A Family of Non-Self Maps Satisfying i -Contractive Condition and Having Unique Common Fixed Point in Metrically Convex Spaces *

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1 Advaces Pure Matheatcs htt://dxdoorg/0436/a04036 Publshed Ole July 0 (htt://wwwscrporg/oural/a) A Faly of No-Self Mas Satsfyg -Cotractve Codto ad Havg Uque Coo Fxed Pot Metrcally Covex Saces * Yoge Pao Dogzhe Pao Deartet of Matheatcs College of Scece Yaba Uversty Ya Cha Eal: y66@hotalco Receved February 0; revsed March 9 0; acceted March 6 0 ABSRAC Class of 5-desoal fuctos was troduced ad a coverget sequece detered by o-self ags satsfyg certa -cotractve codto was costructed ad the the lt of the sequece s the uque coo fxed ot of the ags was roved Fally several ore geeral fors were gve Our a results geeralze ad ufy ay sae tye fxed ot theores refereces Keywords: Metrcally Covex Sace; 5-Desoal Fuctos ; Colete -Cotractve Codto; Coo Fxed Pot; Itroducto here have aeared ay fxed ot theores for sgle-valued self-a of closed subset of Baach sace However ay alcatos the ag uder cosderatos s a ot self-ag of closed sets 96 Assad [] gave suffcet codto for such sgle-valued ag to obta a fxed ot by rovg a fxed ot theore for Kaa ags o a Baach sace ad uttg certa boudary codtos o the ag Slar results for ult-valued ags were resecttvely gve by Assad [] ad Assad ad Kr [3] Later soe authors geeralzed the sae tye results o colete etrcally covex etrc saces see [4-8] hose above results were dscussed uder soe cotractve codtos or certa boudary codto Recetly the author dscussed uque coo fxed ot theores for a faly of cotractve or quas-cotractve tye ags o etrcally covex saces or -etrc saces see [9-3] these results rove ay ow coo fxed ot theores I order to geeralze ad ufy further these results ths ote we shall dscuss ad obta soe uque coo fxed ot theores for a faly of ore geeral o-self as satsfyg - cotractve codto o closed subset of a colete etrcally covex etrc sace We eed the followg defto ad lea the * hs aer s suorted by the Foudato of Educato Mstry Jl Provce Cha (No 0 [343]) sequel Defto ([45]) A etrc sace X d s sad to be etrcally covex f for ay x y X wth x y there exsts z X z x z y such d x z d z y d x y Lea ([34]) K s a oety closed subset of a colete etrcally covex etrc sace X d the for ay x K ad y K there exsts z K such d x z d z y dx y Let deotes a faly of ags such each : s cotuous ad creasg each coordate varable ad also t tt tttt for every t 0 where t 0 Obvously here exst ay fuctos whch belogs to : 5 Exale 3 Let : 5 be defed by t t t3 t4 t5 tt t3 t4 t5 he obvously Exale 4 Let : 5 be defed by ttt3t4t t 5 t t3 t4 t5 l he obvously s cotuous ad creasg each coordate varable ad Coyrght 0 ScRes

