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

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.