Approximation for a Common Fixed Point for Family of Multivalued Nonself Mappings

Transcription

1 Amerca Joural of Aled Mathematcs ad Statstcs 07 Vol 5 o Avalable ole at htt://ubssceubcom/aams/5/6/ Scece ad Educato Publshg DOI:069/aams-5-6- Aroxmato for a Commo Fxed Pot for Famly of Multvalued oself Mags Mollalg Hale Taele * B Krsha Reddy Deartmet of Mathematcs College of Scece Bahr Dar Uversty Ethoa Deartmet of Mathematcs Uversty College of Scece Osmaa Uversty Ida *Corresodg author: mollalgh@bdueduet Abstract I ths aer we troduce Ma tye teratve method for fte ad fte famly of multvalued oself ad o exasve mags real uformly covex Baach saces We exted the result to the class of quas o exasve mags real Uformty covex Baach saces We also exted for aroxmatg a commo fxed ot for the class of multvalued strctly seudo cotractve ad geeralzed strctly seudo cotractve oself mags real Hlbert saces We rove both wea ad strog covergece results of the teratve method Keywords: fxed ot oself mag oexasve mag strctly seudo cotractve geeralzed strctly seudo cotractve mags multvalued mag Ma tye teratve method uformly covex Baach sace Cte Ths Artcle: Mollalg Hale Taele ad B Krsha Reddy Aroxmato for a Commo Fxed Pot for Famly of Multvalued oself Mags Amerca Joural of Aled Mathematcs ad Statstcs vol 5 o x (07): do: 069/aams-5-6- Itroducto Fxed ot theory for mult-valued mags becomes very terestg for umerous researchers of the feld because of ts may real world alcatos covex otmzato game theory ad dfferetal clusos Mult-valued mags are also mortat solvg crtcal ots otmal cotrol ad other roblems (Agarwal et al [] 88) I sgle valued case for examle studyg the oerator equato Au = 0 (whe the mag A s mootoe) f K s a subset of a Hlbert sace H the A: K H s mootoe mag f Ax Ay x y 0 x y K Browder [5] troduced a ew oerator T defed by T = I A where I s the detty mag o the Hlbert sace H the oerator s called seudo cotractve oerator ad the solutos of Au = 0 are the fxed ots of the seudo cotractve mag T ad vce versa Cosder a mag A: K H ad the Varatoal equalty * * Ax x x 0 x K whch the roblem s to * fd x K satsfyg the equalty ths roblem s the Varatoal equalty roblem arses covex otmzato dfferetal clusos Let f : K R be covex cotuously dfferetable fucto Thus * * f( x ) x x 0 x K s Varatoal equalty for A= f ths equalty s otmalty codto for mmzato roblem m f ( x ) whch x K aears may areas A examle of a mootoe oerator otmzato theory s the mult-valued mag of the sub dfferetal of the fuctoal f f : D( f) H H ad s defed by { } f( x) = g H : x y g f( x) f( y) y K () ad 0 f( x) satsfes the codto x y0 = 0 f( x) f( y) y K I artcular f f : K R s covex cotuously dfferetable fucto the A= f the gradet s a sub dfferetal whch s sgle valued mag ad the codto f( x) = 0 s oerator equato ad f( x) x y 0 s Varatoal equalty ad both codtos are closely related to otmalty codtos Thus fdg fxed ot or commo fxed ot for Mult valued mag s mortat may ractcal areas Let K be a o emty subset of a real ormed sace E the CB( E) deotes the set of o emty closed ad bouded subsets of E We say K s roxmal f for every x E there exsts some y K such that x y = f { x z z K} We deote the famly of oemty roxmal bouded subsets of K by Prox(K) We observe that Hlbert saces by roecto theorem every o emty closed ad covex subset of H s roxmal Also Agarwal et al [] reseted that every oemty closed ad covex subset of a uformly covex Baach sace s roxmal For AB CB( E ) we

2 76 Amerca Joural of Aled Mathematcs ad Statstcs defe the Housdorff dstace betwee A ad B CB( E) by D( A B) = Max su d( x A)su d( x B) x B x A where dxa ( ) f { x a a A} = Kuratows [9] reseted that ( CB( E) D ) s comlete f E s comlete A mag T : K E s o self multvalued mag geeral ad the set of fxed ot of T s F = F( T ) = x K : x Tx defed as { } As Chdume et al [8] roosed we gve the defto of mult valued verso for cotractve mags o a o emty subset K of a real Baach saces E whch s a geeralzato of sgle valued case as follows Defto The mag T : K E s sad to be a) cotracto f there s 0 α < such that D( Tx Ty) x y for all xy K b) L-Lschtza f D( Tx Ty) L x y for some L > 0 ad for all xy K c) oexasve f D( Tx Ty) x y for all xy K whe L = d) Quas o exasve mag f F( T ) ad D( Tx T) x for all FT ( ) x K () I real Hlbert sace H f K s oemty subset of H T : K CB( H ) s sad to be e) Pseudo cotractve f D ( Tx Ty ) x y ( x u ( y v ) u Tx v Ty x for all y K f) Hem cotractve real Hlbert sace f FT ( ) Ø ad D ( Tx T) x y d ( x Tx) for all FT ( ) x DT ( ) g) -strctly seudo cotractve mag Hlbert saces f there exsts (0) such that D ( Tx Ty ) x y ( x u ( y v ) u Tx v Ty (3) holds h) Dem cotractve FT ( ) Ø ad there exsts (0) such that D ( Tx T) x d ( x Tx) FT ( ) x DT ( ) holds O the other had Chdume ad Oala [9] troduced geeralzed -strctly seudo cotractve multvalued mag whch s defed as follow Defto Let K be a o emty subset of a real Hlbert sace ad the the mag T : K CB( H ) s sad to be a) geeralzed strctly seudo cotractve mag f there exsts (0) such that D ( Tx Ty) x y D ( x Tx y Ty) x y DT ( ) (4) holds; b) Geeralzed Hem cotractve real Hlbert sace f F( T ) ad D ( Tx T) x D ( x Tx) (5) for all FT ( ) x DT ( ) It ca be see that the class of geeralzed - strctly seudo cotractve mags cludes the class of - strctly seudo cotractve mags Thus the class of cotracto as well as o exasve mags are subset of the class of Lschtza ad the class of strctly Pseudo cotractve mags ad hece the geeralzed -strctly seudo cotractve mags Furthermore the class of quas o exasve mags cludes the class of o exasve mags Thus the class of -geeralzed strctly seudo cotractve mags s more geeral tha the class of o exasve mags ad the class of strctly seudo cotractve mags The study of fxed ots of o exasve ad cotractve tyes of Mult valued mags s very mortat ad more comlex ts alcatos covex otmzato otmal cotrol theory dfferetal equatos ad others ( ) Examle Let T :[0 ) be gve by Tx = [ x0] for all x [0 ) The for all xy [0 ) D( Tx Ty) = x y hece T s o exasve ad o self mag R Examle Let T : [0] be gve by 4 Tx = 0 4 x The T s oself multvalued 3 -strctly seudo cotractve mag but ot o FT ( ) = 0 exasve tye (see [35]) wth { } ( ) Examle 3 Let T :[0 ) be defed by 4 Tx = 0 x 3 4 D( Tx Ty) = x y thus D ( Tx Ty) = x y = x y x y 9 9 The T s oself whch s ot oexasve mag Mar [3] was the frst who reseted the wor o fxed ots for mult-valued (oexasve) mags by the alcato of Hausdorff metrc ad followg hs wor a extesve wor was doe by adler [4] sce the exstece of fxed ots ad ther aroxmatos for mult-valued cotracto ad oexasve mags ad ther geeralzatos have bee studed by several authors [ ] To meto a few 005 Sastry ad Babu [7] costructed Ma ad Ishawa-tye teratos as gve bellow Let T : K Prox (K) be a mult-valued mag ad let FT ( ) Ø the the sequece of Ma-tye terates gve by

