## Convergence theorems for generalized projections and maximal monotone operators in Banach spaces.(English)Zbl 1045.47041

Let $$E$$ be a smooth strictly convex reflexive Banach space and $$E^{\ast}$$ its dual space. Let $$V:E\times E\rightarrow\mathbb{R},$$ $$V\left( x,y\right) =\left\| x\right\| ^{2}-2\langle J\left( x\right) ,y\rangle+\left\| y\right\| ^{2},$$ where $$J$$ is the normalized duality mapping and $$C\subset E$$ is a closed convex set. Then for every $$x\in E$$ there exists a unique $$y_{x}\in C$$ such that $$V\left( x,y_{x}\right) =\min_{y\in C}V\left( x,y\right)$$. The mapping $$\Pi_{C}:E\rightarrow C,\;\Pi_{C}\left( x\right) =y_{x}$$ is called a generalized projection on $$C$$.
Within this framework, the authors establish the following results: (1) If $$\left\{ C_{n}\right\} _{n\geq1}$$ is a sequence of nonempty closed convex subsets of $$C$$ and if there exists the limit in the Mosco sense $$C_{0}=M-\lim C_{n}\neq\emptyset$$, then $$C_{0}$$ is closed and convex and, for every $$x\in C$$, $$\Pi_{C_{n}}\left( x\right)$$ is weakly convergent to $$\Pi_{C_{0} }\left( x\right)$$. (2) If the norm of $$E^{\ast}$$ is Fréchet differentiable then, for every $$x\in C$$, $$\Pi_{C_{n}}\left( x\right)$$ converges strongly to $$\Pi_{C_{0}}\left( x\right)$$. (3) If the norm of $$E$$ is Fréchet differentiable and $$\lim\Pi_{C_{n}}\left( x\right) =\Pi_{C_{0}}\left( x\right)$$ for every $$x\in C,$$ then $$C_{0} =M-\lim_{n}C_{n}$$.
Using the above results, the authors prove weak convergence and strong convergence theorems for the resolvents of a sequence of maximal monotone operators.

### MSC:

 47H05 Monotone operators and generalizations 49J53 Set-valued and variational analysis 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 46B10 Duality and reflexivity in normed linear and Banach spaces 54B20 Hyperspaces in general topology
Full Text: