Iteratioal Refereed Joural of Egieerig ad Sciece (IRJES ISSN (Olie 239-83X (Prit 239-82 Volume 2 Issue 4(April 23 PP.22-28 Uique Commo Fixed Poit Theorem for Three Pairs of Weakly Compatible Mappigs Satisfyig eeralized Cotractive Coditio of Itegral Type Kavita B. Bajpai 2 Majusha P. adhi Karmavir Dadasaheb Kaamwar College of Egieerig Nagpur Idia Yeshwatrao Chava College of Egieerig Waadogri Nagpur Idia Abstract: We prove some uique commo fixed poit result for three pairs of weakly compatible mappigs satisfyig a geeralized cotractive coditio of Itegral type i complete -metric space.the preset theorem is the improvemet ad extesio of Vishal upta ad Navee Mai [5] ad may other results existig i literature. Keywords: Fixed poit Complete - metric space -Cauchy sequece Weakly compatible mappig Itegral Type cotractive coditio. I. Itroductio eeralizatio of Baach cotractio priciple i various ways has bee studied by may authors. Oe may refer Beg I. & Abbas M.[2] Dutta P.N. & Choudhury B.S.[3] Kha M.S. Swaleh M. & Sessa S.[9] Rhoades B.E.[2] Sastry K.P.R. & Babu.V.R.[3] Suzuki T.[5]. Alber Ya.I. & uerre-delabriere S. [] had proved results for weakly cotractive mappig i Hilbert space the same was proved by Rhoades B.E.[2] i complete metric space. Jugck.[6] proved a commo fixed poit theorem for commutig mappigs which is the extesio of Baach cotractio priciple. Sessa S.[4] itroduced the term Weakly commutig mappigs which was geeralized by Jugck.[6] as Compatible mappigs. Pat R.P.[] coied the otio of R-weakly commutig mappigs whereas Jugck.& Rhoades B.E. [8] defied a term called weakly compatible mappigs i metric space. Fisher B. [4] proved a importat Commo Fixed Poit theorem for weakly compatible mappig i complete metric space. Mustafa i collaboratio with Sims [] itroduced a ew otatio of geeralized metric space called - metric space i 26. He proved may fixed poit results for a self mappig i - metric space uder certai coditios. Now we give some prelimiaries ad basic defiitios which are used through-out the paper. Defiitio.: Let X be a o empty set ad let followig properties: ( ( if x y z ( 2 y for all y X with x y ( 3 y for all z X with y z : X X X R be a fuctio satisfyig the ( 4 z y z x (Symmetry i all three variables ( 5 a a a for all z a X (rectagle iequality The the fuctio is called a geeralized metric space or more specially a - metric o X ad the pair (X is called a metric space. Defiitio.2: Let ( X be a - 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 ( x x ad we say that the sequece { x } m is - coverget to x or { x } -coverges to x. { m 22 Page
Thus x x i a - metric space ( X if for ay there exists k N such that x x x for all m k ( m Propositio.3: Let ( X be a - metric space. The the followig are equivalet: i { x } is - coverget to x x x x ii as iii x x as iv x xm x as m Propositio.4 : Let ( X be a - metric space. The for ay x y z a i X it follows that i If ( the x y z ii y iii y 2 x iv a a v 2 a a a 3 vi a a a a z a a Defiitio.5: Let ( X be a - metric space. A sequece x } is called a - Cauchy sequece if for ay there exists N m m l that is ( x x m x l as m l. Propositio.6: Let ( X be a - metric space.the the followig are equivalet: { k such that x x x for all k ( l i The sequece { x } is - Cauchy; ii For ay there exists k N such that x x x for all m k ( m m Propositio.7: A - metric space ( X is called -complete if every -Cauchy sequece is - coverget i ( X. Propositio.8: Let (X be a - metric space. The the fuctio is joitly cotiuous i all three of its variables. Defiitio.9 : Let f ad g be two self maps o a set X. Maps f ad g are said to be commutig if fgx gfx for all x X Defiitio. : Let f ad g be two self maps o a set X. If fx gx for some x X the x is called coicidece poit of f ad g. Defiitio.: Let f ad g be two self maps defied o a set X the f ad g are said to be weakly compatible if they commute at coicidece poits. That is if fu gu for some u X the fgu gfu. The mai aim of this paper is to prove a uique commo fixed poit theorem for three pairs of weakly compatible mappigs satisfyig Itegral type cotractive coditio i a complete metric space. The result is the extesio of the followig theorem of Vishal upta ad Navee Mai [5]. II. Theorem Let S ad T be self compatible maps of a complete metric space (X d satisfyig the followig coditios i S( X T( X ii d ( S Sy ( t ( t d ( T Ty d ( T Ty ( t for each y X where : is a cotiuous ad o decreasig fuctio ad : is a lower semi cotiuous ad o decreasig fuctio such that ( t ( t ad oly if : t also is a Lebesgue-itegrable fuctio which is summable o each if 23 Page
