The purpose is to present an iterative scheme for finding a common element of the set of solutions of the variational inclusion problem with multi-valued maximal monotone mapping and inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space.

A strong convergence theorem for quasi variational inclusion problem with multi-valued maximal monotone mapping and inverse-strongly monotone mapping and fixed point problems with nonexpansive mappings in Hilbert spaces were discussed.

The concept of set-valued monotone mapping in Linear 2-normed space was introduced,several new propositions concerning were obtained.

We study some classes of generalized vector equilibrium problems with pseudomonotone set-valued bifunctions in FC-spaces.

Under the conditions of compatibility or sub-compatibility between a set-valued mappingand a single-valued mapping, the existence of cotnnlon fixed points for two set-valued mapping and a single-valued mapping in complete metric space is discussed.

The sufficient and necessary conditions of two single-valued mapping and a set-valued mapping with the only common fixed point in complete metric space are discussed.

With the use of some conditions of compatibility or sub-compatibility of a single-valued mapping with a set-valued mapping,the common fixed point theorems of single-valued mapping pair and set -mapping pair in complete metric spaces are deduced.