Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales
© Tongxing Li et al. 2009
Received: 5 March 2009
Accepted: 24 August 2009
Published: 11 October 2009
The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis in 1988 in order to unify continuous and discrete analysis (see Hilger ). Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al.  and references cited therein. A book on the subject of time scales, by Bohner and Peterson , summarizes and organizes much of the time scale calculus; we refer also to the last book by Bohner and Peterson  for advances in dynamic equations on time scales. For the notation used below we refer to the next section that provides some basic facts on time scales extracted from Bohner and Peterson .
In recent years, there has been much research activity concerning the oscillation of solutions of various equations on time scales, and we refer the reader to Erbe , Saker , and Hassan . And there are some results dealing with the oscillation of the solutions of second-order delay dynamic equations on time scales [8–22].
In this work, we will consider the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation of the form
The motivation originates from Kulenović and Hadžiomerpahić  and Zhu and Wang . In , the authors established some sufficient conditions for the existence of positive solutions of the delay equation
Recently,  established the existence of nonoscillatory solutions to the neutral equation
Neutral equations find numerous applications in natural science and technology. For instance, they are frequently used for the study of distributed networks containing lossless transmission lines. So, we try to establish some sufficient conditions for the existence of equations of (1.1). However, there are few papers to discuss the existence of nonoscillatory solutions for neutral delay dynamic equations on time scales.
As usual, by a solution of (1.1) we mean a continuous function which is defined on and satisfies (1.1) for A solution of (1.1) is said to be eventually positive (or eventually negative) if there exists such that (or ) for all in A solution of (1.1) is said to be nonoscillatory if it is either eventually positive or eventually negative; otherwise, it is oscillatory.
2. Main Results
Endowed on with the norm ( ) is a Banach space (see ). Let we say that is uniformly Cauchy if for any given there exists such that for any for all .
Also, we need the following auxiliary results.
Lemma 2.1 (see [24, Lemma ]).
Lemma 2.2 (see [25, Kranoselskii's fixed point theorem]).
Throughout this section, we will assume in (1.1) that
We will give the following example to illustrate our main results.
The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have lead to the present improved version of the original manuscript. This research is supported by the Natural Science Foundation of China (60774004), China Postdoctoral Science Foundation Funded Project (20080441126), Shandong Postdoctoral Funded Project (200802018), Shandong Research Funds (Y2008A28, Y2007A27), and also supported by the University of Jinan Research Funds for Doctors (B0621, XBS0843).
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