Existence properties for solutions to nonlinear boundary value problems involving ordinary differential equations

Abstract:

The area of differential equations and associated "boundary value problems" (BVPs) continue to captivate researchers due to wide modeling capabilities which span the fields of science, engineering, and technology.

In this talk, we will address a basic mathematical question: How can we guarantee that certain nonlinear BVPs actually do have solution(s)? This is known as an "existence" question. For a differential equation to be useful in the context of modeling, it must have a solution. Ensuring that the existence property holds may be thought of as a first step in the analysis of solutions to BVPs, underpinning other interesting approaches such as numerical treatments.

Our methods use a classical fixed-point approach due to J. Schauder (U. Lwow), combined with differential inequalities which ensure an a priori bound on all possible solutions. The ideas extend the work of Philip Hartman (Johns Hopkins U.).

This talk will be suitable for non-specialists, including graduate students.