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Convergence theorems for a maximal monotone operator and a \(V\)-strongly nonexpansive mapping in a Banach space. (English) Zbl 1368.47071

Summary: Let \(E\) be a smooth Banach space with a norm \(\|\cdot\|\). Let \(V(x,y)=\| x\|^2 + \| y \|^2 - 2\langle x,Jy\rangle\) for any \(x, y \in E\), where \(\langle \cdot,\cdot\rangle\) stands for the duality pair and \(J\) is the normalized duality mapping. With respect to this bifunction \(V(\cdot,\cdot)\), a generalized nonexpansive mapping and a \(V\)-strongly nonexpansive mapping are defined in \(E\). In this paper, using the properties of generalized nonexpansive mappings, we prove convergence theorems for common zero points of a maximal monotone operator and a \(V\)-strongly nonexpansive mapping.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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