**G. Kharatishvili, T. Tadumadze, and N. Gorgodze**

## Continuous Dependence and Differentiability of Solution with Respect to Initial Data
and Right-Hand Side for Differential Equations
with Deviating Argument

**abstract:**

Non-linear differential equations with variable
delay and quasi-linear neutral differential equations are considered
in the case where at the initial moment of time the value of the initial
function, generally speaking,
does not coincide with the initial value of the trajectory
(discontinuity at the initial
moment). Theorems on continuity of solution of the Cauchy problem with respect
to initial data and right-hand side are proved. The perturbations of the initial
data, i.e., of the initial function and the initial values (the initial moment,
the initial
value of the trajectory) are small in the uniform and Euclidean norms,
respectively.
The pertrurbation of the right-hand side of the equation is small in the
integral sense.
Representation formulas of the differential of solution are obtained, when
pertrurbations are small in the Euclidean topology. If the effect of discontinuouty
at the initial moment influences upon the right-hand side of the equation, then,
in contrast to earlier obtained formulas, representation formulas of the
differential contain a new term.

**Mathematics Subject Classification:**
34K15, 34K40.

**Key words and phrases:**
Differential equation with deviating argument, delay
differential equation, neutral type differential equation, continuous dependence
of the solution, differentiability of the solution.