Several Classes Of Generalized Monotone Maps & Generalized Convex Functions And Their Application To Variational Inequalities

The generalized monotone maps have compactly connection with the generalized convex functions and the variational inequality problems. In this paper, we mainly study some kinds of the relations of the generalized monotone maps with the generalized convex functions and generalized monotone maps with the generalized variational inequality problems. This graduation paper contain two aspects, the first part is the theory, and the second part is the algorithm. In the section of the theory, we do our work by three direction. Firstly, on the base of the strictly monotone map, we provide the definition of strictly invariant quasi-monotone map, it is the extent of the strictly monotone mapping, at the same time, it is the especial case of the invariant quasi-monotone map, too. It is supported by some examples. On the condition of the differential, relationship between the strictly invariant quasi-monotonicity of the gradientmap and the strictly pre-quasi-invexity of the underlying function are established. Secondly, we extend a kind of generalized monotone map-strongly monotone map. We introduce strongly invariant monotone map and strongly G-monotone map. At the same time, strongly invariant monotone map is the special case of the invariant monotone map. It is supported by some examples. On the condition of the differential, the equality relationship between the strongly invariant monotonicity of the gradientmap and the strongly pre-invexity of the underlying function, and the relationship betweent the strongly G-monotone mapping and the strongly G-convex function are established. The end, we introduce a generalized quasi-monotone multivalued operator. On the condition of the Banach space, we require the exist condition of the solution of the generalized variational inequality problem for generalized quasi-monotone multivalued operator. On the section of the algorithm, we do our work on the two aspects. Firstly, we extend the iterative scheme u_n+1 =u_n- γ(u -P_K[u- ρTu]) of the fixed relationship u = P_K [u -ρTu] ,which is equally to the variational inequality problem ≥ 0, to the condition, which T is multivalued operator. Secondly, we conside some kinds of projection iterative schemes for variational inequality problem ≥0, which divide to twoaspects, one is S is the single-valued operator, the other is S is the multi-valued.