In this paper, four smoothness concepts for the exact solution \( x^\dagger \in X \) of an ill-posed problem \( A x = y^0 \in \mathcal{R}(A) \) are considered. Here, \( A: X \to Y \) is an injective bounded linear operator between Hilbert spaces \( X \) and \( Y \), with a non-closed range \( \mathcal{R}(A) \neq \overline{\mathcal{R}(A)} \). Those four smoothness concepts for \( x^\dagger \) are source conditions, approximate source conditions, variational inequalities, and approximate variational inequalities. The main concern of the paper is to consider their connections and to present new results for variational inequalities. In the beginning of the paper, the four concepts are introduced, and for convenience of the reader we recall one of them here. In fact, the solution \( x^\dagger \) satisfies a variational inequality, if \( \beta \| x-x^\dagger \|^2 \leq \| x \|^2 - \| x^\dagger \|^2 + \varphi(\| \psi(A^* A)(x-x^\dagger) \|) \) holds for all \( x \in X \). Here, \( \beta > 0 \) is fixed, and \( \varphi:[0,\infty) \to [0,\infty) \) and \( \psi:[0,\infty) \to [0,\infty) \) are given index functions, i.e., they are continuous, strictly increasing functions that vanish at the origin \(0\), respectively.
For each approach, under very general conditions, convergence rates for \( \| x_\alpha - x^\dagger \| \) are given, where \( x_\alpha \in X \) denotes the minimizer of the Tikhonov functional \( T_\alpha(x) = \| Ax -y^0 \|^2 + \alpha \| x \|^2, \, x \in X \). For each smoothness concept, also conditions for the involved index functions are formulated to ensure the special convergence rate \( \| x_\alpha - x^\dagger \| = \mathcal{O}(\alpha^\mu) \;(\alpha \to 0, \, 0 < \mu \leq 1{}^{ \text{constant}}) \), respectively.
Particular emphasis is put on variational inequalities. For this smoothness concept, some necessary conditions for the constant \( \beta \) and the modifier function \( \varphi \) are presented. In addition, some necessary and sufficient conditions for variational inequalities are given for the special case \( \varphi(t) = at^\kappa \) with \( a > 0 \) and \( \kappa \in (0,1] \). In particular, in the case \( \kappa = 1 \) and \( \beta \leq 1 \), the solution \( x^\dagger \) satisfies a variational inequality if and only if a source condition \( x^\dagger = \psi(A^*A) w \) is satisfied for some \( w \in X\), \(\| w \| \leq a/2 \).
Then new results are presented which clarify the connections between approximate source conditions, variational inequalities, and approximate variational inequalities in Hilbert spaces. In particular, it turns out that these three concepts are equivalent. Fenchel duality is used here as a mathematical tool.