Research Article Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces

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1 Iteratioal Scholarly Research Network ISRN Mathematical Aalysis Volume 2011, Article ID , 13 pages doi: /2011/ Research Article Covergece Theorems for Fiite Family of Multivalued Maps i Uiformly Covex Baach Spaces Yekii Shehu Departmet of Mathematics, Uiversity of Nigeria, Nsukka, Nigeria Correspodece should be addressed to Yekii Shehu, deltaougt2006@yahoo.com Received 20 April 2011; Accepted 9 Jue 2011 Academic Editors: A. Levy ad O. Miyagaki Copyright q 2011 Yekii Shehu. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We itroduce a ew iterative process to approximate a commo fixed poit of a fiite family of multivalued maps i a uiformly covex real Baach space ad establish strog covergece theorems for the proposed process. Furthermore, strog covergece theorems for fiite family of quasi-oexpasive multivalued maps are obtaied. Our results exted importat recet results. 1. Itroductio Let D be a oempty, closed, ad covex subset of a real Hilbert space H.ThesetD is called proximial if for each x H, there exists y D such that x y d x, D, where d x, D if{ x z : z D}.LetCB D,K D,adP D deote the families of oempty, closed ad bouded subsets, oempty compact subsets, ad oempty proximial bouded subsets of D, respectively. The Hausdorff metric o CB D is defied by H A, B max { sup x A d x, B, sup d ( y, A )}, y B 1.1 for A, B CB D. A sigle-valued map T : D D is called oexpasive if Tx Ty x y for all x, y D. A multivalued map T : D CB D is said to be oexpasive if H Tx,Ty x y for all x, y D. A elemet p D is called a fixed poit of T : D D resp., T : D CB D if p Tp resp., p Tp. The set of fixed poits of T is deoted by F T. A multivalued map T : D CB D is said to be quasi-oexpasive if H Tx,Tp x p for all x D ad for all p F T.

2 2 ISRN Mathematical Aalysis A T : D CB D is said to satisfy Coditio (I) if there is a odecreasig fuctio f : 0, 0, with f 0 0,f r > 0forr 0, such that d x, Tx f d x, F T, 1.2 for all x D. The fixed poit theory of multivalued oexpasive mappigs is much more complicated ad difficult tha the correspodig theory of sigle-valued oexpasive mappigs. However, some classical fixed poit theorems for sigle-valued oexpasive mappigs have already bee exteded to multivalued mappigs. The first results i this directio were established by Marki 1 i Hilbert spaces ad by Browder 2 for spaces havig weakly cotiuous duality mappig. Dozo 3 geeralized these results to a Baach space satisfyig Opial s coditio. I 1974, by usig Edelstei s method of asymptotic ceters, Lim 4 obtaied a fixed poit theorem for a multivalued oexpasive self-mappig i a uiformly covex Baach space. Theorem 1.1 Lim 4. Let D be a oempty, closed covex, ad bouded subset of a uiformly covex Baach space E ad T : D C E a multivalued oexpasive mappig. The, T has a fixed poit. I 1990, Kirk ad Massa 5 gave a extesio of Lim s theorem provig the existece of a fixed poit i a Baach space for which the asymptotic ceter of a bouded sequece i a closed bouded covex subset is oempty ad compact. Theorem 1.2 Kirk ad Massa 5. Let D be a oempty, closed covex, ad bouded subset of a Baach space E ad T : D CB E a multivalued oexpasive mappig. Suppose that the asymptotic ceter i E of each bouded sequece of E is oempty ad compact. The, T has a fixed poit. Baach cotractio mappig priciple was exteded icely multivalued mappigs by Nadler 6 i Below is stated i a Baach space settig. Theorem 1.3 Nadler 6. Let D be a oempty closed subset of a Baach space E ad T : D CB D a multivalued cotractio. The, T has a fixed poit. I 1953, Ma 7 itroduced the followig iterative scheme to approximate a fixed poit of a oexpasive mappig T i a Hilbert space H: x 1 α x 1 α Tx, 1, 1.3 where the iitial poit x 0 is take arbitrarily i D ad {α } 1 is a sequece i 0, 1. However, we ote that Ma s iteratio has oly weak covergece; see, for example, 8. I 2005, Sastry ad Babu 9 proved that the Ma ad Ishikawa iteratio schemes for a multivalued map T with a fixed poit p coverge to a fixed poit q of T uder certai coditios. They also claimed that the fixed poit p may be differet from q. I 2007, Payaak 10 exteded the results of Sastry ad Babu to uiformly covex Baach spaces ad proved the followig theorems.

