Principle of non-existence of nonlinear operator equation solutions

The main part of article deals with the following non-existence principle: if an operator $A$ has a fixed point $x_0$ and satisfied the variable Lipschitz condition then there exists a ball $B(x_0,r)$ without other fixed points; moreover, it is possible to give the lower estimates for the radius $r$ of this non-existence ball. Also it is shown that similar results can be obtained for Minty–Browder monotonic mappings. There are also several examples of nonlinear integral equations demonstrating the efficiency of presented results.