International Journal of Mathematical Archive-7(6), 2016, Available online through ISSN

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1 Iteratioal Joural of Mathematical Archive-7(6, 06, 04-0 Available olie through ISSN COMMON FIED POINT THEOREM FOR FOUR WEAKLY COMPATIBLE SELFMAPS OF A COMPLETE G METRIC SPACE J. NIRANJAN GOUD*, M. RANGAMMA Departmet of Mathematics, Osmaia Uiversity, Hyderabad , Telagaa state, Idia. (Received O: ; Revised & Accepted O: ABSTRACT I the preset paper we prove a commo fixed poit theorem for four weakly compatible self maps of a complete G metric space 00 Mathematics Subject Classificatio: 54H5, 47H0. Key words: G Metric space, weakly Compatible mappigs, Fixed poit, Associated sequece of a poit relative to four self maps, α -property.. INTRODUCTION I a attempt to geeralize fixed poit theorems o a metric space, Gahler [, ] itroduced the otio of -metric spaces while Dhage [] iitiated the otio of D - metric spaces. Subsequetly several researchers have proved that most of their claims made are ot valid. As a probable modificatio to D - metric spaces Shaba Sedghi, Nabi Shobe * ad Haiyu Zhou [5] itroduced D metric spaces. I 006, Zead Mustafa ad Brailey Sims [7] iitiated G - metric spaces.of these two geeralizatios, the G -metric space eviced iterest i may researchers. The purpose of this paper is to prove a commo fixed poit theorem for four weakly compatible self maps of a complete G -metric space. Now we recall some basic defiitios ad lemmas which will be useful i our later discussio.. PRELIMINARIES We begi with Defiitio.: ([7], Defiitio Let be a o-empty set ad G: [0, be a fuctio satisfyig: (G Gxyz (,, = 0 if x= y = z (G 0 < Gxxy (,, for all x, y with x y (G Gxxy (,, Gxyz (,, for all x, y,z with z y (G4 Gxyz (,, = G( σ ( xyz,, for all x, y,z, where σ ( xyz,, { x, y, z } ad (G5 Gxyz (,, Gxww (,, Gwyz (,, for all x, y, z, w. The G is called a G - metric o ad the pair (, G is called a G - metric Space. Defiitio.: ([7], Defiitio 4 A G-metric Space (, G is said to be symmetric if (G6 Gx (, y, y = Gxx (,, y for all x, y Correspodig Author: J. Niraja Goud*, Departmet of Mathematics, Osmaia Uiversity, Hyderabad , Telagaa state, Idia. is a permutatio of the set Iteratioal Joural of Mathematical Archive- 7(6, Jue 06 04

2 The example give below is a o-symmetric G-metric space. Example.: ([7], Example: Let = { ab, }. Defie G: [0, by Gaaa (,, = Gbbb (,, = 0; Gaab (,, =, Gabb (,, = ad exted G to all of a G - metric space. Sice Gaab (,, Gabb (,,, the space (, G is o-symmetric, i view of (G6. Example.4: Let (, d be a metric space. Defie G : [0, by d Gs ( x, y, z = [ d( x, y d( y, z d( z, x ] for x, y, z d s by usig (G4. The it is easy to verify that (, G is.the (, d G s is a G-metric Space. Lemma.5: ([7], p.9 If (, G is a G-metric space the Gxyy (,, Gyxx (,, for all xy, Defiitio.6: Let (, G be a G-metric Space. A sequece { x } i is said to be G-coverget if there is a x0 such that to each 0. ε > there is a atural umber N for which G( x, x, x0 < ε for all N Lemma.7: ([7], Propositio 6 Let (, G be a G-metric Space, the for a sequece{ x } ad poit x followig are equivalet. ( { x } is G- coverget to x. ( dg( x, x 0 as (that is { x } coverges to x relative to the metric d G ( G( x, x, x 0 as (4 Gx (, xx, 0 as (5 Gx (, x, x 0 as m, m the Defiitio.8: ([7], Defiitio 8 Let (, G be a G-metric space, the a sequeces { x } if for each > 0, there exists a atural umber N such that Gx (, xm, xl < ε for all ml,, Note that every G-coverget sequece i a G-metric space (, G is G-Cauchy. is said to be G-Cauchy N. Defiitio.9: ([7], Defiitio 9 A G-metric space (, G is said to be G-complete if every G -Cauchy sequece i (, G is G-coverget i (, G. The otio of weakly compatible mappigs as a geeralizatio of commutig maps is itroduced by Gerald Jugck [4]. We ow give the defiitio of weakly compatibility i a G-metric space Defiitio.0: Suppose f ad g are self maps of a G-metric space (, G. The pair f ad g is said to be weakly compatible pair if G( fgx, gfx, gfx G( fx, gx, gx for all x, Defiitio.: Let ( G, be a G -metric space ad f, hg,ad, p be selfmaps of such that f( g(, h ( p (. For ay x0, there is a sequece{ x } i such that fx = gx ad hx = px for 0, the the sequece{ } x relative to self maps f, hg,, ad p or simply. a associated sequece of x0 Defiitio.: Let : (i is associative ad commutative (ii is cotiuous x is called a associated sequece of 0 be a biary operatio satisfyig the followig coditios Defiitio.: ([6], Defiitio. The biary operatio is said to satisfy α - property if there exists a positive real umberα such that a b α max {, ab} for all ab, 06, IJMA. All Rights Reserved 05

