A Generalized Topology Approach to Trajectory Convergence in Nonautonomous Evolution Equations With Monotone Operators

This paper investigates the application of β ‐open sets to the convergence analysis of nonautonomous evolution equations governed by maximal monotone operators in Hilbert spaces. β ‐open sets are a class of generalized open sets introduced by Njåstad (1965), which coincides with the class of semiopen sets by Levine (1963). We first examine whether the properties of β ‐open sets, which form a generalized topology (not a classical topology), can offer a more flexible framework for studying trajectory convergence. And then we discuss potential advantages in relaxing certain coercivity conditions in contrast with analyses in standard metric or weak topologies, while addressing the non‐Hausdorff nature and limited intersection closure of β ‐open sets. Examples from optimization problems (variational inequalities and sparse regression) and numerical insights from image denoising applications are utilized to illustrate the benefits of the approach. The paper highlights key challenges and outlines directions for further theoretical and computational development.