Projection Methods for Variational InequalitiesGoverned by Inverse Strongly MonotoneOperators

• Main Authors: Yen-Ru Lin, 林晏如
• Other Authors: Hong-Kun Xu
• Format: Others
• Language: en_US
• Published: 2010
• Online Access: http://ndltd.ncl.edu.tw/handle/50405590232216593365

碩士 === 國立中山大學 === 應用數學系研究所 === 98 === Consider the variational inequality (VI) x* ∈C, ‹Fx*, x - x* ›≥0, x∈C (*) where C is a nonempty closed convex subset of a real Hilbert space H and F : C→ H is a monotone operator form C into H. It is known that if F is strongly monotone and Lipschitzian, then VI (*) is equivalently turned into a fixed point problem of a contraction; hence Banach''s contraction principle applies. However, in the case where F is inverse strongly monotone, VI (*) is equivalently transformed into a fixed point problem of a nonexpansive mapping. The purpose of this paper is to present some results which apply iterative methods for nonexpansive mappings to solve VI (*). We introduce Mann''s algorithm and Halpern''s algorithm and prove that the sequences generated by these algorithms converge weakly and respectively, strongly to a solution of VI (*), under appropriate conditions imposed on the parameter sequences in the algorithms.