3 ISRN Mathematical Aalysis 3 Theorem 1.4 Payaak 10. Let E be a uiformly covex Baach space, D a oempty closed bouded covex subset of E, ad T : D P D a multivalued oexpasive mappig that satisfies coditio I. Assume that (i) 0 α < 1 ad (ii) Σ 1 α. Suppose that F T a oempty proximial subset of D. The, the Ma iterates {x } defied by x 0 D, x 1 α y 1 α x, α a, b, 0 <a<b<1, 0, 1.4 where y Tx such that y u d u,tx ad u F T such that x u d x,f T, coverges strogly to a fixed poit of T. Theorem 1.5 Payaak 10. Let E be a uiformly covex Baach space, D a oempty compact covex subset of E, ad T : D P D a multivalued oexpasive mappig with a fixed poit p. Assume that (i) 0 α,β < 1; (ii) β 0 ad (iii) Σ 1 α β. The, the Ishikawa iterates {x } defied by x 0 D, y β z ( 1 β ) x, β 0, 1, 0, 1.5 z Tx such that z p d p, Tx, ad x 1 α z 1 α x, α 0, 1, / 0, 1.6 z Ty such that z p d p, Ty coverges strogly to a fixed poit of T. Later, Sog ad Wag 11 oted there was a gap i the proofs of Theorem 1.5 above ad of 9, Theorem 5. They further solved/revised the gap ad also gave the affirmative aswer Payaak 10 questio usig the Ishikawa iterative scheme. I the mai results, the domai of T is still compact, which is a strog coditio see 11, Theorem 1 ad T satisfies coditio I see 11, Theorem 1. Recetly, Shahzad ad Zegeye 12 proved the followig theorems for quasioexpasive multivalued map ad multivalued map i uiformly covex Baach space. Theorem 1.6 Shahzad ad Zegeye 12. Let E be a uiformly covex real Baach space ad D a oempty, closed ad covex subset of E. LetT : D CB D be a quasi-oexpasive multivalued map with F T / for which Tp {p}, for all p F T.Let{x } 1 be a sequece defied iteratively by x 0 D, y β z ( 1 β ) x, β 0, 1, 0, 1.7 z Tx, ad x 1 α z 1 α x, α 0, 1, / 0, 1.8 z Ty. Assume that T satisfies coditio (I) ad α,β a, b 0, 1. The, {x } 1 coverges strogly to a fixed poit of T.

4 4 ISRN Mathematical Aalysis Theorem 1.7 Shahzad ad Zegeye 12. Let E be a uiformly covex real Baach space ad D a oempty, closed, ad covex subset of E. LetT : D P D be a multivalued map with F T / such that P T is oexpasive. Let {x } 1 be a sequece defied iteratively by x 0 D, y β z ( 1 β ) x, β 0, 1, 0, 1.9 z P T x, ad x 1 α z 1 α x, α 0, 1, / 0, 1.10 z P T y. Assume that T satisfies coditio (I) ad α,β a, b 0, 1. The, {x } 1 coverges strogly to a fixed poit of T. More recetly, Abbas et al. 13 itroduced the followig oe-step iterative process to compute commo fixed poits of two multivalued oexpasive mappigs. x 1 D, x 1 a x b y c z, Usig 1.11, Abbas et al. 13 proved weak ad strog covergece theorems for approximatio of commo fixed poit of two multivalued oexpasive mappigs i Baach spaces. Motivated by the ogoig research ad the above metioed results, we itroduce a ew iterative scheme for approximatio of commo fixed poits of fiite family of multivalued maps i a real Baach space. Furthermore, we prove strog covergece theorems for approximatio of commo fixed poits of fiite family of multivalued maps i a uiformly covex real Baach space. Next, we prove a ecessary ad sufficiet coditio for strog covergece of our ew iterative process to a commo fixed poit of fiite family of multivalued maps. Fially, we itroduce a ew iterative scheme ad prove strog covergece theorems for fiite family of quasi-oexpasive multivalued maps i a uiformly covex real Baach space. Our results exted the results of Sastry ad Babu 9, Payaak 10, Shahzad ad Zegeye 12, ad Sog ad Wag Prelimiaries Let E be Baach space ad dim E 2. The modulus of covexity of E is the fuctio δ E : 0, 2 0, 1 defied by { δ E ɛ : if 1 x y 2 : x y 1; ɛ x y }. 2.1 E is uiformly covex if for ay ɛ 0, 2, there exists a δ δ ɛ > 0 such that if x, y E with x 1, y 1ad x y ɛ, the 1/2 x y 1 δ. Equivaletly, E is uiformly covex if ad oly if δ E ɛ > 0 for all ɛ 0, 2.

