The Fixed Point Property of Mean Non-expansive Mapping

Transcription

1 Vol. 3 No. 4 JOURNAL OF HARBIN UNIVERSITY OF SCIENCE AND TECHNOLOGY Aug Oil Bch X Grci-Flset T Grci-Flset DOI /j. jhust O77. A The Fixed Poit Proerty of Me No-exsive Mig CUI Yu-WANG Jig-che Dertmet of MthemticsHrbi Uiversity of Sciece d TechologyHrbi 50080Chi Abstrct I this erit mily discussed the fixed oit roerties of the me o-exsive mig. First of ll it roved the wekly comct covex subset with Oil roerties for me o-exsive mig tht hs wek fixed oit roerty. Secodly it discussed the wekly comct covex subset of reflexive Bch sce X which hs symtotic orml structure tht hs fixed oit for me o-exsive mig Fillyit roved tht whe the Grci-Flset costt stisfied secific iequlitythe me o-exsive mig T hs fixed oit. Keywords me o-exsive mig fixed oit symtotic orml structure Grci-Flset costt X Bch T X X k ( 0 ) x y X Tx - Ty k x - y T 6 - Bch T x 0 X Tx 0 = x 0 x 0 T X A E-mil @ qq. com.

2 4 3 4 X Bch T X X x { } X Tx - x 0 # { x } X Bch C X { x } T X X lim x < lim x + x x 0 Tx - Ty x - y + b x - Tx + c x - Ty # # + b < c RT C Bch X Oil x Cx + = Tx Bch x + - x = Tx - Tx - C C 0 x - x - + b x - Tx + c x - Tx - = x C 0 x - x - + b x + - x su { x - y y C 0 } < δ( C0 ) = ( - b ) x + - x x - x - su { x - y x y C 0 } C + b < x + - x - b x - x Bch { x } Cuchy N C C 0 x + - x x + - x x + - x C 0 { x } x + - x 0 ( # ) x C 0 ( - ) + - b - b + + [ ( - b) ] x - x 0 < limif x - x < δ( C0 ) k # C - k x - x 0 0 k = Cuchy b x { } Grci-Flset R Cuchy ( X) X Bch x 0 C # x X x x 0 C T ( x ) = - ( B X) { x } 4 limsu # D[ ( x ) ] = limsu( limsu x - x m ). m # # 0 K Bch 9 z - x + b + c R( X ) = su{ limif x + x x x B( X) x 0 }. x x 0 T Tx 0 = x 0 # X Bch C X 7 9 Bevides ( R X) T X X R( X) Tx - Ty x - y + b x - Tx + c x - Ty R( X ) = su{ limif x + x } + b = T C Tx + x 0 = T C C T K K b c 0 T ( x) - T ( y ) = - Tx - Ty + b + c Tx - Ty x - y + b x - Tx + c x - Ty K T ( - ) ( x - y + b x - Tx + c x - Ty ) = dim( K ) = { x } T x - y + b x - Tx + c x - Ty ε > 0K { z } = ( - ) b = ( - ) b c = ( - ) c { z } K z + b = N z > - ε 3 m N z m - z + b = ( - ) + ( - ) b = + b + c - ( ) ( + b ) =

3 4 3 - < x = T x 0 x + - x 0 ( # ) x C 0 lim x - x = δ ( C0 ) = rc x y C. x C y C 0 s = limsu x - T x = x y Tx - x = Tx - T x = D = { x C 0 limsu x - x s} Tx - ( - ) Tx - x 0 = D Tx - x = Tx - Tx - Tx - x 0 0 x - x - + c x - Tx - + c x - - Tx ( + c ) { x } T C x - x + c x - Tx + ( + c ) x - x 3 X Bch T X X C - X + b X Tx - x + c - c x - x + + c - c x Oil - x - = Tx - Ty x - y + b x - Tx x 0 C C x { } C x D = C 0 { x i } lim x i - y x 0 Tx 0 = x 0 = s ' C 0 zx { } { x } Tx 0 x 0 x - x 0 0lim x - x 0 < lim x - x 0 + ( x 0 - Tx0 ) = # # lim x + Tx - Tx - Tx 0 = # lim Tx - Tx 0 # x - x 0 + b x - Tx lim x - x 0 # Tx -Ty x -y +b { x -Tx + y -Ty } + c { x -Ty + y -Tx } x y C c 0 0 b < + b + c T C b = c = 0 Billo Schoeberg ε b > 0 c > 0 Bogi b > 0 c = 0 Gregus Tω - v + u - v + ε b = 0 c > 0 Zor C C 0 k λ i Tω - Tv i + ε i = T 7 C 0 C 0 x - x + x - x - D T C 0 lim x j - z = t E = ' { x C 0 limsu x j - x mi{ t s }} E = C 0 y z C 0 x C 0 lim x j s ' i = E t = s ' - x F = { u C 0 su { u -x x C 0 } s ' } F x C 0 x i - x s lim x - x 0 x - z s ' z F s ' < rf C 0 # C 0 Tx 0 = x 0 F T F 4 C Bch ω C T C C su { Tω - x x C 0 } = s s s ' s > s ' ε 0 < ε < ( + c) ( s - s ' ) / u C 0 s < Tω - u + ε C 0 TC 0 C 5 0 v = k λ itv i v i C 0 λ i > 0 k λ i = u - v i = i = s < Tω - u + ε k λ i i = T ε x 0 C 0 k ( s ' + cs ' + cs ) + ε = i = λ i { ω - v i + c ω - Tv i + c v i - Tω } + j