compact subset of uique commo fixed poit. R oegative ad such that for each ( t. The S ad T have a III. MAIN RESULT Theorem 2. : Let ( X be a complete -metric space ad L M N P Q R : X X be mappigs such that i L( X P( X M( X Q( X N( X R( X L M N P Q Rz P Q Rz ii ---------------------(2.. for all z X where : is a cotiuous ad o-decreasig fuctio : is a lower semi cotiuous ad o-decreasig fuctio such that ( t ( t if ad oly if f : is a Lebesgue itegrable fuctio which is summable o each compact subset of t also R o egative ad such that for each iii The pairs ( L P ( M Q ( N R are weakly compatible. The L M N P Q R have a uique commo fixed poit i X. Proof : Let x be a arbitrary poit of X ad defie the sequece x i X such that y Lx Px y Mx Qx2 y 2 Nx2 Rx 3 Cosider y y Lx Mx 2 Nx2 Px Qx Rx2 Px Qx Rx2 --------------(2..2 Sice is cotiuous ad has a mootoe property 2 ( t 2 f ------------------(2..3 Let us take the it follows that is mootoe decreasig ad lower bouded sequece of umbers. Therefore there exists k such that k as. Suppose that k Takig limit as o both sides of (2..2 ad usig that is lower semi cotiuous we get ( k ( k ( k (k which is a cotradictio. Hece k. 2 This implies that as i.e. as.----------------(2..4 24 Page
Now we prove that y is a - Cauchy sequece. O the cotrary suppose it is ot a - Cauchy sequece. There exists ad subsequeces y ad y such that for each positive iteger i (i is i (i ( i i i ( i i i ( i ym ( i ym ( i ( i ym ( i ym ( i ym ( i ym ( i ym ( i ym ( i ym ( i ym ( i miimal i the sese that y y y ad y y y Now y Let Takig ----------------(2..5 y ( i i i i i i y y y i ad usig (2..4 we get Now usig rectagular iequality we have y y y y y y lim i y ( i y y y i y i -------------(2..6 y y y y y y (2..7 y ( i m ( i m ( i ( i ( i ( i ( i m ( i m ( i m ( i m ( i m ( i y y y y y y y y y y y ( i m ( i m ( i ( i ( i ( i ( i m ( i m ( i m ( i m ( i m ( i (2..8 y y y y y y y y y y y y ad ( i i i ( i ( i ( i ( i i i i i i y y y ( i ( i ( i ( i y i y i ym ( i y i y i ( i i i Takig limit as i ad usig (2..4 (2..6 we get y y y y y y ( i i i ( i i i This implies that lim i y ( i y i y i ----------------------(2..9 Now from (2.. we have ( i y i y i ( i y i y i ( i y i y i Takig limit as i ad usig (2..6 (2..8 we will have ( ( ( ( which is a cotradictio. Hece we have. Hece y is a - Cauchy sequece. Sice ( X is a complete -metric space there exists a poit u X such that lim u i.e. y lim Lx lim Px u lim Mx lim Qx2 u lim Nx2 lim Rx 3 u Lx u ad Px u h such that Qh u. As therefore we ca fid some X Lx Mh Mh Lx Mh Nx Px Qh Rx Px Qh Rx Mh Mh O takig limit as we get ( ( ---- ---- 25 Page
Mh Mh which implies that Mh u. Hece Mh Qh u i.e. h is the poit of coicidece of M ad Q. Sice the pair of maps M ad Q are weakly compatible we write MQh QMh i.e. Mu Qu. Also Mx u ad Qx 2 u we ca fid some v X such that Pv u. Lv Mx Mx Lv Mx Nx Pv Qx Rx Pv Qx Rx O takig limit as 2 2 Lv u ( ( 2 we get Lv u which implies that Lv u. Hece we have Lv Pv u i.e. v is the poit of coicidece of L ad P. Sice the pair of maps L ad P are weakly compatible we ca write LPv PLv i.e. Lu Pu. Agai Nx 2 u ad Rx 3 u therefore we ca fid some w X such that Rw u. Lx Mx Nw Px Qx Rw Px Qx Rw Nw O takig limit as we get ( ( Nw i.e. which implies that Nw u. Thus we get Nw Rw u i.e. w is the coicidece poit of N ad R. Sice the pair of maps N ad R are weakly compatible we have NRw RNw i.e. Nu Ru Now we show that u is the fixed poit of L. L u L Mh Nw Pu Qh Rw Pu Qh Rw Cosider i.e. L u L u L u L u L u L u L u L u i.e. which is a cotradictio. we get Lu u Lu Pu u i.e. u is fixed poit of L ad P. Now we prove that u is fixed poit of M. Mu L M Nw Pu Q Rw Pu Q Rw Cosider Mu M u M u 26 Page
