We introduce \(r\)-Fock space \(\mathscr{F}_{r}\) which generalizes some previously known Hilbert spaces, and study the \(r\)-derivative operator \(\frac{\mbox{d}^r}{\mbox{d}z^r}\) and the multiplication operator by \(z^r\). A general uncertainty inequality of Heisenberg-type is obtained. We also consider the extremal functions for the \(r\)-difference operator \(D_r\) on the space and obtain approximate inversion formulas.

Active lock-in options are a class of complex derivatives characterized by pronounced path dependence and optimal decision making features, and they possess significant application value in the design of structured financial products and risk management. This paper investigates the pricing of active lock-in call options under a stochastic volatility framework. The lock-in decision is formulated as an optimal stopping problem and is further reformulated as a partial differential equation with obstacle constraints. By introducing a linear complementarity problem formulation, the structural properties of the option value function and the optimal lock-in boundary are systematically characterized. From a numerical perspective, an IMEX time discretization scheme is employed to transform the continuous problem into a sequence of time-layered discrete complementarity systems. These systems are efficiently solved using the projected successive over relaxation (PSOR) algorithm. Numerical experiments are conducted to analyze the structural features and economic interpretations of the value function and the associated free boundary surface.

In this paper, we extend the classical logistic law by incorporating autonomously evolving, time-dependent coefficients that allow both the intrinsic growth rate \(\gamma(t)\) and the carrying capacity \(K(t)\) to vary over time according to logistic modulated dynamics. In particular, the carrying capacity is modeled as a logistic process with intrinsic growth rate \(\alpha\) and saturation parameter \(\beta\), yielding an asymptotic level of \(\frac{\alpha}{\beta}\). The objective is to investigate how temporal variability in the governing coefficients influences both transient and asymptotic regimes of the population dynamics and to assess the extent to which the system behavior can be controlled through a reduced set of key parameters. Analytical results are derived in closed form, expressed in terms of hypergeometric functions, and compared with numerical integrations for validation purposes. It is shown that the model admits a long-term equilibrium determined by the ratio \(\frac{\alpha}{\beta}\), independently of the initial population size \(S_0\), while short- and medium-term dynamics are strongly shaped by the interplay between \(S_0\) and the non-autonomous logistic evolution of the carrying capacity \(K(t)\). These results illustrate how analytically tractable non-autonomous logistic models with internally generated coefficient trajectories can enhance the qualitative understanding of population dynamics and provide reliable benchmarks for numerical simulations, with potential applications in sustainable resource management, aquaculture, and ecological modeling.

Some new classes of inverse variational inequalities, which can be viewed as a novel important special case of general variational equalities, are investigated. Projection method, auxiliary principle and dynamical systems coupled with finite difference approach are used to suggest and analyzed a number of new and known numerical techniques for solving inverse variational inequalities. Convergence analysis of these methods is investigated under suitable conditions. One can obtain a number of new classes of inverse variational inequalities by interchanging the role of operators. Some important special cases are highlighted. Several open problems are suggested for future research.

We introduce the concept of projective suprametrics and provide new part suprametrics in a normed vector space ordered by a cone. We then examine how the convergence of the underlying norm relates to that of the projective and given suprametrics, and we establish sufficient conditions for the completeness of certain subsets. Moreover, we prove a version of Krein-Rutman theorem via a fixed point theorem in suprametric spaces, and study spectral properties of positive linear operators. Furthermore, we show that operator equations involving some concave or convex operators satisfy a Geraghty contraction and therefore have solutions. As an application, we prove a Perron-Frobenius theorem for a tensor eigenvalue problem.

We characterize the boundedness and compactness of generalized integration operators acting between Fock spaces and apply these characterizations to investigate the path-connected components and the singleton of path-connected components of the space of bounded generalized integration operators. Moreover, we describe the essential norm of these operators.