3 Amerca Joural of Aled Mathematcs ad Statstcs 77 x0 K x = αy ( ) x α [0] 0 y Tx such that y = Tx F( T ) Ad the sequece of Ishawa-tye terates such that x0 K y = ( β) x βu u Tx x = ( α) x αv v Ty α β [0] u = Tx F( T ) ad v = Ty F( T ) (6) (7) (8) ad they roved strog covergece of the teratve methods to some ots F(T) assumg that K s comact ad a covex subset of a real Hlbert sace H T s oexasve mag wth FT ( ) the arameters α β satsfyg certa aled codtos Payaa [4] cosequetly Sog ad Wag [3] wth addtoal aled codto T = { } F( T ) exteded the result of Sastry ad Babu [7] to more geeral saces uformly covex Baach saces deed they roved the covergece results of Ishawa-tye teratve method Moreover Shahzad ad Zegeye [9] exteded the above results to multvalued quasoexasve mags ad removed the comactess assumto o K They also costructed a ew teratve scheme to relax the strog codto T = { } F( T ) the Sog ad Wag [3] cosequetly Dtte ad See [4] costructed the Ishawa tye teratve method for mult-valued ad Lschtz seudo cotractve mag they also roved covergece wth more restrctos I addto Chdume ad Oala[9] costructed teratve method of Ma ad Ishawa tye for aroxmatg fxed ots for geeralzed strctly seudo cotractve Multvalued mag later o Oala[5] modfed the terato for three ste Ishawa teratve method for aroxmatg fxed ots for Hem cotractve mags However all the above results were for self mags o the other had ractcal areas there are cases of whch we must cosder o self mag or famly of o self mags For aroxmatg fxed ots of oself sgle-valued mags several Ma ad Ishawa-tye teratve schemes have bee studed va roecto for suy oexasve retracto [ ] However recetly Colao ad Maro [] reseted that the comutato for suy o exasve retracto s costly ad they roosed the method wth lowerg the requremet of metrc roecto Motvated by the wor of Colao ad Maro [] may authors reseted teratve methods for aroxmatg a fxed ot ad a commo fxed ot for both fte ad fte famly of sgle valued mags wthout the requremet of metrc roecto [3435] More recetly Tufa ad Zegeye [37] troduced a Ma-tye teratve scheme for aroxmatg fxed ots for mult-valued oexasve oself sgle mag real Hlbert sace whch geeralzes the result of Colao ad Maro [] to the class of multvalued mags ad they roved covergece wth the assumto that the mag satsfes ward codto the followg theorem Defto Let K be a oemty subset of a real Baach sace E a mag T : K E s sad to be ward f for each x K { } Tx IK( x) = x c( w x) c w K Examle 3 Cosderg examle let u Tx = [ x0] The u = t( x) ( t)0 0 t thus we have u = u x x = x (( t x) x) t x x x = x x t = x x x = x c( v x) t c = v = x [0 ) HeceT s ward mag fact FT ( ) = { 0} Thus T s oself oexasve ward mag Theorem TZ [37] (Tufa ad Zegeye; Theorem 3) Let K be a oemty closed ad covex subset of a real Hlbert H ad let T : K Prox(H) be a ward oexasve mag wth FT ( ) ad T = { } F( T ) Let { x } be a sequece of Ma-tye gve by x K x = αx ( ) u α [0] 0 u Tx such that u = Tx α FT ( ) u u D( Tx Tx ) = max h ( ) u x α max { α hu ( x ) } = { λ λ λ } h ( x ) = f [0] : x ( ) u K u The { x } wealy coverges to a fxed ot of T Moreover f ad K s strctly covex ( α ) < = the the covergece s strog It has bee observed that the exstece of the sequece { u } satsfyg the codto u u D( Tx Tx ) s guarateed by lemma 3 [7] whch s stated our relmary secto Authors [37] also exteded the result for quasoexasve tye mag a real uformly covex Baach sace E wth some arorate restrctos Defto A uformly covex sace E s a ormed sace E for whch for every 0< ε < there s a δ > 0 such x y S = x E : x = f x y > ε ( x y) that for every { } x y the δ