5 ISRN Mathematical Aalysis 5 A family {T i : D CB D, i 1, 2,...,m} is said to satisfy Coditio (II) if there is a odecreasig fuctio f : 0, 0, with f 0 0,f r > 0forr 0, such that d x, T i x f ( d ( x, m i 1 F T i )), 2.2 for all i 1, 2,...,mad x D. The mappig T : D CB D is called hemicompact if for ay sequece {x } i D such that d x,tx 0as, there exists a subsequece {x k } of {x } such that x k p D.WeotethatifD is compact, the every multivalued mappig T : D CB D is hemicompact. Let D be a oempty, closed, ad covex subset of a real Baach space E.LetT : D P D be a multimap ad P T x : {u x Tx : x u x d x, Tx }. 2.3 The, P T x : D P D is oempty ad compact for every x D. Furthermore, we observe that P T y {y} if y is a fixed poit of T. A mappig T : D P D is -oexpasive 14 if for all x, y D ad u x Tx with d x, u x if{d x, z : z Tx}, there exists u y Ty with d y, u y if{d y, w : w Ty} such that d ( u x,u y ) d ( x, y ). 2.4 It is kow that -oexpasiveess is differet from oexpasiveess for multimaps. There are some -oexpasive multimaps which are ot oexpasive ad some oexpasive multimaps which are ot -oexpasive 15, 16. By the defiitio of Hausdorff metric, we obtai that if a multimap T : D P D is -oexpasive, the P T is oexpasive. Throughout this paper, we write x x to idicate that the sequece {x } coverges strogly to x. Also, this followig lemma will be used i the sequel. Lemma 2.1 Schu 17. Suppose that E is a uiformly covex Baach space ad 0 <p t q<1 for all positive itegers. Also, suppose that {x } ad {y } are two sequeces of E such that lim sup x r, lim sup y r ad lim t x 1 t y r hold for some r>0. The, lim x y Mai Results We ow itroduce the followig iteratio scheme. Let E be a real ormed space ad D a oempty subset of E. LetT 1,T 2,...,T m be multivalued maps of D ito P D with F : m i 1 F T i / such that P T1,P T2,...,P Tm are oexpasive ad {α i } 1,i 0, 1,...,m

6 6 ISRN Mathematical Aalysis a sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let {x } 1 be a sequece defied iteratively by x 1 D, x 1 α 0 x α 1 y 1 α m, 3.1 where y i P Ti x,i 1, 2,...,m. Lemma 3.1. Let E be a real ormed space ad D a oempty subset of E. LetT 1,T 2,...,T m be multivalued maps of D ito P D with F : m i 1 F T i / such that P T1,P T2,...,P Tm are oexpasive. Let {α i } 1, i 0, 1,...,ma sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let{x } 1 be a sequece defied iteratively by 3.1. The, lim d x,t i x 0, i 1, 2,...,m. 3.2 Proof. Let x m i 1 F T i. The, from 3.1, we have the followig estimates: x 1 x α 0 x x α 1 y 1 ( α 0 x x α 1 d y 1 x αm x,p T1 x ) ( α m d,p Tm x ) α 0 x x α 1 H P T1 x,p T1 x α m H P Tm x,p Tm x 3.3 α 0 x x α 1 x x α m x x x x. Thus, lim x x exists. Let lim x x c, 3.4 for some c 0. The, c lim x 1 x ( α0 lim x x α 1 y 1 x ) ( α m x ) lim 1 α 0 [ α1 1 α 0 ( y 1 x ) α ( m 1 α 0 x )] α 0 x x. 3.5 Sice P Ti,i 1, 2,...,mis oexpasive mappig ad m i 1 F T i /, we have y i x d ( y i,p Ti x ) H P Ti x,p Ti x x x, 3.6

