Fixed points and approximate fixed points in product spaces

TAIWANESE JOURNAL OF MATHEMATICS Vol. 5, No. 2, pp. 405-416, June 2001 This pape is a ailable online a h p://www.ma h.n hu.edu. w/ jm/ FIXED POINTS AND APPROXIMATE FIXED POINTS IN PRODUCT SPACES R. Espínola and W. A. Ki k Abs ac . The pape deals wi h he gene al heme o wha is known abou he exis ence o ixed poin s and app oxima e ixed poin s o mappings which sa is y geome ic condi ions in p oduc spaces. In pa icula i is shown ha i Xand Ya e me ic spaces each o which has he ixed poin p ope y o nonexpansi e mappings, hen he p oduc space (X×Y)∞has he ixed poin p ope y o nonexpansi e mappings sa is ying a ious con ac i e condi ions. I is also shown ha he p oduc space H=(M×K)∞has he app oxima e ixed poin p ope y o nonexpansi e mappings whene e Mis a me ic space which has he app oxima e ixed poin p ope y o such mappings and Kis a bounded con ex subse o a Banach space. 1. INTRODUCTION The s udy o ixed poin heo y o nonexpansi e mappings in p oduc spaces is an ou g ow h o i s analog o con inuous mappings. A opological space is said o ha e he ixed poin p ope y i e e y con inuous sel -map o he space has a ixed poin . I has been known o some ime ha i bo h Xand Yha e he ixed poin p ope y o con inouus mappings, hen i need no be he case ha X×Yhas he ixed poin p ope y o mappings :X×Y→X×Ywhich a e con inuous ela i e o he p oduc opology. Indeed, an example is gi en in [4] o a me ic space Xwhich has he ixed poin p ope y, ye he space X×X ails o ha e he ixed poin p ope y. See, o example, [6] (speci ically, Theo em 4.9) o a mo e ex ensi e discussion. Recei ed Feb ua y 9, 2000; e ised Decembe 15, 2000. Communica ed by M.-H. Shih. 2001 Ma hema ics Subjec Classi ica ion: 54H25, 47H09; Seconda y 47H10. Key wo ds and ph ases: Nonexpansi e mapping, p oduc space, ixed poin , app oxima e ixed poin p ope y. This esea ch was conduc ed while he i s au ho was isi ing he Uni e si y o Iowa. He acknowl- edges he kind hospi ali y o he Uni e si y o Iowa and also he suppo o DGICYT esea ch p ojec PB96-1338-C02-01. 405 406 R. Espíinola and W. A. Ki k In 1968, Nadle [17] ini ia ed a s udy o ixed poin p ope ies o mappings T:X×Y→X×Y, whe e Xis a opological space wi h he ixed poin p ope y, Yis a me ic space, and Tis a con inuous mapping which is also a local con ac ion in i s second coo dina e. Fo a con inues his app oach in [7]. The abo e esul s lead na u ally o he ques ion o wha happens i bo h Xand Ya e me ic spaces wi h he con ac i e condi ions placed di ec ly on T. Fo his discussion we need o ix some e minology. A mapping o a me ic space (M,d) in o a me ic space (N, )is said o be nonexpansi e i ( (x), (y)) ≤d(x, y) o all x, y ∈M. I ( (x), (y)) <d(x, y) o all x, y ∈Mwi h x6=y, hen is said o be s ic ly con ac i e. A mapping is said o be a gene alized con ac ion i o each x∈M he e exis s α(x)∈(0,1) such ha o each y∈M, ( (x), (y)) ≤α(x)d(x, y).I αis a cons an map, hen o cou se is a con ac ion mapping in he sense o Banach. We shall use ix ( ) o deno e he se o ixed poin s o a mapping :M→M. I (X, ρ)and (Y,d)a e me ic spaces, hen he me ic d∞on X×Yis de ined in he usual way: