Abstract
Let T:K→H be a nonlinear mapping from a nonempty
closed invex subset K of an infinite-dimensional Hilbert space
H into H. Let f:K→R be proper, invex, and lower
semicontinuous on K and let h:K→R be continuously Fréchet-differentiable on K with h′, the gradient of h, (η,α)-strongly monotone, and (η,β)-Lipschitz continuous on K. Suppose that
there exist an x*∈K, and numbers a>0, r≥0, ρ(a<p<α) such that for all t∈[0,1]
and for all x∈K∗, the set S∗ defined by S∗={(h,η):h′(x∗+t(x−x∗))(x−x∗)≥〈h′(x∗+tη(x,x∗)),η(x,x∗)〉} is nonempty, where K∗={x∈K:‖x−x∗‖≤r} and η:K×K→H is (λ)-Lipschitz continuous with the following assumptions. (i)
η(x,y)+η(y,x)=0,η(x,y)=η(x,z)+η(z,y), and ‖η(x,y)‖≤r. (ii) For each fixed y∈K, map x→η(y,x) is sequentially continuous from the weak
topology to the weak topology. If, in addition, h′ is continuous from H equipped with weak topology to H equipped with strong topology, then the sequence {xk} generated by the general auxiliary problem principle converges to a solution x∗ of the variational inequality problem (VIP): 〈T(x∗),η(x,x∗)〉+f(x)−f(x∗)≥0 for all x∈K.