Mu Mu i.e. which is a cotradictio. we get Mu u Hece Mu Qu u i.e. u is fixed poit of M ad Q. At last we prove that u is fixed poit of N. Nu L M Nu Pu Q Ru Pu Q Ru Cosider i.e. Nu Ru Ru Nu Ru Nu Nu i.e. which meas as Nu Ru. Which implies that Nu u. Hece we get Nu Ru u. i.e. u is fixed poit of N ad R. Thus u is the commo fixed poit of L M N P Q ad R. Now we prove that u is the uique commo fixed poit of L M N P Q ad R. If possible let us assume that is aother fixed poit of L M N P Q ad R. L M N Pu Q R Pu Q R Thus u is the uique commo fixed poit of Corollary 2.2: Let ( be mappigs such that i ( X P( X ( X P( X i.e. which is agai a cotradictio. Hece fially we will have u. L M N P Q ad R. X be a complete -metric space ad L M N P : X X L M N( X P( X L M N P Py Pz P Py Pz ii for all z X where : is a cotiuous ad o-decreasig fuctio : is a lower semi cotiuous ad o-decreasig fuctio such that ( t ( t if ad oly if f : is a Lebesgue itegrable fuctio which is summable o each compact subset of t also R o egative ad such that for each iii The pairs ( L P ( M P ( N P are weakly compatible. The L M N P have a uique commo fixed poit i X. Proof : By takig P Q R i Theorem 2. we get the proof. Corollary 2.3: Let ( X be a complete -metric space ad L P : X X be mappigs such that 27 Page
i L( X P( X L L L P Py Pz P Py Pz ii for all z X where : is a cotiuous ad o-decreasig fuctio : is a lower semi cotiuous ad o-decreasig fuctio such that ( t ( t if ad oly if f : is a Lebesgue itegrable fuctio which is summable o each compact subset of t also R o egative ad such that for each iii The pair ( L P is weakly compatible. The L P have a uique commo fixed poit i X. Proof: By substitutig L M N ad P Q R i Theorem 2. we get the proof. Remark: The Corollary 2.3 is the result proved by Vishal upta ad Navee Mai [5] i complete metric space. IV. Refereces [] Alber Ya.I. & uerre-delabriere S. (997. Priciple of weakly cotractive maps i Hilbert spaces New Results i Operator theory ad its applicatios i I.ohberg ad Y.Lyubich (Eds. 98 Operator Theory: Advaces ad Applicatios(7-22. Birkhauser Basel Switzerlad. [2] Beg I. & Abbas M.Coicidece poit ad ivariat approximatio for mappigs satisfyig geeralized weak cotractive coditio. Fixed poit theory ad Appl. article ID 7453-7. [3] Dutta P.N. & Choudhury B.S. (28. A geeralizatio of cotractio priciple i metric spaces. Fixed poit theory ad Appl. article ID46368-8. [4] Fisher B. Commo Fixed Poit of Four Mappigs Bull. Ist. of.math.academia. Siicia (983 3-3. [5] upta V. & Mai N. A Commo Fixed Poit Theorem for Two Weakly Compatible Mappigs Satisfyig a New Cotractive Coditio of Itegral Type Mathematical Theory ad Modelig Vol. No. 2 [6] Jugck. (976. Commutig mappigs ad fixed poits. Amer.Math.Mothl 83 26-263. [7] Jugck. (986. Compatible mappigs ad commo fixed poits. Iterat. J. Math. Sci. 9 43-49. [8] Jugck. & Rhoades B.E. (998. Fixed poits for set valued fuctios without cotiuity. Idia J.Pure.Appl.Math. 29 No. 3 227-238. [9] Kha M.S. Swaleh M.&Sessa S. (984. Fixed poit theorems by alterig distaces betwee the poits Bull.Austral.Math.Soc. 3-9. [] Mustafa Z. Sims B. A ew approach to geeralized metric spaces J.Noliear Covex Aal. 7 (26 289-297. [] Pat R.P. (994. Commo fixed poits of o commutig mappigs. J.Math.Aal.Appl. 88 436-44. [2] Rhoades B.E. (2. Some theorems o weakly cotractive maps Noliear Aalysis:Theory. Methods&Applicatios47 (4 2683-2693. [3] Sastry K.P.R. Naidu S.V.R. Bab.V.R. & Naid.A. (2. eeralizatios of commo fixed poit theorems for weakly commutig mappigs by alterig distaces. Tamkag J.Math. 3 243-25. [4] Sessa S. (982. O weak commutative coditio of mappigs i fixed poit cosideratios. Publ.Ist.Math. N.S. 32 o.46 49-53. [5] Suzuki T. (28. A geeralized Baach cotractio priciple that characterizes metric completeess. Proc.Amer.Math.soc.36 (5 86-869. 28 Page