Journal of Applied Research and Technology ISSN: Centro de Ciencias Aplicadas y Desarrollo Tecnológico.

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1 Joural of Applied Research ad Techology ISSN: Cetro de Ciecias Aplicadas y Desarrollo Tecológico México Shahi, Priya; Kaur, Jatiderdeep; Bhatia, S. S. Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces Joural of Applied Research ad Techology, vol., úm., juio-, 04, pp Cetro de Ciecias Aplicadas y Desarrollo Tecológico Distrito Federal, México Available i: How to cite Complete issue More iformatio about this article Joural's homepage i redalyc.org Scietific Iformatio System Network of Scietific Jourals from Lati America, the Caribbea, Spai ad Portugal No-profit academic project, developed uder the ope access iitiative

2 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces Priya Shahi*, Jatiderdeep Kaur ad S. S. Bhatia School of Mathematics ad Computer Applicatios Thapar Uiversity, Patiala-47004, Idia * priya.thaparia@gmail.com ABSTRACT I this paper, we establish a commo fixed poit theorem for expasive mappigs by usig the cocept of weak compatibility i the settig of G -metric spaces. This result geeralizes the result of Ahmed [] from -metric spaces to G -metric spaces by removig the coditio of sequetial cotiuity of the mappigs. Further, we geeralize ad exted the theorem of Şahi ad Telci [0] to G -metric spaces ad thereby extedig the theorem of Wag et al. [] for a pair of mappigs to G -metric spaces. Some comparative examples are costructed which illustrate the obtaied results. Keywords: Commo fixed poit, G -metric spaces, Weakly compatible, Expasive mappig.. Itroductio Fixed poit theory has gaied impetus, due to its wide rage of applicability, to resolve diverse problems emaatig from the theory of oliear differetial equatios ad itegral equatios [4], game theory relevat to military, sports ad medicie as well as ecoomics []. A metrical commo fixed poit theorem is broadly comprised of coditios o commutativity, cotiuity, completeess ad cotractio besides suitable cotaimet of rage of oe map ito the rage of the other. For provig ew results, the researchers of this domai are required to improve oe or more of these coditios. With a view to accommodate a wider class of mappigs i the cotext of commo fixed poit theorems, Sessa [] itroduced the otio of weakly commutig mappigs which was further geeralized by Jugck [4] by defiig compatible mappigs. After this, there came a host of such defiitios which are scattered throughout the recet literature whose survey ad illustratio (upto 00) is available i Murthy []. A miimal coditio merely requirig the commutativity at the set of coicidece poits of the pair called weak compatibility was itroduced by Jugck [6] i 996. This ew otio was extesively utilized to prove ew results. Mustafa ad Sims [6] itroduced the G-metric spaces as a geeralizatio of the otio of metric spaces. Mustafa et al. ( []-[5], [7]) obtaied some fixed poit theorems for mappigs satisfyig differet cotractive coditios. Abbas ad Rhoades [] iitiated the study of commo fixed poit i G-metric spaces. I 984, Wag et al. [] preseted some iterestig work o expasio mappigs i metric spaces which correspod to some cotractive mappigs i [8]. Rhoades [9] ad Taiguchi [] geeralized the results of Wag [] for pair of mappigs. Later, Kha et al. [9] i 986 geeralized the result of [] by makig use of the fuctios. Kag [7] geeralized these results of Kha et al. [9], Rhoades [9] ad Taiguchi [] for expasio mappigs. I 009, Ahmed [] established a commo fixed poit theorem for expasive mappigs by usig the cocept of compatibility of type (A) i -metric spaces. The theorem proved by Ahmed [] was the geeralizatio of the result of Kag et al. [8] for expasive mappigs. Recetly, Şahi ad Telci [0] preseted a commo fixed poit theorem for expasio type mappigs i complete coe metric Joural of Applied Research ad Techology 607

3 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces, A Priya Shahi et al. / spaces which geeralizes ad exteds the theorem of Wag et al. [] for a pair of mappigs to coe metric spaces. The purpose of this paper is to geeralize the results of Ahmed [] to G -metric spaces by removig the coditio of sequetial cotiuity of the mappigs. I order to prove the results, a more geeralized cocept of weak compatibility i G - metric spaces have bee used istead of compatibility of type (A) used by Ahmed [] i - metric spaces. Also, we exted the results of Şahi ad Telci [0] to G -metric spaces thereby extedig the theorem of Wag et al. [] for a pair of mappigs to G -metric spaces.. Prelimiaries Cosistet with Mustafa ad Sims [6], the followig defiitios ad results will be eeded i the sequel. Defiitio.: ( G -Metric Space [6]). Let X be a oempty set ad let G: X X X R be a fuctio satisfyig the followig properties: ( ) G x, y, z 0 if x y z, () 0 < G x, x, y for all x, y X with x y, () G x, x, y G x, y, z for all x, y, zx with z y, 4 G x, y, z G x, z, y G y, z, x... symmetry i all three variables (5) G x, y, z G x, a, a Ga, y, z for all x, y, z, a X rectagle iequality. The the fuctio G is called a G -metric o X, ad the pair ( X, G ) is called a G -metric space. Defiitio.: ([6]). Let ( X, G ) be a G -metric space ad let { x } be a sequece of poits of X, a poit x X is said to be the limit of the sequece { x } if lim Gx (, x, xm) 0 ad we m, say that the sequece { x } is G -coverget to x. Thus, if x x i a G -metric space X, G, the for ay 0, there exists a positive iteger N G x, x, x <, for all, m N. such that m It has bee show i [6] that the G -metric iduces a Hausdorff topology ad the covergece described i the above defiitio is relative to this topology. The topology beig Hausdorff, a sequece ca coverge at most to oe poit. Propositio.: ([6]). Let ( X, G ) be a G -metric space, the the followig are equivalet: x is G coverget to x. Gx (, x, x) 0 as. Gx (, xx, ) 0 as. 4 Gx (, x, x) 0as. m Defiitio.: ([6]). Let ( X, G ) be a G -metric space, a sequece { x } is called G -Cauchy if for every ε 0, there is a positive iteger N such that Gx (, x, x), for all ml,, N, that is, if m l Gx (, x, x) 0, as, m, l. m l Propositio.: Let ( X, G ) be a G -metric space. The the followig statemets are equivalet: () The sequece { x } is G -Cauchy, () For ay 0, there exists N such that Gx (, x, x), for all m, N. m m Defiitio.4: ([6]). Let X, G, X ', G ' be ' two G -metric spaces. The a fuctio f : X X is G -cotiuous at a poit x X if ad oly if it is G -sequetially cotiuous at x, that is, wheever f x is G - x is G -coverget to x, coverget to f ( x ). Defiitio.5: ([6]). A G -metric space X, G is called symmetric G -metric space if G x, y, y G y, x, x for all x, y X. 608 Vol., Jue 04

