Carathéodory Solutions for a Nonlinear Coupled Two-Dimensional Implicit Differential System with Nonlocal Boundary Constraints

Abstract

We study the existence and uniqueness of solutions for a system consisting of two coupled nonlinear first-order differential equations involving implicit derivatives. The study focuses on systems described under multipoint nonlocal boundary conditions in two-dimensional, bounded and convex domains. Using an appropriate transformation, such implicit differential system is reduced to a corresponding integral form to analyze its fixed points in a product Lebesgue space. We follow closely Schauder’s approach employed to establish a rigorous existence result; in particular, we verify the compactness of the operator via the Kolmogorov compactness criterion. Uniqueness of solutions is further investigated by adapting certain conditions due to Banach contraction technique. More notably, we prove that despite the derivatives being defined in the integrable space, the solutions exist in the product space of absolutely continuous functions. The relaxed problem involving nonlocal integral conditions of the coupled system type is also successfully treated by the same framework. Finally, numerical simulations are performed in MATLAB to support our theoretical findings.