Methods of Functional Analysis
and Topology

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.

In this paper, a new class of $H$-monotone in Banach spaces is considered and studied. The resolvent operator and Cayley approximation operator associated with the $H$-monotone are defined, and the Lipschitz continuity of Cayley approximation operator is also established. An application involves the solvability of a class of generalized Cayley inclusions with $H$-monotone in Banach spaces. By utilizing the technique of resolvent, an iterative algorithm is developed for solving such a class of generalized Cayley inclusions in Banach spaces. The convergence of the iterative sequence generated by the algorithm is proven under certain suitable conditions. The results are justified by means of a numerical example analytically and graphically using Python(matplotlib).

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.

$(\tau_1,\tau_2)$-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).