4 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces, A Priya Shahi et al. / Defiitio.6: ([6]). A G -metric space X, G is said to be G -complete (or complete G -metric space) if every G -Cauchy sequece i X, G is coverget i X. Matkowski [0] cosidered the set of all real fuctios :[ 0, ) [ 0, ) satisfyig the followig coditios: (i) is o-decreasig ad upper-semicotiuous from the right at 0, (ii) t t for each t 0 ad t 0t 0. Lemma.: ([0]). Let :[ 0, ) [ 0, ) be a fuctio satisfyig the coditios () i ad ( ii ). The lim t 0, t deotes the m, where compositio of () t with -times. Ahmed [] proved the followig commo fixed poit theorem for expasive mappigs i -metric spaces: Theorem.4. Let A, B, S ad T be mappigs of a complete -metric space X, d ito itself such that A ad B are surjective. Suppose that oe of the mappigs A, B, S, T is sequetially cotiuous ad the pairs A, S ad B, T are compatible mappigs of type (A). If there exists such that the iequality ( d( AxBya,, ) dsxtya (,, ) holds, the A, B, S ad T have a uique commo fixed poit. Later, Şahi ad Telci [0] i 00 proved the followig commo fixed poit theorem for expasio type of mappigs i complete coe metric spaces. Theorem.5. Let X, d be a complete coe metric space ad P be a coe. Let f ad g be surjective self-mappigs of X satisfyig the followig iequalities: d( gfx, fx) ad( fx, x) d( fgx, gx) bd( gx, x) for all x i X, where a, b. If either f or g is cotiuous, the f ad g have a commo fixed poit. Defiitio.7. (Compatible Mappigs[4]) Two self mappigs f ad g o a metric space X, d are said to be compatible if lim d( fgx, gfx ) 0, Wheever x is a sequece i X such that lim fx lim gx t for some t X. Defiitio.8. (Compatible Mappigs of type (A) [5]). Two self mappigs f ad g o a metric space X, d are said to be compatible of type (A) if lim d( fgx, ggx ) 0 ad lim d( gfx, ffx ) 0, wheever x is a sequece i X such that lim fx lim gx t for some t X. Defiitio.9. (Weakly compatible mappigs [6]). Two self mappigs f ad g o a metric space ( X, d ) are said to be weakly compatible if they commute at coicidece poits. Compatible maps are weakly compatible but the coverse is ot true. The followig lemma asserts that the cocept of weak compatibility is more geeral tha the cocept of compatibility of type (A). So, i our result we shall make use of this more geeralized otio of compatibility called as weak compatibility. Lemma.6. [5] Let A ad S be self-mappigs of compatible of type (A) of a metric space ( X, d ). If Ax Sx for some x X, the ASx SAx. Joural of Applied Research ad Techology 609

5 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces, A Priya Shahi et al. / Mai Results Let A, BS, ad T be mappigs from G -metric space ( X, G ) ito itself satisfyig the coditios: A ad B are surjective, () ( GAxByBy (,, ) GSxTyTy (,, ) () for each x, y X, where. Sice, A ad B are surjective, oe ca choose a poit x i X for a arbitrary poit x 0 i X such that Ax Tx0 y0 For a poit x, there exists a poit x i X such that Bx Sx. y Iductively, oe ca defie a sequece ( y ) i X such that Ax Tx y, Bx Sx y, {0} where is the set of all positive itegers We, ow prove a commo fixed poit theorem for expasive mappigs i G -metric spaces by usig the cocept of weak compatibility. Theorem.. Let A, BS, ad T be mappigs of a complete symmetric G -metric space ( X, G ) ito itself satisfyig the coditio (). Suppose that the pairs { A, S } ad { BT, } are weakly compatible mappigs. If there exists such that the iequality () holds, the A, BS, ad T have a uique commo fixed poit. Proof. First, we have to prove that the sequece ( y ) defied above is a G - Cauchy sequece. Clearly, we have G( y,, ) y y ( G( y0, y, y)) Therefore, by lemma., Gy (, y, y ) 0 as () Cosider, G( y, ym, ym) G( y, y, y) G( y, y, y)... G( y, y, y ) ( G( y, y, y )) ( G( y, y, y ))... ( (,, )) m m m 0 0 m G y0 y y Usig Lemma., we obtai Gy (, ym, ym) 0 as m,. Thus, { y } is a G Cauchy sequece. Sice ( X, G ) is a complete G-metric space, it yields that ( y ) ad hece ay subsequece thereof, coverge to z X. ( Ax ),( Bx ),( Sx ) So, ad ( Tx ) coverge to z X. Sice A( X ) is a complete G -metric space, so there exists a poit p X such that Ap z. Now, usig iequality (), we have ( G( Ap, Bx, Bx)) G( Sp, Tx, Tx) lettig,, we have ( GApzz (,, )) GSpzz (,, ) That is, 0 (0) GSpzz (,, ). This proves that Sp z. Sice A ad S are weakly compatible mappigs, therefore ASp SAp Az Sz. Now, sice BX ( ) is also a complete G metric space. So, there exists a poit p X such that Bp z. Now, cosider G Ax Bp Bp G Sx Tp Tp ( (,, )) (,, ) lettig,, we get ( GzBp (,, Bp)) GzTp (,, Tp) Recallig that Bp z, we obtai 0 (0) G( z, Tp, Tp ) This implies that Tp z. Sice B ad T are weakly compatible mappigs, therefore, BTp TBp Bz Tz. 60 Vol., Jue 04

6 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces, A Priya Shahi et al. / Now, we have to show that Az z ad Bz z. Let us suppose that GAzzz (,, ) 0. So, usig the iequality () ad the fact that () t t for all t 0, we have GAzzz (,, ) ( GAzzz (,, )) GSzzz (,, ) GAzzz (,, ), which is a cotradictio. So, GAzzz (,, ) 0. That is, Az Sz z. Similarly, let us suppose that GBzzz (,, ) 0. Therefore, by usig the fact that the G metric space X is symmetric ad Az Sz z, we get G( Bz, z, z) ( G( Bz, z, z)) (G( Az, Bz, Bz)) which is GSzTzTz (,, ) G( AzBzBz,, ) GBzzz (,, ), a cotradictio. So, GBzzz (,, ) 0. This proves that Bz Tz z. Therefore, Az Bz Sz Tz z. It follows that z is the uique commo fixed poit of A, BS, ad T. As a corollary of the previous theorem, we have the followig. Corollary.. Let A, BS, ad T be mappigs of a complete symmetric G metric space ( X, G ) ito itself satisfyig the coditio (). Suppose that the pairs { A, S } ad { BT, } are weakly compatible mappigs. Assume that there exists h such that GAxByBy (,, ) hgsxtyty (,, ) (4) for all x, y X. The A, BS, ad T. have a uique commo fixed poit. t Proof. By takig () t, where h i h Theorem., we get the proof of the Corollary.. Corollary.. Let A ad B be mappigs of a complete symmetric G metric space ( X, G ) ito itself satisfyig the coditio (). If there exists such that the iequality ( GAxByBy (,, )) Gxyy (,, ) Holds, the A ad B have a uique commo fixed poit. Proof. If we put S T ix (the idetity mappig o X ) i Theorem., we obtai the proof of the Corollary.. Corollary.4. Let A ad B be mappigs of a complete symmetric G metric space ( X, G ) ito itself satisfyig the coditio (). Assume that there exists h s.t GAxByBy (,, ) hgxyy (,, ) xy, X The, A ad B have a uique commo fixed poit. t Proof. By takig () t, where h i Corollary h., we obtai the proof of the Corollary.4. Remark.. Theorem. is more geeral tha Corollary. as show i the followig example. Example.. Let X {( a, b) : a, b [0,]} ad G be the G metric o X defied by Gxyz (,, ) max{ x y, xz, yz }, where d is the metric o X defied by d(( a, b ),( a, b )) a a b b for all ( a, b),( a, b) X. Defie A, BST,, : X X Aab (, ) Bab (, ) ( a,0), by Sab (, ) Tab (, ) a a,0 for each ( ab, ) X. The it is easily see that A ad S are weakly compatible mappigs. Cosider, t t if 0 t, () t t if t the. Further, we see that Joural of Applied Research ad Techology 6

