Grant: Discovery Grants Program - Individual

In this proposed research program, one of the research directions is to explore and establish new fixed point theory (FPT) or nonzero fixed point theorems (FPTs) and coexistence FPTs for nowhere normal-outward maps (not necessarily self-maps) in product Banach spaces. These FPTs will be established based on the fixed point index theory for such maps in a single Banach space established by myself and my collaborator in 2016 and my study on coexistence fixed points for self-maps in product Banach spaces in 2020. The aim of establishing the theoretical results in product Banach spaces is to build a new bridge between the  FPT in product Banach spaces and  steady-state solutions (SSS) for population models of multispecies governed by systems of one or higher dimensional parabolic partial differential equations (PPDEs). The PPDEs are often used to model various population densities in population dynamics, which play important roles in modern applicable mathematics, and contain reaction-diffusion equations in bounded domains. The SSS of systems of the parabolic PDEs are solutions of systems of the corresponding elliptic boundary value problems (BVPs)  The elliptic BVPs contain Laplace's equations and diffusion advection models as special cases.  Another research direction is to study  nonlinear first and higher order fractional differential equations (FDEs). I studied the equivalences between linear first and higher order FDEs and fractional integral equations (FIEs), and compactness of fractional integral operators during 2020-2022. These equivalences and compactness results established theoretical foundations for further studying solutions of nonlinear FDEs via FIEs and nonlinear problems  arising in mathematical biology and physical problems. The objectives  include (a): to search for new ideas and approaches to improve and generalize the existing fixed point theorems or theories to make these new theorems and theories applicable to population models of multispecies; (b) to generalize some classical results in real and functional analysis; (c) to study solutions of systems of one or higher dimensional elliptic BVPs and elliptic models of multispecies by using the new theoretical results obtained in (a)-(b); (d) to apply the newly obtained results on elliptic BVPs to study solutions of a variety of population models, and provide better understanding of the spatial and temporal behaviors of interacting multispecies, and (e) to establish theories for initial value problems and BVPs for nonlinear first or higher order FDEs and provide their applications. The expected theoretical results  will contribute to nonlinear analysis and enrich graduate texts. The expected applications  will enrich and develop both modern PDEs, ODEs, FDEs and population dynamics, which  will be of interest to researchers and spur new problems in  these fields. Our results will lead to publishing strong journal and conference papers and to the training of HQP.