Common Fixed point of two multivalued quasi-nonexpansive mappings by two-step iterative scheme.doc

1、1Common Fixed point of two multivalued quasi-nonexpansive mappings by two-step iterative schemeAbstract. In this paper,we introuduce a new two-step iterative process to approximate common Fixed point of two multivalued quasi-nonexpansive mappings in a real uniformly convex Banach Space.Furthermore,

2、we also prove some strong and weak convergence theorems in uniformly convex Banach Space. Key words: nonexpansive mapping; fixed point ;two-step iterative scheme 1. Introduction Let be a real Banach Space.A subset of called proximinal，if for each ，there exists an element , such that . It is well kno

3、wn that weakly compact convex subsets of a Banach Space and closed convex subsets of a uniformly cinvex Banach Space are proximinal. We shall denote the family of nonempty bounded proximinal subsets of by ,the family of nonempty compact subsets of by ,and the family of nonempty closed bounded subset

4、s by . Let be the Hausdorff metric induced by the metric of 2and given by For .It is obivious that . A point is called a fixed point of a multivalued mapping if .We denote the set of all fixed points of by . A multivalued mapping is said to be (i) nonexpansive if for all ; (ii)quasi nonexpansive if

5、and for all every . It is well known that a multivalued nonexpansive mapping has a fixed point if is nonempty closed bounded convex subset of a uniformly convex Banach Space . Throughout the paper denotes the set of positive integers. The mapping is called hemicompact, if for any sequence in such th

6、at as ,there exists a subsequence of ,such that . We note that if is compact, then every multi-valued mapping is hemicompact. is said to satisfy Condition (I), if there is a nondecreasing function with for such that for all . 3The theory of multivalued nonexpansive mapping is harder than the corresp

7、onding theory of single valued nonexpansive mappings. Different iterative processes have been used to approximate fixed points of multivalued nonexpansive mappings. Sastry and Babu considered the following. Let be a mapping with , The sequence of Mann iterates is defined (1) where is such that and i

8、s a sequence in satisfying .Danyanak modified the iteration scheme,the sequence of Ishikawa iterates is defined by (2) where are such that and and , are real sequences of numbers with satisfying and . On the other hand,Mujahid Abbas obtained common fixed point of two nonexpansive mapping satisfying

9、certain condition to achieve this ,they employed the following iterative process (3) where and , such that ,where 4is fixed point of any one of the mappings and , are sequence of niumbers in satisfying . Hu et al propose a Ishikawa iteratives defined by (4) where are in satisfying certain conditions

10、. In this paper, we introuce a new Ishikawa iterstive process to approximate common fixed points of two multivalued quasi-nonexpansive mapping. Let be a nonempty convex of a Banach space and with ,the sequence of Ishikawa iterates is defined by (5) Where . 2. Preliminaries We shall make use of the f

11、ollowing result of Xu Lemma2.1 Let be a fixed number ,then is a uniformly convex if and only if there exists a continuous strictly increasing and convex function with such that for all and . 3. Main results 5In order to prove some strong convergence theorems.We need the following Lemmas by means of

12、iterative(5), we shall prove the following lemmas. Lemma3.1 Let be a uniformly convex Banach space, be a nonempty bounded convex subset of . be a quasi-nonexpansive multi-valued map with and for which for each , let be the Ishikawa iterates defined by (5), then exists for each . Proof. Let , it foll

13、ows from (5) that Consequently,the sequence is decreasing and bounded below, thus exists for all .Also is bounded. Theorem3.1 Let be a uniformly convex Banach space, be a non empty closed convex subset of . be a quasi-nonexpansive multi-valued map with and for each , let be the Ishikawa iterates def

14、ined by (5), Assume that satisfies condition ( ) and , then converges strongly to a common fixed point of and . Proof. Let ,then , is bounded.Therefore,there exists , such that , for all .Applying Lemma 2.1, we have Since ,we have .Since is continuous at and strictly 6incresing,we have ,so .Since sa

15、tisfies condition , .Since is strctly increasing,we have ,thus there exists a subsuquence and a ,such that for all .By Lemma3.1,we obtain ,This shows that is a cauchy sequence in ,let convergence to as . Since and , thus which implies that .Similarly that Since exists,it follows that convergence str

16、ongly to . 4. Acknowledgements (1) Project of Education Department of Hebei Province,Project number: 2011169 (2) Langfang Teachers College projectProject number: LSZQ201008 References 1 Geobel K, Kirk W A. A fixed point thoerem for asymptotically nonexpansive mappingJ. Proc Amer Math Soc,1972,35(1):

17、171-174. 2 Wittmann R, Appoximation of fixed points of 7nanexpansive mappingJ. Arch Math, 1992, 58:486-491. 3 Jean philippe,Chancelier, Iterative schemes for computing fixed point of nanexpansive mappings in Banach spaceJ.Jour Comp Appl Math,2009,353:141-153. 4 Benlong Xu, Fixed point iteratios for

18、asymptotically nonexpansive mapping in Banach spaceJ.Jour Comp Appl Math,2002,267:444-453. 5 Mujahid Abbas,Safeer H Khan. Common fixed points of two multivalued nonexpansive mappings by one-step iterative schemeJ. Appl Math Lett,2011,24:97-102. 6 Moha Esla,Ali Abkar.One-step iterative process for a

19、finite family of multivalued mappingsJ. Math and Comp Mode,2011,54:105-111. 7 Naoki Shioji,Wataru Takahashi. Strong convergence of approximated sequences for nonexpansive mappings in Banach spaceJ.Amer Math Soci, 1997,125(2):3641-3645. 8 S.H.Khan,M.abbas,B.e.Rhoades.A new One-step iterative process

20、for approximating Common fixed points of two multivalued nonexpansive mappingsJ.Rend.Circ.Mat,2010,59:149-157. 9 K.Aoynama,Y.Kimura,W.Takahashi,M.Toyoda.Approximation of commom fixed points of countable family of nonexpansive 8mappings in a Banach spacesJ.Nonl.Anal,2007,67:2350-2360. 10 K.Nakajo,K.Shimoji,W.Takahashi.Strong convergence theor- ems by the hybrid method forfamilies of nonexpansive mappings in a Hilbert spacesJ.Taiwanese J.Math,2006,10:339-360.