Applications of Topological Fixed Point Theory to Nonlocal Differential Equations with Convolution Coefficients

Abstract

In this talk I will discuss how a nonstandard cone together with topological fixed point theory can be used to deduce existence results for boundary value problems involving a nonlocal differential equation. A model case is the equation:

-A((b*(g∘u))(1)) u^'' (t)=f(t,u(t)), 0<t<1

subject to some boundary conditions. The notation * denotes the finite convolution operator, and in this way a variety of nonlocal coefficients can be accommodated – for example, fractional integrals and derivatives. I will discuss the various assumptions imposed on the functions A, b, and g, and how these assumptions are affected by the use of the nonstandard cone. A significant part of the talk will be devoted to explaining the history behind this problem and its connections to applications in modeling.