Partial relaxed monotonicity and general auxiliary problem principle with applications

✍ Scribed by R.U. Verma

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 401 KB
• Volume: 16
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/s0893-9659(03)00084-3

✦ Synopsis

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 r* E K such that (T(zc'), z -2*) f f(z) -f(z*) 2 0, for all z E K, where T : K -Rn is a mapping from a nonempty closed convex subset K of R" 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 zk E K and for a parameter p > 0, determine &+l such that (~3' (z") + h' (I~+') -h' (z"> ,z -zk+l) + p [f(s) -f (z"")] 2 (-ok), for all z E K, where h : K + R is m-times continuously Frechet-differentiable on K and uk > 0 is a number