Nonlinear Tikhonov existence and stability — Examples and Counterexamples

Given: $X=Y=\mathbb{R}$ with the standard real inner product and norm, $U=\mathbb{R}$, $F(t)=t^2$, $x_0=0$, $y=1$, and $\alpha =1$.

The standard real inner product induces the usual norm, and $\mathbb{R}$ is complete for it; hence it is a real Hilbert space (Real and complex inner-product spaces and their induced length, The reals are complete, Hilbert space).

Weak convergence is convergence under every bounded real-linear functional; in particular the identity functional shows that weak convergence in $\mathbb{R}$ implies ordinary convergence (Weak convergence of nets and sequences).

A weakly closed forward map has weakly closed domain and sequentially weak-to-weak closed graph (Weakly closed nonlinear forward maps on Hilbert spaces).

The nonlinear Tikhonov functional is $\frac{1}{2}\mathrm{\parallel }F(t)-y{\mathrm{\parallel }}^2+\frac{\alpha }{2}\mathrm{\parallel }t-x_0{\mathrm{\parallel }}^2$ (The nonlinear Tikhonov functional).

A nonlinear forward map is a map from a nonempty norm-open subset of a real Hilbert space to a real Hilbert space; no linearity is assumed (Nonlinear forward maps between Hilbert spaces).

Given: The ultrafilter lemma, DC, and HB; the finite set $\mathrm{\Omega }=\{0,1\}$ with $\mu (A)=\mathrm{\mid }A\mathrm{\mid }\mathrm{/}2$; the maps $g_s(0)=s$, $g_s(1)=s^2$; and $X=Y=D=\mathbb{R}$ with the usual real Hilbert structure.

The ultrafilter lemma extends every filter to an ultrafilter (The ultrafilter extension principle (UL/BPI)).

DC supplies an infinite sequence following an everywhere-extendible relation (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain).

HB is the real dominated-extension principle (The real dominated-extension principle as an additional hypothesis over ZF).

A measure is zero on the empty set and countably additive on disjoint measurable sets (Measures on sigma-algebras); a measure space consists of a set, sigma-algebra, and measure (Measure spaces).

A measurable simple Banach-valued function $g={\sum }_j{x}_j{\mathbf{1}}_{A_j}$ with finite-measure nonzero level sets is integrable and has integral $\int gd\mu ={\sum }_j\mu (A_j)x_j$ (Banach-valued simple function and integral).

An integrable simple Banach-valued function is Bochner integrable, with its Bochner integral equal to the simple integral (Bochner-integrable function, The Banach-valued simple integral is well defined).

The real numbers with their standard inner product form a real Hilbert space (Real and complex inner-product spaces and their induced length, The reals are complete, Hilbert space).

Weak convergence in a normed space is convergence under every bounded real-linear functional (Weak convergence of nets and sequences).

The open-domain nonlinear forward-map interface permits any map $F:U\to Y$ on a nonempty norm-open set, without assuming linearity (Nonlinear forward maps between Hilbert spaces).

Under the assumptions of [A1]–[A3], a real-Hilbert weakly closed forward map on a nonempty weakly closed domain has a global minimizer for every positive parameter and datum (Existence of a global nonlinear Tikhonov minimizer).

The Tikhonov functional is $\frac{1}{2}\mathrm{\parallel }F(s)-y{\mathrm{\parallel }}^2+\frac{\alpha }{2}\mathrm{\parallel }s-x_0{\mathrm{\parallel }}^2$ on the domain (The nonlinear Tikhonov functional).

DC implies countable choice, the principle that every countable family of nonempty sets admits a choice function (Dependent choice implies countable choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

A weakly closed forward map has a weakly closed domain and sequentially weak-to-weak closed graph (Weakly closed nonlinear forward maps on Hilbert spaces).