7 ISRN Mathematical Aalysis 7 for each x m i 1 F T i. Takig lim sup o both sides, we obtai lim supy i x c, i 1, 2,...,m. 3.7 Next, lim sup α ( 1 y 1 x ) α ( 2 y 2 x ) α ( m 1 α 0 1 α 0 1 α 0 x ) [ α1 y 1 lim sup x α 2 y 2 x α m x ] 1 α 0 1 α 0 1 α α 1 α 2 α m lim sup x x c. 1 α 0 Usig 3.5, 3.8, ad Lemma 2.1,weobtai lim α ( 1 y 1 x ) α ( 2 y 2 x ) α ( m 1 α 0 1 α 0 1 α 0 x ) x x This yields α 1 0 lim y 1 α 2 y 2 α m x 1 α 0 1 α 0 1 α 0 ( 1 ) α1y lim 1 α 2 y 2 α m 1 α 0 x 1 α 0 ( ) 1 lim x 1 x. 1 α Thus, lim x 1 x 0. Furthermore, c lim x 1 x ( α0 lim x x α 1 y 1 x ) ( α m x ) [ α0 lim 1 α 1 x x α ( 2 y 2 x ) α ( m x )] 1 α 1 1 α 1 1 α 1 α 1 ( y 1 x ),

8 8 ISRN Mathematical Aalysis lim sup α 0 x x α ( 2 y 2 x ) α ( m x ) 1 α 1 1 α 1 1 α 1 [ α0 lim sup x x α 2 y 2 x α m x ] 1 α 1 1 α 1 1 α 1 α 0 α 2 α m lim sup x x c. 1 α Usig 3.11 ad Lemma 2.1,weobtai lim α 0 x x α ( 2 y 2 x ) α ( m 1 α 1 1 α 1 1 α 1 x ) ( y 1 x ) This yields α 0 0 lim x α 2 y 2 α m 1 α 1 1 α 1 1 α 1 ( 1 ) α0x lim α 2y 2 α m 1 α 1 ( 1 ) x 1 lim y 1. 1 α 1 y 1 1 α 1 y Thus, lim x 1 y 1 0. So, x y 1 x 1 x x 1 y 1 0, The, d x,t 1 x d x,p T1 x x y 1 0, I a similar way, we ca show that lim d x,t i x 0, i 2, 3,...,m This completes the proof. Theorem 3.2. Let E be a uiformly covex real Baach space ad D a oempty, closed, ad covex subset of E.LetT 1,T 2,...,T m be multivalued maps of D ito P D with F : m i 1 F T i / such that P T1,P T2,...,P Tm are oexpasive ad {T i } m i 1 satisfyig coditio (II). Let {α i} 1,i 0, 1,...,m

