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 satisfy pointwise asymptotic nonexpansive and contractive conditions in the modular sense, and the results can also be applied directly to Banach spaces. Utilizing the framework of regular and super-regular 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 the potential of future research opportunities.