First,the notions of the measure of noncompactness and condensing set-valued mappings were introduced in locally FCuniform spaces without convexity structure.

Under several suitable transformations, the problem of positive solutions for set-valued condensing mapping equation in an ordered locally convex topological space is studied by some homotopy method.

The existence of the nonzero solution for set-valued condensing mapping equation in Hilbert space;

Fixed point theorems for condensing mappings and the solvability of equation Ax+Bx+Cx=x;

Some results for condensing mapping perturbations with toplogy degree theory are established with some extended results of compact perturbations.

Altman s Theorem is further improved and some new fixes-point theorems for condensing mapping or P 1-compact mappings are set up by topology theory.

A system of generalized variational inclusions for set-valued mapping;

The auxiliary principle technique is extended to study a class of generalized strongly nonlinear mixed variational-like inequalities for set-valued mappings without compact values.

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.