A new kind of functional differential equations

Abstract

In this paper we introduce and investigate a new kind of functional (including ordinary and evolutionary partial) differential equations. The main goal of this paper is to explore our new philosophy by some examples on functional ODEs and PDEs. For some typical examples, we prove the global existence of smooth solutions, analyze some interesting properties enjoyed by these solutions, and illustrate the differences between this new class of equations and the traditional ones. This kind of functional differential equations is a new and powerful tool to study some problems arising from both mathematics and physics, more applications in particular to differential geometry and fundamental physics can be expected.