\documentclass{amsart}
\title{Mosco convergence of closed convex subsets and resolvents of
monotone operators}
\author{Yasunori Kimura*}
\address[Yasunori Kimura]
{Department of Mathematical and Computing Sciences,
Tokyo Institute of Technology,
2-12-1, O-okayama, Meguro, Tokyo 152-8552, Japan}
\email{yasunori@is.titech.ac.jp}
\thanks{*Presenting author}
\author{Wataru Takahashi}
\address[Wataru Takahashi]
{Department of Mathematical and Computing Sciences,
Tokyo Institute of Technology,
2-12-1, O-okayama, Meguro, Tokyo 152-8552, Japan}
\email{wataru@is.titech.ac.jp}
\keywords{maximal monotone operator, resolvent, Mosco convergence,
metric projection, generalized projection}
\subjclass[2010]{Primary 47H05; Secondary 41A65}
\begin{document}
\maketitle
The theory of maximal operators in Banach spaces has been deeply studied
and applied to various areas of mathematics such as differential
equations, variational inequalities, minimization problems, and so on.
On the other hand, the theory of set-convergence also has wide variety
of applications. Especially, the Mosco convergence is one of the most
important concepts for reflexive Banach spaces. We consider a sequence
of maximal monotone operators and study a relation between convergence
of their resolvents and Mosco convergence of their zeros. We also study
equivalent conditions of strong convergence for a sequence of
resolvents.
In this talk, we first consider one type of resolvents, that is, $(I +
\lambda J^{-1}A)^{-1}$, and prove an analogous result to the theorem
proved by Ibaraki, Kimura, and Takahashi. Next, we study strong
convergence of this kind of sequence and obtain equivalent conditions
for both types of resolvents, that is, $(I + \lambda J^{-1}A)^{-1}$ and
$(J + \lambda A)^{-1}J$. Finally, we show a result concerning the
relation between strong convergence of these two types of resolvents.
\def\thefootnote{}\footnote{Topics: Fixed Point Theory and its
Applications, Set-Valued Analysis}
\end{document}