Generalized ( ψϕ-weak, ) Contractions In 0-complete Partial Metric Spaces

Transcription

1 Joural of Mathematical Scieces Alicatios, 06, Vol 4, No, 4-9 Available olie at htt://ubsscieubcom/jmsa/4//3 Sciece Educatio Publishig DOI:069/jmsa-4--3 Geeralized ( ϕ-wea, Cotractios I 0-comlete Partial Metric Saces Mehmet Ali Atur,*, Esra Yolaca Istabul Uiversit, Facult of Egieerig, Deartmet of Egieerig Scieces, Avcilar Camus-3430, Istabul, Ture Reublic of Ture Miistr of Natioal Educatio, Mathematics Teacher, Toat, Ture *Corresodig author: mehmetaliatur@excom Abstract I this aer, we rove some commo fixed oit theorems i 0-comlete artial metric saces Our results exted geeralize ma existig results i the literature Some examles are icluded which show that the geeralizatio is roer Kewords: artial metric sace, wea cotractio, fixed oit Cite This Article: Mehmet Ali Atur, Esra Yolaca, Geeralized ( ϕ, -wea Cotractios I 0-comlete Partial Metric Saces Joural of Mathematical Scieces Alicatios, vol 4, o (06: 4-9 doi: 069/jmsa-4--3 Itroductio Prelimiaries Partial metric saces were itroduced b Matthews i [9] as a art of the stud of deotatioal sematics of dataow etwors I fact, it is widel recogized that artial metric saces la a imortat role i costructig models i the theor of comutatio [0,,,3,4] Defiitio [9] A artial metric o a oemt set is a fuctio : such that for all xz,, x = x, x = x, =,, (ms ( ( ( (ms ( x, x ( x,, (ms3 ( x, = ( x,, (ms4 ( x, ( x, z ( z, ( z, z The air ( is called a artial metric sace If is a artial metric o, the the fuctio s s : give b ( x, = x (, xx (,, is a metric o Each artial metric o ( iroduces a T 0 toolog τ o which has as a base the famil of oe balls D ( x, ε = { c : ( x, c < ( x, x ε} for all x ε > 0 be a artial metric sace, Defiitio [9] Let (, let { x } be a sequece i x The (a a sequece { x } is coverget to x with resect to τ, if lim ( x, x = ( x, x ; (b a sequece { x } is a Cauch sequece i ( if lim ( x, x m, m exists is fiite; (c ( is called comlete if for ever Cauch sequece { x } i there exists x such that lim ( x, x ( x, x m, m = I 00, Romaguera roved i [4-Theorem 3] that a is 0-comlete if ol if artial metric sace ( s ever -Caristi maig o has a fixed oit Sice the several aers have dealt with fixed oit theor for sigle-valued multi-valued oerators i 0-comlete artial metric sace (see [-8] refereces therei be a artial metric sace A Defiitio 3 [4] Let ( sequece { x } i is called a 0-Cauch sequece if lim m, ( x, xm = 0 The sace ( is said to be 0-comlete if ever 0-Cauch sequece i coverges with resect to τ to a oit x such that ( xx, = 0 Remar [,6] Let ( be a artial metric sace If ( x, z ( z, z = 0 as, the ( x, ( z, as for all Lemma [] Let ( be a artial metric sace let { } be a sequece i such that lim (, = 0 ( If { } is ot 0-Cauch sequece i (, ε > two sequeces { m } { } there exists 0 ositive itegers such that m sequeces ted to ε as :, the of > > the followig

