**Zdzisław Kamont, Adam Nadolski**

## Functional Differential
Inequalities with Unbounded Delay

**Abstract:**

We prove that a function of several variables satisfying a functional
differential inequality with unbounded delay can be estimated by a solution of a
suitable initial problem for an ordinary functional differential equation. As a
consequence of the comparison theorem we obtain a Perron-type uniqueness result
and a result on continuous dependence of solutions on given functions for
partial functional differential equations with unbounded delay. We consider
classical solutions on the Haar pyramid.

**Keywords:**

Maximal solutions, initial problems, unbounded delay, nonlinear estimates of the
Perron type, comparison result.

**MSC 2000:** 35R10, 34K12