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The main strategy of this paper is intended to speed up the convergence of the inertial Mann iterative method and further speed up it through the normal S-iterative method for a certain class of nonexpansive-type operators that are linked with variational inequality problems. Our new convergence theory permits us to settle down the difficulty of unification of Korpelevich's extragradient method, Tseng's extragardient method, and subgardient extragardient method for solving variational inequality problems through an auxiliary algorithmic operator, which is associated with the seed operator. The paper establishes an interesting the fact that the relaxed inertial normal S-iterative extragradient methods do influence much more on convergence behaviour. Finally, the numerical experiments are carried out to illustrate that the relaxed inertial iterative methods; in particular, the relaxed inertial normal S-iterative extragradient methods may have a number of advantages over other methods in computing solutions to variational inequality problems in many cases. [ABSTRACT FROM AUTHOR]

We consider a new system of generalized variational inequalities (SGVI) defined on two closed convex subsets of a real Hilbert space. To find the solution of considered SGVI, a parallel Mann iteration process and a parallel S-iteration process have been proposed and the strong convergence of the sequences generated by these parallel iteration processes is discussed. Numerical example illustrates that the proposed parallel S-iteration process has an advantage over parallel Mann iteration process in computing altering points of some mappings. [ABSTRACT FROM AUTHOR]

In this paper, by combining a modified extragradient scheme with the viscosity approximation technique, an iterative scheme is developed for computing the common element of the set of fixed points of a sequence of asymptotically nonexpansive mappings and the set of solutions of the variational inequality problem for an α-inverse strongly monotone mapping. We prove a strong convergence theorem for the sequences generated by this scheme and give some applications of our convergence theorem. [ABSTRACT FROM AUTHOR]