Multiple positive solutions of parabolic systems with nonlinear, nonlocal initial conditions

TL;DR

This paper investigates the existence and multiplicity of positive solutions for parabolic systems with nonlinear, nonlocal initial conditions using fixed point index theory, extending previous abstract frameworks.

Contribution

It extends an abstract fixed point theory to analyze positive solutions of parabolic systems with nonlocal initial conditions, including non-existence results and illustrative examples.

Findings

Established conditions for existence and multiplicity of solutions

Provided a non-existence result for certain cases

Included an example illustrating the theoretical results

Abstract

In this paper we study the existence, localization and multiplicity of positive solutions for parabolic systems with nonlocal initial conditions. In order to do this, we extend an abstract theory that was recently developed by the authors jointly with Radu Precup, related to the existence of fixed points of nonlinear operators satisfying some upper and lower bounds. Our main tool is the Granas fixed point index theory. We also provide a non-existence result and an example to illustrate our theory.