We prove, for Orlicz spaces ${\mathbf{L}}_A\left({ℝ}^N\right)$ such that $A$ satisfies the ${\Delta }_2$ condition, the nonresolvability of the $A$-Laplacian equation ${\Delta }_{A}u+h=0$ on ${ℝ}^N$, where $\int h\ne 0$, if ${ℝ}^N$ is $A$-parabolic. For a large class of Orlicz spaces including Lebesgue spaces ${\mathbf{L}}^p$ ($p>1$), we also prove that the same equation, with any bounded measurable function $h$ with compact support, has a solution with gradient in ${\mathbf{L}}_A\left({ℝ}^N\right)$ if ${ℝ}^N$ is $A$-hyperbolic.

We prove two minimax principles to find almost critical points of $C^1$ functionals restricted to globally defined $C^1$ manifolds of codimension $1$. The proof of the theorems relies on Ekeland variational principle.

We consider the applications of the theory of condensing set-valued maps, the theory of set-valued linear operators, and the topological degree theory of the existence of mild solutions for a class of degenerate differential inclusions in a reflexive Banach space. Further, these techniques are used to obtain the solvability of general boundary value problems for a given class of inclusions. Some particular cases including periodic problems are considered.

We discuss the operator transforming the argument of a function in the ${\text{L}}^2$-setting. Here this operator is unbounded and closed. For the approximate solution of ill-posed equations with closed operators, we present a new view on the Tikhonov regularization.