4 78 Amerca Joural of Aled Mathematcs ad Statstcs Hlbert saces the sequeces sace l the Lebsgue sace L ( < < ) are examles of Uformly covex Baach saces The above results so far dscussed were alcable for a sgle o exasve or quas o exasve mag o the other had may ractcal areas we may face famly of mags ad a more geeral class of mags the so called the class of strctly seudo cotractve mags Thus motvated by the ogog research wor artcular the result of Tuffa ad Zegeye [37] our questo s that s t ossble to aroxmate a commo fxed ot for the famly of oself multvalued ad o exasve ad strctly seudo cotractve mags real Hlbert saces ad real uformly covex Baach saces? Thus t s the urose of ths aer to costruct Ma tye teratve method for aroxmatg a commo fxed ot of both fte ad fte famly of oself multvalued oexasve mags ad quas oexasve mags as well ad to exted the result to the class of strctly seudo cotractve mags whch s a ostve aswer to our questo Prelmary Cocets We use the followg otatos ad deftos; Defto Let K be a o emty subset of a real Baach sace E ad let T : K E be multvalued mag I T s dem closed at 0 f for ay sequece { x } K coverges wealy to ad d( x Tx) 0 the T Moreover I T s dem closed at 0 s strogly dem closed at 0 f for ay sequece { x } K coverges strogly to ad d( x Tx) 0 the d( T ) = 0 Lemma ([8] lemma 6) Let K be a oemty closed ad covex subset of a real Hlbert sace H ad let T : K Prox(H) be a oexasve mult-valued mag The I T s dem closed at zero Defto A Baach sace E s sad to satsfy Oal s codto f for ay sequece { x } E x coverges wealy to some x E mles lmf for all y E y x x x < lmf x y Defto 3 A sequece { x } K s sad to be Feer mootoe wth resect to a subset F of K f x F x x x x Lemma [4] Let E be a real Baach sace The f A B CB( E) ) ad a A the for every γ > 0 there exsts b B such that b a DAB ( ) γ Lemma 3 [7] Let E be a real Baach sace The f AB Prox (E) ad a A the there exsts b B such that b a DAB ( ) Lemma 4 (Xu [4]) Let > R > be two fxed umbers ad E s a real Baach sace The E s uformly covex f ad oly f there exsts a cotuous strctly creasg ad covex fucto g :[0 ) [0 ) wth g (0) = 0 such that ( ) λx ( λ) y λ x ( λ) y W ( λ) g x y R 0 where W ( λ) = λ ( λ) λ( λ) Lemma 5 [4] I real Hlbert sace H for all x H ad α [0] for such that α = the equalty for all xy B (0) = { x X: x< R} ad λ [ ] α x = α x αα = x x holds Lemma 6 (Browder [7] Ferrera-Olvera [3]) Let E be a comlete metrc sace ad K E a oemty subset If x s { x } s Feer mootoe wth resect to K the { } bouded Furthermore f a cluster ot x of { x } belogs to K the { x } coverges strogly to x I the artcular case of a Hlbert sace gve the set of all wealy cluster ots of { x } ω w( x) = { x : x x weaely} { x } Coverges wealy to a ot x K f ad oly f ωw( x) K Lemma 7 (See for examle Zedler [43 ] 484) Let E be a real uformly covex Baach sace { x} { y } E be two sequeces f there exsts a costat r 0 such that lm su x r lm su y r ad lm λx ( λ) y = r for { λ} [ ε ε] (0) for some ε (0) the lm x y = 0 Lemma 8 [8]: Let K be a oemty subset of a real Hlbert sace H ad let T : K CB( K) be a multvalued -strctly seudo cotractve mag The T s Lschtz wth Lchtz costat Lemma 9 [38] Let H be a real Hlbert sace Suose K s a closed covex oemty subset of H Assume that T : K CB( K) s seudo cotractve mult-valued mag wth F(T) s o emty The F(T) s closed ad covex Lemma 0 [38] Let H be a real Hlbert sace Suose K s a closed covex oemty subset of H Assume that T : K CB( K) s Lschtz seudo cotractve

5 Amerca Joural of Aled Mathematcs ad Statstcs 79 mult-valued mag The I T s dem closed at zero Lemma Let K be a oemty subset of a real Hlbert sace HH ad let T : K Prox(H) be a multvalued -strctly seudo cotractve mag The T s Lschtza wth Lschtz costat ad hece I T s dem closed at 0 (Proof ca be doe wth lemma 3 lemma 8 ad lemma 0) Defto 4 Let F K be two closed ad covex oemty sets a Baach saces E ad F K For ay sequece { x} K f { x } coverges strogly to a elemet x K \ F x x mles that { x } s ot Feer-mootoe wth resect to the set F K we say the ar (F K) satsfes S-codto F = 0 K = [ ] The the ar ( FK ) Examle Let { } satsfes S- codto = Defto 5 Let { T } : K rox( E) be sequece of mags wth oemty commo fxed ot set F= FT ( ) The the famly { T } s sad to be = = uformly wealy closed f for ay coverget sequece x K such that lm dx ( Tx) = 0 the the wea { } cluster Pots of { x} K belog to F Lemma [9]: Let K be a oemty subset of a real Hlbert sace H ad T : K CB( K) be a multvalued geeralzed -strctly seudo cotractve mag The T s Lschtz wth Lschtz costat ad F(T) s closed ad covex Lemma 3 [9] Let K be a oemty ad closed subset of a real Hlbert sace HH ad let T : K CB(K) be a multvalued geeralzed -strctly seudo cotractve mag The T s Lschtza wth Lschtz costat ad I T s strogly dem closed at 0 Defto 6 Let K be a oemty ad closed subset of a real Hlbert sace HH The a ma T : K CB(H) s sad to be Hem comact f for ay sequece { x } K such d( x Tx) 0 the there exsts a sub sequece { x } of { } x such that { } x coverges strogly to K Remar: Ay mag o a comact doma s Hem comact Lemma5 [36] Let{ a } be a sequece of o egatve real umbers such that a a δ δ = < the { a } coverges ad f addto the sequece { a } has a subsequece whch coverges to 0 the the orgal sequece { a } coverges to 0 The followg lemma ca be foud [9] Lemma 6 [9] Let E be a ormed lear sace A B CB( E) ad xy E The the followg hold; a) Dx ( Ax B) = DAB ( ); ; b) D( A B) = DAB ( ); c) D( x A y B) x y D( A B); d) D { x} A = Su{ x a } ( ) ; a A e) D( { x} A) = D(0 x A) Cosequetly from (d) the followg was obtaed [9] Lemma 7 [9] Let K be a o emty ad closed subset of a real Hlbert sace H ad let T : K CB( H ) be geeralzed - strctly seudo cotractve mag The for ay gve { x } K there exsts u Tx such that D ({ x} Tx) x u I artcular f Tx s roxmal there exsts u Tx u x = D ( x Tx) x u 3 Ma Results Let T T TK : K Prox(E) be famly of o self ad multvalued mags o a o-emty closed covex subset K of a real uformly covex Baach sace E our obectve s to troduce a teratve method for commo fxed ot of the famly ad determe codtos for covergece of the teratve method We use the codto that mags to be ward stead of metrc roecto whch s comutatoally exesve may cases ad we rove both wea ad strog covergece of the teratve method Thus we shall have the followg lemma Lemma 3 Let K be a oemty closed ad covex subset of a real Baach sace E T T T : K CB( E) or Prox(E) be multvalued mags u Tx Defe hu : K R by { λ λ λ } h ( x) = f [0] : x ( ) u K u The for ay x K the followg hold: ) hu ( x) [0] ad hu ( x ) = 0 f ad oly f u K ; ) If β [ hu ( x)] the βx ( β) u K ; 3) If T s ward mag hu ( x ) < ; 4) If u K the h u ( ) ( u ( )) x x h x u K where K s the boudary of K The roof of ths lemma follows from lemma 3 of Taele ad Reddy [3] Calo ad Marao [] ad Tuffa ad Zegeye [37] Theorem 3: Let T T T : K Prox(H) be famly of o self mult valued oexasve ad ward mags o a o-emty closed ad covex subset K of

