is said to be conditionally commuting if f and g commute on a nonempty subset of the set of coincidence points

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1 6 Vol 03, Issue03; Sep-Dec 01 PUBLICATIONS OF PROBLEMS & APPLICATION IN ENGINEERING RESEARCH - PAPER ISSN: ; e-issn: Commo Fxe Pots for Par of Cotoally Commutg a Asorg No-self Mappgs PRAKASH C MATHAPL 1, MAHESH C JOSHI 1 Departmet of Mathematcs, Natoal Isttute of Techology, Uttarakha, INDIA Departmet of Mathematcs, D S B Campus, Kumau Uversty, Natal, INDIA 1 mathpal.p@gmal.com, mcjosh69@gmal.com ABSTRACT I ths paper we aopt the efto of cotoally commutg mappgs gve y Pat a asorg mappgs gve y Gopal, for o-self settgs a estalshe some commo fxe pot theorems. Keywors: Asorg mappgs, commo fxe pots, cotoally commutg mappgs a o-self mappg. 1. INTRODUCTION Durg last two-three ecaes, a lot of research has ee mae to vestgate the exstece of commo fxe/ cocece pots for pars of mappgs y mposg certa cotos o the mappgs vz; weak commutatvty, compatlty, R-weakly commutatvty a cocetly commutatvty. O the other ha, these cocepts are also ee use o o-self mappgs to estalsh the fxe pot theorems. I a terestg paper Pat et al. [7] gave a cocept of cotoally commutg mappgs a prove some results o commo fxe pots for cotoally o compatle self mappgs of a metrc space. I smlar way, we efe cotoally commutg par of o-self mappgs as followg. 1.1 Defto Let K e a oempty suset of a metrc space X, a f, g : K X. The the par f, g s sa to e cotoally commutg f f a g commute o a oempty suset of the set of cocece pots prove fx K. Ima a Kumar [6] efe cocetally commutg o self mappgs y takg f a g commutatve for all cocece pots of f a g. It s remarkale that all cocetally commutg mappgs are cotoally commutg ut coverse s ot true. Ths s llustrate y followg example. 1. Example, 6 X, 0 a e usual metrc o X. Defe f, g : K X y Let K a x or 6 x fx a gx 3 x x 6 x x 6 Here f g a f 3 6 g3 f a g commute oly at oe cocece pot.e. fg gf fg3 gf 3. hece a 3 are cocece pots of f a g ut. However, fg 3, gf 3 1,mples I a recet paper, Gopal et al. [3] gave the cocept of asorg mappgs a prove that the oto of asorg mappgs s epeet of compatlty a pot wse R-weak commutatvty. Here we exte the cocept of asorg mappgs for par of o-self mappg. 1.3 Defto Let K e a oempty suset of a metrc space X, a f, g : K X ( f g). The f wll e calle g-asorg f there exsts some real umer R 0 such that gx, gfx R fx, gx for all x K prove fx K. Smlarly g wll e calle f -asorg f there exsts some real umer R 0 such that fx, fgx R fx, gx for all x K prove fx K. The map f wll e calle pot wse g-asorg f gve x K, there exsts R 0 such that gx, gfx R fx, gx for all x K. Smlarly, we ca efe pot wse f -asorg map. Followg example shows that asorg mappgs o ot ecessarly commute at ther cocece pot. 1.4 Example 1 Let K 0, a X 0,1 a e usual metrc o X. Defe f, g : K X y IJPAPER Iexg Process - EMBASE, EmCARE, Electrocs & Commucato Astracts, SCIRUS, SPARC, GOOGLE Dataase, EBSCO, NewJour, Worlcat, DOAJ, a other major ataases etc.,