7 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces, A Priya Shahi et al. / ( G( Ax, By, By)) G( Ax, By, By) G( Ax, By, By)) xy xy for xy xy G( Sx, Ty, Ty) all x ( x, x), y ( y, y) X. Therefore, all the hypothesis of Theorem. are satisfied. However, coditio (4) is ot satisfied. Ideed, for x (0,0), y ( a,0), 0 a ad h, we have G( Ax, By, By) a hg( Sx, Ty, Ty) ha a. This implies that h, which yields a cotradictio. Now, we prove a commo fixed poit theorem for expasive mappigs i G metric spaces, which i tur exteds the results of Şahi ad Telci [0]. Theorem.5. Let ( X, G ) be a complete G metric space. Let f ad g be surjective self mappigs of X satisfyig the followig iequalities: ( G( gfx, fx, fx) G( fx, x, x) (5) ( G( fgx, gx, gx) G( fx, x, x) (6) for all x X, where. If either f or g is cotiuous, the f ad g have a commo fixed poit. Proof. Let x0 X be arbitrary. Due to the reaso that the mappigs f ad g are surjective mappigs, there exists poits x f ( x0) ad x g ( x). Cotiuig i this way, we obtai the sequece { x } with x f ( x) ad x g ( x ). (5) (6) Note that if x x for some, the x is a commo fixed poit of f ad g. Ideed, if x x for some 0, the x is a fixed poit of f.. O the other had, we have from iequality (6) that 0 ( Gx (, x, x )) ( G( fx, gx, gx)) ( G( fgx, gx, gx)) G( gx, x, x ) This implies that Gx (, x, x) 0. So, by the property of a G metric, we have x x. Therefore, x is a commo fixed poit (5), we obtai x is a commo fixed poit of f ad g. So, let us suppose that x x for all. Clearly, we have Gx (, x, x ) ( Gx (, x, x)). 0 Now, we have to prove that { x } is a G Cauchy sequece. For this, we eed to show that, ) 0as Gx (, xm xm m,. Let m, with m, Gx (, xm, xm) Gx (, x, x ) Gx (, x, x)... Gx (, x, x) ( G( x, x, x )) ( G( x, x, x ))... ( (,, )) m m m 0 0 m Gx0 x x Usig lemma., we obtai Gx (, xm, xm) 0 as m,. Hece, { x } is a G Cauchy sequece i X. Sice X is G complete, there exists a poit z X such that lim x z. Now, we cosider that f is cotiuous. Sice x fx, so we have z lim x lim fx fz ad so z is a fixed poit of f. Sice g is surjective, therefore there exists y X such that gy z. Now, usig iequality (6), we have 6 Vol., Jue 04

8 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces, A Priya Shahi et al. / ( G( fzgygy,, )) ( G( fgygygy,, )) Ggyyy (,, ) Gzyy (,, ). This implies that Gzyy (,, ) 0which further implies that z y. Therefore, z is a commo fixed poit of f ad g. Similarly, by cosiderig the cotiuity of g, it ca be proved that f ad g have a commo fixed poit ad this completes the proof. Corollary.6. Let ( X, G ) be a complete G - metric space. Let f ad g be surjective selfmappigs of X satisfyig the followig iequalities G( gfx, fx, fx) ag( fx, x, x) G( fgx, gx, gx) bg( gx, x, x) for all x i X, where ab,. If either f or g is cotiuous, the f ad g have a commo fixed poit. t Proof. Take () t, where h hmax{ a, b} i iequalities (5) ad (6) of Theorem.5, we get the proof. Corollary.7. Let ( X, G ) be a complete G metric space. Let f be a surjective self-mappig of X satisfyig the followig iequality G f x fx fx (,, ) kg( fx, x, x) for all x i X, where k. If f is cotiuous, the f has a fixed poit. Proof. Puttig f g ad k mi{ a, b} i Corollary.6, we get Corollary.7. Remark..Takig G( x, y, z) max{ d( x, y), d( y, z), d( z, x)} for usual metric space ( X, d ) i the corollary.7, we obtai the followig result of Wag et al. []. Corollary.8. Let ( X, d ) be a complete metric space ad let f be a surjective self-mappig of X satisfyig the followig iequality d f x fx (, ) kd( fx, x) for all x i X, where k. If f is cotiuous, the f has a fixed poit. We ow give the followig example i support of our Theorem.5. Example.. Let X [0, ) ad :[0, ) [0, ) t defied by () t. Defie a G metric o X by Gxyz (,, ) max{ x y, xz, y z } Defie the surjective self mappigs f, g: X X by f ( x) 8x ad g( x) 6x for all x i X. The we have ( G( gfx, fx, fx)) ( G(48 x,8 x,8 x)) 0x 7 x G( fxxx,, ) ad ( G( fgx, gx, gx)) ( G(48 x,6 x,6 x) x 5 xggxxx (,, ) hold for all x i X. Thus, iequalities (5) ad (6) are satisfied ad hece all the hypothesis of Theorem.5 are satisfied. Clearly, x 0 is a commo fixed poit of f ad g. Ackowledgmet The first author, Juior Research Fellow (No. F. - 6/0 (SA-I) gratefully ackowledges the Uiversity Grats Commissio, Govermet of Idia for fiacial support. Joural of Applied Research ad Techology 6