d∞((x, u),(y, )) = max{ρ(x, y),d(u, )} o (x, u),(y, )∈X×Y. We shall con ine ou sel es o he me ic d∞in his pape , al hough all o he esul s, indeed in some ins ances e en s onge ones, seem o hold o he me ics dp,p∈[1,∞), dp((x, u),(y, )) = [(ρ(x, y))p+(d(u, ))p]1/p. A basic ques ion now becomes: I (X, ρ)and (Y,d)ha e he ixed poin p ope y o nonexpansi e mappings and i T:X×Y→X×Yis nonexpansi e ela i e o he me ic d∞, hen does Tnecessa ily ha e a ixed poin ? Al hough sha p esul s ha e been ob ained, he ull answe o his ques ion emains open. In he nex sec ion we summa ize wha is known abou me ic ixed poin heo y in p oduc spaces. In Sec ion 3 we p o e some new esul s o mappings sa is ying `con ac i e' condi ions. In Sec ion 4 we p o e a new esul abou he exis ence o `app oxima e ixed poin s' o nonexpansi e mappings in p oduc spaces by ap- plying a well-known esul abou asymp o ic egula i y o `a e aged' nonexpansi e mappings. 2. OVERVIEW We begin by summa izing he esul s o Nadle and Fo a. He e and h oughou we use P1( esp., P2) o deno e he na u al coo dina e p ojec ion o X×Yon o X Fixed Poin s and App oxima e Fixed Poin s in P oduc Space 407 ( esp., on o Y). Ve sion (C) o his esul is due o Nadle ; e sion (C0) o Fo a. Al e na e p oo s o hese esul s a e gi en in [11]. Theo em 2.1. Suppose Xis a opological space which has he ixed poin p ope y wi h espec o con inuous mappings,suppose Yis a comple e me ic space,and suppose T:X×Y→X×Yis a con inuous mapping which sa is ies (C) o each x∈X, he e exis s a numbe λ(x)∈(0,1) such ha o all u, ∈Y, d(P2◦T(x, u),P 2◦T(x, )) ≤λ(x)d(u, ). Then Thas a ixed poin i ei he (a) Tis uni o mly con inuous;o (b) Yis locally compac . Assump ions (a) and (b) can be d opped i condi ion (C) is s eng hened o (C0) o each x∈X, he e exis s a numbe λ(x)∈(0,1) and a neighbo hood Vx such ha o each w∈Vxand all u, ∈Y, d(P2◦T(w, u),P 2◦T(w, )) ≤λ(x)d(u, ). We now u n o nonexpansi e mappings in p oduc spaces. In [15], i was shown ha i a bounded closed con ex subse Ho a Banach space has he ixed poin p ope y o nonexpansi e mappings, and i Kis a bounded closed con ex subse o ei he a uni o mly con ex o uni o mly smoo h Banach space, hen e e y nonexpansi e T:H×K→H×Khas a ixed poin . This esul led o a sequence o gene aliza ions, culmina ing in a ema kable esul o T. Kuczumow [16]. In o de o desc ibe Kuczumow's esul , we need some addi ional ac s. I is known ha , in gene al, a weakly compac con ex subse o a Banach space need no ha e he ixed poin p ope y o nonexpansi e mappings (Alspach [1]), bu a he same ime weak compac ness (o e lexi i y o he unde lying space) in conjunc ion wi h a a ie y o o he geome ic condi ions (e.g., see [9]) does in ac assu e ha any closed con ex se has he ixed poin p ope y o nonexpansi e mappings. In iew o his, he ollowing de ini ion is qui e na u al. De ini ion 2.1. A closed con ex subse Kis said o ha e he gene ic ixed poin p ope y ( o nonexpansi e mappings)i o e e y nonexpansi e T:K→K and e e y T-in a ian nonemp y closed con ex H⊆K, ix (T)∩H6=∅. Kuczumow used a e ac ion app oach based on a me hod o B uck [3] o p o e he ollowing. Theo em 2.2. Le Xbe a Banach space. Suppose K⊆Xis