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Volume 2019

A new iteration procedure for multivalued nonexpansive maps in uniformly convex Banach spaces -by G. V. R. Babu and G. Satyanarayana

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Abstract: Let $K$ be a nonempty closed convex and bounded subset of a uniformly convex Banach space $X$, $\mathcal{K}(K)$ be the set of all nonempty compact subsets of $K$ and $T:K\to\mathcal{K}(K)$ be a multi-valued nonexpansive map. In this paper, we introduce a new iteration procedure for multivalued maps namely $BS$ iteration procedure through which we define Noor and Picard-S iteration procedures for multivalued maps with slight modifications. We prove that these iteration procedures converges strongly to an end point of $T$ under the hypotheses that $T$ satisfies condition $\textit{(J)}$ or $T$ is semicompact. Also, we prove the weak convergence of these iteration procedures when $X$ satisfies Opial condition. Further, we provide an example in support of the validity of our main result. Our results unify the results that are available in the existing literature.

Keywords: Uniformly convex Banach space, multivalued nonexpansive map, end point, condition $\textit{(J)}$, semicompact, strong convergence, Opial condition, weak convergence.

DOI: 10.30697/rfpta-2019-032

How to cite this article: G. V. R. Babu and G. Satyanarayana, A new iteration procedure for multivalued nonexpansive maps in uniformly convex Banach spaces, Results in fixed point theory and applications, Vol. 2019, Article ID 2019032, 14 pages, 2019. DOI: https://doi.org/10.30697/rfpta-2019-032

Copyright © 2019 SUMA Publishing Group., This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.