Partial relaxed monotonicity and general auxiliary problem principle with applications

AbstractFirst, a general framework for the auxiliary problem principle is introduced and then it is applied to the approximation-solvability of the following class of nonlinear variational inequality problems (NVIP) involving partially relaxed monotone mappings. Find an element x∗ ϵ K such that 〈T(x∗)x−x∗〉+f(x)−f(x∗)≧0,for all xϵK, where T : K → Rn is a mapping from a nonempty closed convex subset K of Rn into Rn, and f : K → R is a continuous convex functional on K. The general class of the auxiliary problem principles is described as follows: for a given iterate xk E K and for a parameter ϱ > 0, determine xk+1 such that 〈T(x∗), x−x∗〉 + f(x∗) ≧ 0, for all x ϵ K, where T : K → Rn is a mapping from a nonempty closed convex subject K of Rn into Rn, and f : K → R is a continuous convex functional of K. The general calss of the auxillary problem principles is described as follows: for a given iterate xk ϵ K and for a parameter ρ > 0, determine xk+1 such that 〉ρT (xk) + h′ (xk+1) − h′(xk), x − xk+1〉 + ρf(x) − f(xk+1) ≧ (− φk), for all x ϵ K, where h : K → R is m-times continuously Frechet-differentiable on K and σk > 0 is a number