Global existence theory for fractional differential equations: New advances via continuation methods for contractive maps

Abstract

Abstract The aim of this article is to form new existence theory for global solutions to nonlinear fractional differential equations. Traditional approaches to existence, uniqueness and approximation of global solutions for initial value problems involving fractional differential equations have been unwieldy or intractable due to the limitations of previously used methods. This includes, for example, certain invariance conditions of the underlying local fixed point strategies. Herein we draw on an alternative tactics, applying the more modern ideas of continuation methods for contractive maps to fractional differential equations. In doing so, we shed new light on the situation, producing these new perspectives through a range of novel theorems that involve sufficient conditions under which global existence, uniqueness, approximation and location of solutions are ensured.