2 Joural of Mathematical Scieces Alicatios ( m,, ( m,, ( m ( m,,, ( Defiitio 4 [7] Let f g be self mas of a set If w = fx = gx for some x the x is called a coicidece oit of f g, w is called a oit of coicidece of f g The air f, g of self mas is weal comatible if the commute at their coicidece oits Proositio [7] Let f g be weal comatible self mas of a set If f g have a uique oit of coicidece w = fx = gx, the w is the uique commo fixed oit of f g Mai Results Deote b Ψ the set of fuctios : [ 0, [ 0, satisfig the followig coditios: is cotiuous odecreasig; ( i ( ii ( t < t for all t > 0 ( 0 < 0 Deote b Φ the set of fuctios : [ 0, [ 0, satisfig the followig coditios: ϕ ϕ is a lower semi-cotiuous fuctios; ( i ( ϕ ii ( t ϕ < t for all t > 0 ϕ ( 0 = 0 Theorem Let ( be a 0-comlete artial metric saces Suose maigs f, g: satisf ( ( fx, f ( M ( x, ( M ( x, ϕ ( where Ψ ϕ Φ ( ( ( ( ( gx, g, gx, f, g, f, M( x, = max gx, f g, fx ( for all x, If the rage of g cotais the rage of f f ( or g ( is a closed subset of the f g have a uique oit of coicidece i Moreover, if f g are weal comatible, the f g have a uique commo fixed oit z ( v, v = 0 = ( fz, fz = ( gz, gz Proof First, we rove that f g have a uique oit of coicidece (if it exists If c with fa = ga = c c with fa = ga = c, we assume c 6= c Usig ( (, we have ( ( c, c = ( ( fa, fa ( ga ga ( ga fa ( ga fa max ( ga, fa ( ga, fa,,,,,, ( ga, ga, ( ga, fa, ( ga, fa, max ( ga, fa ( ga, fa ( ( ( ( ( c, c, c, c, c, c, = max c, c c, c ( c, c, ( c, c, ( c, c, max ( c, c ( c, c = < ( ( c, c ϕ( ( c, c ( b ( ms ( ( c, c, which is a cotradictio Thus (, c c, that is, c = c Thus, the oit of coicidece of f g is uique (if it exists We costruct a sequece { } as follows: Let x0 Choose a oit x such that fx0 = gx0 = This ca be doe, as the rage of g cotais the rage of f Cotiuig i the same wa, havig chose x we get x such that fx = gx = (sa Therefore, we get the sequece { } = { gx } such that fx = gx = for all x Cosider the two ossible cases: (i (, = 0 for some I this case fx = gx = is a oit of coicidece the the roof is fiished (ii (, = 0 for ever From ( (, usig roerties of fuctios ϕ, we obtai ( (, = ( ( fx, fx ( M ( x, x ( M ( x, x ( M ( x, x which imlies that The, we have ( (, M x, x (, x ( gx gx ( gx fx ( gx fx ( gx fx ( gx fx M x,,,,,, = max,, (,, (,, (,, = max (, (, (,, (,, max (, (, = max { (,, (, } If (, > (,, the M ( x, x (, 0 = > Furthermore, it imlies that

3 6 Joural of Mathematical Scieces Alicatios ( (, ( (, ( (, ϕ which is a cotradictio Therefore, we have ( ( (, M x, x, (3 It follows from (3 that the sequece ( gx, gx is oicreasig Therefore, { } ( ( * lim, = lim M x, x = 0 Lettig i iequalit ( (, ( M ( x, x ( M ( x, x ϕ we obtai ( * ( * ϕ( * ( lim, 0 * = 0 Thus = (4 We ext rove that { fx} = { gx } = { } is a 0-Cauch sequece i the sace ( It is sufficiet to show that { fx } is a 0-Cauch sequece Suose the oosite The usig Lemma, we see that there exist 0 of ositive ε > two sequeces { m } { } itegers sequeces ( m,, (,, m ( m ( m,,, ( all ted to ε, whe Usig ( (, we get that ( ( m, = ( ( fxm, fx ( gxm, gx, ( gxm, fx, m max ( gx, fx, ( gxm, fx ( gx, fx m ( gxm, gx, ( gxm, fxm, max ( gx, fx, ( gxm, fx ( gx, fx m ( m,, ( m, m, = max (,, ( m, (, fx m ( m,, ( m,, m max (,, (6 ( m, (, fx m Usig (4 (, we obtai max,, fx ε as ( m,, (,, m m ( ( m, (, m Lettig i (6, we get that ( ε ( ε ϕ( ε which is a cotradictio if ε > 0 This show that { fx } is a 0-Cauch sequece i the { fx } is a 0-Cauch sequece i the sace (, sace ( If g ( is closed i (, such that v = gz the there exist zv, ( ( ( lim, v = lim, m = v, v = 0 m, Now, uttig x = x, = z, gz = v = fx = gx i ( ( we have ( ( fx, fz ( gx gz ( gx fx ( gz fz max ( gx, fz ( gz, fx,,,,,, (7 ( gx, gz, ( gx, fx, ( gz, fz, max ( gx, (, fz gz fx Lettig i (7 b Remar, we obtai ( ( gz, fz ( ( gz, fz ϕ( ( gz, fz This imlies ( gz, fz = 0, that is, gz fz = Hece, f g have a uique oit of coicidece B Proositio, f g have a uique commo fixed oit the roof similar Whe f ( is closed set i ( Corollar Let ( be a 0-comlete artial metric saces Suose maig f : satisf ( ( fx, f ( M ( x, ( M ( x, ϕ (8 where Ψ ϕ Φ

4 Joural of Mathematical Scieces Alicatios 7 ( ( ( ( ( x,, x, fx,, f, M( x, = max x, f, fx (9 for all x, The f has a uique fixed oit v ( vv, = 0 Proof Taig g = I (the idetit maig of, alog the lies of the roof of Theorem, we get the desired results I view of the aalog, we si the details of the roof be a 0-comlete artial metric Corollar Let ( saces Suose maigs f, g: satisf where ϕ Φ (, (, ϕ ( (, P fx f M x M x (0 ( ( ( ( ( gx, g, gx, fx, g, f, M( x, = max gx, f gx, fx ( for all x, If the rage of g cotais the rage of f f ( or g ( is a closed subset of the f g have a uique oit of coicidece i Moreover, if f g are weal comatible, the f g have a uique commo fixed oit z ( = = ( = ( v, v 0 fz, fz gz, gz Proof To rove the above corollar it suffices to tae ( t = t i Theorem Corollar 3 Let (; be a 0-comlete artial metric saces Suose maig f :! satisf where ϕ Φ (, (, ϕ ( (, fx f M x M x ( ( ( ( ( ( x,, x, fx,, f, M( x, = max x, f, fx (3 for all x, The f has a uique fixed oit v ( vv, = 0 Proof Taig g = I i Corollar, we have desired results Corollar 4 [] Let ( be a 0-comlete artial metric saces Suose maigs f, g: satisf where [ 0, (, M ( x, fx f (4 ( ( ( ( ( gx, g, gx, fx, g, f, M( x, = max gx, f g, fx ( for all x, If the rage of g cotais the rage of f f ( or g( is a closed subset of the f g have a uique oit of coicidece i Moreover, if f g are weal comatible, the f g have a uique v, v = 0 = fz, fz commo fixed oit z ( ( = ( gz, gz Proof To rove the above corollar it suffices to tae ϕ t = t i Corollar ( ( Corollar Let ( be a 0-comlete artial metric saces Suose maig f : satisf where [ 0, (, M ( x, fx f (6 ( ( ( ( ( x,, x, fx,, f, M( x, = max x, f, fx (7 for all ( x, The f has a uique fixed oit v ( vv, = 0 Corollar 6 [8] Let ( be a 0-comlete artial metric saces Suose maig f : there exist oegative costats bi satisfig bi < such i= that, for each x, (, (, (, 3 (, b4 ( x, f b (, fx ( fx f b x b x fx b f (8 The f has a uique fixed oit v vv, = 0 Corollar 6 is a simle cosequece of