Partial relaxed monotonicity and general auxiliary problem principle with applications Original Research Article

First, 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 View the MathML source , 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 xkϵK and for a parameter ρ > 0, determine xk+1 such that View the MathML source , where h : K → R is m-times continuously Frechet-differentiable on K and σk > 0 is a number.