9 ISRN Mathematical Aalysis 9 a sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let{x } 1 be a sequece defied iteratively by 3.1. The, {x } 1 coverges strogly to a commo fixed poit of {T i} m i 1. Proof. Sice {T i } m i 1 satisfies coditio II, we have that d x, m i 1 F T i 0as. Thus, there is a subsequece {x k } of {x } ad a sequece {p k } m i 1 F T i such that for all k. By Lemma 3.1,weobtai xk p k < 1 2 k, 3.17 xk 1 p k xk p k < 1 2 k We ow show that {p k } is a Cauchy sequece i D. Observe that pk 1 p k pk 1 x k 1 xk 1 p k < 1 2 k k < 1 2 k This shows that {p k } is a Cauchy sequece i D, ad thus coverges to p D. Sice d ( p k,t i p ) d ( pk,p Ti p ) H ( P Ti p, P Ti p k ) p pk, 3.20 ad p k p as k, it follows that d p, T i p 0, ad thus p m i 1 F T i, ad{x k } coverges strogly to p. Sice lim x p exists, it follows that {x } coverges strogly to p. This completes the proof. Corollary 3.3. Let E be a uiformly covex real Baach space ad D a oempty, closed, ad covex subset of E. LetT 1,T 2,...,T m be -oexpasive multimaps of D ito P D with F : m i 1 F T i / ad {T i } m i 1 satisfyig coditio (II). Let {α i} 1,i 0, 1,...,ma sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let{x } 1 be a sequece defied iteratively by 3.1. The, {x } 1 coverges strogly to a commo fixed poit of {T i} m i 1. Theorem 3.4. Let E be a uiformly covex real Baach space ad D a oempty, closed ad covex subset of E.LetT 1,T 2,...,T m be multivalued maps of D ito P D with F : m i 1 F T i / such that P T1,P T2,...,P Tm are oexpasive ad T i is hemicompact ad cotiuous for each i 1, 2,...,m.Let {α i } 1,i 0, 1,...,m a sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let {x } 1 be a sequece defied iteratively by 3.1. The, {x } 1 coverges strogly to a commo fixed poit of {T i } m i 1. Proof. Sice lim d x,t i x 0, for all i 1, 2,...,m ad T i is hemicompact for each i 1, 2,...,m, there is a subsequece {x k } of {x } such that x k p as k for some p D.

10 10 ISRN Mathematical Aalysis Sice T i is cotiuous for each i 1, 2,...,m, we have d x k,t i x k d p, T i p. Asaresult, we have that d p, T i p 0, for all i 1, 2,...,m,adso,p m i 1 F T i. Sice lim x p exists, it follows that {x } coverges strogly to p. This completes the proof. Corollary 3.5. Let E be a uiformly covex real Baach space ad D a oempty, closed, ad covex subset of E. LetT 1,T 2,...,T m be -oexpasive multimaps of D ito P D with F : m i 1 F T i / ad T i is hemicompact ad cotiuous for each i 1, 2,...,m.Let{α i } 1, i 0, 1,...,masequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let{x } 1 be a sequece defied iteratively by 3.1. The, {x } 1 coverges strogly to a commo fixed poit of {T i} m i 1. Theorem 3.6. Let E be a uiformly covex real Baach space ad D a oempty compact covex subset of E.LetT 1,T 2,...,T m be multivalued maps of D ito P D with F : m i 1 F T i / such that P T1,P T2,...,P Tm are oexpasive. Let {α i } 1,i 0, 1,...,ma sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let{x } 1 be a sequece defied iteratively by 3.1. The, {x } 1 coverges strogly to a commo fixed poit of {T i} m i 1. Proof. From the compactess of D, there exists a subsequece {x k } k of {x } 1 such that lim k x k q 0 for some q D.Thus, d ( q, T i q ) d ( q, P Ti q ) xk q d xk,p Ti x k H ( P Ti x k,p Ti q ) 2 xk q d xk,p Ti x k 0 as k Hece, q m i 1 F T i. Now, o takig q i place of x, we get that lim x q exists. This completes the proof. The followig result gives a ecessary ad sufficiet coditio for strog covergece of the sequece i 3.1 to a commo fixed poit of {T i } m i 1. Theorem 3.7. Let D be a oempty, closed, ad covex subset of a real Baach space E. Let T 1,T 2,...,T m be multivalued maps of D ito P D with F : m i 1 F T i / such that P T1,P T2,...,P Tm are oexpasive. Let {α i } 1,i 0, 1,...,m a sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let{x } 1 be a sequece defied iteratively by 3.1. The, {x } 1 coverges strogly to a commo fixed poit of {T i } m i 1 if ad oly if lim if d x,f 0. Proof. The ecessity is obvious. Coversely, suppose that lim if d x, F 0. By 3.3, we have x 1 x x x This gives d x 1,F d x,f Hece, lim d x,f exists. By hypothesis, lim if d x,f 0, so we must have lim d x,f 0.

