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Let $H$ be a real Hilbert space and $F:H\rightarrow H$ be a mapping such that $$ (A)\qquad\qquad(u_n\rightharpoonup u_*, F(u_n)\rightarrow F(u_*))\; \Longrightarrow\;(u_n\rightarrow u_*) $$ We are easy to verify that if $F$ is strongly monotone, i.e., there exists $\alpha>0$ such that $$ \langle F(u)-F(v), u-v\rangle\geq\alpha\|u-v\|^2\quad \forall u,v\in H, $$ then the property $(A)$ is satisfied.

I have some difficulties in the following questions:

  1. If $F$ satisfies the property $(A)$ what do we call $F$?

  2. Could we find a class of mappings $F$ such that the property $(A)$ is satisfied but $F$ is not strongly monotone.

  3. I do not know any references about the mapping $F$ satisfying $(A)$.

Thank you for your kind help.

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  • $\begingroup$ Ad 2: You could take mappings, such that $-F$ is strongly monotone (i.e. negative identity). Moreover, if $F$ is strongly monotone, you have even $F(u^n) \to F(u^*) \Rightarrow u^n \to u^*$. $\endgroup$ – gerw Mar 26 '13 at 9:32
  • $\begingroup$ Thank you for your interesting comment. Besides class of strongly monotone mappings I would like to ask whether we can construct another class of mappings that have property $(A)$. I am waiting your kind reply. $\endgroup$ – blindman Mar 26 '13 at 9:44
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    $\begingroup$ Take $F(u) = \|u\|^p \, e$, where $e \in H$ is a nonzero vector and $p > 0$. I think property $(A)$ is to weak, to say something about the operators satisfying it. $\endgroup$ – gerw Mar 26 '13 at 9:50
  • $\begingroup$ Thanks. Do you know the name of mappings $F$ that have property $(A)$. $\endgroup$ – blindman Mar 26 '13 at 9:55
  • $\begingroup$ I don't think they have a name. $\endgroup$ – gerw Mar 26 '13 at 10:24
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Due to the comment of gerw, you can not expect too much from property $(A)$. However, gerw's comment reminded me of the notion of a norm (or functional) with the Radon-Riesz property (also called Kadets-Klee property). A function $f:H\to\mathbb{R}\cup\{\infty\}$ has this property, if $$ u^n\rightharpoonup u,\quad\text{and}\quad f(u^n)\to f(u) $$ implies that $$ u^n\to u. $$ And in the spirit of gerw's comment every function $F(u)=f(u)e$ with any non-zero element $e\in H$ has property $(A)$.

For example, a lot of functionals $f:\ell^2\to\mathbb{R}\cup\{\infty\}$ of the form $$ f(u) = \begin{cases} \sum_k \phi(u_k) & \text{if the sum exists}\\ \infty & \text{else} \end{cases} $$ with appropriate $\phi$ have the Radon-Riesz property.

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