Fractional equations and generalizations of Schaefer’s and Krasnoselskii’s fixed point theorems

Abstract

We study a fractional differential equation of Caputo type by first transforming it into an integral equation with an L1[0,∞) kernel and then applying fixed point theory of Banach and Schauder type using a weighted norm to avoid stringent compactness conditions. It becomes clear that tedious construction of mapping sets and boundedness conditions can be avoided if we use fixed point theorems of Schaefer and Krasnoselskii type. The weighted norm then produces open sets so large that it is difficult to show that mappings are compact. This then leads us to generalize both Schaefer’s and Krasnoselskii’s fixed point theorems which yield simple and direct qualitative results for the fractional differential equations. The weight, g, yields compactness, but it does much more. The generalized fixed point theorems now yield growth properties of the solutions of the fractional differential equations.