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In this work, using a new geometrical approach we study to the existence ofthe fixed-point of mappings that independence of the smoothness, and also oftheir single-values or multi-values. This work proved the theorems thatgeneralize in some sense the Brouwer and Schauder fixed-point theorems, andalso such type results in multi-valued cases. One can reckon this approach isbased on the generalization of the one theorem Cauchy and on the convexityproperties of sets. As the used approach is based on the geometry of the imageof the examined mappings that are independent of the topological properties ofthe 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 andinclusions in VTS, and also by applying these results are investigateddifferent concrete nonlinear problems. Here provided also sufficient conditionsunder which the conditions of the theorems will fulfill.

Some general fixed-point theorems for nonlinear mappings connected with one Cauchy theorem