We speculate on the role of relativistic versions of delayed differential equations in fundamental physics. Relativistic invariance implies that we must consider both advanced and retarded terms in the equations, so we refer to them as shifted equations. The shifted Dirac equation has some novel properties. A tentative formulation of shifted Einstein-Maxwell equations naturally incorporates a small but nonzero cosmological constant.
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.
We characterize stability under composition, inversion, and solution of ordinary differential equations for ultradifferentiable classes, and prove that all these stability properties are equivalent.