On fixed point theorems for single and multivalued mappings and applications to differential problems

Abstract

Fixed point theory is an intersting field of mathematics that have applications in many sciences like economy, computer sciences, medecine and game theory. It provides tools applied in other domains with no immediate connection with mathematics at first sight. Due to their importance, a chapter of this work is destinated to discuss most important theorems with their generalisations and applications. Furthermore, extensions of Darbo fixed point theorem were given; some of them are done by weaking the set contractive condition while others by weakening the space working on. As application, the existence of mild solutions for an impulsive differential equation with nonlocal conditions was investigated. Moreover, the existence of common fixed points for commutative set contraction mappings was garanteed. The obtained results were used to solve an integral equation. Finally, in virtue of the measure of noncompactness, the existence of fixed points for multivalued mappings under different type of conditions was established and as example, using these results, the existence of mild solutions for a nonlocal evolution differential inclusion was inquired.