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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 618058, 13 pages
http://dx.doi.org/10.1155/2012/618058
Research Article

Existence for Eventually Positive Solutions of High-Order Nonlinear Neutral Differential Equations with Distributed Delay

1College of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China
2School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Received 14 February 2012; Accepted 12 March 2012

Academic Editor: Mingshu Peng

Copyright © 2012 Huanhuan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the existence for eventually positive solutions of high-order nonlinear neutral differential equations with distributed delay. We use Lebesgue's dominated convergence theorem to obtain new necessary and sufficient condition for the existence of eventually positive solutions.

1. Introduction and Preliminary

In this paper, we consider the high-order nonlinear neutral differential equation: and the associated inequality:(1)where is a positive integer, ,(2), (3), ,(4) are continuously nondecreasing real function with respect to defined on R such that , for , for is positive odd integer; are continuously decreasing real function with respect to defined on such that , for , for is positive even integer.

Recently, there has been a lot of activities concerning the existence of eventually positive solutions for nonlinear neutral differential equations. See [18]. In [1], Liu et al. have studied the even-order neutral differential equation: and the associated differential inequality: They have obtained that the existences of eventually positive solutions of (1.3) and (1.4) are equivalent. In [2], Ouyang et al. has studied the odd-order neutral differential equation: and the associated differential inequality: He has obtained that the existences of eventually positive solutions of (1.5) and (1.6) are equivalent.

As usual, a solution of (1.1) is a continuous function defined on such that is times differentiable and (1.1) holds for all . Such a solution is called an eventually positive solution if there is , such that , for . