Vol. 32 (2026), no. 2 (Current Issue)

In this paper, we introduce the notion of a generalized modified $F-$con\-traction via rectangular $G-\alpha-$admissible mapping and establish some new fixed point results under Wardowski's $F-$contractive condition $(F1)$ together with an auxiliary function $\phi$ in the setting of $G-$metric spaces. Our new fixed point results improve, generalize and unify several related result in the existing literature. Moreover, we give some non-trivial examples to verify our results. Finally, we derive the existence of a solution to an integral equation to support our main findings.

Let us consider the scattering theory of one-dimensional Schrödinger equation defined by piecewise constant potentials with compact support. We construct the asymptotics of reflection coefficient by iterating the transition matrices. The jump points of the step potential appear in the asymptotics of reflection coefficient. Accordingly, we investigate the modulus of reflection coefficient. The modulus is connected to the zero set of reflection coefficient via Nevanlinna-Levin type of representation theorem. The growth theory of zero set of the reflection coefficient plays a role.

Let $\mathcal{L}(H)$ denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space $H$ into itself. Given $A,B\in \mathcal{L}(H)$, the generalized derivation $\delta_{A,B}\in \mathcal{L}(\mathcal{L}(H))$ is defined by $\delta_{A,B}(X)=AX-XB$. For hyponormal operators $A$ and $B$, L.A. Fialkow gave necessary conditions for which the range of $\delta_{A,B}$ is norm closed in $\mathcal{L}(H)$ [13]. In this paper, we present necessary or sufficient conditions under which the generalized derivation $\delta_{A,B}$ has closed range, where $A$ and $B$ are $p$-hyponormal operators. We give a characterization for the range of $\delta_{A,B}$ to have closed range in the case when $A$ and $B$ are cyclic subnormal operators.

In this paper, we prove that the space of strongly \((r, p, \sigma)\)-summable sequences is the topological dual of the space of \((p, \sigma)\)-weakly summable sequences. As an application, we provide a new proof that an operator \( T: E \to F \) is \((p, \sigma)\)-absolutely continuous if and only if its adjoint \( T^*: F^* \to E^* \) is strongly \((p^*, \sigma)\)-continuous.

In the paper, we introduce and analyze the notion of $\lambda$-slant Hankel operator and $(\lambda,\mu)$-slant Hankel operator on the Lebesgue space $L^2(\mathbb{T})$. Certain basic properties of these operators, connection between them and their co-relations with certain existing operators are also discussed.

In this manuscript we define $A$-double density of a given point in $\mathbb{R}$ for double sequences. We state the relation between double density of a point and spliced double sequences and also examine limit points in view of density of a point.

In this paper two lemmas for arbitrary general functions are given and then, by iteration,we express them as a continued fraction after that by specializing the parameters the continued fraction representation for the generalized tenth order mock theta functions are obtained.

We prove sufficient conditions for the faithfulness of the family of cylinders generated by $\widetilde{Q}$ -expansions for the Hausdorff-Besicovitch dimension calculation. We also represent the conjecture about necessary and sufficient conditions for such a family of cylinders to be faithful. Based on these new results we study fine fractal properties of random variables with independent $\widetilde{Q}$-digits and prove exact formulae for the calculation of the Hausdorff dimension of the corresponding probability measure.

We introduce a concept of antisymmetric Arens regularity of Banach spaces and prove that a Banach space is Arens regular if and only if it is both symmetrically Arens regular and antisymmetrically Arens regular. We show that if a Banach space is nonreflexive, then it is not antisymmetrically Arens regular and provide an example of a symmetrically Arens regular Banach space which is not antisymmetrically Arens regular.