6 80 Amerca Joural of Aled Mathematcs ad Statstcs a real Hlbert sace H wth T T = ( Mod ) for all Let { x} F = FT ( ) o emty F FT ( ) = T { } K ( ) = be a sequece of Ma tye defed by the teratve method gve by { u } x K u Tx α= max α h ( x ) α > 0 x = αx ( ) u u Tx u u DT ( x T x ) α = max { α hu ( x ) } hu ( x) = f { λ 0 : λx ( λ) u K} s well-defed ad f { α} [ ε ε] (0) ε > 0 the the sequece { } for some x coverges wealy some elemet of F= FT ( ) Moreover f ( α ) < = ad (FK) satsfes S-codto the the covergece s strog x s well-defed ad s K Proof: By lemma 3 { } thus to rove the theorem frst we rove { x } s feer mootoe wth resect to F to do so let F the we have the followg equalty; x = αx ( α) u x ( ) u T x ( ) DT ( x T) = α x ( ) x = x (3) Thus the sequece { x } s feer mootoe wth resect to F Sce x s decreasg ad bouded below t coverges ad hece { x } ad { } That s u are bouded u = u u DT ( x T ) x M for some M 0 Also we have the followg equalty x = α x ( ) u x ( α) u T = α ( α x ( ) D( Tx T) ( x ( α) x ( α α = x ( (3) Suose{ α } [ ε ε] the α( ) x u = x x = ( ) < Hece lm x u = 0 ad whch mles that x x = ( α) x u lm x x = lm ( ) x u = 0 Thus by ducto ad tragle equalty we have Thus (33) lm x x = 0 for all 0 (34) lm x u lm x x lm x u = 0 Thus by defto of fmum ad d we have d( x T x ) x u 0 as 0 d( x T x) x x d( x T x ) D( T x T x) 0 Thus lm d( x T x ) 0 = Sce { } (35) x s bouded t has a coverget subsequece { x } such that x x wealy sce K s closed ad covex x K ad l(mod ) l ad for each = for some { } { } l there s some 0 such that = l(mod ) Thus d( x Tx ) 0 as d ( x Tx ) 0 l l Sce T l s dem closed we have x FT ( l ) ad sce l s arbtrary we have x FT ( ) Sce H satsfes oal s codto ad coverget we get x x wealy x x s Thus the sequece { x } coverges wealy some elemet of F= FT ( ) Moreover f ( α ) < the = x x = ( α) x u < M ( α) < = = =

7 Amerca Joural of Aled Mathematcs ad Statstcs 8 Hece the sequece { x } s strogly Cauchy thus t s Cauchy ad coverges to some elemet x K Moreover sce T s ward the hu ( x ) < hece for every β [ hu ( x)) we have βx ( β) u K artcular sce lm α = there s a subsequece { } x of { } x such that lm hu ( x ) = h ( ) ( ( )) u x x h u x u K whose lmt s x K Thus x K ad sce the ar (F K) satsfes S- codto x F x coverges strogly to some elemet Thus { } FT ( ) Theorem 33: Let T T T : K Prox (E) be famly of o self mult valued oexasve ad ward mags o a o-emty closed ad covex subset K of a real Uformly covex Baach sace E satsfyg oal s codto wth F= FT ( ) o emty T = T ( Mod ) for all = ( ) for F FT each TK ( ) = { } ad suose I T s dem closed at 0 let { x} be a sequece of Ma tye defed by the teratve method { u } x K u Tx α= max α h ( x ) α > 0 x = αx ( ) u u Tx u u DT ( x T x ) α = max { α hu ( x ) } hu ( x) = f { λ 0 : λx ( λ) u K} The the sequece s well-defed ad f α ε ε for some ε > 0 ad E satsfes { } [ ] (0) oal s codto the the sequece { x } coverges wealy some elemet of ( α ) < = covergece s strog F = FT ( ) Moreover f ad (FK) satsfes S-codto the the Proof: By lemma 3 { x } s well-defed ad s K thus to rove the theorem frst we rove { x } s feer mootoe wth resect to F to do so let F the we have the followg equalty; x = αx ( α) u x ( ) u T x ( ) DT ( x T ) = α x ( ) x = x (36) Thus the sequece{ x } s feer mootoe wth resect to F Sce x s decreasg ad bouded below thus t coverges ad hece { x } ad { } That s u = u u u are bouded DT ( x T ) x M for some M 0 Suose { α} [ ε ε] the α( ) x u = ( x x ) < = ca be show by Lemma 4 Xu [38] sce E s uformly covex Baach sace for > R > real umbers there exsts a cotuous strctly creasg ad covex fucto g :[0 ) [0 ) wth g (0) = 0 such that λx ( λ) y λ x ( λ) y W ( λ) g( x y ) for all xy B (0) = { x E: x< R} ad λ [ ] R where W ( λ) = λ ( λ) λ( λ) (37) 0 Sce { x } s bouded R ca be chose so that { x } B (0) If = > we have the equalty R ( ) x ( ) y x ( ) y W ( ) g x y λ λ λ λ λ Thus for λ = α x = x y = u we get Whch mles x = αx ( ) u = α x ( ) u T ( ) g x u x ( ) D( Tx T ) ( ) g x u x ( ) x ( ) g x u = x ( ) g x u (38) α( ) g x u x x < = = cacellato of terms ad covergece of 0< α < ad hece x wth W ( α ) = α ( ) ε > 0 for

