OSCILLATORY PROPERTIES AND ASYMPTOTIC BEHAVIOR OF THE SOLUTIONS OF A CLASS OF OPERATOR-DIFFERENTIAL

In the present paper an operator-differential equation is 
investigated. Sufficient conditions for the presence of Kneser's properties 
are found.


INTRODUCTION
In the present paper sufficient conditions are obtained for the presence of Kneser's properties for the operator-differential equation considered.Conditions are also found which guarantee the existence of nonoscillating solutions, and some of their asymptotic properties are investigated.Sufficient conditions for finding the number of the zeros of a given solution of this equation in a finite closed interval are given.
Analogous results for ordinary differential equations are obtained in [1].
The consideration of an operator-differential equation allows us by means of a single approach to investigate the properties of the solutions of a number of little investigated classes of differential equations.1Received: July 1993.Revised" November 1993.
H2" A: AC"-1(.t., ) --' loc( /' )" H3" If the function x AC"-I(N+, N) is eventually nonzero and with a constant sign, then the function Ax Lo(+, ) is also eventually nonzero and with a constant sign, and they have the same sign.
H6: The operator A is linear.
Introduce the following notation: then If s o is a zero of the function v: R+--, R with multiplicity no, and m , o for no _< m A.,(V,So) m for no>m By #,(v,s0) (#',(V,So)) denote the number of indices (i = 1, ..., m-1) for which Let to +. Denote by Eto the set of all numbers t (t > to) for which there exists a solution x of equation ( 1) such that x(to)= x(t)= 0 and x(t)>_ 0 for t [to, t].Introduce the notation r,(to, p) = sup E o" Lemma 1.
Let the following conditions hold: 1. Conditions H1-H3 are met.
Lemma 1 follows from the Lemma of Kiguradze [1] and conditions H2 and H3.

MAIN RESULTS
Theorem 1.
Let the following conditions hold: 1. Conditions H1-H3 are met.
Then from (4)it follows that A,(x; [to, t])_< n. 2. Let an integer i exist (1 _< i _< n-1) such tha x (0 in the interval [to, t ] has infinitely many zeros.From condition H1 it follows that for each closed interval there exist a finite number of intervals Ti (which can be also points-for instance Toj) such that x()(t)=_ 0 for t Tij and there exists an e- neighborhood T of the interval Tj such that for t .T'iy\Ty, z(O(t) O.
Moreover, if Tij and Tt are wo subintervals which do no degenerate into points, hen either Tii = Tt or Tii T = O.