, Idia J. Pure Al. Mah., 9(998), [9] Sigh B., Jai S., Semi-comaibiliy ad fixed oi heorems i Meger sace usig imlici relaio, Eas Asia Mah. J. (), 005, [0] Sigh B., Jai S., A fixed oi heorem i Meger sace hrough weak comaibiliy, J. Mah. Aal. Al., 30(005), [] Jugck G., Rhoades B.E., Fixed oi heorems for occasioally weakly comaible maigs, Fixed oi heory 7, (006) [] Serve Kuukcu, A fixed oi heorem i Meger saces, Ieraioal Mahemaical Forum, (3), 006, [3] Al-Thagafi M. A., Shahzad N., Geeralized I-o exasive selfmas ad ivaria aroximaios, Aca. Mah. Siica (Eglish series) 4, 5(008) [4] Chauha S., Kumar S., Pa B. D., Commo fixed oi heorems for occasioally weakly comaible maigs i Meger saces, J. Adv. Research Pure Mah. 3, 4(0) 7-3. [5] Ariha Jai ad Basa Chaudhary, O commo fixed oi heorems for semi-comaible ad occasioally weakly comaible maigs i Meger sace, IJRAS, 4(3), 03, [6] Yumam Rohe ad Th. Chharaji, Semicomaible maigs ad fixed oi heorems i coe meris sace, Wulfeia Joural, (4), 04, 6-4.