Krasnoselskii-type Theorems for Monotone Operators in Ordered Banach Algebra with Applications in Fractional Differential Equations and Inclusion

Abstract

This chapter discusses Krasnoselskii-type fixed point results for monotone operators. It is well known that the monotone operators are not continuous on the whole domain, so we will find the solutions of discontinuous operator equations and inclusions. The presented fixed point results may be considered as variants of the Krasnoselskii fixed point theorem in a more general setting. The results of Darbo, Schauder and Bohnentblust-Karlin are also generalized. We prove these results for the case of single-valued and set-valued monotone operators. We use our main result for single-valued operators to obtain the existence of solutions of anti-periodic ABC fractional BVP. The fixed point result for set-valued monotone operators is used to discuss the existence of solutions of a given fractional integral inclusion in ordered Banach spaces.