4 4 5 ( + c) s ' + cs + ε < s F C 0 C 0 F T R( X) 5 R( X ) = su{ limif x + x x B # R( X) su{ limif x + x x B( X) # x 0 x - x m x S( X) } ε > 0 { x } B ( X) x S ( X) D ( x ) R( X) - ε limif x + x # D( x ) = limsu( limsu x - x m ) # m # ε N 3 > s. t. # i = 3 4 m N m = mx{ m 0 m } m k m k > m x - x m < + ε z = x - = # ( - ) = + ε x { } i = 0 { z } + + # - i = i = 0 { z } B( X) z - z m + i = # z + x = i = i = + i = 3 + x + x + ε + ε + ε x x + x - ε + ε + ε x R( X) - ε - ε + ε + ε = x + ε + x = R( X) - ε + ε su{ limif x + x x B( X) x 0 # x - x m x S( X) } R( X) - ε + ε ε ε 0 su{ limif x + x x B( X) x 0 # x - x m x S( X) } R( X) su{ limif x + x x B( X) x 0 # x - x m x S( X ) } = R( X) R( X ) = su{ limif x + x x B( X) x 0 # x - x m x S( X) } m N X l ( < < #) R ( X ) = { x } { } { } { } m { } ( X) x lim 0 x - x m x S( X) m N } x m k # k = lim x m k k = # { x } 0 7 { } B( X) 0 i = ε > 0 Nm 0 Ns. t. m k > m 0 i = 4 > s. t. # i = 4 { } m N s. t. m k > m i = 4 i = + i - 3 i = + i - # - - # i = + i = - i = + 3 i = i = + - ε - ε i = i = 3 + = i = + + R( l ) = + m N 4 Nm k > m 4 + x i = x + # i = x = + x + = R( l ) = R( l ) R( l ) + - ε 6 X Bch K X T K K Tx - Ty x - y + b x - Tx + c x - Ty + b + c R( X) < + b + c

5 6 3 x 0 KT ( x0 ) = x 0 ~ 4 K { z } z ω = + b + c = ω = z + b + c ω - ω 0 ω - ω lim ω - ω m ω - ω m m # ω = ω - ω + ω R( X) z R( X) + b + c + b + c < = + b + c 0 T K x 0 KT ( x0 ) = x 0. M Soc M. 6BELLUCE LPKIRK WASTEINER EF Bch J M JUNG JS. Itertive Aroches to Commo Fixed Poits of Noexsive Migs i Bch ScesJ. Jourl of Mthemticl Alysis & Alictios KIRK WA. A Fixed Poit Theorem for Migs Which do Not Icrese DistceJ. Americ Mthemticl Mothly BROWDER FE. Noexsive Noliere Oertors i Bch SceJ. Proceedigs of the Americ Mthemticl Society SIMS B. Orthogolity d Fixed Poits of Noexsive Ms J. Proc. Cetre. Mth. Al. Nt. Uiv. Austrl BAE JS. Fixed Poit Theorems of Geerlized Noexsive Ms J. Jourl of the Kore Mthemticl Society KANNAN RFixed Poit Theorems i Reflexive Bch Sces J. Proceedigs of the Americ Mthemticl Society J SCHAUDER J. Der Fixuktstz i Fuktiolrume J. Studi Mthemtic BANACH S. Sur Les Oértios Ds Les Esembles Abstrits et Leur Alictio Aux qutios Itégrles J. Fudmet Mthemtice KIRK WASIMS B. Hdbook of Metric Fixed Poit Theory M. Sriger Netherlds OPIAL Z. Wek Covergece of the Sequece of Successive Aroxi-mtios for Noexsive MigsJ. Bull Amer. Mth. Bch Sce Norml Structur Pcific Jourl of Mthemtics BENAVIDES TDN. Wek Uiform Norml Structur Directio Sum ScesJ. Studi Mthemtic GARICIA-FALSET J. Stbility d Fixed Poits for Noexsive Migs J. Husto Mth DOM NGUEZ T Bevides. A Geometricl Coefficiet Imlyig the Fixed Poit Proerty d Stbility ResultsJ. Housto Jourl of Mthemtics WANG JMCHEN LLCUI YA. The Fixed Poit Proerty of Me No-exsive MigJ. Jourl of Nturl Sciece of Heilogjig Uiversity