My research interests are partial differential equations (PDEs), nonlinear analysis, image processing, and scientific computing. Particularly, I am interested in models described by PDEs with applications in biomedical image analysis, ecology, and population studies. I also study new geometric properties of Banach spaces and approximation methods with application to convex feasibility problems, convex optimization problems, variational inequality problems, equilibrium problems, generalized mixed equilibrium problems, and Hammerstein integral equations.
partial differential equations
Most studies on population systems typically assume that for different species competing for shared resources, each species has similar characteristics among its kind. In reality, species' characteristics like birth and dispersal rates may differ for different subgroups of the same population. By developing new analytic tools to examine the effects of dispersal rates on the principal spectrum points of positive operators on Banach lattices, we obtained some interesting results in studying the effect of dispersal rates on the dynamics of a class of two-stage age-structured population models. See, e.g., the following sample publication:
Dynamics of Classical Solutions of a Two-Stage Structured Population Model with Nonlocal Dispersal, (with M. A. Onyido, R. B. Salako, and C. I. Udeani) Mathematics, 11(4)(2023), 925
Asymptotic limits of the principal spectrum point of a nonlocal dispersal cooperative system and application to a two-stage structured population model, (with M. A. Onyido, R. B. Salako, and C. I. Udeani) Journal of Differential Equations, 388 (2024), 357–402
nonlinear analysis
Monotone operators were first introduced in real Hilbert spaces to aid in the abstract study of electrical networks and later studied in the setting of partial differential equations. The concept of monotone operators was used to establish the existence of periodic solutions of a general class of nonlinear evolution equations in Hilbert spaces. These operators appear in a wide variety of contexts since they can be found in many functional equations. Many of them also appear in the calculus of variations as subdifferential of convex functions. Monotone operators are also known to have a strong connection with optimization. I am interested in new methods for approximating solutions of nonlinear inclusions, integral equations of Hammerstein-type, fixed points problems, equilibrium problems, and variational inequality problems, which involve monotone-type operators. Our results indicate that if the locations favoring the growth of the juveniles significantly overlap with that favoring the reproduction rates of adults, then low diffusion rates benefit species survival. However, low diffusion rates may harm species survival if these two locations do not overlap. Inspired by this, I study our integro-differential model under the different conditions. See, e.g., the following sample publications:
A Novel Hybrid Method for Equilibrium Problem and A Countable Family of Generalized Nonexpansive-type Maps, with Applications, (with E. E. Otubo and M. A. Onyido) Fixed Point Theory, 22(1)(2021), 359–376
A Strong Convergence Theorem for an Iterative Method for Finding Zeros of Maximal Monotone Maps with Applications to Convex Minimization and Variational Inequality Problems, (with C. E. Chidume, M. I. Uzochukwu, E. E. Otubo, and K. O. Idu) Proceedings of Edinburgh Mathematical Society, 62(2019), 241–257
image processing
My research in image processing and computer vision seek to furnish developmental biologists and immunologists with foundational mathematical and computational methods for studying cells in developing embryos. I am interested in the design of novel mathematical models described by PDEs, the numerical scheme for the solution of the models, efficient (parallel) implementation of the numerical scheme & their application to cell microscopy images. See, e.g., the following sample publications:
4D Segmentation Algorithm with Application to 3D+time Image Segmentation, (with K. Mikula and S. Park) Japan Journal of Industrial and Applied Mathematics, 40(2022), 109–139
3D Image Segmentation Supported by a Point Cloud, (with B. Kosa, K. Mikula, A. Weberling, N. Christodoulou, and M. Zernicka-Goetz) Discrete and Continuous Dynamical Systems - Series S (DCDS-S), 14(3)(2021), 971–985