Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales

Abstract

We employ Kranoselskii's fixed point theorem to establish the existence of nonoscillatory solutions to the second-order neutral delay dynamic equation on a time scale T. To dwell upon the importance of our results, one interesting example is also included.

1. Introduction

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 [1]). Several authors have expounded on various aspects of this new theory; see the survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of time scales, by Bohner and Peterson [3], summarizes and organizes much of the time scale calculus; we refer also to the last book by Bohner and Peterson [4] 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 [3].

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 [5], Saker [6], and Hassan [7]. 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

(1.1)

on a time scale (an arbitrary closed subset of the reals).

The motivation originates from Kulenović and Hadžiomerpahić [23] and Zhu and Wang [24]. In [23], the authors established some sufficient conditions for the existence of positive solutions of the delay equation

(1.2)

Recently, [24] established the existence of nonoscillatory solutions to the neutral equation

(1.3)

on a time scale

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.

Since we are interested in the nonoscillatory behavior of (1.1), we assume throughout that the time scale under consideration satisfies and

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

In this section, we establish the existence of nonoscillatory solutions to (1.1). For let and Further, let denote all continuous functions mapping into and

(2.1)

Endowed on with the norm () is a Banach space (see [24]). Let we say that is uniformly Cauchy if for any given there exists such that for any for all .

is said to be equicontinuous on if for any given there exists such that for any and with

Also, we need the following auxiliary results.

Lemma 2.1 (see [24, Lemma ]).

Suppose that is bounded and uniformly Cauchy. Further, suppose that is equicontinuous on for any Then is relatively compact.

Lemma 2.2 (see [25, Kranoselskii's fixed point theorem]).

Suppose that is a Banach space and is a bounded, convex, and closed subset of Suppose further that there exist two operators such that

• (i) for all

• (ii) is a contraction mapping;

• (iii) is completely continuous.

Then has a fixed point in

Throughout this section, we will assume in (1.1) that

, , = , = ,  ,, , , = , and there exists a function such that = , =

Theorem 2.3.

Assume that holds and Then (1.1) has an eventually positive solution.

Proof.

From the assumption we can choose large enough and positive constants and which satisfy the condition

(2.2)

such that

(2.3)

(2.4)

(2.5)

(2.6)

Furthermore, from we see that there exists with such that for

Define the Banach space as in (2.1) and let

(2.7)

It is easy to verify that is a bounded, convex, and closed subset of

Now we define two operators and as follows:

(2.8)

Next, we will show that and satisfy the conditions in Lemma 2.2.

We first prove that for any Note that for any For any and in view of (2.3), (2.4) and (2.6), we have

(2.9)

Similarly, we can prove that for any and Hence, for any

• (ii) We prove that is a contraction mapping. Indeed, for we have

  

  (2.10)

for and

(2.11)

for Therefore, we have

(2.12)

for any Hence, is a contraction mapping.

We will prove that is a completely continuous mapping. First, by we know that maps into

Second, we consider the continuity of Let and as then and as for any Consequently, by (2.5) we have

(2.13)

for So, we obtain

(2.14)

which proves that is continuous on

Finally, we prove that is relatively compact. It is sufficient to verify that satisfies all conditions in Lemma 2.1. By the definition of we see that is bounded. For any take so that

(2.15)

For any and we have

(2.16)

Thus, is uniformly Cauchy.

The remainder is to consider the equicontinuous on for any Without loss of generality, we set For any we have for and

(2.17)

for

Now, we see that for any there exists such that when with

(2.18)

for all This means that is equicontinuous on for any

By means of Lemma 2.1, is relatively compact. From the above, we have proved that is a completely continuous mapping.

By Lemma 2.2, there exists such that Therefore, we have

(2.19)

which implies that is an eventually positive solution of (1.1). The proof is complete.

Theorem 2.4.

Assume that holds and Then (1.1) has an eventually positive solution.

Proof.

From the assumption we can choose large enough and positive constants and which satisfy the condition

(2.20)

such that

(2.21)

Furthermore, from we see that there exists with such that for

Define the Banach space as in (2.1) and let

(2.22)

It is easy to verify that is a bounded, convex, and closed subset of

Now we define two operators and as in Theorem 2.3 with replaced by The rest of the proof is similar to that of Theorem 2.3 and hence omitted. The proof is complete.

Theorem 2.5.

Assume that holds and Then (1.1) has an eventually positive solution.

Proof.

From the assumption we can choose large enough and positive constants and which satisfy the condition

(2.23)

such that

(2.24)

Furthermore, from we see that there exists with such that for

Define the Banach space as in (2.1) and let

(2.25)

It is easy to verify that is a bounded, convex, and closed subset of

Now we define two operators and as in Theorem 2.3 with replaced by The rest of the proof is similar to that of Theorem 2.3 and hence omitted. The proof is complete.

We will give the following example to illustrate our main results.

Example 2.6.

Consider the second-order delay dynamic equations on time scales

(2.26)

where , , , , , , , Then , , Let It is easy to see that the assumption holds. By Theorem 2.3, (2.26) has an eventually positive solution.