Abstract
In this paper we generalize the integral inequality of Gronwall and study the continuous dependence of the solution of the initial value problem for nonlinear impulsive integro‐differential equations of Volterra type on the initial conditions.
Highlights
In the present paper analogues of Gronwall’s inequality for piecewise continuous functions are introduced
In this paper we generalize the integral inequality of Gronwall and study the continuous dependence of the solution of the initial value problem for nonlinear impulsive integro-differential equations of Volterra type on the initial conditions
The results obtained for these inequalities are applied to finding sufficient conditions for continuous dependence on the initial conditions of the solutions of the initial value problem for nonlinear impulsive integro-differential equations
Summary
In the present paper analogues of Gronwall’s inequality for piecewise continuous functions are introduced. In this paper we generalize the integral inequality of Gronwall and study the continuous dependence of the solution of the initial value problem for nonlinear impulsive integro-differential equations of Volterra type on the initial conditions. The results obtained for these inequalities are applied to finding sufficient conditions for continuous dependence on the initial conditions of the solutions of the initial value problem for nonlinear impulsive integro-differential equations. Lemma 1: (Theorem 16.4, [1]) Let for t > to the inequality
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