A Review on Classical and Generalized Fixed Point Theorems and their Applications to Fractional Differential Equations

P. Thirupathi Rao, Adilakshmi Gorantla

Abstract: Fixed point theory seems pretty central in nonlinear analysis these days. It shows up in differential equations, optimization, economics and a few other spots too. Banach started the main idea with his contraction principle in 1922, and after that the whole area grew fast. People like Kannan, Chatterjea, Reich and others each added their own contractive conditions, which let the theory cover more mappings than before. It feels like those changes made fixed points useful in places it did not reach earlier. Some of the newer versions use rational expressions or theta type mappings, and those have turned up in fractional differential equations along with market models. I think the review tries to trace how the ideas changed over time and lines up the different assumptions. It also looks at recent work and how it connects to fractional calculus. The comparisons between all the conditions can get a bit messy though, and not every detail ends up completely settled.

Keywords: Fixed point theorem, Banach contraction, Kannan contraction, Reich contraction, Fractional differential equations, ?-contractions