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Weak and strong convergence theorems for maximal monotone operators in a Banach space. (English) Zbl 1078.47050
Let $E$ be a smooth and uniformly convex Banach space with the normalized duality mapping $J:E\to {E}^{*}$ and $T\subset E×{E}^{*}$ a maximal monotone operator with ${T}^{-1}0\ne \varnothing$. Let ${J}_{r}={\left(J+rT\right)}^{-1}J$, $r>0$, and $P\left(x\right)=\phantom{\rule{4.pt}{0ex}}{\text{argmin}}_{y\in {T}^{-1}0}{\left(\parallel x\parallel }^{2}-2〈x,Jy〉+{\parallel y\parallel }^{2}\right)$. The authors study the iterative process ${x}_{n+1}={J}^{-1}\left({\alpha }_{n}J\left({x}_{n}\right)+\left(1-{\alpha }_{n}\right)J\left({J}_{{r}_{n}}{x}_{n}\right)\right)$, ${\alpha }_{n}\in \left[0,1\right]$, ${r}_{n}\in \left(0,\infty \right)$ $\left(n=1,2,\cdots \right)$ and prove that the sequence $\left\{P\left({x}_{n}\right)\right\}$ converges strongly to an element of ${T}^{-1}0$. The results are applied to the convex minimization problem and the variational inequality problem.

MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H05 Monotone operators (with respect to duality) and generalizations 90C48 Programming in abstract spaces 90C25 Convex programming 47N10 Applications of operator theory in optimization, convex analysis, programming, economics
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