FROM EQUILIBRIUM PROBLEMS AND FIXED POINTS PROBLEMS TO MINIMIZATION PROBLEMS

Abstract

Algorithms approach to equilibrium problems and fixed points problems have been extensively studied in the literature. The purpose of this paper is devoted to consider the minimization problem of finding a point $x^†$ with the property \begin{equation*} x^†\in \Omega\quad {\rm and}\quad \|x^†\|^2=\min_{x\in \Omega}\|x\|^2, \end{equation*} where $\Omega$ is the intersection of the solution set of equilibrium problem and the fixed points set of nonexpansive mapping. For this purpose, we suggest two algorithms: \begin{eqnarray*} F(z_t,y)+\frac{1}{\lambda}\Big{\langle} y-z_t,z_t-\Big{(}(1-t)I-\lambda A\Big{)}Sz_t\Big{\rangle}\ge 0,\;\forall y\in C. \end{eqnarray*} and \begin{eqnarray*} \begin{cases} F(z_n,y)+\langle Ax_n,y-z_n\rangle+\frac{1}{\lambda_n}\Big{\langle}y-z_n,z_n-(1-\alpha_n)x_n\Big{\rangle}\ge 0,\;\forall y\in C,\\ x_{n+1}=\beta_nx_n+(1-\beta_n)Sz_n,\; n\ge 0. \end{cases} \end{eqnarray*} It is shown that under some mild conditions, the net $\{z_t\}$ and the sequences $\{z_n\}$ and $\{x_n\}$ converge strongly to $\tilde{x}$ which is the unique solution of the above minimization problem. It should be point out that our suggested algorithms solve the above minimization problem without involving projection.