Corollar be a 0-comlete artial metric Corollar 7 Let ( saces Suose maig f : satisf for each x, fixed oit v ( fx, f ( x, [ 0, ( vv, = 0 (9 The f has a uique Proof It follows from Corollar 6 Coclusio Our theorems corolaries which iclude the corresodig results aouced i Bod Wog [9] (969, Rhoades [0] (977 as secial cases fudametall imrove geeralize the results of Ahmad et al [] (0 Radeović [8] (03 Taig b = b4 = b = 0 b b3 = λ 0, i Corollar 6, we obtai extesio of Kaa Theorem o a 0-comlete artial metric saces 3 Taig b4 = b = 0 b b b3 [ 0, i Corollar 6, we obtai extesio of Reich Theorem o a 0-comlete artial metric saces 4 Taig b = b = b3 = 0 b 4 = b = λ 0, i Corollar 6, we obtai extesio of Chatterjea Theorem o a 0-comlete artial metric saces Now, we give a examle which illustrate Theorem Examle Let = { 0,,,3}, le : be defied b (, max {, } x = x x for all

5 8 Joural of Mathematical Scieces Alicatios x, The, (, sace Defie f, g: is a 0-comlete artial metric f0 = 0, f = 0, f = 0, f3 =, g0 = 0, g =, g =, g3 = 3 t Tae ( t = t ϕ ( t = for each t 0 We distiguish five cases: Case : If (x = 0 = 0 or (x = 0 = or (x = 0 = or (x = = or (x = = or (x = =, we have where ( ( fx, f = 0 ( M ( x, ( M ( x, ϕ ( ( ( (, (, gx, g, gx, fx, g, f, M( x, = max gx f g fx Case : If x = 0 = 3, we have ( ( f f ( 0, 3 = 0, = ( M ( 0,3 ( M ( 0,3 ( g g ( g f ( g f max ( g0, f3 ( g3, f0 0, 3, 0, 0, 3, 3, = ( ( ( ( ( g0, g3, g0, f0, g3, f3, max g0, f3 g3, f0 ( 0, ( 3, 0 = max ( 0,3, ( 0,0, ( 3,, ( 0, ( 3, 0 max ( 0,3, ( 0,0, ( 3,, 6 6 = max 6,0,, max 6,0,, = 6 3 = 3 Hece, ( ( f f ( M ( ( M ( 0, 3 = 0, 3 ϕ 0, 3 = 3 Case 3: If x = = 3, we have ( ( f f (, 3 = 0, = ( M (, 3 ( M (, 3 ( g g ( g f ( g f max ( g, f3 ( g3, f, 3,,, 3, 3, = ( ( ( ( ( g, g3, g, f, g3, f3, max g, f3 g3, f (, ( 3, 0 = max (, 3, (, 0, ( 3,, (, ( 3, 0 max (, 3, (, 0, ( 3,, 6 6 = max,,, max,,, = = Thus, ( ( f, f 3 = ( M (, 3 ϕ( M (, 3 = Case 4: If x = = 3, we have ( ( f f (, 3 = 0, = ( M (,3 ( M (,3 ( g g ( g f ( g f max ( g, f3 ( g3, f, 3,,, 3, 3, = ( ( ( ( ( g, g3, g, f, g3, f3, max g, f3 g3, f (, ( 3, 0 = max (,3, (,0, ( 3,, (, ( 3, 0 max (,3, (,0, ( 3,, = max 4, 4,, max 4, 4,, = = Thus, ( ( f, f 3 = ( M (, 3 ϕ( M (, 3 = Case : If x = 3 = 3, we have ( ( f f ( 3, 3 =, = ( M ( 3,3 ( M ( 3,3 ( g g ( g f ( g f max ( g3, f3 ( g3, f3 3, 3, 3, 3, 3, 3, = ( ( ( ( ( g3, g3, g3, f3, g3, f3, max g3, f3 g3, f3 ( 3, ( 3, = max ( 3,3, ( 3,, ( 3,,

6 Joural of Mathematical Scieces Alicatios 9 ( 3, ( 3, max ( 3,3, ( 3,, ( 3,, = max 3,,, max 3,,, = = Thus, ( ( f3, f 3 = ( M ( 3, 3 ϕ( M ( 3, 3 = It is obvious that all the coditio of Theorem is satisfied Therefore, we al Theorem f g have a uique commo fixed oit, ie 0 The followig is a examle which illustrate our results that the geeralizatios are roer Examle Let = [ 0, ], let : be defied b x (, max { x, } x = for all, The, ( is a 0-comlete artial metric sace, but it is ot comlete artial metric sace Defie f, g: b 0 if x =, if x =, fx = x gx = otherwise x otherwise The all the coditios of Theorem