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The purpose of this paper is by using the modified block iterative method to propose an algorithm for finding a common element in the intersection of the set of common fixed points of an infinite family of quasi- ϕ-asymptotically nonexpansive and the set of solutions to an equilibrium problem and the set of solutions to a variational inequality. Under suitable conditions some strong convergence theorems are established in 2-uniformly convex and uniformly smooth Banach spaces. As applications we utilize the results presented in the paper to solving the convex feasibility problem (CFP) and zero point problem of maximal monotone mappings in Banach spaces. The results presented in the paper improve and extend the corresponding results announced by many authors. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a general inertial proximal point algorithm (PPA) for the mixed variational inequality (VI) problem. Based on our knowledge, without stronger assumptions, a convergence rate result is not known in the literature for inertial type PPAs. Under certain conditions, we are able to establish the global convergence and nonasymptotic O(1/k) convergence rate result (under a certain measure) of the proposed general inertial PPA. We then show that both the linearized augmented Lagrangian method (ALM) and the linearized alternating direction method of multipliers (ADMM) for structured convex optimization are applications of a general PPA, provided that the algorithmic parameters are properly chosen. Consequently, global convergence and convergence rate results of the linearized ALM and ADMM follow directly from results existing in the literature. In particular, by applying the proposed inertial PPA for mixed VI to structured convex optimization, we obtain inertial versions of the linearized ALM and ADMM whose global convergence is guaranteed. We also demonstrate the effect of the inertial extrapolation step via experimental results on the compressive principal component pursuit problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iterative algorithm for a simplified friction problem which is formulated as an elliptic variational inequality of the second kind. We approximate the simplified friction problem by a discrete system with the finite element method. Based on the use of the linearized technique and by constructing a particular function, we put forward the new algorithm to get the discrete solution. This algorithm is attractive due to its simple proof of convergence and easy implementation. A linear equation is solved in each iteration. Numerical results confirm that our algorithm is efficient and mesh independent. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a subgradient extragradient method for solving multi-valued variational inequality. It is showed that the method converges globally to a solution of multi-valued variational inequality, provided the multi-valued mapping is pseudomonotone with nonempty compact convex values. Convergence rate is analyzed. Preliminary computational experience is also reported. [Copyright &y& Elsevier]

The main purpose of this paper is to study the split fixed point and equilibrium problems which includes fixed point problems, equilibrium problems, and variational inequality problems as special cases. A damped algorithm is presented for solving this split common problem. Strong convergence analysis is shown. [ABSTRACT FROM AUTHOR]

This paper deals with design algorithms for the split variational inequality and equilibriumproblems. Strong convergence theorems are demonstrated. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to consider a new scheme by the hybrid extragradient-like method for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of solutions of a variational inequality, and the set of fixed points of an infinitely family of strictly pseudocontractive mappings in Hilbert spaces. Then, we obtain a strong convergence theorem of the iterative sequence generated by the proposed iterative algorithm. Our results extend and improve the results of Issara Inchan 2010 and many others. [ABSTRACT FROM AUTHOR]

Under the condition that the values of mapping F are evaluated approximately, we propose a proximal analytic center cutting plane algorithm for solving variational inequalities. It can be considered as an approximation of the earlier cutting plane method, and the conditions we impose on the corresponding mappings are more relaxed. The convergence analysis for the proposed algorithm is also given at the end of this paper. [ABSTRACT FROM AUTHOR]

This paper discusses the monotone variational inequality over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence theorem for the proposed algorithm to the solution is guaranteed under some suitable assumptions. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the system of mixed variational inequalities is introduced and considered in Banach spaces, which includes some known systems of variational inequalities and the classical variational inequalities as special cases. Using the projection operator technique, we suggest some iterative algorithms for solving the system of mixed variational inequalities and prove the convergence of the proposed iterative methods under suitable conditions. Our theorems generalize some known results shown recently. [Copyright &y& Elsevier]

Abstract: In this paper, we suggest and analyze two projection methods (one implicit and one explicit) for finding a particular solution of a mixed equilibrium problem in a real Hilbert space. Furthermore, we prove that the proposed projection methods converge strongly to a particular solution of the mixed equilibrium problem. [ABSTRACT FROM AUTHOR]