Methods of Functional Analysis
and Topology

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.

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.

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.

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.