8 8 Amerca Joural of Aled Mathematcs ad Statstcs some ε > 0 we get g x u < = Sce g s cotuous strctly creasg ad covex fucto g x u 0 as Also by lemma 7 [40] x u 0 as Thus x x = ( ) x u whch mles that lm x x = lm ( ) x u = 0 Thus by ducto ad tragle equalty we have lm x x = 0 for all 0 Thus lm x lm x x lm x u = 0 u (39) Thus by defto of fmum ad d we have d( x T x ) x u 0 as 0 d( x T x ) x x d( x T x ) D( T x T x ) 0 Thus lm d( x T x) 0 = Sce { } x s bouded t has a coverget subsequece { x } such that x x wealy sce K s closed ad covex x K ad l(mod ) l ad for each = for some { } { } l there s some 0 such that = l(mod ) Thus d( x Tx ) 0 as l Thus d ( x l ) 0 Tx as sce T l s dem closed we have x FT ( l ) ad sce l s arbtrary we have x FT ( ) Sce E satsfes oal s codto ad coverget we get x x wealy x x s Thus the sequece { x } coverges wealy to some elemet of F= FT ( ) Moreover f ( α ) < the = x x = ( α) x u < M ( α) < = = = Hece the sequece { x } s strogly Cauchy thus t s Cauchy ad coverges to some elemet x K Moreover sce T s ward the hu ( x ) < hece for every β [ hu ( x)) we have βx ( β) u K artcular sce lm α = there s a subsequece { } x of { } x such that lm hu ( x ) = h ( ) ( ( )) u x x h u x u K whose lmt s x K Thus x K ad sce the ar (F K) satsfes S- codto x F x coverges strogly to some elemet Thus { } FT ( ) Theorem 34 Let K be a covex closed ad oemty subset of a real Hlbert sace H ad let { T } : Pr ( ) K = ox H be a uformly wealy closed coutable famly of o self mult valued ad oexasve mags wth F= FT ( ) s o emty F= FT ( ) { } ad for all T x be a sequece defed by the Ma tye teratve method = Let { } { u } x K u Tx α= max α h ( x ) α > 0 x = αx ( ) u u Tx u DTx ( ) u u DT ( x T x ) α = max { α hu ( x ) } hu ( x) = f { λ 0 : λx ( λ) u K} The the sequece { x } s well-defed ad f { α } [ ε ε] (0) for some ε > 0 the the sequece { x } coverges wealy some elemet of Moreover f ( α ) < = F = FT ( ) ad (FK) satsfes S- codto the the covergece s strog Proof let F ad by lemma 3 [7] there s a sequece{ u } u Tx satsfyg u u DT ( x T x ) thus we have the followg equalty; x = αx ( α) u x ( ) u T x ( ) DT ( x T) = α x ( ) x = x Thus { x } s feer mootoe wth resect to F Sce x (3) s decreasg ad bouded below t coverges ad hece { x } ad { } u are bouded

9 Amerca Joural of Aled Mathematcs ad Statstcs 83 That s u = u u DT ( x T ) x M for some M 0 Also we have the followg equalty = α ( α) x x u x ( α) u T = α ( α x ( ) D( Tx T) ( x ( α) x ( α α = x ( Suose{ α } [ ε ε] the α( ) x u = ( x x ) < = (3) Hece lm x u = 0 Thus by defto of fmum ad d we have dx ( Tx ) x u 0 as Sce { x } s bouded t has a coverget subsequece { x } such that x x wealy sce K s closed ad covex x K d( x T x ) 0 sce { T } = x FT ( ) s uformly wealy closed x F Sce H satsfes oal s codto ad coverget we get x x wealy that s x x s Thus the sequece { x } coverges wealy to some elemet of F= FT ( ) Moreover f ( α ) < the = x x = ( ) x u = = < M ( α ) < = Hece the sequece { x } s strogly Cauchy thus t s Cauchy ad coverges to some elemet x K Moreover sce T s ward the hu ( x ) < hece for every β [ hu ( x)) we have βx ( β) u K artcular sce lm α = there s a subsequece { } x of { } x such that lm hu ( x ) = h ( ) ( ( )) u x x h u x u K whose lmt s x K Thus x K ad sce the ar (F K) satsfes S- codto x F x coverges strogly to some elemet Thus { } FT ( ) Theorem 35 Let K be a covex closed ad oemty subset of a real Uformly covex Baach sace E satsfyg oal s codto ad let { T } : ( ) K = Prox E be a uformly wealy closed coutable famly of o self mult valued ad oexasve(quas o exasve) mags wth F= FT ( ) s o emty ad for all F= FT ( ) { } T x be a sequece defed by the Ma tye teratve method = Let { } { u } x K u Tx α= max α h ( x ) α > 0 x = αx ( ) u u Tx u u DT ( x T x ) α = max { α hu ( x ) } hu ( x) = f { λ 0 : λx ( λ) u K} The the sequece { x } s well-defed ad f { α} [ ε ε] (0) for some ε > 0 ad E satsfes oal s codto the the sequece { x } coverges wealy some elemet of Moreover f F= FT ( ) ( α ) < = ad (FK) satsfes S-codto the the covergece s strog Proof ca be made smlar way as theorem 33 ad 34 Theorem 36 Let K be a strctly covex closed ad oemty subset of a real Hlbert sace H ad let T : K Pr ox( H ) be a o self mult valued ad -strctly seudo cotractve ad ward mag wth F= FT ( ) s o emty ad for each x K Tx s closed ad T = { } for all FT ( ) Let { x } be a sequece defed by the teratve method

10 84 Amerca Joural of Aled Mathematcs ad Statstcs { u } x K u Tx α = max α h ( x ) < α < x = αx ( ) u u Tx u u D( Tx Tx ) α = max { α hu ( x ) } hu ( x) = f { λ 0 : λx ( λ) u K} The the sequece { x } s well-defed ad the sequece { x } coverges wealy to some elemet of F= FT ( ) Moreover f covergece s strog ( α ) < = the the Proof By lemma 3 { x} s well-defed ad s K thus to rove the theorem frst we rove { x } s feer mootoe wth resect to F to do so let F the the followg holds; = α ( α) x x u x ( α) u = α ( α x ( ) D ( Tx T) ( x α x x u ( )[ ] ( α)( ) = x a x u x Thus { x } s feer mootoe wth resect to F Sce x (33) s decreasg ad bouded below t coverges ad hece { x } ad sce T s Lschtza by lemma 8 [8] { u } are bouded That s u = u u D( Tx T) x M for some M 0 We also have the followg equalty; = α ( α) x x u x ( α) u = α ( α x ( ) D ( Tx T) ( x α x x u ( )[ ] α = x ( )( a Thus ( α )( ) x u = ( x x ) < = (34) Suose ( α ) = sce α > there exsts = ε > 0 such thatα > ε thus ( α )( α) = = hece lmf x u = 0 also from the method of roof of Marao ad Trombetta [] t ca be see{ x u } s decreasg as x x = ( ) x u ad x u = αx ( ) u u = α( x u ) ( )( u u = α x x x u ( ) u u ( ) x u x x x u α x x x u ( ) x u ( ) D ( Tx Tx ) ( ) x u x u α ( x u ( ) x u ( )[( ) x u [ x u ( x u x u x u ] ( α ( )) x u ( α)( α) x u ( α )( ) x u x u Lettg γ = x u we have the followg; γ ( α ) γ α γ γ solvg the equalty we get whch gves γ { γ } = { x u } s decreasg ad hece coverges to lmf x u = 0 thus lm x u = 0 as a result 0 lm d( x Tx) lm x u = 0 whch mles that d( x Tx) 0 as O the other had sce the sequece{ x } s bouded t has a wealy coverget subsequece { x } such that γ