3 Example.4: (i if a b= a b for each ab, the for α we have a b α max { ab}, Example.5: (ii if ab a b= for each ab, the for α we have a b α max { ab}, max{ ab,,}. MAIN RESULTS We ow state our mai theorem. Theorem.: Let (, G be a complete G- metric space such that satisfies α - property withα > 0. Let f, hg, ad p be self maps of satisfyig the followig coditios. (.. f( g (, h ( p ( ad g( or p ( is a closed subset of (.. G( fx, hy, hy k( G( px, gy, gy G( fx, px, px k( G( px, gy, gy G( hy, gy, gy G( px, hy, hy G( fx, gy, gy k( G( px, gy, gy for all xy,, where k, k, k > 0ad 0 < α( k k k < (.. the pairs ( f, p ad ( h, g are weakly compatible The f, hgad, p have a uique commo fixed poit i Proof: Suppose f, hgad, p be self maps of for which the coditio (.. holds. Let x0 the we defie a associated sequece { y } i such that (..4 y= fx = gx a y = hx = px for 0 Now we claim that the sequece { y } is Cauchy sequece By (.. we have G( y, y, y = G( fx, hx, hx k ( G( px, gx, gx G( fx, px, px k ( G( px, gx, gx G( hx, gx, gx G( px, hx, hx G( fx, gx, gx k ( G( px, gx, gx = k ( Gy (, y, y Gy (, y, y k ( Gy (, y, y Gy (, y, y Gy (, y, y Gy (, y, y k( Gy (, y, y kα max{( Gy (, y, y, Gy (, y, y } k α max Gy ({, y, y, Gy (, y, y} Gy (, y, y Gy (, y, y kα max{ Gy (, y, y, } kα max{ Gy (, y, y, Gy (, y, y} k α max{ Gy (, y, y, Gy (, y, y } Gy (, y, y Gy (, y, y kα max{ Gy (, y, y, } 06, IJMA. All Rights Reserved 06

4 Now if G( y, y, y > G( y, y, y, we have G( y, y, y kαg( y, y, y kg( y, y, y kαg( y, y, y < kαgy (, y, y kgy (, y, y kαgy (, y, y = α ( k k k Gy (, y, y < Gy (, y, y Sice α ( k k k < which is a cotradictio. G( y, y, y G( y, y, y (..5 Similarly G( y, y, y G( y, y, y (..6 From (..5 ad (..6 we have (..7 G( y, y, y G( y, y, y for = 0,,,... Usig (..7 we get G( y, y, y α( k k k G( y, y, y = kg( y, y, y k = α( k k k < Where So G( y, y, y kg( y, y, y kgy (, y, y k Gy ( 0, y, y 0 k as Sice 0 If m> the Gy (, y, y Gy (, y, y Gy (, y, y... Gy (, y, y m m m m m m kgy ( 0, y, y k Gy ( 0, y, y... k Gy ( 0, y, y k = G( y0, y, y 0 as, m k Showig that the sequece { y } is a Cauchy, ad by the completeess of, sequece{ y } coverges to z lim fx = lim hx = lim gx = lim px = z (..8 If g( is closed subset of the there exists a v such that gv = z If hv z the by (.. we get G( fx, hv, hv k ( G( px, gv, gv G( fx, px, px k ( G( px, gv, gv G( hv, gv, gv (..9 G( px, hv, hv G( fx, gv, gv k( G( px, gv, gv 06, IJMA. All Rights Reserved 07