2 Y J PIAO D Z PIAO 8 Hece l 6 t 6 l t t t t t t t tt ttt t Exale 5 Let : 5 be defed by t t t3 t4 t5 arcta tarcta t arcta t arcta t arcta t he obvously s cotuous ad creasg each coordate varable ad t tt ttt arcta t arcta t arcta t arcta tarcta t 4 arcta t arcta t 4t tt Hece Uque Coo Fxed Pot Here we wll dscuss uque exstece robles of coo fxed ots for a faly of o-self ags satsfyg certa -cotractve codto ad certa boudary codto colete etrcally covex saces heore Let K be a oety closed subset of a colete etrcally covex etrc sace X d ad : K X a faly of o-self as such for each x y K ad wth y dx y dy x dx y d x y q d x x d y where 0 q ad for each Further f K K for each the has a uque coo fxed ot K Proof ae a x0 K We wll costruct two sequeces x ad x the followg aer: Defe x x 0 x K ut x x ; f x K the by Lea there exsts x K such dx 0 x d x x dx0 x Defe x x x K ut x x ; f x K the by Lea there exsts x K such dx x dx x dx x Cotug ths way we obta x ad x : ) x ; x ) f x K the x x ; ) f x K the there exsts x K such dx x x dx x dx by Lea () ad Q x x: x x Let P x x : x x the sce K K for all t s easy to show x Q les x x P () Now we wsh to estate dx x We ca dvde the roof to three cases vew of () Case I x x P we have q dx x dx x dx x dx x dx x q dx x dx x dx x 0 dx x q dx x dx x dx xdx x 0 dx x d x x d x x d x x d x x d x x the (3) becoes dx x dx x dx x < qd x x (3) Whch s a cotradcto sce q hece we have d x x d x x I ths case (3) further becoes the followg dx x dx x dx x qd x x Case II x P x have (4) Q the by ) ad () we d x x d x x dx x dx x dx x dx x q dx x dx x dx x dx x dx x qdx x dx x dx x 0 dx x q dx x dx x dx x dx x 0 dx x d x x d x x the we obta fro (5) (5) ( dx x dx x dx x < qd x x Coyrght 0 ScRes

3 8 Y J PIAO D Z PIAO Whch s a cotradcto sce q hece dx dx x x I ths case we obta fro (5) q dx x dx dx x dx x qd x x d x x d x x x d x x Case III x Q x P By () ad ) we ow x P ad we obta dx x dx x d x x dx xdx x dx x dx x dx x dx x dx x dx x dx x dx x q dx x dx x dx x dx x dx x dx x q dx x dx x dx x dx x dx x Sce dx x dx x dx x dx x dx x d x x d x x ad hece () ca be restated the followg (6) () x dx x q dx x ) dx x dx x dx x dx x dx x q dx x dx x dx x dx x dx x dx x dx dx x x q d x x d x x dx x dx x dx x d x dx x dx x the (8) becoes dx x dx x dx x dx x dx x qdx x d x x d x x q d x x hece dx dx x x q ad therefore by (6) Case II we obta (8) q dx x dx x dx x q q the (8) becoes d x x dx x q dx x dx x dx x dx x dx x dx x qdx x qdx x d x x d x x By (6) Case II aga we obta qqdx x d x x q d x x ( ) (9) (0) hus two stuatos we obta fro (9) ad (0) q dx x ax qqdx x q Hece all three cases (see (4) (6) (9) ad (0)) we fd d x x q ax q qqax dx x dx x q > q M q q q q 0 q ad we have Let ax sce the 0 M ax d x x M d x x d x x for all wth Ad therefore 0 d x x M ax d x x d x x for all wth Let M dx0 x dx x ax the d x x M for all wth Hece for N 0 N N N d x x d x x M as Whch eas x s a Cauchy sequece Let be a lt of x the K sce K s closed ad x K for all Fro () we are easy to ow there exsts a fte subsequece Coyrght 0 ScRes

4 Y J PIAO D Z PIAO 83 x of x such x P x x x For ay fxed we ca tae such d x d x q d dx x d x d x d x q d dx x d x d x d x Let becoes the d d x d Hece x d x d x the by the cotuty of the above d d d d d d d ( ) q d d d q d d qd Sce 0 q d 0 e hs coletes s the coo fxed ot of u ad v are coo fxed ots of the du v q 00 duv duv duv q duv duv duv duv qd u v v 0 d u v duv q d uu d vv d uv d v u d uv Hece d u sce 0q ; ths s u v hs coletes has a uque coo fxed ot he followg s the very artcular for of heore : Corollary Let K be a oety closed subset of a colete etrcally covex etrc sace X d ad : K X a faly of o-self as such for each x y K ad wth dxy arcta dxx arcta dyy 5 arcta d x y arcta d y x arcta d x y Further f K for each the has a uque coo fxed ot K K Proof Let q ad be Exale 3 5 for each the q ad satsfy all codtos of heore hece has a uque coo fxed ot K by heore Fro heore we ca obta ore geeral fors tha heore heore 3 Let K be a oety closed subset of a colete etrcally covex etrc sace X d ad : K X a faly of as ad a faly of ostve tegers such for each x y K ad wth dx y dy x dx y d x y q d x x d y y where 0 q ad for each Further f K K for all the has uque coo fxed ot K S Proof Let for each the S satsfy all codtos of heore hece S has a uque coo fxed ot K Next we rove s also a uque coo fxed I fact for ay fxed ot of S S hece s a fxed ot of S for each ay wth For q d S d S d S d S d qd S d S d S d S d S q d S d S d S S Sce 0 q d S 0 whch les S Slarly for ay wth we ca obta d S qd S hece S herefore s a coo fxed ot of for all By the uqueess of S Coyrght 0 ScRes

