Oscillation and nonoscillation for impulsive dynamic equations on certain time scales
OSCILLATION AND NONOSCILLATION FOR IMPULSIVE DYNAMIC EQUATIONS ON CERTAIN TIME SCALES
MOUFFAK BENCHOHRA
SAMIRA HAMANI
JOHNNY HENDERSON
We discuss the existence of oscillatory and nonoscillatory solutions for first-order impulsive dynamic equations on time scales with certain restrictions on the points of impulse. We will rely on the nonlinear alternative of Leray-Schauder type combined with a lower and upper solutions method. Ik ∈ C(R, R), tk ∈ T, 0 = t0 < t1 < · · · < tm < tm+1 < · · · < ∞, y(tk+) = limh→0+ y(tk + h) and y(tk−) = limh→0+ y(tk − h) represent the right and left limits of y(t) at t = tk in the sense of the time scale; that is, in terms of h > 0 for which tk + h, tk − h ∈ [t0, ∞) ∩ T, whereas if tk is left-scattered (resp., right-scattered), we interpret y(tk−) = y(tk) (resp., y(tk+) = y(tk)). Impulsive differential equations have become important in recent years in mathematical models of real processes and they rise in phenomena studied in physics, chemical technology, population dynamics, biotechnology and economics. There have been significant developments in impulse theory also in recent years, especially in the area of impulsive differential equations with fixed moments; see the monographs of Bainov and Simeonov [5], Lakshmikantham et al. [22], Samo˘ılenko and Perestyuk [25], and the references therein. In recent years, dynamic equations on times scales have received much attention.
1. Introduction
This paper is concerned with the existence of oscillatory and nonoscillatory solutions of
first-order impulsive dynamic equations on certain time scales. We consider the problem
yΔ(t) = f t, y(t) , t ∈ JT := [0, ∞) ∩ T, t = tk, k = 1, . . . ,
y tk+ = Ik y tk− , k = 1, . . . ,
We refer the reader to the books by Bohner and Peterson [10, 11], Lakshmikantham et
al. [23], and the references therein. The time scale calculus has tremendous potential for
applications in mathematical models of real processes, for example, in physics, chemical
technology, population dynamics, biotechnology and economics, neural networks, social
sciences; see the monographs of Aulbach and Hilger [4], Bohner and Peterson [10, 11],
Lakshmikantham et al. [23], and the references therein. The existence of solutions of
boundary value problem on a measure chain (i.e., time scale) was recently studied by
Henderson [20] and Henderson and Tisdell [21]. The question of existence of solutions
to some classes of impulsive dynamic equations on time scales was treated very recently
by Henderson [19] and Benchohra et al. in [1, 7, 8]. The aim of this paper is to initiate the
study of oscillatory and nonoscillatory solutions to impulsive dynamic equations on time
scales. For oscillation and nonoscillation of impulsive differential equations, see, for
instance, the monograph of Bainov and Simonov [5] and the papers of Graef et al. [16, 17].
The purpose of this paper is to give some sufficient conditions for existence of oscillatory
and nonoscillatory solutions of the first-order dynamic impulsive problem (1.1) on time
scales. There has been, in fact, a good deal of research already devoted to oscillation
questions for dynamic equations on time scales; see, for example, [2, 9, 12, 14, 15, 24]. For the
purposes of this paper, we will rely on the nonlinear alternative of Leray-Schauder type
combined with a lower and upper solutions method. Our results can be considered as
contributions to this emerging field.
2. Preliminaries
We will briefly recall some basic definitions and facts from time scale calculus that we will
use in the sequel.
A time scale T is an closed subset of R. It follows that the jump operators σ, ρ : T → T
defined by
(supplemented by inf ∅ := sup T and sup ∅ := inf T) are well defined. The point t ∈ T is
left-dense, left-scattered, right-dense, right-scattered if ρ(t) = t, ρ(t) < t, σ(t) = t, σ(t) >
t, respectively. If T has a right-scattered minimum m, define Tk := T − {m}; otherwise,
set Tk = T. If T has a left-scattered maximum M, define Tk := T − {M}; otherwise, set
Tk = T. The notations [a, b], [a, b), and so on will denote time scales intervals
[a, b] = {t ∈ T : a ≤ t ≤ b},
Definition 2.1. Let X be a Banach space. The function g : T → X will be called rd−
continuous provided it is continuous at each right-dense point and has a left-sided limit at
each point, and write g ∈ Crd(T) = Crd(T, X). For t ∈ Tk, the Δ derivative of g at t,
denoted by gΔ(t), is the number (provided it exists) such that for all ε > 0, there exists a
neighborhood U of t such that
A function g : T → R is called regressive if
∀t ∈ T,
where μ(t) = σ(t) − t which is called the graininess function. The set of all rd−continuous
functions g that satisfy 1 + μ(t)g(t) > 0 for all t ∈ T will be denoted by +.
The generalized exponential function ep is defined as the unique solution y(t) =
ep(t, a) of the initial value problem yΔ = p(t)y, y(a) = 1, where p is a regressive
function. An explicit formula for ep(t, a) is given by
ep(t, s) = exp
if h = (...truncated)