weakly compac con ex and has he gene ic ixed poin p ope y,and suppose (Y,d)is a me ic 408 R. Espíinola and W. A. Ki k space which has he ixed poin p ope y o nonexpansi e mappings. Then e e y nonexpansi e T:(K×Y)∞→(K×Y)∞has a ixed poin . Kuczumow obse ed ha i Xis a conjuga e space, hen he weak opology in he abo e esul can be eplaced by he weak∗ opology. B uck's pape [3] is ema kably ich in ideas, and in ac a di e en app oach ound in he same pape can be modi ied o p o e he ollowing esul . The de ails a e ound in [12]. Theo em 2.3. Le Ebe a Banach space. Suppose X⊆Eis a sepa able closed con ex subse o Ewhich has he gene ic ixed poin p ope y,and suppose (Y,d) is a sepa able me ic space which has he ixed poin p ope y o nonexpansi e mappings. Then e e y nonexpansi e T:(X×Y)∞→(X×Y)∞has a ixed poin . While i s me hod o p oo is di e en , i is no clea o wha ex en , i any, Theo em 2.3 is ac ually quali a i ely mo e gene al han Theo em 2.2. This is because he e is no known example o a closed con ex subse o a Banach space which has he gene ic ixed poin p ope y ye ails o be weakly compac . The e a e pe haps wo addi ional esul s which should be men ioned. While we a e basically in e es ed he e in he case p=∞,i is qui e easy o p o e he ollowing o 1≤p<∞. Theo em 2.4. Le Eand Fbe Banach spaces. Suppose X⊆Eand Y⊆ Fbo h ha e he ixed poin p ope y o nonexpansi e mappings. Then e e y nonexpansi e T:(X×Y)p→(X×Y)phas a ixed poin o 1≤p<∞. A p oo o he abo e esul is gi en in [14], based on an a gumen gi en o he ollowing esul in [11]. Theo em 2.5. Le Eand Fbe Banach spaces. Suppose X⊆Eand Y⊆ Fbo h ha e he ixed poin p ope y o gene alized con ac ions. Then e e y gene alized con ac ion T:(X×Y)p→(X×Y)phas a ixed poin o 1≤p≤∞. This comple es an o e iew o wha appea o be he mos impo an known esul s. We now u n o some new obse a ions. 3. CONTRACTIVE MAPPINGS IN PRODUCT SPACES I he assump ion o nonexpansi eness is s eng hened, hen i is possible o p o e addi ional esul s in a ai ly di ec manne . Fixed Poin s and App oxima e Fixed Poin s in P oduc Space 409 Theo em 3.1. Le (X, ρ)and (Y,d)be me ic spaces. Suppose Yhas he ixed poin p ope y o nonexpansi e mappings and suppose Xhas he ixed poin p ope y o s ic ly con ac i e mappings,and suppose T:(X×Y)∞→(X×Y)∞ is a nonexpansi e mapping which sa is ies he addi ional condi ion ρ(P1◦T(x, u),P 1◦T(y, )) <d ∞((x, u),(y, )) o all (x, u),(y, )∈X×Ysa is ying ρ(x, y)6=d(u, ).Then Thas a ixed poin . Theo em 3.2. Le (X, ρ)and (Y,d)be me ic spaces,each o which has he ixed poin p ope y o s ic ly con ac i e mappings. Then e e y s ic ly con ac- i e mapping T:(X×Y)∞→(X×Y)∞has a ixed poin . P oo o Theo em 3.1. Fix u∈Yand de ine Tu:X→Xby se ing Tu(x)=P1◦T(x, u),x∈X. Then i x6=y, i ollows ha ρ(x, y)6=d(u, u)=0,and we ha e ρ(Tu(x),T u(y))= ρ(P1◦T(x, u)),ρ(P1◦T(y,u)) <d ∞((x, u),(y,u)) =ρ(x, y). Thus Tuis s ic ly con ac i e and by assump ion has a unique ixed g(u)∈X. Now de ine ϕ(u)=P2◦T(g(u),u). We show ha ϕis nonexpansi e. No e ha since Tu(g(u)) = g(u)and T (g( )) = g( ),we ha e g(u)=P1◦T(g(u),u); g( )=P1◦T(g( ), ), and, mo eo e , i ρ(g(u),g( )) 6=d(u, ), hen ρ(g(u),g( ))= ρ(P1◦T(g(u),u),P 1◦T(g( ), )) <d ∞((g(u),u),(g( ), )) = max{ρ(g(u),g( )),d(u, )} =d(u, ). The e o e, ρ(g(u),g( )) ≤d(u, ) o all u, ∈Y. I ollows ha d(ϕ(u),ϕ( )) = d(P2◦T(g(u),u),P 2◦T(g( ), )) ≤max{ρ(P1◦T(g(u),u),P 1◦T(g( ), )),d(P2◦T(g(u),u),P 2◦T(g( ), ))} =d∞(T(g(u),u),T(g( ), )) ≤d∞((g(u),u),(g( ), )) = max{ρ(g(u),g( )),d(u, )}=d(u, ). 