Oscillation of nonlinear impulsive differential equations with piecewise constant arguments

Existence and uniqueness of the solutions of a class of first order non- linear impulsive differential equation with piecewise constant arguments is studied. Moreover, sufficient conditions for the oscillation of the solutions are obtained.


Introduction
In this paper, we consider an impulsive differential equation with piecewise constant arguments of the form with the initial conditions where a : [0, ∞) → R, f : R → R are continuous functions, d n : N → R − {1}, ∆x (n) = x (n + ) − x (n − ) , x (n + ) = lim t→n + x (t) , x (n − ) = lim t→n − x (t) , [.] denotes the greatest integer function, and x −1 , x 0 are given real numbers.Since 1980's differential equations with piecewise constant arguments have attracted great deal of attention of researchers in mathematical and some of the others fields in science.Piecewise constant systems exist in a widely expanded areas such as biomedicine, chemistry, mechanical engineering, physics, etc.These kind of equations such as Eq.( 1) are similar in structure to those found in certain sequential-continuous models of disease dynamics [1].In 1994, Dai and Sing [2] studied the oscillatory motion of spring-mass systems with subject to piecewise constant forces of the form f (x[t]) or f ([t]).Later, they improved an analytical and numerical method for solving linear and nonlinear vibration problems and they showed that a function f ([N (t)]/N ) is a good approximation to the given continuous function f (t) if N is sufficiently large [3].
This method was also used to find the numerical solutions of a non-linear Froude pendulum and the oscillatory behavior of the pendulum [4].In 1984, Cooke and Wiener [5] studied oscillatory and periodic solutions of a linear differential equation with piecewise constant argument and they note that such equations are comprehensively related to impulsive and difference equations.After this work, oscillatory and periodic solutions of linear differential equations with piecewise constant arguments have been dealt with by many authors [6,7,8] and the references cited therein.But, as we know, nonlinear differential equations with piecewise constant arguments have been studied in a few papers [9,10,11].On the other hand, in 1994, the case of studying discontinuous solutions of differential equations with piecewise continuous arguments has been proposed as an open problem by Wiener [12].Due to this open problem, the following linear impulsive differential equations have been studied [13,14]: and Now, our aim is to consider the Wiener's open problem for the nonlinear problem ( 1)- (3).In this respect, we first prove existence and uniqueness of the solutions of Eq. ( 1)-( 3) and we also obtain sufficient conditions for the existence of oscillatory solutions.Finally, we give some examples to illustrate our results.

Existence of solutions
EJQTDE, 2013 No. 49, p. 2 where y(n) = x (n) and the sequence {y(n)} n≥−1 is the unique solution of the difference equation with the initial conditions Proof.Let x n (t) ≡ x(t) be a solution of ( 1)-( 2) on n ≤ t < n + 1. Eq. ( 1)-( 2) is rewritten in the form From ( 8), for n ≤ t < n + 1 we obtain On the other hand, if x n−1 (t) is a solution of Eq.( 1)-( 2) on n − 1 ≤ t < n, then we get Using the impulse conditions (2), from ( 9) and ( 10), we obtain the difference equation Considering the initial conditions (7), the solution of Equation ( 6) can be obtained uniquely.Thus, the unique solution of ( 1)-( 3) is obtained as (5).

Oscillatory solutions
where N is sufficiently large.Otherwise, the solution is called nonoscillatory.6) is said to be oscillatory if the sequence {y n } n≥−1 is neither eventually positive nor eventually negative.Otherwise, the solution is called non-oscillatory.Theorem 3. Let x (t) be the unique solution of the problem ( 1)-( 3) on [0, ∞) .If the solution y(n), n ≥ −1, of Eq. ( 6) with the initial conditions ( 7) is oscillatory, then the solution x (t) is also oscillatory.

Remark 1. According to Definition 2, a piecewise continuous function
Remark 2. We note that even if the solution y(n), n ≥ −1, of the Eq. ( 6) with the initial conditions ( 7) is nonoscillatory, the solution x(t) of ( 1)-(3) might be oscillatory.
In the following theorem give a necessary and sufficient condition for the existence of nonoscillatory solution x(t), when the solution of difference equation ( 6)-( 7) is nonoscillatory.Theorem 4. Let {y n } n≥−1 be a nonoscillatory solution of Eq. ( 6) with the initial conditions (7).Then the solution x(t) of the problem (1)-( 3) is nonoscillatory iff there exist a N ∈ N such that Proof.Without loss of generality we may assume that y which is a contradiction to (11).
Theorem 5. Suppose that 1 − d n > 0 for n ∈ N and there exist a Then, all solutions of Eq. ( 6) are oscillatory.
which is a special case of (4).In this case, condition (12) reduces to the following condition which is stated in [13] for b(t) ≡ b > 0. Now, consider following nonimpulsive equation where a : [0, ∞) → R, f : R → R are continuous functions.
Corollary 2. Assume that there exists a constant then all solutions of Eq. ( 16) are oscillatory. and Then, all solutions of Eq. ( 6) are oscillatory.
Proof.Let y(n) be a solution of Eq. ( 6).Assume that y(n) > 0, y(n − 1) > 0 for n > N, where N is sufficiently large.From Eq. ( 6), we have Let w n = y(n) y(n−1) .Since w n > 0, we consider two cases: Taking the inferior limit on both sides of inequality (21), we get which is a contradiction to the lim n→∞ inf w n = ∞.So, we consider the second case; 20) by y(n − 1), we have which yields Let lim Taking the inferior limit on both sides of inequality ( 22), we have Now, from (18), there are two subcases: then we obtain a contradiction from (23).
which contradicts to (19).So Eq. ( 6) cannot have an eventually positive solution.Similarly, existence of an eventually negative solution leads us a contradiction.Thus all solutions of (6) are oscillatory.
Consider the following equation.