Analysis of fractional integro-differential equations of thermistor type

Abstract

We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results as regards fractional integral and differential equations of thermistor type. Several nonlocal problems are considered: problems concerned with Riemann–Liouville, Caputo, and time-scale fractional operators. The existence and uniqueness of positive solutions are obtained through suitable fixed-point theorems in proper Banach spaces. Additionally, existence and continuation theorems are given, ensuring global existence.