410 R. Espíinola and W. A. Ki k The e o e, ϕ:Y→Yis nonexpansi e. Since Yhas he ixed poin p ope y o nonexpansi e mappings, he e exis s u∈Ysuch ha ϕ(u)=u;whence u=ϕ(u)=P2◦T(g(u),u). Since by assump ion g(u)∈ ix (Tu),we ha e Tu(g(u)) = P1◦T(g(u),u). P oo o Theo em 3.2. The a gumen ollows he p e ious one, excep in his case we mus show ha ϕis s ic ly con ac i e. The ac ha Tis s ic ly con ac i e assu es ha max{ρ(P1◦T(x, u),P 1◦T(y, )),d(P2◦T(x, u),P 2◦T(y, ))} <max{ρ(x, y),d(u, )} i x6=yo u6= . Following he p e ious a gumen s ep-by-s ep, we conclude ha o u∈Yand x6=y, he mapping Tuis s ic ly con ac i e and has a unique ixed poin g(u). Also, i u6= we ha e d(ϕ(u),ϕ( ))= d(P2◦T(g(u),u),P 2◦T(g( ), )) ≤d∞(T(g(u),u),T(g( ), )) <d ∞((g(u),u),(g( ), )) =d(u, ). The conclusion now ollows as in Theo em 3.1. The ollowing is a a ian o Theo em 3.1. The assump ions on he mapping T do no seem o be compa able. Theo em 3.3. Le (X, ρ)and (Y,d)be me ic spaces,each o which has he ixed poin p ope y o nonexpansi e mappings,and suppose T:(X×Y)∞→ (X×Y)∞is a nonexpansi e mapping which sa is ies he addi ional condi ion ρ(P1◦T(x, u, P1◦T(y, )) <d ∞((x, u),(y, )) o all (x, u),(y, )∈X×Ysa is ying u6= and x6=y. Then Thas a ixed poin . Theo em 3.3 has he ollowing immedia e co olla y. Co olla y 3.1. Le (X, ρ)and (Y,d)be me ic spaces,each o which has he ixed poin p ope y o nonexpansi e mappings,and suppose T:(X×Y)∞→ (X×Y)∞is a nonexpansi e mapping which is quasi-con ac i e in he sense ha d∞(T(x, u),T(y, )) <d ∞((x, u),(y, )) Fixed Poin s and App oxima e Fixed Poin s in P oduc Space 411 o all (x, u),(y, )∈X×Ysa is ying u6= and x6=y. Then Thas a (unique) ixed poin . No e ha he condi ion o he co olla y is weake han he con ac i e condi ion o Theo em 3.2. In exchange, a li le mo e is assumed abou he spaces; speci ically ha Xhas he ixed poin p ope y o nonexpansi e mappings. P oo o Theo em 3.3. Fix u∈Yand as be o e de ine Tu:X→Xby se ing Tu(x)=P1◦T(x, u),x∈X. Then ρ(Tu(x),T u(y)) ≤max{ρ(P1◦T(x, u),P 1◦T(y,u)),d(P2◦T(x, u),P 2◦T(y,u))} =d∞(T(x, u),T(y,u)) ≤d∞((x, u),(y,u)) =ρ(x, y). Thus Tuis nonexpansi e and by assump ion has a nonemp y ixed poin se ix (Tu)⊆X. Le gbe any selec ion o he mapping u7→ ix (Tu) and de ine ϕas in Theo em 3.1. We show ha ϕis nonexpansi e. Since g(u)∈ ix (Tu)and g( )∈ ix (T ), we ha e g(u)=P1◦T(g(u),u); g( )=P1◦T(g( ), ). Now le u, ∈Y. The e a e wo cases. 1. I g(u)=g( ), hen ob iously ρ(g(u),g( )) ≤d(u, )and we ha e d(ϕ(u),ϕ( )) = d(P2◦T(g(u),u),P 2◦T(g( ), )) ≤max{ρ(P1◦T(g(u),u),P 1◦T(g( ), )),d(P2◦T(g(u),u),P 2◦T(g( ), ))} =d∞(T(g(u),u),T(g( ), )) ≤d∞((g(u),u),(g( ), )) ≤d(u, ). 