11 Amerca Joural of Aled Mathematcs ad Statstcs 85 x x wealy sce K s closed ad covex x K sce I T s dem closed at 0 x Tx Sce every Hlbert sace satsfes oal s codto x x wealy for some x Tx Moreover f ( α ) < = the x x = ( α) x u < M ' ( α) < = = = Hece the sequece { x } s strogly Cauchy thus t s Cauchy ad coverges to some elemet x K Moreover sce T s ward the hu ( x ) < hece for every β [ hu ( x)) we have βx ( β) u K artcular sce lm α = there s a subsequece { } x of { } x such that lm hu ( x ) = h ( ) ( ( )) u x x h u x u K whose lmt s x K thus x K The cotuty of Lschtz mag T gves u u D( Tx Tx ) x x thus there s u H such that u u as Sce T s cotuous ad wth each Tx s closed the followg holds; d( u Tx) d( u Tx) D( Tx Tx) 0 as u Tx thus for all β [ hu ( x)) we have βx ( β) u K as a result t ca be show that βx ( β) u K Sce K s strctly covex smlar fasho (see [37]) t ca be see that βx ( β) u = x hece u x Tx x coverges strogly to some elemet FT ( ) Theorem 37 Let K be a strctly covex closed ad oemty subset of a real Hlbert sace H ad let T : K Pr ox( H ) be a o self mult valued ad geeralzed -strctly seudo cotractve ad ward mag wth F= FT ( ) s o emty ad for all FT ( ) T = { } for each x K Tx s closed Let = Thus the sequece { } { x } be a by the teratve sequece defed method { u } x K u Tx α = max α h ( x ) α > x = αx ( ) u u Tx u u D( Tx Tx ) D ({ x} Tx) x u α = max α hu ( x ) ; ( ) hu ( x) = f { λ 0 : λx ( λ) u K} The the sequece { x } s well-defed ad the sequece { x } coverges strogly to some elemet of F= FT ( ) Proof By lemma 3 { x } s well-defed ad s K Let F Sce u D( Tx T) ad by lemma 6 we have the followg equalty; x = αx ( α) u = α x ( ) u ( ) x u x ( ) D ( Tx T) ( ) x u x ( )[ x D ( x Tx 0] ( ) a x u { } = x ( ) D ( x Tx ( ) x u x ( ) x u ( ) x u ( ) x (35) Thus by lemma 5 we have the sequece { x } coverges to some r 0 Thus the sequece{ x } ad hece { u } are bouded Sce ( α ) < < = = ( ) x x = ( α) x u < M ' ( α) < = = = for some M 0 the we have Hece the sequece { } x s strogly Cauchy thus t s Cauchy ad coverges to some elemet x K Moreover sce T s ward the hu ( x ) < hece for every β [ hu ( x)) we have βx ( β) u K Sce lm α such that lm hu ( x ) = = there s a subsequece { } x of { x } h ( ) ( ( )) u x x h u x u K whose lmt s x K thus x K Sce T s Lschtz mag u u D( Tx Tx ) x x hece { u } s Cauchy sequece thus there s u H such that u u as Sce T Lchtz cotuous we have d( u Tx) d( u Tx) D( Tx Tx) 0 as

12 86 Amerca Joural of Aled Mathematcs ad Statstcs sce Tx s closed u Tx hece for all β [ hu ( x)) we have βx ( β) u K as a result t ca be show that βx ( β) u K Sce K s strctly covex smlar fasho (see [37]) t ca be see that u = x Tx Thus the x coverges strogly to some elemet FT ( ) sequece { } Theorem 38 Let K be a strctly covex closed ad oemty subset of a real Hlbert sace H ad let T : K CB( H ) be a o self mult valued ad geeralzed -strctly seudo cotractve ad ward mag wth F= FT ( ) s o emty ad for all FT ( ) T = { } Let { x } be a by the teratve sequece defed method x K u Tx α = max { α hu ( x ) } α > x = αx ( α) u u Tx γ [ 0 ) γ < = u u D( Tx Tx ) γ D ({ x} Tx) x u α = max α hu ( x ) ( ) hu ( x ) f { 0 : ( ) } = λ λx λ u K The the sequece { x } s well-defed ad the sequece { } Proof By lemma 3 { x } s well-defed ad s K x coverges strogly to some elemet of F= FT ( ) Let F The alyg lemma 5 ad lemma 6 we have x = αx ( α) u = α x ( ) u ( ) x u x ( ) D ( Tx T) ( ) x u γ x ( )[ x D ( x Tx 0] ( ) a x u γ { } = x ( ) D ( x Tx ( ) x u γ x ( ) x u ( ) x u γ ( ) x ( )( ) x u ( ) x γ (36) Thus by lemma 5 we have the sequece { x } coverges to some r 0 Thus the sequece{ x } ad hece { u } are bouded Sce ( α ) < < = = ( ) x x = ( α) x u < M ' ( α) < the we have = = = Hece the sequece { } x s strogly Cauchy thus t s Cauchy ad coverges to some elemet x K Moreover sce T s ward the hu ( x ) < hece for every β [ hu ( x)) we have βx ( β) u K Sce lm α such that lm hu ( x ) = = there s a subsequece { } x of { x } h ( ) ( ( )) u x x h u x u K whose lmt s x K Sce T s Lschtz mag u u D( Tx Tx ) x x hece { u } s Cauchy sequece thus there s u H such that u u as Sce T s Lschtz cotuous we have d( u Tx) d( u Tx) D( Tx Tx) 0as Sce Tx s closed u Tx hece for all β [ hu ( x)) we have βx ( β) u K as a result t ca be show that βx ( β) u K Sce K s strctly covex smlar fasho (see [37]) t ca be see that u = x Tx Thus the sequece { x } coverges strogly to some elemet FT ( ) Theorem 39 Let K be a strctly covex closed ad oemty subset of a real Hlbert sace H ad let T : K CB( H ) = be a o self mult valued ad geeralzed -strctly seudo cotractve ad ward mag wth F= FT ( ) s o emty ad for all T { } = Let u Tx x u D ( x Tx ) x u & u u DTx ( Tx ) γ γ < = Let { δ } (0) such that lm 0 δ = Let { } teratve method f δ > ad x be a sequece defed by the

