By using some properties of generalized maximal monotonicty mapping and the resolvent operator technique, in Hilbert space, the author studied a class of variational inclusions with generalized maximal monotonicty mapping and constructed an algorithm for approximating the solution of this class of variational inclusions with generalized maximal monotonicty mapping.

It is especially noteworthy that the generalized monotonicity of sequence is independent of O-regularly varying sequence.

Relations tetween generalized inverse,maximum generalized inverse and reciprocal generalized inverse were obtained.

In addition, under the generalized convexity and generalized monotonicity, solution existence for implicit vector equilibrium problems is investigated.

The existence of the solutions of the variational inequality problem with a maximal monotone set-valued map on a reflexive a Banach space under a general convex constrained set is discussed in this paper.

Firstly, we combine the theories of monotone operator and maximal monotone operator’s Yosida approaching with domain method to prove the solution’s existence of single input-output equation in R n in the similar economical backgrounds in present articles and then gain the solution’s continuity by exterior approaching method,at last we give the responding eco.