On the existence of fixed points in completely continuous operators in $F$-space

This work is dedicated to the development of the theory of fixed points of completely continuous operators. We prove existence of new theorems of fixed points of completely continuous operators in $F$-space (Frechet space). This class of spaces except Banach includes such important space as a countably normed space and $L_p(0<p<1),$ $l_p(0<p<1).$