Convergence analysis of the proximal point algorithm for pseudo-monotone equilibrium problems

In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators.