The essential aim of the present paper is to consider a new iterative scheme to study the approximate solvability problem for a system of generalized nonlinear variational inequalities in the framework of uniformly smooth and strictly convex Banach spaces by using the generalized projection approach. Here the authors consider some special cases of the problem (the so-called system of generalized nonlinear variational inequalities problem, SGNIVP): find (a nonempty closed convex subset of real strictly convex Banach space) such that for all ( are the nonlinear mappings)
(1), , .
(I) If and are bifurcations from to (the dual space) with ( is the normalized duality mapping) and , where is a mapping, , then the problem (1) reduces to find such that for all :
(2) , , ;
where and are two positive constants.
(II) If and both are univariate mappings then the problem (2) reduces to the (SGNVIP). Find such that for all ;
(3) , .
Main result: If is a real smooth and strictly convex Banach space with Kadec-Klee property, is a nonempty closed and convex subset of with and be continuous mappings satisfying the condition: There exist a compact subset and constants , , such that , where , and , , , then problem (1) has a solution and , , , and the sequences converge strongly to a unique solution .
The problem (1) is a more general system of generalized nonlinear variational inequality problem, which includes many kinds of well-known systems of variational inequalities as its special case.