NONAUTONOMOUS DIFFERENTIAL EQUATIONS OF ALTERNATELY RETARDED AND ADVANCED TYPE

We obtain a solution formula of the differential equation ẋ(t)+a(t)x(t)+ b(t)x(g(t)) = f(t). At the same time, we study its oscillation and asymptotic stability properties. 2000 Mathematics Subject Classification. 34K11.


Introduction and preliminary.
In this paper, we investigate the global asymptotic behavior as well as oscillation of equations with piecewise constant argumenṫ subject to the initial condition where a(t), b(t), and f (t) are locally integrable functions on [0, ∞), g(t) is a piecewise constant function defined by g(t) = np for t ∈ np − l, (n + 1)p − l (n ∈ N), (1.3) where p and l are positive constants satisfying p > l.
Since the argument deviation of (1.1), namely is negative in [np − l, np) and positive in [np, (n + 1)p − l), equation (1.1) is said to be of alternately advanced and retarded type. Equations with piecewise constant argument (EPCA) deviation were investigated in many papers (see [1,2,3,4,5,6,7,8,9]). Since EPCA combine the features of both differential and difference equations, their asymptotic behavior as t → ∞ resembles in some cases the solution growth of differential equations, while in others it inherits the properties of difference equations. So this makes EPCA more interesting. (1.1) and (1.2) if the following conditions hold:  A solution of (1.1) and (1.2) is oscillatory if it has no last zero. Let [·] denote the greatest integer function. This paper was motivated by [7] in which the equatioṅ was investigated, where A and B are r × r matrices, x is an r -vector and f (t) is a locally integrable function on [0, ∞).
2. The case a(t) ≡ 0. In this case, (1.1) becomeṡ To simplify the notation, define f (s) ds, In addition, if b(t) and f (t) are integrable on (−∞, 0], this solution can be continued backwards on (−∞, 0] and is given by Proof. We use the notation given in (2.2).
In each interval of the type I n , (2.1) becomeṡ which has a unique solution whenever a preassigned value for x(np) is given. The solution of (2.1), with x(np) = x n , is and with x((n + 1)p) = x n+1 is Continuity of the solution at t = (n + 1)p − l requires so that from which it follows that  Similarly, F j = jp (j−1)p f (s)ds → 0 as j → ∞. Hence, given ε > 0, choose P 1 such that |F j | < K if j < P 1 and |F j | < ε(1−α)B 3 /2B 2 for j ≥ P 1 , choose P 2 so that if n > P 2 then (2.14) where we define   Example 2.6. Consider the equatioṅ
3. The case (1.1). To simplify the notation, define a(s) ds F g(t), t , where B(a, b) and F (a, b) are defined in (3.1).
In addition, if a(t), b(t), and f (t) are integrable on (−∞, 0], this solution can be continued backwards on (−∞, 0] and is given by a(s) ds F g(t), t . In summary, equations with piecewise constant argument are interesting in their own right, and have some curious and unpredictable properties. The systems of nonautonomous differential equations of alternately retarded and advanced type can be studied in similar ways.