Proximal Point Method for Quasi-Equilibrium Problems in Banach Spaces
Behzad Djafari Rouhani, Vahid Mohebbi · Numerical Functional Analysis and Optimization · 2020
We study the proximal point method for solving quasi-equilibrium problems in Banach spaces, which generalizes the proximal point method for equilibrium problems and quasi-variational inequalities. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the quasi-equilibrium problem, under standard assumptions on the problem without assuming neither pseudo-monotonicity nor any weak continuity assumption of the bifunction in its arguments that in many well-known methods have been used. Also, we show that the boundedness of the generated sequences implies that the solution set of the quasi-equilibrium problem is nonempty and prove the strong convergence of the generated sequence to a solution of the quasi-equilibrium problem when these conditions are satisfied.