Iterative Approximation of Endpoints for Multivalued Mappings in Banach Spaces (original) (raw)
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In this article, we first give a multivalued version of an iteration scheme of Agarwal et al. We use an idea due to Shahzad and Zegeye which removes a "strong condition" on the mapping involved in the iteration scheme and an observation by Song and Cho about the set of fixed points of that mapping. In this way, we approximate fixed points of a multivalued nonexpansive mapping through an iteration scheme which is independent of but faster than Ishikawa scheme used both by Song and Cho, and Shahzad and Zegeye. Thus our results improve and unify corresponding results in the contemporary literature. : 47H10; 54H25.
Arabian Journal of Mathematics
In this paper, we propose a new iteration process, called multi-valued F-iteration process, for the approximation of fixed points. We introduce a new class of multi-valued generalized nonexpansive mappings satisfying a B_{\gamma ,\mu }$$ B γ , μ property. Moreover, we establish certain weak and strong convergence theorems in uniformly convex Banach spaces. We also discuss the stability of the modified F-iteration process. Furthermore, a numerical example is presented to illustrate the superiority of our results.
Fixed-Point Iterations for Asymptotically Nonexpansive Mappings in Banach Spaces
Journal of Mathematical Analysis and Applications, 2002
In this paper, we suggest and analyze a three-step iterative scheme for asymptotically nonexpansive mappings in Banach spaces. The new iterative scheme includes Ishikawa-type and Mann-type interations as special cases. The results obtained in this paper represent an extension as well as refinement of previous known results. 2002 Elsevier Science (USA)
Approximating Fixed Points of Generalized Nonexpansive Mappings in Banach Spaces
International Journal of Analysis and Applications, 2014
In this paper, we prove a fixed point theorem for the selfmaps of a closed convex and bounded subset of the Banach space satisfying a generalized nonexpansive type condition. Some results concerning the approximations of fixed points with Krasnoselskii and Mann type iterations are also proved under suitable conditions. Our results include the well-known result of Kannan (1968) and Bose and Mukherjee (1981) as the special cases with a different and constructive method. 2010 Mathematics Subject Classification. 47H10.
Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces
Journal of Mathematical Analysis and Applications, 2009
Let X be a real Banach space with a normalized duality mapping uniformly norm-to-weak ⋆ continuous on bounded sets or a reflexive Banach space which admits a weakly continuous duality mapping JΦ with gauge φ. Let f be an α-contraction and {Tn} a sequence of nonexpansive mapping, we study the strong convergence of explicit iterative schemes xn+1 = αnf (xn) + (1 − αn)Tnxn
In this manuscript, we establish a novel class of nonlinear mappings called β-enriched Suzuki generalized multivalued non-expansive mappings and prove a new existence of common fixed points for a commuting pair comprising an enriched single-valued and an enriched multivalued mapping both satisfying condition (C) in the setup of a uniformly convex Banach space. Further, weak and strong convergence results are obtained for an infinite family of this new class of mappings in the framework of uniformly convex Banach spaces. A nontrivial numerical example is presented to validate the main results of this paper. Our results extend, improve and generalize many well known results currently in literature.
On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces
Nonlinear Analysis: Theory, Methods & Applications, 2009
Keywords: Quasi-nonexpansive multimap Nonexpansive multi-valued map Fixed point Strong convergence Banach space a b s t r a c t We prove strong convergence theorems for the Ishikawa iteration scheme involving quasinonexpansive multi-valued maps. We also construct an iteration scheme which removes a restrictive condition in Song and Wang results [Y. Song, H. Wang, Erratum to ''Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces'' [Comput. Math. Appl. 54 (2007) 872-877], Comput. Math. Appl. 55 (2008) 2999-3002]. Our results provide an affirmative answer to Panyanak's question [Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl., 54 (2007), 872-877], in a more general setting.