Abstract: Many methods have been proposed in the literature for solving the split variational inequality problem. Most of these methods either require that this problem is transformed into an equivalent variational inequality problem in a product space, or that the underlying operators are co-coercive. However, it has been discovered that such product space transformation may cause some potential difficulties during implementation and its approach may not fully exploit the attractive splitting nature of the split variational inequality problem. On the other hand, the co-coercive assumption of the underlying operators would preclude the potential applications of these methods. To avoid these setbacks, we propose two new relaxed inertial methods for solving the split variational inequality problem without any product space transformation, and for which the underlying operators are freed from the restrictive co-coercive assumption. The methods proposed, involve projections onto half-spaces only, and originate from an explicit discretization of a dynamical system, which combines both the inertial and relaxation techniques in order to achieve high convergence speed. Moreover, the sequence generated by these methods is shown to converge strongly to a minimum-norm solution of the problem in real Hilbert spaces. Furthermore, numerical implementations and comparisons are given to support our theoretical findings.

Key words: split variational inequality problems, relaxation technique, inertial extrapolation, minimum-norm solutions, product space formulation, half-spaces