A Hybrid Proximal Point Algorithm for Resolvent operator in Banach Spaces

Equilibrium problems have many uses in optimization theory and convex analysis and which is why different methods are presented for solving equilibrium problems in different spaces, such as Hilbert spaces and Banach spaces. The purpose of this paper is to provide a method for obtaining a solution to the equilibrium problem in Banach spaces. In fact, we consider a hybrid proximal point algorithm using the resolvent of a maximal monotone operator in Banach space. Under appropriate conditions, we prove the strong convergence of the generated sequence by the algorithm to the zero of the maximal monotone operator. As an application of the main result, and using proved theorems, we can provide a maximal monotone operator for any monotone bifunction so that the zero of the maximal monotone operator is the solution to the equilibrium problem. The results of this paper generalize or improve the obtained results in the various papers.