In this paper, we introduce a new method for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. The iterative process is based on two well-known projection method and the hybrid (or outer approximation) method. However we do not use an extrapolation step in the projection method. The absence of one projection in our method is explained by slightly different choice of sets in hybrid method. We prove a strong convergence of the sequences generated by our method.2010 Mathematics Subject Classification: 47J20

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We propose a modified extragradient method with dynamic step size adjustment to solve variational inequalities with monotone operators acting in a Hilbert space. In addition, we consider a version of the method that finds a solution of a variational inequality that is also a fixed point of a quasi-nonexpansive operator. We establish the weak convergence of the methods without any Lipschitzian continuity assumption on operators.

A modification of Korpelevich's method with one metric projection onto the feasible set at an iteration step is proposed to solve monotone equilibrium problems. The weak convergence of the modified method is proved.

Aims and ScopeOptimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences.The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multiobjective programming, description of software packages, approximation techniques and heuristic approaches.