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The purpose of this paper is to solve the hierarchical variational inequality with the constraint of a general system of variational inequalities in a uniformly convex and 2-uniformly smooth Banach space. We introduce implicit and explicit iterative algorithms which converge strongly to a unique solution of the hierarchical variational inequality problem. Our results improve and extend the corresponding results announced by some authors. [ABSTRACT FROM AUTHOR]

The present paper deals with the common solution method for finding a fixed point of a nonexpansive mapping and a solution of split hierarchical Minty variational inequality problems. We discuss the weak convergence of the sequences generated by the proposed method to a common solution of a fixed point problem and a split hierarchical Minty variational inequality problem. An example is presented to illustrate the proposed algorithm and result. [ABSTRACT FROM AUTHOR]

In this paper, we introduce one composite implicit relaxed extragradient-like scheme and another composite explicit relaxed extragradient-like scheme for finding a common solution of a finite family of generalized mixed equilibrium problems (GMEPs) with the constraints of a system of generalized equilibrium problems (SGEP) and the hierarchical fixed point problem (HFPP) for a strictly pseudocontractive mapping in a real Hilbert space. We establish the strong convergence of these two composite relaxed extragradient-like schemes to the same common solution of finitely many GMEPs and the SGEP, which is the unique solution of the HFPP for a strictly pseudocontractive mapping. In particular, we make use of weaker control conditions than previous ones for the sake of proving strong convergence. Utilizing these results, we first propose the composite implicit and explicit relaxed extragradient-like schemes for finding a common fixed point of a finite family of strictly pseudocontractive mappings, and then we derive their strong convergence to the unique common solution of the SGEP and some HFPP. Our results complement, develop, improve, and extend the corresponding ones given by some authors recently in this area. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new viscosity approximation method by using the shrinking projection algorithm to approximate a common fixed point of a countable family of nonlinear mappings in a Banach space. Under quite mild assumptions, we establish the strong convergence of the sequence generated by the proposed algorithm and provide an affirmative answer to an open problem posed by Maingé (Comput. Math. Appl. 59:74-79, 2010) for quasi-nonexpansive mappings. In contrast with related processes, our method does not require any demiclosedness principle condition imposed on the involved operators belonging to the wide class of quasi-nonexpansive operators. As an application, we also introduce an iterative algorithm for finding a common element of the set of common fixed points of an infinite family of quasi-nonexpansive mappings and the set of solutions of a mixed equilibrium problem in a real Banach space. We prove a strong convergence theorem by using the proposed algorithm under some suitable conditions. Our results improve and generalize many known results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of a general system of variational inequalities, the set of solutions of a split feasibility problem (SFP), and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest-descent method and gradient-projection method (GPM) with regularization. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality. [ABSTRACT FROM AUTHOR]

In this paper, we propose a descent direction method for solving variational inequalities. A new iterate is obtained by searching the optimal step size along a new descent direction which is obtained by the linear combination of two descent directions. Under suitable conditions, the global convergence of the proposed method is studied. Two numerical experiments are presented to illustrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a relaxed extragradient iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of solutions of finitely many variational inclusions, and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. The iterative algorithm is based on Korpelevich's extragradient method, the viscosity approximation method, Mann's iterative method, and the strongly positive bounded linear operator approach. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical minimization. [ABSTRACT FROM AUTHOR]

The purpose of this article is to propose a modified viscosity implicit-type proximal point algorithm for approximating a common solution of a monotone inclusion problem and a fixed point problem for an asymptotically nonexpansive mapping in Hadamard spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved. Our results extend and complement some recent results in this direction. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to propose a modified proximal point algorithm for solving minimization problems in Hadamard spaces. We then prove that the sequence generated by the algorithm converges strongly (convergence in metric) to a minimizer of convex objective functions. The results extend several results in Hilbert spaces, Hadamard manifolds and non-positive curvature metric spaces. [ABSTRACT FROM AUTHOR]