  Sufficient conditions are given for the relation $\lim_{t\to\infty}y(t) = 0$
to hold, where $y(t)$ is a continuous nonnegative function on $[0,1)$
satisfying some nonlinear inequalities. The results are used for a study of
large time behavior of the solutions to nonlinear evolution equations. Example
of application is given for a solution to some evolution equation with a
nonlinear partial differential operator.