13 Amerca Joural of Aled Mathematcs ad Statstcs 87 { u } x K α = max α h ( x ) < α < ( ) x = αx α u u = δ u α = max { α hu ( x ) } hu ( x) = f { λ 0 : λx ( λ) u K} Suose T s hem comact ad ( α = ) = the { x } coverges to some FT ( ) Ad f K s strctly covex δ δ < ad ( α ) < = = the { x } coverges to some FT ( ) Proof Frst we see that for ay x K sce each T s ward the hu ( x ) < deed for u = δ u u = x c ( w x) c & w K we have w u = δ u = δ x c x) = = = c = x c ( w x) c w K Thus hu ( x) < c < Let ( ) FT Thus alyg lemma 5 ad lemma 6 we have x = αx ( ) u = α( x ) ( )( u ) = α x ( ) u ( ) x u ( ) x δ u ( ) x δ u δ u x ( ) δ δ u u < δ x u ( ) δ δ u u < ( ) x δ D ( TTx ) ( ) δ δ u u ( ) δ x u α( ) δ δ u u x ( ) δ { x D ( x Tx)} ( ) δ δ u u ( ) δ x u α( ) δ δ x ( )( ) α δ x u ( ) ( ) δ δ u u x u u Thus by lemma 5 we have { x } some r 0 hece the sequece { } x { } bouded From (37) we have ( )( α ) δ x u ( )( ) α δ x u x x (37) coverges to u ad { } Sce lmf δ = m > 0 for some m > 0 we have ( )( α mx ) u ( )( ) α δ x u x x u are Case suose ( α ) = ad T s hem comact sce α > 0 let by Archmedea roerty of real umbers α = ε > 0 we have ( )( α ) = ad

14 88 Amerca Joural of Aled Mathematcs ad Statstcs ( )( α mx ) u ( )( ) α δ x u ( x x ) < (38) Thus for each lmf x u = 0 hece there exsts a subsequece { x u } of { x } u such that lm x u = 0 thus as Sce s hem comact there exsts a subsequece { } x of { } x such that x q K Moreover f we tae u Tx satsfyg x ( ) u d x Tx ad lschtz roerty T of we have d( qtq ) q x x u d( u Tq ) q x d( x Tx ) DTx ( Tq ) x q d( x Tx ) x 0 q (39) Thus d( qtq ) = 0 hece q FT ( ) sce the result s true for ay q Sce for ay q FT ( ) FT ( ) { x q } coverges hece the sequece { x } coverges strogly to q FT ( ) Case Suose K s strctly covex ad ( α ) < the = x x = ( ) x u = = < M ' ( α ) < = Hece the sequece { x } s strogly Cauchy thus t s Cauchy ad coverges to some elemet x K Moreover sce T s ward the hu ( x ) < hece for every β [ hu ( x)) we have βx ( β) u K Sce lm α = there s a subsequece { } x of { x } such that lm h ( x ) = u h ( ) ( ( )) u x x h u x u K whose lmt s x K thus x K Sce T s Lschtz mag u u DTx ( Tx ) x x hece { } u s Cauchy sequece thus there s u H such that u u as Sce { δ } s strogly Cauchy t coverges hece there exsts δ > 0 such that δ δ let u = δ u the we have u u δ u u u δ δ 0 (30) T s Lschtz cotuous we have d( u Tx) d( u Tx ) DTx ( Tx) 0 as sce Tx s closed u Tx hece for all β [ hu ( x)) we have βx ( β) u K as a result t ca be show that βx ( β) u K Sce K s strctly covex smlar fasho (see [37]) t ca be see that u = x Tx Thus the sequece { x } coverges strogly to some elemet FT ( ) Remar: I the above dscussos f we cosder famly of strctly seudo cotractve or geeralzed strctly seudo cotractve mags we ca use = max{ } theorem 39 Examle 3 ow we gve a examle of sequece of multvalued mags{ T } = x Let T : [0] rox( R ) be defed by Tx = 0 The D( Tx Ty) = x y x y Thus T s oexasve multvalued oself mag x For each x [0] let u Tx u 0 the x u = t ( t)0 0 t ad u = u x x thus ( tx ) u= x x x cv ( x) = ( t) c = & v = x [0] hece T s ward mag

