In this article, we introduce and consider a new system of general nonconvex variational inequalities defined on uniformly prox-regular sets. We establish the equivalence between the new system of general nonconvex variational inequalities and the fixed point problems to analyze an explicit projection method for solving this system. We also consider the convergence of the projection method under some suitable conditions. Results presented in this article improve and extend the previously known results for the variational inequalities and related optimization problems. MSC (2000): 47J20; 47N10; 49J30.

In this paper, we introduce and consider a system of variational inequalities involving two different operators in q-uniformly smooth Banach spaces. We suggest and analyze a new explicit projection method for solving the system under some more general conditions. Our results extend and unify the results of Verma (Appl. Math. Lett. 18:1286Lett. 18: -1292Lett. 18: , 2005 and Yao, Liou and Kang (J. Glob. Optim., 2011, doi:10.1007/s10898-011-9804-0) and some other previously known results. MSC: 47H05; 47H10; 47J25

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We modify the relaxed hybrid steepest-descent methods to the case of variational inequality for finding a solution over the set of common fixed points of a finite family of strictly pseudocontractive mappings. The strongly monotone property defined on cost operator was extended to relaxed cocoercive in convergence analysis. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

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