Vol. 32 (2026), no. 1 (Current Issue)

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, 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.

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.

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 introduce generalized multivalued $F-$proximal contraction mappings within the partial metric spaces framework and establish best proximity point results for such mappings. The best proximity point theorem for multivalued $F-$proximal contraction mappings involving $\alpha-$admissibility is also obtained. Several related results in the literature are unified and generalized by our new best proximity point results. We also provide nontrivial examples to support our findings. Finally, we derive an existence of a solution to an integral equation that validates our finding.

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

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.

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).

This paper examines the spectral properties of specific classes of positive operators arising from matrices associated with the linear complementarity problem. Such operators occupy a central position in diverse domains of mathematics and physics, including operator theory, functional analysis, and quantum mechanics. A thorough understanding of their spectral behavior is fundamental for exploring the dynamics and stability of systems governed by these operators. P-matrix is one of the important types of matrices appearing in linear complementarity problems. In this paper, with the help of spectral results we have given a factorization for P-matrices, as the product of two non-trivial P-matrices. We also focus on elucidating spectral properties such as eigenvalues, approximate eigenvalues and spectral values associated with certain positive operators.