Saddle Point Problems, Bilevel Problems, and Mathematical Program with Equilibrium Constraint on Complete Metric Spaces

We apply an existence theorem of variational inclusion problem on metric spaces to study optimization problems, set-valued vector saddle point Adaptor Handlebar problems, bilevel problems, and mathematical programs with equilibrium constraint on metric spaces.We study Pate these problems without any convexity and compactness assumptions.Our results are different from any existence results of these types of problems in topological vector spaces.