Strong convergence for asymptotical pseudocontractions with the demiclosedness principle in banach spaces
Tóm tắt
The aim of this article is to give an answer to an interesting question proposed in Zhou. At the end of his article, he remarked that it was of great interest to extend his results to certain Banach spaces. So in this article, we extend the demiclosedness principle from Hilbert spaces to Banach spaces. A strong convergence theorem for asymptotical pseudo-contractions in Banach spaces is established. The approaches are based on the extended demiclosedness principle, and the generalized projective operator, and the hybrid method in mathematical programming. Our results extend the previous known results from Hilbert spaces to Banach spaces. MSC: 47H10; 47H09; 47H05.
Tài liệu tham khảo
Zhou HY: Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces. Nonlinear Anal 2009, 70: 3140–3145.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Am Math Soc 1972, 35: 171–174.
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. J Math Anal Appl 1991, 158: 407–413.
Liu QH: Convergence theorems of the sequence of iterates for asymptotically demicon-tractive and hemicontractive mappings. Nonlinear Anal 1996, 26: 1835–1842.
Kim TH, Xu HK: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal 2006, 64: 1140–1152.
Cioranescu I: Geometry of Banach Spaces. In Duality Mappings and Nonlinear Problems. Kluwer Academic Publishers Dordrecht; 1990.
Li J: The generalized projection operator on reflexive Banach spaces and its applications. J Math Anal Appl 2005, 306: 55–71.
Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim 2002, 13: 938–945.
Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Op-erator of Accretive and Monotone Type,Lecture Notes in Pure and Appl Math. Volume 178. Edited by: Kartsatos AG. Marcel Dekker New York; 1996:15–50.
Wang YH, Zeng LC: Convergence of generalized projective modified iterative methods in Banach spaces (in Chinese). Chin Ann Math 2009, 30A: 55–62.
Ceng LC, Al-Homidan S, Ansari QH, Yaod JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J Comput Appl Math 2009, 223: 967–974.