presented in Korea

1. Weak and strong convergence to common fixed points of a countable family of multivalued mappings in Banach spaces WatctarapornCholamjiak, Chiang Mai University, Thailand SuthepSuantai, Chiang Mai University, Thailand Yeol Je Cho, Gyeongsang National University, Korea

2. Definition The metric projection operator (the nearest point projection) PD defined on E is a mapping from E to 2Dsuch that If PDx for every x in E, then D is called proximinalset. If PDx is a singleton, for every x in E, then D is said to be a Chebyshev set. Let D be a nonempty closed convex subset of a strictly convex and reflexive Banachspace E and . Then there exists a unique element such that .

3. Definition: Hausdorff Metric Let CB(D) = the families of nonempty closed bounded subsets K(D) = the families of nonempty compact subsets P(D) = the families of nonempty proximinal bounded subsets of D The Hausdorff metric on CB(D) is defined by for A,B CB(D)

4. Example: Hausdorff Metric In Real number

5. Definition: MultivaluedNonexpansive Mapping A single valued mapping T:D->D is called nonexpansive if * p=Tp , p F(T)=set of all fixed point of T A multivalued mapping T:D-> CB(D) is callednonexpansiveif * p Tp , p F(T)=set of all fixed point of T

6. Example:MultivaluedNonexpansive mapping Example 1. Consider D=[0,1]x[0,1] with the usual norm. Define T:D->CB(D) by T(x,y)={(x,0),(0,y)}. For x=(x1,y1) , y=(x2,y2) D ,we have H(Tx,Ty)=max{|x1-x2|,|y1-y2|} Example 2. Consider D=[0,1]x[0,1] with the usual norm. Define T:D->CB(D) byT(x,y)={x}x[ ,1 ]. For x=(x1,y1) ,y=(x2,y2) D ,we have H(Tx,Ty)=

7. Definition: the best approximation operator Let T:D->P(D), the best approximation operator PTx defined by

8. Fixed point theory 1. the existence and uniqueness of fixed points 2. the structure of the fixed point sets 3. the approximation of fixed points

9. Mann Iterations for Multivalued Mappings In 2005, Sastry and Babu(Hilbert Spaces) Let T:D-> P(D) be a multi-valued map and fix p in F(T), where such that In 2007, Panyanak(uniformly convex Banach spaces) Let T:D-> P(D) be a multi-valued map and fix p in F(T), where such that

10. Ishikawa iterates for multivaluednonexpansive mappings In 2009, Shahzad and Zegeye (Banach Spaces) Let T:D-> P(D) be a multivaluednonexpansivemappina and . The Ishikawa iterates is Defined by , Where and

11. NST-condition:a family of nonlinear mappings In 2007, Nakajo, Shimoji and Takahashi Let {Tn} and be two families of nonlinear mappings of D into itself with , where is the set of all fixed points of Tn and is the set of common fixed point of . The family {Tn} is said to satisfy the NST-condition with respect to if, for each bounded sequence {zn} in D,

12. SC-condition:a family of multivalued mappings Let {Tn} and be two families of multivalued mappings from D into 2D with , where is the set of all fixed points of Tn and is the set of common fixed point of . The family {Tn} is said to satisfy the SC-condition with respect to if, for each bounded sequence {zn} in D and ,

13. Condition I:a multivalued mapping Let T be a multivalued mapping from D into 2D with . The mapping T is said to satisfy Condition I if there is a non-decresing function with f(0)=0, f(r)>0 for such that for all . In 1974, Senter and Dotson Lemma: Let D be a bounded closed subset of a Banach space E. Suppose that a nonexpansivemultivalued mapping T:D->P(D) has a nonempty fixed point set. If I-T is closed, then T satisfies Condition I on D.

14. Condition A:a family of multivalued mappings Let {Tn} and be two families of multivalued mappings from D into 2D with , where is the set of all fixed points of Tn and is the set of common fixed point of . The family {Tn} is said to satisfy Condition(A) if there is a nondecreasing function with f(0)=0, f(r)>0 for such that for all .

15. Example: the SC-condition and Condition A Let E be a real Banach space , D a nonempty closed convex subset of E , a family of nonexpansivemultivalued mappings of D into CB(D) , such that for all . We define a mapping Sn :D->2D as follows: where the identity mapping.

16. Example: T is not nonexpansive, but PT is nonexpansive Consider D=[0,1] with the usual norm. Define T:D->K(D) by Since , T is not Nonexpansive. However, PT is nonexpansive. Case 1, if then . Case2, if and then Case3, if then

17. Motivation: the modified Mann iteration Let E be a Banachspace, D a nonempty closed convex subset of E, a family of multivalued mappings from D into 2D ,The sequence of the modified Mann iteration is defined byand (1)

18. Motivation: the modified Mann iteration Step1 Step2

19. Motivation: the modified Mann iteration Step3

20. Motivation: Weak convergence Theorem 1 Let D be a closed and convex subset of a uniformly convex Banach space E which satisfies Opial’s property. Let and be two families of multivalued mappings from D into P(D) with . Let be a sequence in (0,1) such that . Let be the sequence generated by (1). Assume that (A1) for each (A2) I-T is demi-closed at 0 for all .If satisfies the SC-condition, then the sequence converges weakly to an element in .

21. Motivation: Weak convergence Remark: If the space satisfies Opial’s property, then I-T is demi-closed at 0, where T:D->K(D) is nonexpansivemultivalued mapping. Corollary2 Let D be a closed and convex subset of a uniformly convex Banach space E which satisfies Opial’s property. Let and be two families of nonexpansivemultivalued mappings from D into K(D) with . Let be a sequence in (0,1) such that . Let be the sequence generated by (1). Assume that for eachIf satisfies the SC-condition, then the sequence converges weakly to an element in .

22. Motivation: Strong convergence Theorem 3 Let D be a closed and convex subset of a uniformly convex Banach space E which satisfies Opial’s property. Let and be two families of multivalued mappings from D into P(D) with . Let be a sequence in (0,1) such that . Let be the sequence generated by (1). Assume that (B1) for each (B2) the best approximation operator is nonexpansive for every ; (B3) is closed. If satisfies the SC-condition, then the sequence converges strong to an element in .

23. Motivation: Weak convergence Remark: If T is a quasi-nonexpansivemultivalued mapping, then is closed Corollary4 Let D be a closed and convex subset of a uniformly convex Banach space E which satisfies Opial’s property. Let and be two families of nonexpansivemultivalued mappings from D into P(D) with . Let be a sequence in (0,1) such that . Let be the sequence generated by (1). Assume that for each and the best approximation operator is nonexpansive for every .If satisfies the SC-condition, then the sequence converges strong to an element in .

25. The Graduate School of Chiang Mai University