5 84 Y J PIAO D Z PIAO coo fxed ot of we have S for all ths eas s a coo fxed ot of u ad v are coo fxed ots of the u ad v are also coo fxed ots of S hece aga by uqueess of coo fxed ots of S we have u v hs co letes s the uque coo fxed ot of heore 4 Let K be a oety closed subset of a colete etrcally covex etrc sace X d ad K X a faly of as ad a : faly of ostve tegers such for each x y K ad wth y y dx x dx y y d y d x y q d x x d where 0 q ad for each Further f ) K K for all ; ) for all wth the has a uque coo fxed ot K Proof For ay fxed has a uque coo fxed ot K by ) ad heore 3 Now we rove for all I fact for each wth sce hece ad ad therefore ) hs eas of for all But by s a coo fxed ot has a uque coo fxed hece for all whch les s a coo fxed ot of hece sce s the uque co- o fxed ot of the uque fxed ot of roof Let * * the s hs coletes our [] N A Assad Fxed Pot heores for Set-Valued rasforatos o Coact Sets Boll U Math Ital Vol No [3] N A Assad ad W A Kr Fxed Pot heores for Set-Valued Mags of Cotractve ye Pacfc Joural of Matheatcs Vol [4] M S Kha H K Patha ad M D Kha Soe Fxed Pot heores Metrcally Covex Saces Georga Joural of Matheatcs Vol No [5] Y J Pao A New Geeralzed Fxed Pot heore Metrcally Covex Metrc Saces Joural of Yaba Uversty (Natural Scece Edto) Vol 9 No [6] S K Chatterea Fxed Pot heores CR ACad Bulgare Sc Vol [] Y J Pao A Fxed Pot heroe for No-Self-Mag Metrcally Covex Metrc Saces Joural of Jl Noral Uversty (Natural Scece Edto) Vol 4 No [8] O Hadzc A Coo Fxed Pot heore for a Faly of Mags Covex Metrc Saces Uv U Novo Sadu Zb Rad Prrod Mat Fa Ser Mat Vol 0 No [9] Y J Pao ad D Z Pao Uque Coo Fxed Pot heores for a Faly of No-Self Mas Metrcally Covex Saces Matheatca Acata Vol No [0] Y J Pao Uque Coo Fxed Pot for a Faly of Quas-Cotractve ye Mas Metrcally Covex Saces Acta Matheatca Sceta Vol 30A No (I Chese) [] Y J Pao Uque Coo Fxed Pot for a Faly of Self-Mas wth Sae ye Cotractve Codto -Metrc Saces Aalyss heory ad Alcatos Vol 4 No do:000/s [] Y J Pao Uque Coo Fxed Pot for a Faly of Self-Mas wth Sae Quas-Cotractve ye Codto -Metrc Sace Joural of Nag Uversty Matheatcal Bquarterly Vol No (I Chese) [3] Y J Pao Uqueess of Coo Fxed Pots for a Faly of Mags wth -Cotractve Codto -Metrc Saces Aled Matheatcs Vol 3 No 0 3- do:0436/a030 REFERENCES [] N A Assad O Fxed Pot heore of Kaa Baach Saces akag Joural of Matheatcs Vol Coyrght 0 ScRes