11 ISRN Mathematical Aalysis 11 Next, we show that {x } 1 is a Cauchy sequece i D. Letɛ>0begive, ad sice lim if d x,f 0, there exists 0 such that for all 0, we have d x,f < ɛ I particular, if{ x 0 p : p F} <ɛ/4 so that there must exist a p F such that Now, for m, 0, we have x0 p < ɛ x m x x m p x p 2 x0 p ( ɛ < 2 ɛ. 2) 3.26 Hece, {x } is a Cauchy sequece i a closed subset D of a Baach space E, ad therefore, it must coverge i D. Let lim x p. Now, for each i 1, 2,...,m,weobtai d ( p, T i p ) d ( p, P Ti p ) d ( p, x ) d x,p Ti x H ( P Ti x,p Ti p ) d ( p, x ) d x,p Ti x d ( x,p ) 0 as 3.27 gives that d p, T i p 0, m i 1 F T i /. i 1, 2,...,m which implies that p T i p. Cosequetly, p F All the results we have obtaied so far ca be established for fiite family of quasioexpasive multivalued maps. Let T 1,T 2,...,T m be quasi-oexpasive multivalued maps of D ito CB D such that F : m i 1 F T i / for which T i p {p}, for all p m i 1 F T i. Let {α i } 1,i 0, 1,...,ma sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let {x } 1 be a sequece defied iteratively by x 1 D, x 1 α 0 x α 1 y 1 α m, 3.28 where y i T i x, i 1, 2,...,m. Thus, we obtai the followig theorems usig iterative process Theorem 3.8. Let E be a uiformly covex real Baach space ad D a oempty, closed, ad covex subset of E. LetT 1,T 2,...,T m be quasi-oexpasive multivalued maps of D ito CB D such that F : m i 1 F T i / for which T i p {p}, for all p m i 1 F T i ad {T i } m i 1 satisfyig coditio (II). Let {α i } 1,i 0, 1,...,ma sequece i ɛ, 1 ɛ,ɛ 0, 1 such that m i 0 α i 1 for all 1. Let {x } 1 be a sequece defied iteratively by The, {x } 1 coverges strogly to a commo fixed poit of {T i } m i 1.

12 12 ISRN Mathematical Aalysis Theorem 3.9. Let E be a uiformly covex real Baach space ad D a oempty, closed, ad covex subset of E. LetT 1,T 2,...,T m be quasi-oexpasive multivalued maps of D ito CB D such that F : m i 1 F T i / for which T i p {p}, for all p m i 1 F T i ad T i is hemicompact ad cotiuous for each i 1, 2,...,m.Let{α i } 1,i 0, 1,...,m a sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1. Let{x } 1 be a sequece defied iteratively by The, {x } 1 coverges strogly to a commo fixed poit of {T i } m i 1. Theorem Let E be a uiformly covex real Baach space ad D a oempty compact covex subset of E. LetT 1,T 2,...,T m be quasi-oexpasive multivalued maps of D ito CB D such that F : m i 1 F T i / for which T i p {p}, for all p m i 1 F T i. {α i } 1, i 0, 1,...,ma sequece i ɛ, 1 ɛ, ɛ 0, 1 such that m i 0 α i 1 for all 1.Let{x } 1 be a sequece defied iteratively by The, {x } 1 coverges strogly to a commo fixed poit of {T i} m i 1. Corollary 3.11 Abbas et al. 13. Let E be a uiformly covex real Baach space satisfyig Opial s coditio. Let D be a oempty, closed, ad covex of E. LetT, S be multivalued oexpasive mappigs of D ito K D such that F : F T F S /. Let{a } 1, {b } 1, ad {c } 1 be sequece i 0, 1 satisfyig a b c 1. Let{x } 1 be a sequece defied iteratively by x 1 D, x 1 a x b y c z, 1, 3.29 where y Tx,z Sx such that y p d p, Tx ad z p d p, Sx wheever p is a fixed poit of ay oe of mappigs T ad S. The, {x } 1 coverges weakly to a commo fixed poit of F T F S. Corollary 3.12 Abbas et al. 13. Let E be a real Baach space ad D a oempty, closed, ad covex subset of E. LetT, S be multivalued oexpasive mappigs of D ito K D such that F : F T F S /. Let{a } 1, {b } 1 ad {c } 1 be sequece i 0, 1 satisfyig a b c 1. Let {x } 1 be a sequece defied iteratively by x 1 D, x 1 a x b y c z, 1, 3.30 where y Tx, z Sx such that y p d p, Tx ad z p d p, Sx wheever p is a fixed poit of ay oe of mappigs T ad S. The, {x } 1 coverges strogly to a commo fixed poit of F T F S if ad oly if lim if d x,f 0. Remark Our results exted the results of Sastry ad Babu 9, Payaak 10, ad Sog ad Wag 11 from approximatio of a fixed poit of a sigle multivaued oexpasive mappig to approximatio of commo fixed poit of a fiite family of quasi-oexpasive multivaued mappigs. Remark Our results exted the results of Shahzad ad Zegeye 12 from approximatio of a fixed poit of a sigle quasi-oexpasive multivaued mappig ad sigle multivalued map to approximatio of commo fixed poit of a fiite family of quasi-oexpasive multivaued mappigs ad a fiite family of multivalued maps.