5 O lettig i (..9 ad usig (..8, we get G(, z hv, hv k( G( z, z, z G( z, z, z k( G( z, z, z G( hv, z, z G(, z hv, hv G(, z z, z k ( Gzzz (,, By usig α -property, we get G(, z hv, hv G(, z hv, hv kαg( hv, z, z kα < α ( k k G(, z hv, hv < G(, z hv, hv Sice α ( k k < which is a cotradictio, hece hv = z (..0 hv = gv = z sice the pair ( hg, is weakly compatible the we have hgv = ghv ad so (.. hz = gz Now if hz z the by (.. we get (.. G( fx, hz, hz k ( G( px, gz, gz G( fx, px, px k( G( px, gz, gz G( hz, gz, gz G( px, hz, hz G( fx, gz, gz k ( G( px, gz, gz O lettig i (.. ad usig (..8, (.., we get G( z, hz, hz k ( G( z, hz, hz G( z, z, z k ( G( z, hz, hz G( gz, gz, gz G( z, hz, hz G( z, hz, hz k ( G( z, hz, hz By α -property, we get G(, z hz, hz α ( k k k G(, z hz, hz < G(, z hz, hz Sice α ( k k k < which is a cotradictio, hece hz = z (.. hz = gz = z Sice h ( p ( there exists u such that hz = pu = z If fu z by (.. we get (..4 G( fu, hz, hz k ( G( pu, gz, gz G( fu, pu, pu k ( G( pu, gz, gz G( hz, gz, gz G( pu, hz, hz G( fu, gz, gz k ( G( pu, gz, gz k ( Gzzz (,, G( fuzz,, k ( Gzzz (,, Gzzz (,, G( z, z, z G( fu, z, z k ( Gzzz (,, By α -property, we get G( fu, z, z α ( k k G( fu, z, z < G( fu, z, z Sice α ( k k < which is a cotradictio, hece fu = z 06, IJMA. All Rights Reserved 08

6 (..5 fu = pu = z Sice the pair ( f, p is weakly compatible the fpu = pfu so fz = pz If fz z the by (.. we get (..6 G( fz, z, z = G( fz, hz, hz k( G( fz, z, z G( fz, fz, fz k( G( fz, z, z G( z, z, z G( fz, z, z G( fz, z, z k ( G( fz, z, z By α -property, we get G( fzzz,, α ( k k k G( fzzz,, < G( fzzz,, Sice α ( k k k <, which is a cotradictio, hece fz = z fz = gz = hz = pz = z Showig that z is a commo fixed poit for self maps f, hg, ad p The proof is similar whe p ( is a closed subset of with appropriate chages We ow prove the uiqueess of the commo fixed poit If possible let w be ay other commo fixed poit for self maps f, hg, ad p The from the coditio (.., we have (..7 G( z, w, w = G( fz, hw, hw k ( (,, (,, G pz gw gw G fz pz pz k ( (,, (,, G pz gw gw G hw gw gw G( pz, hw, hw G( fz, gz, gz k ( G( pz, gw, gw = k ( Gzww (,, Gzz (,, z k ( Gzww (,, Gwww (,, Gzww (,, Gzzz (,, k ( Gzww (,, by usig α -property, we get Gzww (,, α ( k k k Gz (, ww, < Gzww (,, Sice α ( k k k <, which leads to a cotradictio if z w, hece z = w. z is a uique commo fixed poit for self maps f, hg, ad p Corollary.: Let (, G be a complete G- metric space. Let f, hgad, p be self maps of satisfyig the followig coditios. (.. f( g (, h ( p ( ad g( or p ( is a closed subset of (.. G( fx, hy, hy k( G( px, gy, gy G( fx, px, px k( G( px, gy, gy G( hy, gy, gy G( px, hy, hy G( fx, gy, gy k( G( px, gy, gy for all xy,, where k, k, k > 0ad 0 < ( k k k < 4 06, IJMA. All Rights Reserved 09

7 (.. the pairs ( f, p ad ( h, g are weakly compatible The f, hgad, p have a uique commo fixed poit i Proof: Defie a b= a b for each ab, the for α we have a b α max{ ab, } Takig α = poit i all the coditios of the Theorem (. hold. f, hgad, p have a uique commo fixed REFERENCES. B.C.Dhage, Geeralized metric space ad mappig with fixed poit, Bull.Cal.Math.Soc.84 (99, S.Gahler,- metrische raume ad ihre topologische structure,math.nachr,6 ( S.Gahler, Zur geometric -metrische raume, Revue. Roumaie, Math pures Appl, ( G.Jugck, Commo fixed poits for o-cotiuous o- selfmaps o o metricspaces, Far East J.Math. Sci 4 ( (996, Shaba Sedghi, Nabi Shobe, Haiyu Zhou., A commo fixed poit theorem i D*- metric spaces, Fixed poit Theory Appl 007, S.Sedghi ad N.Shobe, Commo Fixed Poit Theorems For Four Mappigs I Complete Metric space Iraia Mathematical Society vol,(007, Zead Mustafa ad Brailey Sims, A ew approach to geeralized metric spaces, Joural of No liear ad Covex Aalysis,7, (006, Source of support: Nil, Coflict of iterest: Noe Declared [Copy right 06. This is a Ope Access article distributed uder the terms of the Iteratioal Joural of Mathematical Archive (IJMA, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.] 06, IJMA. All Rights Reserved 0