  This study delves into equilibrium problems, focusing on the identification
of finite solutions for feasible solution sequences. We introduce an innovative
extension of the weak sharp minimum concept from convex programming to
equilibrium problems, coining this as weak sharpness for solution sets.
Recognizing situations where the solution set may not exhibit weak sharpness,
we propose an augmented mapping approach to mitigate this limitation. The core
of our research is the formulation of augmented weak sharpness for the solution
set, a comprehensive concept that encapsulates both weak sharpness and strong
non-degeneracy within feasible solution sequences. Crucially, we identify a
necessary and sufficient condition for the finite termination of these
sequences under the premise of augmented weak sharpness for the solution set in
equilibrium problems. This condition significantly broadens the scope of
existing literature, which often assumes the solution set to be weakly sharp or
strongly non-degenerate, especially in the context of mathematical programming
and variational inequality problems. Our findings not only shed light on the
termination conditions in equilibrium problems but also introduce a less
stringent sufficient condition for the finite termination of various
optimization algorithms. This research, therefore, makes a substantial
contribution to the field by enhancing our understanding of termination
conditions in equilibrium problems and expanding the applicability of
established theories to a wider range of optimization scenarios.
