PhD Application Research Proposal. Title: Unique Solutions to Non-Local Partial Differential Equations (ODE's/PDE's) Using Various Fixed Point Theories Introduction Non-local ODE's/PDE's are a significant area of study in mathematical and physical sciences due to their ability to model complex phenomena that local ODE's/PDE's cannot capture. However, finding unique solutions to these equations remains a challenge. This research aims to explore the application of various fixed point theories to address this problem. Literature Review Numerous studies have explored the application of fixed point theories to local ODE's/PDE's, but their application to non-local ODE's/PDE's is less explored. This research will build on the foundational work of Dr Christopher S. Goodrich, extending his work to the realm of non-autonomous, non local ODE's/PDE's. Objectives The primary objectives of this research will be: to develop and analyse new methods for finding unique solutions to non-local ODE's/PDE's using fixed point theories. Determining the existence of positive solution(s) to boundary value problems for nonlocal ODEs (and, possibly, PDEs). These will involve theoretical development and may involve computational testing of these methods. Methodology This research will begin with a thorough mathematical analysis of existing fixed point theories and their applications to non-local ODE's/PDE's. Techniques such as the Banach and Schauder fixed point theorems will