Pseudocharacters on free groups, invariant with respect to some types of endomorphisms

We investigate methods for constructing nontrivial pseudocharacters on a free group F _n invariant with respect to certain types of endomorphisms. We find some conditions for endomorphisms of the free group under which there is a nontrivial pseudocharacter that is invariant with respect to these endomorphisms. We consider free products $ R = tilde{R}*prodlimits_{{i = k}}^n {leftlangle {{r_i}} rightrangle } $ , where one factor is F _n , and the other factor is a group on which there is a pseudocharacter. For such products we obtain a similar result about the conditions of existence of nontrivial pseudocharacters invariant with respect to certain endomorphisms.