Oscillatory and Periodic Solutions of Impulsive Differential Equations with Piecewise Constant Argument
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Oscillatory and Periodic Solutions of Impulsive Differential Equations with Piecewise Constant Argument Fatma Karakoc · Huseyin Bereketoglu · Gizem Seyhan
Received: 13 November 2008 / Accepted: 21 January 2009 / Published online: 6 February 2009 © Springer Science+Business Media B.V. 2009
Abstract We study the existence of oscillatory and periodic solutions of a class of first order scalar impulsive delay differential equations with piecewise constant argument. Keywords Impulsive delay differential equation · Differential equation with piecewise constant argument · Difference equation · Oscillatory solution · Periodic solution Mathematics Subject Classification (2000) 34K11 · 34K13 · 34K45
1 Introduction In this paper, we consider the first order linear scalar impulsive delay differential equation of the type ⎧ ⎨x (t) + a(t)x(t) + b(t)x([t − 1]) = 0, ⎩x(n ) − x(n ) = dn x(n), +
−
t = n,
n ∈ N = {0, 1, 2, . . .},
(1) (2)
where a, b : R+ → R continuous functions, R+ = [0, ∞), dn ∈ R − {1}, n ∈ N, x(n+ ) = limt→n+ x(t), x(n− ) = limt→n− x(t), and [.] denotes the greatest integer function. Since the early 1980’s, differential equations with piecewise constant arguments have attracted great deal of attention of researchers in mathematical and some of the other fields F. Karakoc () · H. Bereketoglu · G. Seyhan Department of Mathematics, Faculty of Sciences, Ankara University, Ankara, Turkey e-mail: [email protected] H. Bereketoglu e-mail: [email protected] G. Seyhan e-mail: [email protected]
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in science. Piecewise constant systems exists in a widely expanded areas such as biomedicine, chemistry, mechanical engineering, physics, etc. The systematical studies with mathematical models involving piecewise constant arguments were initiated for solving some biomedical problems. These kinds of equations such as (1) are similar in structure to those found in certain sequential-continuous models of disease dynamics. In [4], it is investigated the following system of equations describing the dynamics of the disease for generation n = 1, 2, . . . ⎧ (n) dI ⎪ ⎪ (t) = −c(t)I (n) (t) + k(t)S (n) (t)I (n) (t), n < t ≤ n + 1, ⎨ dt (n) ⎪ ⎪ ⎩ dS (t) = −c(t)S (n) (t) − k(t)S (n) (t)I (n) (t), n < t ≤ n + 1, dt while I (1) (1) = I0 ,
S (1) (1) = S0 ,
where c is the death rate, k is the horizontal transmission factor. These types of models are special cases of the general form dx(t) = F (t, xt ), dt φ[t] = G([t], x[t] ),
[t] < t ≤ [t] + 1, x[t] = φ[t] , [t] ≥ 2; φ1 = H,
which arise naturally in a number of models of epidemic. In 1994, Dai and Sing [6] studied the oscillatory motion of spring-mass systems 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 ) with the argument [N (t)]/N is a good approximation to the given continuous function f (t) with argument t , if the parameter N is sufficiently la
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