are satisfied with t ( t = t ϕ ( t = f g have a uique commo fixed oit, ie 0 Acowledgemet The authors wish to tha the editor referees for their helful commets suggestios Refereces [] Hussai, N, Al-Mezel, S, Salimi, P: Fixed oits for - Grahic Cotractios with Alicatio to Itegral Equatios Abstract Alied Aalsis Volume 03, Article ID 7869, ages [] Ahmad, AGB, Fadail, ZM, Rajić, VĆ, Radeović, S: Noliear Cotractios i 0-comlete artial metric saces Abstract Alied Aalsis Volume 0, Article ID 439, ages [3] Nashie et al: Fixed oit theorems uder Hard-Rogers cotractive coditios o 0-comlete ordered artial metric saces Fixed Poit Theor Al 0 (0, 80 [4] Romaguera, S: A Kir Te characterizatio of comleteess for artial metric saces Fixed Poit Theor Al Volume 00, Article ID 49398, 6 ages [] Shula, S, Radeović, S: Some commo Fixed Poit Theorems for F-Cotractio Te Maigs i 0-comlete artial metric saces Joural of Mathematics Volume 03, Article ID , 7 ages [6] Shula, S, Radeović, S, Vetro, C: Set-Valued Hard-Rogers Te Cotractio i 0-comlete artial metric saces Iteratioal Joural of Mathematics Mathematical Scieces, Volume 04, Article ID 69, 9 ages [7] Shula, S: Set-Valued PREŠIĆ-ĆIRIĆ Te cotractio i 0- comlete artial metric saces Matematiqi Vesi, 66, (04, 78-89, Jue 04 [8] Paesao, D, Vetro, C: Multi-valued F-cotractios i 0-comlete artial metric saces with alicatio to Volterra te itegral equatio Revista de la Real Academia de Ciecias Exactas, Fisicas Naturales Serie A Matematicas Setember 04, Volume 08, Issue, [9] Matthews, SG: "Partial metric toolog", Proc 8 th Summer Coferece o Geeral Toolog Alicatios, A New Yor Acad Sci, 78 ( [0] Hecma, R: Aroximatio of metric saces b artial metric saces, Alied Categorical Structures, vol 7, o:-, 7-83, 999 [] Romaguera, S, Schellees, M: Partial metric mooids semivaluatio saces, Toolog Its Alicatios, vol 3, o:-6, , 00 [] Romaguera, S, Valero, O: A quatitative coutatioal model for comlete artial metric saces via formal balls, Mathematical Structures i Comuter sciece, vol 9, o 3, 4-63, 009 [3] Schellees, M: The Smth comletio: a commo foudatio for deotatioal sematics comlexit aalsis, Electroic Notes i Theoretical Comuter Sciece, Vol, 3-6, 99 [4] Schellees, M: A charecterizatio of artial metrizabilit: domai are quatifiable, Theoretical Comuter Sciece, vol 30, o -3, , 003 [] Abdeljawad, T, Karaıar, E, Taş, K: Existece uiqueess of a commo fixed oit o artial metric saces Al Math Lett 4, (0 [6] Karaıar, E, Erhai IM: Fixed oit theorems for oerators o artial metric saces AlMath Lett 4, (0 [7] Abbas, M, Jugc, J: Commo fixed oit results for ocommutig maigs without cotiuit i coe metric saces J Math Aal Al, 34, (008 [8] Radeovi_c, S: Remars o some couled fixed oit results i Partial metric saces Noliear Fuctioal Aalsis Alicatios, vol 8 No (03, 39-0 [9] Bod, DW, Wog, JSW: O oliear cotractios Proceedigs of the America Mathematical Societ 0 ( [0] Rhoades, BE: comarasio of various de_itios of cotractive maigs, Trasactios of the America Mathematical Societ, 6 (