Iterative process for finding fixed points of quasi-nonexpansive multimaps in CAT(0) spaces
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Abstract
Let $\mathbb{E}$ be a CAT(0) space and $K$ be a nonempty closed convex subset of $\mathbb{E}$. Let $T : K\to \mathcal{P}(K)$ be a multimap such that $F(T) \neq \emptyset $ and $\mathbb{P}_T(x) = \{y \in Tx : d(x,y) = d(x,Tx)\}.$ Define sequence $\{x_n\}$ by $x_{n+1} = (1-\alpha)v_n \bigoplus \alpha w_n, \, y_n = (1-\beta)u_n \bigoplus \beta w_n, \, z_n = (1-\gamma)x_n \bigoplus \gamma u_n$
where $\alpha, \beta, \gamma \in [0;1] ;u_n \in \mathbb{P}_T (x_n); v_n \in \mathbb{P}_T(y_n)$ and $w_n \in \mathbb{P}_T (z_n)$. (1) If $\mathbb{P}_T$ is quasi-nonexpansive, then it is proved that $\{x_n\} $ converges strongly to a fixed point of $T$. (2) If a multimap $T$ satisfies Condition(I) and $\mathbb{P}_T$ is quasi-nonexpansive, then $\{x_n\}$ converges strongly to a fixed point of $T$. (3) Finally, we establish a weak convergence result. Our results extend and unify some of the related results in the literature.
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References
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