$x_0$-minimum-norm solutions on weakly closed domains

Let $X,Y$ be real Hilbert spaces, let $D\subseteq X$ be nonempty and weakly closed, meaning closed in the coarsest topology making every bounded real-linear functional continuous, and let $F:D\to Y$ satisfy this graph condition: whenever $x_n\in D$, $x_n\rightharpoonup x$, and $F(x_n)\rightharpoonup z$, one has $x\in D$ and $F(x)=z$; weak convergence means convergence under every bounded real-linear functional. For $x_0\in X$ and $y\in Y$, a point $x^{†}\in D$ is an $x_0$-minimum-norm solution of $F(x)=y$ when $F(x^{†})=y$ and

$$\mathrm{\parallel }{x}^{†}-x_0{\mathrm{\parallel }}_X\le \mathrm{\parallel }x-x_0{\mathrm{\parallel }}_X\text{for every }x\in D\text{ with }F(x)=y\mathrm{.}$$

This definition asserts neither existence nor uniqueness. The minimum is global over the exact solution set in $D$ and uses distance from the specified reference point $x_0$.

Weakly closed forward maps and their domains

The open-domain version of the same distance condition

Christian Clason, Regularization of Inverse Problems, Chapter 10

Benedikt Wirth, Inverse Problems, Definition 83 and Theorem 85