School of Pure and Applied Sciences

Stability of Semigroups of Linear Operators in Variable Banach Spaces

2026-04-09T15:10:58Z

Stability of Semigroups of Linear Operators in Variable Banach Spaces Obogi, Robert Karieko; Mogoi N, Evans This paper develops a theory for stability analysis of semigroups of linear operators acting on variable Banach spaces—families of Banach spaces {X(t)}t≥0 whose norms may depend on time. We establish generation theorems under appropriate resolvent conditions, characterize exponential stability through Lyapunov-type functionals, and analyze spectral properties of evolution families in variable settings. Our approach systematically extends classical semigroup theory to accommodate time-dependent norms by transporting all objects to a fixed reference space. Applications include non-autonomous parabolic equations and reaction-diffusion systems with time-dependent coefficients. All proofs are provided with full mathematical rigor, addressing technical challenges unique to variable Banach spaces

Duality and Weak Compactness in Generalized Orlicz-Bochner Spaces with Applications to Operator Equations

2026-01-13T08:23:58Z

Duality and Weak Compactness in Generalized Orlicz-Bochner Spaces with Applications to Operator Equations Obogi, Robert Karieko In this paper, we investigate the duality structure and weak compactness properties of generalized Orlicz–Bochner spaces LΦ(X,µ;E), where (X,µ) is a finite measure space, E is a Banach space, and Φ is a convex modular function satisfying a generalized ∆2 -condition. Unlike classical Lebesgue and Bochner spaces, these spaces accommodate variable nonlinearity and non-standard growth, thus providing a more flexible functional analytic framework for studying nonlinear phenomena. We first characterize the dual of LΦ(X,µ;E) under modular convergence and develop criteria for reflexivity and weak compactness based on modular and geometric conditions on Φ and E. Furthermore, we establish sufficient conditions for the compactness and continuity of integral operators acting on these spaces. As an application, we analyze the solvability of nonlinear operator equations and integrodifferential equations with kernel-type operators, where standard Lp or Sobolev methods fail. Our results extend classical duality and compactness theory and open new avenues for solving evolution equations in variable exponent and Orlicz-type frameworks.

Norm attainment and structural properties in Orlicz spaces: A comprehensive study on strict convexity, duality, and optimization

2025-12-16T11:29:11Z

Norm attainment and structural properties in Orlicz spaces: A comprehensive study on strict convexity, duality, and optimization Mogoi, N. Evans; Obogi, Robert We investigate norm attainability and duality properties in Orlicz spaces, extending classical results from Banach and Hilbert spaces to a more gen eral functional framework. We establish 14 fundamental theorems that character ize norm attainment in terms of strict convexity, uniform convexity, and weak con vergence. We explore the duality structure of Orlicz spaces, highlighting key differ ences from Lp spaces and providing a variational characterization of the norm. We also discuss applications in optimization and variational problems, demonstrating the significance of norm-attaining functionals in these settings. Our findings con tribute to a deeper understanding of Orlicz space geometry and its implications for functional analysis and applied mathematics.

Simulation-Based Analysis of Agroforestry Practices in Kisii County Kenya

2025-12-16T11:10:04Z

Simulation-Based Analysis of Agroforestry Practices in Kisii County Kenya Monari, Fred Nyamitago Agroforestry, the integration of trees into agricultural landscapes, is a sustainable practice that enhances biodiversity, improves soil health, and contributes to climate change mitigation. In Kisii County, agroforestry is particularly important due to the region’s reliance on agriculture and the challenges posed by climate change. This study focuses on simulating and analyzing the impact of common agroforestry tree species in Kisii County, including Grevillea robusta, Sesbania sesban, Casuarina equisetifolia, and Markhamia lutea. Using R programming, this study simulates data on tree density, crop yield, livestock density, soil health, biodiversity index, and carbon sequestration. Linear regression models revealed that tree density had a significant negative effect on crop yield (p < 0.001) but positive effects on soil health and carbon sequestration (p < 0.01). The findings suggest trade-offs between tree density and crop productivity that need careful management. This study provides data-driven insights for optimizing agroforestry practices in Kisii County to balance agricultural productivity with environmental benefits.