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# W.cholamjiak

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### W.cholamjiak

1. 1. Weak and strong convergence to common fixed points of a countable family of multivalued mappings in Banach spaces<br />WatctarapornCholamjiak, Chiang Mai University, Thailand<br />SuthepSuantai, Chiang Mai University, Thailand<br />Yeol Je Cho, Gyeongsang National University, Korea<br />
2. 2. Definition<br />The metric projection operator (the nearest point projection) PD defined on E is a mapping from E to 2Dsuch that <br />If PDx for every x in E, then D is called proximinalset. <br />If PDx is a singleton, for every x in E, then D is said <br />to be a Chebyshev set.<br /> 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 .<br />
3. 3. Definition: Hausdorff Metric<br />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<br />The Hausdorff metric on CB(D) is defined by<br />for A,B CB(D)<br />
4. 4. Example: Hausdorff Metric<br />In Real number<br />
5. 5. Definition: MultivaluedNonexpansive Mapping<br />A single valued mapping T:D->D is called nonexpansive if<br />* p=Tp , p F(T)=set of all fixed point of T<br />A multivalued mapping T:D-> CB(D) is callednonexpansiveif <br />* p Tp , p F(T)=set of all fixed point of T<br />
6. 6. Example:MultivaluedNonexpansive mapping<br />Example 1.<br />Consider D=[0,1]x[0,1] with the<br />usual norm. Define T:D->CB(D) by T(x,y)={(x,0),(0,y)}.<br />For x=(x1,y1) , y=(x2,y2) D ,we have<br />H(Tx,Ty)=max{|x1-x2|,|y1-y2|}<br />Example 2.<br />Consider D=[0,1]x[0,1] with the<br />usual norm. Define T:D->CB(D) byT(x,y)={x}x[ ,1 ].<br />For x=(x1,y1) ,y=(x2,y2) D ,we have<br />H(Tx,Ty)=<br />
7. 7. Definition: the best approximation operator<br />Let T:D->P(D), the best approximation operator PTx defined by<br />
8. 8. Fixed point theory <br />1. the existence and uniqueness of fixed points<br />2. the structure of the fixed point sets<br />3. the approximation of fixed points<br />
9. 9. Mann Iterations for Multivalued Mappings <br />In 2005, Sastry and Babu(Hilbert Spaces)<br /> Let T:D-> P(D) be a multi-valued map and fix p in F(T), <br /> where such that <br />In 2007, Panyanak(uniformly convex Banach spaces)<br /> Let T:D-> P(D) be a multi-valued map and fix p in F(T), <br /> where such that <br />
10. 10. Ishikawa iterates for multivaluednonexpansive mappings<br />In 2009, Shahzad and Zegeye (Banach Spaces)<br /> Let T:D-> P(D) be a multivaluednonexpansivemappina and <br /> . The Ishikawa iterates is <br />Defined by ,<br />Where and<br />
11. 11. NST-condition:a family of nonlinear mappings<br /> In 2007, Nakajo, Shimoji and Takahashi<br />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,<br />
12. 12. SC-condition:a family of multivalued mappings<br />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 ,<br />
13. 13. Condition I:a multivalued mapping<br />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 . <br />In 1974, Senter and Dotson<br />Lemma: Let D be a bounded closed subset of a Banach space E.<br />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. <br />
14. 14. Condition A:a family of multivalued mappings<br />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 . <br />
15. 15. Example: the SC-condition and Condition A<br /> 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. <br />
16. 16. Example: T is not nonexpansive, but PT is nonexpansive<br />Consider D=[0,1] with the usual norm. Define T:D->K(D) by<br />Since , T is not <br />Nonexpansive. However, PT is nonexpansive. <br />Case 1, if then .<br />Case2, if and then <br />Case3, if then <br />
17. 17. Motivation: the modified Mann iteration<br />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)<br />
18. 18. Motivation: the modified Mann iteration<br />Step1<br />Step2<br />
19. 19. Motivation: the modified Mann iteration<br />Step3<br />
20. 20. Motivation: Weak convergence<br />Theorem 1<br />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 . <br />
21. 21. Motivation: Weak convergence<br />Remark: If the space satisfies Opial’s property, then I-T is demi-closed at 0, where T:D->K(D) is nonexpansivemultivalued mapping.<br />Corollary2<br />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 . <br />
22. 22. Motivation: Strong convergence<br />Theorem 3<br />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 .<br />
23. 23. Motivation: Weak convergence<br />Remark: If T is a quasi-nonexpansivemultivalued mapping, then<br /> is closed <br />Corollary4<br />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 . <br />
24. 24. Acknowlegement<br />The author would like to thank Prof. Yeol Je Cho and Prof. Shin Min Kang for the helpfulness in Korea and also thank for<br />- Gyeongsang National University, Korea<br /><ul><li>The Center of Excellence in Mathematics, Thailand
25. 25. The Graduate School of Chiang Mai University</li>