OSCILLATION OF THE SOLUTIONS OF A CLASS OF IMPULSIVE DIFFERENTIAL EQUATIONS WITH A DEVIATING ARGUMENT

Sufficient conditions are found for oscillation of all solutions of a class of 
impulsive differential equations with deviating argument.


Introduction
Impulsive differential equations with deviating argument are an adequate mathemati- cal apparatus for simulation of processes which depend on their prehistory and are subject to short-time disturbances.Such processes occur in the theory of optimal con- trol, theoretical physics, population dynamics, pharmacokinetics, biotechnologies, in- dustrial robotics, economics, etc.In spite of numerous possibilities for their applica- tions, the theory of these equations is developing rather slowly due to difficulties of theoretical and technical nature.
In the recent twenty years, a large number of studies devoted to the oscillation were published.To the best of our knowledge, no other publications on this subject have every been published.
In the present paper, we establish sufficient conditions for oscillation of all solu- tions of a class of impulsive differential equations with fixed moments of impulse ef- fect and a deviating argument.
Let us note that in contrast to [1], the present paper deals with the oscillatory properties of more general non-homogeneous impulsive differential-difference equation.
Definition 1: By a solution of the equation (1) with initial function (3), we mean any function x'[-h, oo)+[R, for which the following conditions are valid" 1.

2.
If 0 rl, then x coincides with the solution of the equation x'(t) + q(t)x(t) + v(t)x(t h) O.

3.
If i<tti+l, tiG{vi}={rih}il, then x coincides with the solution of the problem x'(t) + q(t)x(t) + p(t)x(t-h) 0 where the number k is determined from the equality rk."

If
< --< ti + 1, ti {ri}i {rib} =1, then x coincides with the solution of the problem where k is determined from the equality -k." The definition of a solution of the problem (2), ( 3) is 'analogous to Definition 1, where b: [0, oo)--N.
Definition 2: A nonzero solution x of the equation ( 1) is said to be nonoscillating if there exists a point 0 k 0 such that x(t) has a constant sign for t 0. Otherwise, the solution x is said to oscillate.
In work [1], the oscillatory properties of the impulsive differential equation with a deviating argument, are studied.To make this presentation self-contained, we formulate the most signifi- cant results of [1].
There exists a constant T > 0 such that for any lc N we have r>_T.
There exists a constant T > h > O such that for any k GN we have T]z + I Tllc T, There exists a constant M > 0 such that for any k N we have 0 <_ b k <_ M.
Then all solutions of the equation (4) oscillate.
Introduce the following conditions" Ill.There exists a positive constant T such that for any k G N we have rk+a-7k >_ T > h. It2.

Main Results
Theorem 3: Let the following conditions hold: 1.
Conditions tI1 and tI2 arc satisfied.
From (1) it follows that x(t)is a decreasing function on the set (t o + h, rs) U We integrate (1) from r k to rk + h (k _> s) and obtain that and rk+h rk+h r k and x(r k + h) < x('l + 0), then from (5) it follows that x(vk + h) 1+ q(s)ds + x(r k + O) +bk p(s)ds-1 The last inequality contradicts condition 2 of Theorem 3.
Conditions H1 and H2 are satisfied.
Proof: From H3 it follows that > 0. From the condition that p,q l+b k C( +, [ + we conclude that 1 q(t) + l + bk p(t) >0' E + k Then from Theorem 3, it follows that all solutions of the problem (1), (3) oscil- late.
We shall carry out equivalent transformations of impulsive equation (1).Set and substitute it into (1) to obtain z'(t) + Pl(t)z(t-h) O, s rk, (7) where
There exists a positive constant T such that for k N we have r k + 1- rk>_T.
There exists a constant M > 0 such that for any k N, the inequalities 0 <_ btc <_ M are valid.lim inf f p(s) exp q(s 1)ds Ids > + M e t--<x h h Then all solutions of the problem (1), (3) oscillate.
Condition H1 and condition 2 of Theorem 5 are valid. 2.
There exists a function w G cl(fi +, fi + such that w'(t) b(t).
We consider the following two cases" Case 1" Let z(t)> 0 be a solution of (11) for t >_ t 0. It is clear that Z(t-h) 0 for _ tlh.From p C( -F ' + and (11) it follows that z is a de- creasing function on the set (t 1 + h, rs) U[[-J i=s(7i,7i +1)] where the number s is chosen so that r s tl + h < T s.
Theorem 6: Let the following conditions hold: 1.

2.
There exists a constant L > 1 such that r k q(s)ds >_ L, E in k N. [q2 ql] where S coast > 0, 3.