2 7 Vol 03, Issue03; Sep-Dec 01 PUBLICATIONS OF PROBLEMS & APPLICATION IN ENGINEERING RESEARCH - PAPER ISSN: ; e-issn: x 1 x 0 fx a gx 1 0 x 1 x 0 The the map f s g-asorg for ay R 0 ut the par f, g oes ot commute at ther cocece pot x 0. The exstg lterature fxe pot theory cotas umerous fxe pot theorems for self-mappgs metrc a Baach spaces. But practcally there exst may stuatos whe mappg uer examato s ot a selfmap. Therefore fxe pot theorems for o-self mappgs are also worth vestgatg (see [1], [], [4], [5], [8], [9]). Recetly, Ima a Kumar [6] extee the result of Rhoes [8] for a par of cocetally commutg oself mappgs satsfyg followg cotractve coto: gx, gy gx, fy gy, fx fx, fy h max, gx, fx, gy, fy, (1) for all x, y K, 0 h 1, 1 h, where f, g : K X ; K e a oempty suset of a metrc space X. I ths secto we geeralze the aove cotractve coto for o-self mappgs a estalsh some results for the exstece of commo fxe pots of cotoally commutg a asorg mappgs. I the seuel, we use followg efto a Lemma. 1.5 Defto[1] A metrc space X, s sa to e metrcally covex f for ay x, y X wth x y there exsts a pot, z X x y z such that x, z z, y x, y. 1.6 Lemma[6] Let K e a oempty close suset of a metrcally covex metrc space X. If x K a y K, the there exsts a pot z K (Bouary of K ) such that x, z z, y x, y.. MAIN RESULTS.1 Theorem Let X e a metrcally covex metrc space, K a oempty close suset of X, a satsfyg gx, gy gx, fy gy, fx fx, fy a max, gx, fx, gy, fy, max gx, fx, gy, fy for all x, y K, 0 a 1, 0, 1 ( a ), a a 1 a. K gk, fk K gk, f, g : K X (). gx K fx K a. gk s complete. The there exsts a cocece pot v K.If par f, g s cotoally commutg, the f a g have a commo fxe pot K. Proof: Let x K. The there exsts a pot x 0 K such that x gx0 as K gk. Sce gx0 K a gx K fx K, we coclue that fx0 K fk gk. Let x 1 K e such that y1 gx1 fx0 K. Let y fx1. Suppose y K. The y K fk gk, whch mples that there exsts a pot x K such that y gx. Suppose y K. The there exsts a pot p K such that gx, p p, y gx y 1 1, IJPAPER Iexg Process - EMBASE, EmCARE, Electrocs & Commucato Astracts, SCIRUS, SPARC, GOOGLE Dataase, EBSCO, NewJour, Worlcat, DOAJ, a other major ataases etc.,

3 8 Vol 03, Issue03; Sep-Dec 01 PUBLICATIONS OF PROBLEMS & APPLICATION IN ENGINEERING RESEARCH - PAPER ISSN: ; e-issn: Sce p K gk, there exsts a pot x K such that p gx so that aove euato takes the form gx gx, y gx y. 1 1, Let us put y3 fx. Thus, repeatg the foregog argumets, oe otas two seueces x a such that a. y1 fx. y K y gx a c. y K gx K a 1 gx, y gx1 gx y, We eote P gx { gx }: gx y Q gx { gx }: gx y. Ovously, two cosecutve terms caot le Q. Now, we stgush three cases. Case1. If gx 1 P, the gx 1 y, y1 fx1, fx gx1 gx1, fx gx, fx1 a max 1, fx1, gx, fx, max gx1, fx1, gx, fx ( a ) gx 1 Case. If gx P a gx 1 Q, the gx 1 gx 1 gx1, y1 gx, y fx, fx 1 1 From case1, t mples ( a ) gx 1 gx1, gx Case3. If gx Q a gx 1 P, the gx 1 P. Sce gx s covex lear comato of gx1 a y. It follows that gx1 max gx1 1, y, gx1 (3) gx gx y gx, the (a). If 1, 1, 1 gx 1 y 1 fx1, fx gx1, gx a max, fx, gx, fx, max gx, fx, gx, fx gx, fx gx, fx 1 1 y Now y otg that gx gx 1 1, y gx1, gx gx, gx1 gx, y gx, y gx Oe ca coclue that ( a ) gx 1 gx, y a 1 gx IJPAPER Iexg Process - EMBASE, EmCARE, Electrocs & Commucato Astracts, SCIRUS, SPARC, GOOGLE Dataase, EBSCO, NewJour, Worlcat, DOAJ, a other major ataases etc.,

4 9 Vol 03, Issue03; Sep-Dec 01 PUBLICATIONS OF PROBLEMS & APPLICATION IN ENGINEERING RESEARCH - PAPER ISSN: ; e-issn: I vew of Case. y gx gx gx, the from () a (3)., 1 1, 1 (). If gx 1 gx1, gx1 fx, fx gx, gx a max, fx, gx, fx, max By case a usg tragular eualty, we get gx, fx, gx, fx gx gx gx 1 1 gx x I eualty (4) f we take, 1 gx gx a gx gx, 1, 1 Aga eualty (4) f we take gx gx a gx gx, 1, 1 Sce a 1, a cotracto. If the maxmum of the frst term s gx 1 1 a max max gx 1. gx, fx gx, fx (.4) as maxmum oth parts, the y smple calculato we get gx, fx as maxmum oth parts, the we get gx, fx gx fx, gx, fx gx, fx gx, fx, gx, fx Now, takg gx gx, 1 the we have as maxmum a y usg the fact that 1 a a a a We get a 1, 1 gx gx gx, gx1 a a Thus, all cases gx gx k max gx, gx gx gx gx, y gx gx 1, 1, 1 Otherwse, gx gx gx 1, 1 1, a a a where k max, a,, a a Hece, oe ca easly show that 1 gx k max gx, gx gx , whch mples that Sce gk s complete, therefore gx s a Cauchy seuece. elog to P, we get a suseuece gx coverge to a pot u gk. By takg terms of the gx gx k whch s cotae P. Covergece of gx whch mples that IJPAPER Iexg Process - EMBASE, EmCARE, Electrocs & Commucato Astracts, SCIRUS, SPARC, GOOGLE Dataase, EBSCO, NewJour, Worlcat, DOAJ, a other major ataases etc.,

5 10 Vol 03, Issue03; Sep-Dec 01 PUBLICATIONS OF PROBLEMS & APPLICATION IN ENGINEERING RESEARCH - PAPER ISSN: ; e-issn: gx k also coverge tou gk. Let v g u. The u gv suseuece of By () fv, fx k 1 gx also coverge to u gv. gv, gx a max max max fv, gv a max 0, gv, fv k 1, gv, fv, gx, fx,0, gv, fv,0 gv, fv, gx, fx, k 1 k 1 gv, fv Imples that fv, gv ( a ) gv, fv Hece, Takg, par k 1 k 1. It s to e ote that gv, fx gx, fv fv gv whch shows that v s a pot of cocece for f a g. f, g cotoally commutg, two cases arse Case1. f a g commute at v, the u gv fv fu fgv gfv gu. Usg () we have, gu, gv gu, fv gv, fu fu, u fu, fv a max,0,0, max0,0 whch shows that u s a commo fxe pot of f a g. k 1 k 1 fx k 1 Makg eg a k Case. f a g o ot commute at v, the y vrtue of cotoal commutatvty of f a g there exsts a pot w K such that y gw fw fy fgw gfw gy By smlar calculato as Case1, we ca show that y s a commo fxe pot of f a g. Remark1: If we take 0, f a g are cocetally commutg the we get the result of Ima et al. [6].. Theorem Let X e a metrcally covex metrc space, K a oempty close suset of X, a f g : K X f K g K a satsfyg, are ocompatle pot wse g-asorg mappgs such that gx, gy gx, fy gy, fx fx, fy a max, gx, fx, gy, gy, for all (5) max gx, fx, gy, fy c gx, fy gy, fx x, y K, a,, c 0, 1 a, a a c 1. If the rage of f or g e a complete suspace of X, the f a g have a commo fxe pot. Proof: Sce the par f, g s o compatle pot wse g-asorg maps, there exsts a seuece x K such that lm fx lm gx t for some K or o exstet. Suppose X By (5) we have, t ut lm fgx, gfx s ether o zero g s complete suspace of X the there exsts u K such that t gu IJPAPER Iexg Process - EMBASE, EmCARE, Electrocs & Commucato Astracts, SCIRUS, SPARC, GOOGLE Dataase, EBSCO, NewJour, Worlcat, DOAJ, a other major ataases etc.,

6 11 Vol 03, Issue03; Sep-Dec 01 PUBLICATIONS OF PROBLEMS & APPLICATION IN ENGINEERING RESEARCH - PAPER ISSN: ; e-issn: gx, fu gu, fx gx, gu fx, fu a max, fx, gu, fu, Takg lm we max gx, fx, gu, fu c gx, fu gu, fx get, gu, fu gu, fu a max0,0, gu, fu, max0, gu, fu c gu, fu or gu, fu a c gu, fu.e. gu fu. By g-asorg coto, we have gu, gfu R fu, gu gu gfu a gfu ggu fu gu Aga y (5), gu, gfu gu, ffu fu, ffu a max, gu, fu, gfu, ffu, max gu, fu, gfu, ffu c gu, ffu gfu, fu gfu, fu fu, ffu fu, ffu a max 0,0, fu, ffu, max0, fu, ffu c fu, ffu Hece a c fu, ffu fu ffu gfu.e. fu s commo fxe pot of f a g. REFERENCES [1] N A Assa a W A Krk (197), Fxe pots theorems for set-value mappgs of cotractve type, Pacfc J. Math., 43, 3, pp [] N A Assa (1973), Fxe pot theorems for set-value trasformatos o compact sets, Bull. U. Mat. Ital. 4, 7, pp [3] D Gopal, A S Raave a R P Pat(008), Commo fxe pots of asorg maps, Bull. Mar. Math. Soc., 9,1, pp [4] O Hazc a Lj Gajc (1986), Cocece pots for set value mappgs covex metrc spaces, Uv. U. Novom. Sau Z. Ra. Prro, Mat. Fak. Ser. Mat, 16, 1, pp [5] O Hazc (1986), O cocece pots covex metrc spaces, Uv. U. Novom. Sau. Z. Ra. Prro. Mat. Fak. Ser. Mat., 19,, pp [6] M Ima a S Kumar (003), Rhoaes type fxe pot theorems for a par of oselfmappgs, Comp. A Math. Appl., 46, pp [7] R P Pat (010), Commo fxe pot of cotoally commutg maps, Fxe pot theorem A Iteratoal Joural, 11,1, pp [8] B E Rhoaes (1997), A fxe pot theorem for o-self set-value mappgs, Itert. J. Math. &Math. Sc., 0, 1, pp [9] B E Rhoaes (1978), A fxe Pots theorem for o-self mappgs, Math. Japo., 3, 4, pp IJPAPER Iexg Process - EMBASE, EmCARE, Electrocs & Commucato Astracts, SCIRUS, SPARC, GOOGLE Dataase, EBSCO, NewJour, Worlcat, DOAJ, a other major ataases etc.,