Auxiliary problems and algorithm for a system of generalized variational-like inequality problems

In this paper, we consider a system of generalized variational-like inequality problems (SGVLIP) involving nondifferentiable terms, and its related auxiliary problems in real Hilbert spaces. An existence theorem for auxiliary problems is established. Further, by exploiting this theorem, an algorithm for the SGVLIP is constructed. Furthermore, we prove the existence of a unique solution of SGVLIP and discuss the convergence analysis of the algorithm. The technique and results presented in this paper generalize and unify the corresponding techniques and results given in [M.A. Noor, General nonlinear variational inequalities, J. Math. Anal. Appl. 126 (1987) 78-84; M.A. Noor, General variational inequalities, Appl. Math. Lett. 1 (1988) 119-122; M.A. Noor, Mixed variational-like inequalities, Commun. Appl. Nonlinear Anal. 1 (4) (1994) 63-75].