13 ISRN Mathematical Aalysis 13 Refereces 1 J. T. Marki, A fixed poit theorem for set valued mappigs, Bulleti of the America Mathematical Society, vol. 74, pp , F. E. Browder, Noliear operators ad oliear equatios of evolutio i Baach spaces, i Noliear Fuctioal Aalysis (Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968), pp , Amer. Math. Soc., Providece, RI, USA, E. L. Dozo, Multivalued oexpasive mappigs ad Opial s coditio, Proceedigs of the America Mathematical Society, vol. 38, pp , T. C. Lim, A fixed poit theorem for multivalued oexpasive mappigs i a uiformly covex Baach space, Bulleti of the America Mathematical Society, vol. 80, pp , W. A. Kirk ad S. Massa, Remarks o asymptotic ad Chebyshev ceters, Housto Joural of Mathematics, vol. 16, o. 3, pp , S. B. Nadler Jr., Multi-valued cotractio mappigs, Pacific Joural of Mathematics, vol. 30, pp , W. R. Ma, Mea value methods i iteratio, Proceedigs of the America Mathematical Society, vol. 4, pp , A. Geel ad J. Lidestrauss, A example cocerig fixed poits, Israel Joural of Mathematics, vol. 22, o. 1, pp , K. P. R. Sastry ad G. V. R. Babu, Covergece of Ishikawa iterates for a multi-valued mappig with a fixed poit, Czechoslovak Mathematical Joural, vol , o. 4, pp , B. Payaak, Ma ad Ishikawa iterative processes for multivalued mappigs i Baach spaces, Computers & Mathematics with Applicatios, vol. 54, o. 6, pp , Y. Sog ad H. Wag, Erratum to: Ma ad Ishikawa iterative processes for multivalued mappigs i Baach spaces Comput. Math. Appl. vol , o. 6, ; MR by B. Payaak, Computers & Mathematics with Applicatios, vol. 55, o. 12, pp , N. Shahzad ad H. Zegeye, O Ma ad Ishikawa iteratio schemes for multi-valued maps i Baach spaces, Noliear Aalysis: Theory, Methods & Applicatios, vol. 71, o. 3-4, pp , M. Abbas, S. H. Kha, A. R. Kha, ad R. P. Agarwal, Commo fixed poits of two multivalued oexpasive mappigs by oe-step iterative scheme, Applied Mathematics Letters, vol. 24, o. 2, pp , T. Husai ad A. Latif, Fixed poits of multivalued oexpasive maps, Mathematica Japoica, vol. 33, o. 3, pp , N. Shahzad ad H. Zegeye, Strog covergece results for oself multimaps i Baach spaces, Proceedigs of the America Mathematical Society, vol. 136, o. 2, pp , H. K. Xu, O weakly oexpasive ad -oexpasive multi-valued mappigs, Mathematica Japoica, vol. 36, pp , J. Schu, Weak ad strog covergece to fixed poits of asymptotically oexpasive mappigs, Bulleti of the Australia Mathematical Society, vol. 43, o. 1, pp , 1991.

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