2. On he o he hand, i g(u)6=g( ), hen i mus also be he case ha u6= . The e o e, ρ(g(u),g( )) = ρ(P1◦T(g(u),u),P 1◦T(g( ), )) <d ∞((g(u),u),(g( ), )) = max{ρ(g(u),g( )),d(u, )} =d(u, ) 412 R. Espíinola and W. A. Ki k and i ollows ha d(ϕ(u),ϕ( )) = d(P2◦T(g(u),u),P 2◦T(g( ), )) ≤max{ρ(P1◦T(g(u),u),P 1◦T(g( ), )),d(P2◦T(g(u),u),P 2◦T(g( ), ))} =d∞(T(g(u),u),T(g( ), )) ≤d∞((g(u),u),(g( ), )) = max{ρ(g(u),g( )),d(u, )}=d(u, ). The e o e, in ei he case, d(ϕ(u),ϕ( )) ≤d(u, ),and ϕ:Y→Yis nonex- pansi e. The conclusion again ollows as in Theo em 3.1. In he p eceding p oo , he ques ion migh a ise as o whe he ix (Tu)is a single on. Suppose o he wise, and le g1(u)and g2(u)be dis inc choices o he selec ion u7→ ix (Tu).Then acco ding o case 2, o any ∈Y, d(g1(u),g 2(u)) ≤d(g1(u), )+d(g2(u), )<2d(u, ). Ob iously, his can happen only i uis an isola ed poin o Y. To acili a e compa ison, we summa ize he o egoing esul s as ollows: Theo em 3.4. Le (X, ρ)and (Y,d)be me ic spaces,and suppose T:(X× Y)∞→(X×Y)∞is a nonexpansi e mapping. Then Thas a ixed poin i any one o he ollowing condi ions holds. (a) Xand Yha e he ixed poin p ope y o nonexpansi e mappings and T sa is ies d∞(T(x, u),T(y, )) <d ∞((x, u),(y, )) o all (x, u),(y, )∈X×Ysa is ying u6= and x6=y. (b) Yhas he ixed poin p ope y o nonexpansi e mappings, Xhas he ixed poin p ope y o s ic ly con ac i e mappings, and T:(X×Y)∞→ (X×Y)∞sa is ies ρ(P1◦T(x, u),P 1◦T(y, )) <d ∞((x, u),(y, )) o all (x, u),(y, )∈X×Ysa is ying ρ(x, y)6=d(u, ). (c) Xand Yha e he ixed poin p ope y o s ic ly con ac i e mappings and Tis s ic ly con ac i e. Fixed Poin s and App oxima e Fixed Poin s in P oduc Space 413 4. APPROXIMATE FIXED POINTS IN PRODUCT SPACES In his sec ion we p o e an app oxima e ixed poin heo em o nonexpansi e mappings in ce ain p oduc spaces. A me ic space (M,d)is said o ha e he app oxima e ixed poin p ope y i any nonexpansi e mapping T:M→Mhas an app oxima e ixed poin sequence, ha is, a sequence {un}in M o which limnd(un,T(un)) = 0.This o cou se is equi alen o saying in {d(x, T(x)) : x∈M}=0. Ou heo em is based on he ollowing esul , which was p o ed o a single mapping (and o a mo e gene al con e gence p ocess) by Ishikawa [10]. Edels ein and O'B ien [5] showed ha he con e gence is uni o m o e K, and subsequen ly Goebel and Ki k [8] showed ha in ac he con e gence is uni o m o e x0in K and o e he class o all nonexpansi e mappings T:K→K. Ano he p oo o his ac is gi en in [13]. Fo a echnical s udy o he a e o uni o m con e gence and a comp ehensi e e iew o he li e a u e, see [2]. Theo em 4.1. Le Kbe a bounded con ex subse o a Banach space and le ε>0.Then he e exis s N∈Nsuch ha i n≥N, i x0∈K, and i T:K→K is nonexpansi e, hen k n(x0)− n+1(x0)k≤ε, whe e =(1/2)(I+T). An in e es ing ea u e o he p oo gi en below is he ac ha he penul ima e s ep o he p oo equi es he uni o mi y o he con e gence o { n(x0)}in he abo e esul o e he class o all nonexpansi e T:K→K. Theo em 4.2. Suppose Mis a me ic space which has he app oxima e ixed poin p ope y o nonexpansi e mappings and suppose Kis a bounded closed con ex subse o a Banach space X. Le H=(K×M)∞. Then Hhas he app oxima e ixed poin p ope y o nonexpansi e mappings. P oo . Le T:H→Hbe nonexpansi e and le P1and P2deno e he espec i e coo dina e p ojec ions o Hon o Kand M. Fix y∈Mand de ine Ty:K→K by se ing Ty(x)=P1◦T(x, y),x∈K. Now ix x0∈Kand se y=(I+Ty)/2.