INTEGRAL INEQUALITIES OF GRONWALL TYPE FOR PIECEWISE CONTINUOUS FUNCTIONS

In the present paper analogues of Gronwall’s inequality for piecewise continuous functions are introduced. 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. The integral inequalities, established in this paper, can successfully be used in the qualitative theory of the impulsive differential equations. Let us note that the present paper generalizes some results obtained in [2-4].


Introduction
In the present paper analogues of Gronwall's inequality for piecewise continuous func- tions are introduced.The results obtained for these inequalities are applied to finding sufficient conditions for continuous dependence on the initial conditions of the solu- tions of the initial value problem for nonlinear impulsive integro-differential equa- tions.
The integral inequalities, established in this paper, can successfully be used in the qualitative theory of the impulsive differential equations.
Let us note that the present paper generalizes some results obtained in [2-4].
Denote by PC([to, oC),+) the set of all functions u'[t0, oc)+, which are piecewise continuous with discontinuity of the first kind at the points t k (k u(t k + O) u(t k 0) < oc and u(tk) u(t k 0).
Lemma 1: (Theorem 16.4, [1]) Let for t > t o the inequality u(t) <_ a(t) + i g(t,s)u(s)ds + E k(t)u(tk)' to o < k < (1) hold, where ilk(t) (kElP) are nondecreasing functions for t>_to, a E PC([to, C),+) is a nondecreasing function, u PC ([to, cX), +), and g(t,s) is a continuous nonnegative function for t, s t 0 and nondecreasing with respect to t for any fixed s t O.
Then, for t >_ to, the following inequality is valid: to to Proof: Consider the function defined by the equality Thus, (10) follows from inequalities ( 12) and ( 14).
Corollary 1: Let the conditions of Theorem 3 hold for a(t) a const >_ O.
Then, for t >_ to, the following inequality is valid:.
With the aid of the established inequalities we shall analyze the continuous dependence of the solutions of the initial value problem for impulsive integro- differential equations on the initial data.
Theorem 4: Let the following conditions hold: 1.
Then, the solutions of equation ( 15), ( 16) depend continuously on the initial condi- tions, i.e., for any number e >0, there exists a number 5 >0 such that for Xo-Yol < 5 the inequality (t; to, o) (t; to, yo) < holds for t [to,T], T const > to, T < oo.