  In this work, using a new geometrical approach we study to the existence of
the fixed-point of mappings that independence of the smoothness, and also of
their single-values or multi-values. This work proved the theorems that
generalize in some sense the Brouwer and Schauder fixed-point theorems, and
also such type results in multi-valued cases. One can reckon this approach is
based on the generalization of the one theorem Cauchy and on the convexity
properties of sets. As the used approach is based on the geometry of the image
of the examined mappings that are independent of the topological properties of
the space we could to prove the general results for almost every vector space.
The general results we applied to the study of the nonlinear equations and
inclusions in VTS, and also by applying these results are investigated
different concrete nonlinear problems. Here provided also sufficient conditions
under which the conditions of the theorems will fulfill.
