Strong convergence theorems for minimization, variational inequality and fixed point problems for quasi-nonexpansive mappings using modified proximal point algorithms in real Hilbert spaces

In this paper, we investigate the problem of finding a common element of the solution set of convex minimization problem, the solution set of variational inequality problem and the  solution set of fixed point problem with an infinite family of quasi-nonexpansive mappings in  real Hilbert spaces. Based on the well-known proximal point algorithm and viscosity approximation method, we propose and analyze a  new iterative algorithm for computing a common element. Under very mild assumptions, we obtain a strong convergence theorem for the sequence generated by the proposed method. Application to convex minimization and variational inequality problems coupled with inclusion problem  is provided to support our main results.\,Our proposed method is quite general and includes the iterative methods considered in the earlier and recent literature as special cases.

Keywords:

Convex minimization problem , proximal point algorithm , common fixed points , Quasi-nonexpansive mappings , Variational inequality problem