15 Amerca Joural of Aled Mathematcs ad Statstcs 89 Thus the sequece of mags satsfes the codto of the theorem 3 thus the algorthm coverges to a uque commo fxed ot F = { 0} we also see that F = { 0 } [0] = K ad the ar (FK) satsfes S-codto We see the frst four terates as; Let x = α 0 = The Tx = [ 0 ] tag u = thus hu ( x ) = thus α = x = ad 3 4 Tx = 0 8 tag u 0 8 such that x 7 u u DT ( x Tx ) = x = 8 say u = we get hu ( x ) = α 8 = x 3 = ad 3 6 Tx 3 3= 0 48 the same fasho tag u3 = we 48 get hu ( x 3 3) = α 3 = ad x 4 = 4 48 Remar: Let T = T = = T = T : K Prox(H) be o self mult valued oexasve ad ward mag o a o-emty closed ad covex subset K of a real Hlbert sace H wth F= FT ( ) o emty for all F= FT ( ) T( ) = { } Let { x } be a sequece of Ma tye defed by the teratve method { α u } x K u Tx α = max h ( x ) α > 0 x = αx ( α) u u Tx u u D( Tx Tx ) α = max u hu ( x) = f 0 : x ( ) u K { α h ( x ) } { λ λ λ } s well-defed ad f { α} [ ε ε] (0) ε > 0 the the sequece { } for some x coverges wealy to some elemet of F= FT ( ) Moreover f ( α ) < = ad (FK) satsfes S-codto the the covergece s strog 4 Cocluso Our theorems exted may results lterature artcular our theorems [3-35] exted the result of Tufa ad Zegeye [33] to a commo fxed ot for the famly of o exasve mags We also exted the result of [9] ad [5] to aroxmato for a fxed ot ad a commo fxed ot for famly of more geeral class of mags the so called geeralzed -strctly seudo cotractve oself mags Authors Cotrbutos Both authors cotrbuted equally ad sgfcatly wrtg ths artcle Both authors read ad aroved the fal mauscrt Cometg Iterests The authors declare that they have o cometg terests Refereces [] MABBASM YJ CHO Fxed ot results for mult-valued oexasve mags o a ubouded set Aalele Scetfc Ale Uverstat Ovdus Costata 8() (00) 5-4 [] RPAGARWAL DOREGA DRSAHU Fxed Pot Theory for Lschtza tye Mags wth Alcatos Srger ew Yor (009) [3] I BEG MABBAS Fxed-ot theorem for wealy ward multvalued mas o a covex metrc sace DemostrMath 39() (006) [4] TDBEAVIDES PLRAMREZ Fxed ot theorems for multvalued oexasve mags satsfyg wardess codtos J Math Aal Al 9() (004) [5] FEBROWDER olear mags of oexasve ad accretve tye Baach Saces Bull Am Math Soc 73 (967) [6] FEBROWDER oexasve olear oerators a Baach sace Proc at Acad Sc USA 54 (965) [7] FEBROWDER Covergece theorems for sequeces of olear oerators Baach saces Math Zetschr 00 (967) 0-5 [8] CECHIDUME CO CHIDUME DJITTE MSMIJIBIR Covergece theorems for fxed ots of multvalued strctly seudo cotractve mags Hlbert saces Abstract ad Aled Aalyss (03) [9] CECHIDUME MEOKPALA O a geeral class of mult valued strctly seudo cotractve mags Joural of olear Aalyss ad Otmzato 5() (04) 7-0 [0] CECHIDUME M E OKPALA Fxed ot terato for a coutable famly of mult valued strctly seudo cotractve tye mags; Srger Plus (05) [] CE CHIDUME HZEGEYE SHAHZAD Covergece theorems for a Commo fxed ot of a fte famly of oself oexasve mags: Fxed Pot Theory ad Alcatos ; (005) 33-4 [] VCOLAO GMARIO Krasosels Ma method for o-self mags Fxed Pot Theory Al (05) [3] OPFERREIRA PROLIVEIRA Proxmal ot algorthm o Remaa mafolds Otmzato 5() (00) [4] JGARCA-FALSET E LLORES-FUSTER TSUZUKI Fxed ot theory for a class of geeralzed oexasve mags J Math Aal Al 375() (0) [5] KGOEBEL WAKIRK Tocs metrc fxed ot theory Cambrdge Studes Advaced Mathematcs 8 Cambrdge Uversty Press Cambrdge 990 [6] KHUKMIOMURAT ASEZGI O Commo Fxed Pots of Two o-self oexasve mags Baach Saces Chag Ma J Sc; 34(3)(007) 8-88 [7] FO ISIOGUGU MO OSILIKE Covergece theorems for ew classes of multvalued hem cotractve-tye mags Fxed Pot Theory Al (04) [8] SHKHA I YILDIRIM Fxed ots of multvalued oexasve mags Baach saces Fxed Pot Theory Al (0) [9] HKIZILTUC IYILDIRIM O Commo Fxed Pot of oself oexasve mags for Multste Iterato Baach Saces Tha Joural of Mathematcs 6 ( ) (008)

16 90 Amerca Joural of Aled Mathematcs ad Statstcs [0] K KURATOWSKI Toology Academc ress 966 [] G MARIO Fxed ots for multvalued mags defed o ubouded sets Baach saces J Math Aal Al 57() (99) [] G Maro GTrombetta O aroxmatg fxed ots for oexasve mags Ida J Math 34 (99) 9-98 [3] JTMARKI Cotuous deedece of fxed ot sets Proc Am Math Soc 38(973) [4] SBJRADLER Mult-valued cotracto mags Pac J Math 30() (969) [5] M E OKPALA A teratve method for multvalued temered Lschtz hem cotractve mags Afr Mat (07) 8(3-4) [6] BPAYAAK Ma ad Ishawa teratve rocesses for multvalued mags Baach saces Comut Math Al 54(6) (007) [7] KPRSASTRY GVR BABU Covergece of Ishawa terates for a multvalued mag wth a fxed ot Czechoslova Math J 55(4) (005) [8] TW SEBISEBE GS MEGISTU Z HABTU Strog Covergece Theorems for a Commo Fxed Pot of a Fte Famly of Lschtz Hem cotractve-tye Multvalued Mags Advaces Fxed Pot Theory5() (05)8-53 [9] SHAHZAD HZEGEYE O Ma ad Ishawa terato schemes for mult-valued mas Baach saces olear Aal Theory Methods Al 7(3) (009) [30] YSOG R CHE Vscosty aroxmato methods for oexasve oself mags J Math Aal Al 3() (006) [3] YS SOG YJ CHO Averaged terates for o-exasve oself mags Baach saces J Comut Aal Al (009) [3] YSOG HWAG Erratum to Ma ad Ishawa teratve rocesses for multvalued mags Baach saces Comut Math Al 54(007) [33] WTAKAHASHI GE KIM Strog covergece of aroxmats to fxed ots of oexasve oself-mags Baach saces olear Aal Theory Methods Al 3(3) (998) [34] MH TAKELE AD B KREDDY Aroxmato of commo fxed ot of fte famly of oself ad oexasve mags Hlbert sace Ida Joural of Mathematcs ad mathematcal Sceces 3() (07) 77-0 [35] MH TAKELE B KREDDY Fxed ot theorems for aroxmatg a commo fxed ot for a famly of oself strctly seudo cotractve ad ward mags real Hlbert saces Global oural of ure ad aled Mathematcs 3(7) (07) [36] K K Ta ad H K Xu Aroxmatg Fxed Pots of oexasve mags by the Ishawa Iterato Process J Math Aal Al 78() (993) [37] ARTUFA HZEGEYE Ma ad Ishawa-Tye Iteratve Schemes for Aroxmatg Fxed Pots of Mult-valued o- Self Mags MedterrJMath(06) [38] STWOLDEAMAUEL M G SAGAGO H ZEGEYE Strog covergece theorems for a fxed ot of a Lchtz seudo cotractve mult-valued mag Lear olear Aal () (06) [39] HKXU XMYI Strog covergece theorems for oexasve o-self mags olear Aal Theory Methods Al 4() (995) 3-8 [40] HKXu Aroxmatg curves of oexasve oselfmags Baach saces C R Acad Sc Pars Sr I Math 35() (997) 5-56 [4] HKXu Iequaltes Baach saces wth alcatos olear Aal 6 (99) 7-38 [4] HZEGEYE SHAHZAD Covergece of Ma s tye terato method for geeralzed asymtotcally oexasve mags Comut Math Al 6 (0) [43] E ZEIDLERE olear Fuctoal Aalyss ad ts Alcatos I: Fxed-Pot Theorems Srger-Verlag ew Yor Berl Hedelberg Toyo (986)