Developments in Modular Space Fixed Point Theory

This survey article offers a snapshot view of the present state of Fixed Point Theory within modular spaces, highlighting fundamental principles and their applications. The discussion primarily revolves around operators and their semigroups that adhere to pointwise asymp-totic nonexpansive and contractive conditions in the modular sense in a way that they may be directly applied also to Banach spaces. Utilizing the framework of regular and superregular modular spaces, our research generalizes several established results concerning fixed points of nonlinear operators, applicable to both Banach spaces and modular function spaces. The study seeks to identify and discuss current challenges, knowledge gaps, and unresolved questions, providing insights into potential of future research opportunities.