Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

Let be a nonlinear mapping from a nonempty closed invex subset of an infinite-dimensional Hilbert space into . Let be proper, invex, and lower semicontinuous on and let be continuously Fréchet-differentiable on with , the gradient of , -strongly monotone, and -Lipschitz continuous on . Suppose that there exist an , and numbers , , such that for all and for all , the set defined by is nonempty, where and is -Lipschitz continuous with the following assumptions. (i) , and . (ii) For each fixed , map is sequentially continuous from the weak topology to the weak topology. If, in addition, is continuous from equipped with weak topology to equipped with strong topology, then the sequence generated by the general auxiliary problem principle converges to a solution of the variational inequality problem (VIP): for all .