NONOSCILLATION OF THE SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS OF THIRD ORDER MARGARITA BONEVA DIMITROVA

The impulsive differential equations with deviating argument are adequate mathematical models of numerous processes and phenomena in physics, biology and electrical engineering. In spite of wide possibilities for their application, the theory of these equations is developing rather slowly because of considerable difficulties in technical and theoretical character related to their study. In the recent twenty years, the number of investigations devoted to the oscillatory and nonoscillatory behavior of the solutions of functional differential equations has considerably increased. A large part of the works of this subject published in 1977 is presented in [5]. In monographs [2] and [3], published in 1987 and 1991, respectively, the oscillatory and asymptotic properties of the solutions of various classes of functional differential equations were systematically studied. A pioneering work devoted to the investigation of the oscillatory properties of the solutions of impulsive differential equations with deviating argument was rendered by Gopalsamy and Zhang [1]. In the present paper, necessary and sufficient conditions are found for existence of at least one bounded nonoscillatory solution of a class of impulsive differential equations of third order and fixed moments of impulse effect. Some asymptotic properties of the nonoscillating solutions are investigated.


Introduction
The impulsive differential equations with deviating argument are adequate mathematical models of numerous processes and phenomena in physics, biology and electrical engineering. In spite of wide possibilities for their application, the theory of these equations is developing rather slowly because of considerable difficulties in technical and theoretical character related to their study.
In the recent twenty years, the number of investigations devoted to the oscillatory and nonoscillatory behavior of the solutions of functional differential equations has considerably increased. A large part of the works of this subject published in 1977 is presented in [5]. In monographs [2] and [3], published in 1987 and 1991, respectively, the oscillatory and asymptotic properties of the solutions of various classes of functional differential equations were systematically studied. A pioneering work devoted to the investigation of the oscillatory properties of the solutions of impulsive differential equations with deviating argument was rendered by Gopalsamy and Zhang [1].
In the present paper, necessary and sufficient conditions are found for existence of at least one bounded nonoscillatory solution of a class of impulsive differential equations of third order and fixed moments of impulse effect. Some asymptotic properties of the nonoscillating solutions are investigated.
113. f k C(R,R), ufk(u > 0 for u :/= 0 and fk(ul) <_ fk(u2) for Definition 1: A function y C(R+,R) is called a solution of the equation (1) with initial conditions (2) if it satisfies the following conditions" (a) If 0 r 0 __ rl, then the function y coincides with the solution of the equation y'"(t)+f(t,y(t))=O with initial conditions (2). If v k < t <_ 7 k+l, then the function y coincides with the solution of the Let y(t) be a positive and bounded solution of the equation (1) for From condition 1t3, fk(y(-k)) > 0 for rk _> t 1. Then Ay"(-k) < O, 7 k >_ 1.
It follows from the above inequality as t + oc, that limty(t + oc which contradicts the assumption that y is a bounded solution of the equation (1).
Let us suppose c 2<0. Then there exists a constant c a<0 and a point 3>_t 2 such that y'(t) _< c a for >_ 3. Now, we integrate the above inequality from t 3 to t, (t >_ t3) and arrive at the inequality y(t)<_ cat + y(t3). It follows from the above inequality after taking the limit as t + oc, that limt__ + y(t) cx, which contradicts the assumption that y is a positive bounded solution of the equation (1).
(c) To show SY is precompact, we see that (Sy)(t), y e Y, is uniformly bounded. Now we will prove that coy is an equicontinuous family of functions on R+.
For y E Y and t 2 > 1 > 0, we have (Sy)(t2)-(Sy)(tl) < (s-t2)2f(s V(s))ds-(s-1 rk >_T e From (12) and (13), for t >_ T e we get I(Sy)(t)-l < e for all y E Y. Therefore SY is equiconvergent at oe. Lemma 1 implies that the set SY is relatively compact. According to the Schauder fixed point theorem, there exists a y Y such that y-SY. This y is a bounded nonoscillatory solution of (1). The proof is complete.