Methods of Functional Analysis
and Topology

Let \( \mathcal{H} \) be a complex Hilbert space of dimension at least 3, and let \( \mathcal{B}(\mathcal{H}) \) denote the algebra of all bounded linear operators on \( \mathcal{H} \). Based on results by Molnar, this paper revisits the problem addressed in [18], which characterizes surjective maps \( \phi: \mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H}) \) that preserve the set of partial isometric operators in both directions. We focus exclusively on the linear case, rather than the more general additive case. Furthermore, we provide an alternative proof of the main result in [9] from a different point of view. Finally, we propose new directions for exploring maps that preserve higher-order partial isometric operators in both directions.

In this paper, we extends the classical theory of variational inequalities to the hyperbolic scalar setting using the structure of $\mathbb{D}$-Hilbert spaces. We introduce and analyze a new class of variational inequalities, termed general mildly $\mathbb{D}$-nonlinear variational inequalities, which generalize classical formulations by incorporating $\mathbb{D}$-nonlinear and product-type mappings. We characterize these problems in terms of their idempotent components and demonstrate that several known variational inequality problems, including Stampacchia-type and complementarity problems, emerge as special cases.

We show the existence of solutions satisfying Cauchy or terminal boundary conditions for first order differential inclusion $(\phi(x(t)))'\in F(t,x(t))$. We consider the second order problem $(\phi(x'(t)))'\in F(t,x(t))$ with many boundary conditions. The set-valued map $F$ has non-convex values and the function $\phi$ satisfies a weak condition. The resolution method use the topological degree without the method of upper and lower solutions.

In this paper, we study uniform ergodicity for $C_0$-semigroups of universally bounded operators acting on locally convex spaces. Characterizations of uniform ergodic $C_0$-semigroups are given. Importantly, we give a $C_0$-semigroups version of F. Pater, T. Binzar [14] theorem.