Methods of Functional Analysis
and Topology

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.

This paper introduces and investigates the class of $k$-quasi $n$-power posinormal operators in Hilbert spaces, generalizing both posinormal and $n$-power posinormal operators. We establish fundamental properties including matrix representations in $2 \times 2$ block form, tensor product preservation ($T\otimes S$ remains in the class when $T,S$ are), and complete characterizations for weighted conditional type operators $T_{w,u} := wE(uf)$ on $L^2(\Sigma)$. Key theoretical contributions include a structural decomposition theorem for operators with non-dense range, spectral properties, invariant subspace behavior, and interactions with isometric operators. For weighted operators, we derive explicit conditions for $k$-quasi $n$-power posinormality in terms of weight functions $w,u$ and their conditional expectations. The work bridges abstract operator theory with concrete applications, particularly in conditional expectation analysis, while significantly extending posinormal operator theory. The results provide new tools for operator analysis with potential applications in spectral theory, functional calculus and mathematical physics. Concrete examples throughout the paper illustrate the theory and the framework opens new research directions in operator theory and its applications, offering both theoretical insights and practical computational tools for analyzing this important class of operators in Hilbert spaces.

$(\tau_1,\tau_2)$-star-Lindelöfness ensures that for every pair $(\mathcal U_1 \subseteq \tau_1,\mathcal U_2 \subseteq \tau_2)$ of open covers, a countable subcover of $\mathcal U_1$, can spread through $\mathcal U_2 $ via the star operation to cover the entire bitopological space $(X,\tau_1,\tau_2)$. Giving a positive answers to the questions of Choudhury et. al. [12], DCCC and meta-Lindelöf like characterization of star-Lindelöf bitopological spaces are presented in this paper. It has been established that a JDCCC bitopological space is both $(\tau_1,\tau_2)$-2-star-Lindelöf and $(\tau_2,\tau_1)$-2-star-Lindelöf. And if a bitopological space which is both $(\tau_1,\tau_2)$-$n$-star-Lindelöf ($n \in \mathbb N$) and $(\tau_1,\tau_2)$-meta-Lindelöf, then $(X, \tau_2)$ is $(n-1)$-star Lindelöf ($0$-star-Lindelöfness represents Lindelöfness).