RFPTA Recent Articles
Volume 2019

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

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.