9 Commo Fixed Poits of Expasive Mappigs i Geeralized Metric Spaces, A Priya Shahi et al. / Refereces [] M. Abbas ad B.E. Rhoades, Commo fixed poit results for o-commutig mappigs without cotiuity i geeralized metric spaces, Appl. Math. Comput. vol.5, pp. 6-69, 009. [] M. A. Ahmed, A commo fixed poit theorem for expasive mappigs i -metric spaces ad its applicatio, Chaos, Solitos ad Fractals vol. 4, pp , 009. [] K. C. Border, Fixed poit theorems with applicatios to ecoomics ad game theory, Cambridge Uiversity Press, 990. [4] G. Jugck, Compatible mappigs ad commo fixed poits, Iterat. J. Math. Math. Sci., vol.9, pp , 986. [5] G. Jugck, P. P. Marthy ad Y. J. Cho, Compatible Mappigs of Type (A) ad Commo fixed poits, Mathematica Japoica, vol.8, pp. 8-90, 99. [6] G. Jugck, Commo fixed poits for ocotiuous oself maps o ometric spaces, Far East J. Math. Sci., vol.4, pp.99-5, 996. [7] S. M. Kag, Fixed poits for expasio mappigs, Mathematica Japoica, vol. 8, pp. 7-77, 99. [8] S. M. Kag, S. S. Chag ad J. W. Ryu, Commo fixed poits of expasio mappigs, Mathematica Japoica, vol. 4, pp. 7-79, 989. [9] M. A. Kha, M. S. Kha ad S. Sessa, Some theorems o expasio mappigs ad their fixed poits, Demostratio Math., vol. 9, pp , 986. [0] J. Matkowski, Fixed poit theorems for mappigs with cotractive iterate at a poit, Proc. Amer. Math. Soc., vol. 6, pp , 977. [] P. P. Murthy, Importat tools ad possible applicatios of metric fixed poit theory, Noliear Aalysis: Theory, Methods & Applicatios, vol. 47, pp , 00. [] Z. Mustafa, A ew structure for geeralized metric spaces with applicatios to fixed poit theory, Ph.D. thesis, Uiversity of Newcastle, Newcastle, UK, 005. [] Z. Mustafa, M. Khadaqji ad W. Shataawi, Fixed poit results o complete G-metric spaces, Studia Scietiarum Mathematicarum Hugarica, vol. 48, pp. 04-9, 0 doi:0.556/ssc-math [4] Z. Mustafa, H. Obiedat ad F. Awawdeh, Some fixed poit theorem for mappig o complete G-metric spaces, Fixed Poit Theory Appl., Article ID 89870, 008. [5] Z. Mustafa, W. Shataawi ad M. Bataieh, Existece of fixed poit results i G-metric spaces, It. J. Math. Math. Sci., Article ID 808, 009. [6] Z. Mustafa ad B. Sims, A New Approach to Geeralized Metric Spaces, Joural of Noliear ad Covex Aalysis, vol. 7, pp , 006. [7] Z. Mustafa ad B. Sims, Fixed poit theorems for cotractive mappigs i complete G-metric spaces, Fixed Poit Theory Appl. Article ID 9775, 009. [8] B. E. Rhoades, A compariso of various defiitios of cotractive mappigs, Tras. Amer. Math. Soc., vol. 6, pp , 977. [9] B. E. Rhoades, Some fixed poit theorems for pairs of mappigs, Jaabha, vol. 5, pp. 5 56, 985. [0].Şahi ad M. Telci, A theorem o commo fixed poits of expasio type mappigs i coe metric spaces, A. Şt. Uiv. Ovidius Costaƫa, vol. 8, pp. 9-6, 00. [] S. Sessa, O a weak commutativity coditios of mappigs i fixed poit cosideratio, Publicatios De L'istitut Mathématique, vol., pp.49-5, 98. [] T. Taiguchi, Commo fixed poit theorems o expasio type mappigs o complete metric spaces, Mathematica Japoica, vol. 4, pp. 9-4, 989. [] S. Z. Wag, B. Y. Li, Z. M. Gao ad K. Iseki, Some fixed poit theorems o expasio mappigs, Mathematica Japoica, vol. 9, pp. 6-66, 984. [4] E. Zeidler, Noliear Fuctioal Aalysis ad Its Applicatios, New York: Spriger-Verlag, Vol., Jue 04