## Mayumi Hojo, Weak convergence theorems for infinite families of extended generalized hybrid mappings in Banach spaces

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DOI: 10.23952/jnva.3.2019.2.05
Volume 3, Issue 2, 1 August 2019, Pages 159-170

Abstract. Let $E$ be a real Banach space and let $C$ be a nonempty subset of $E$. A mapping $T:C\rightarrow E$ is said to be extended generalized hybrid if there are $\alpha,$ $\beta,$ $\gamma,$ $\delta\in \mathbb{R}$ such that $\alpha+\beta+\gamma+\delta \geq 0$, $\alpha+\beta >0$ and $\alpha \|Tx-Ty\|^2$ $+ \beta \|x-Ty\|^2$ $+ \gamma \|Tx-y\|^2$ $+ \delta \|x-y\|^2 \leq 0$ for all $x,y \in C$. In this paper, we prove a weak convergence theorem of Mann’s type iteration for infinite families of extended generalized hybrid mappings in a Banach space satisfying the Opial’s condition.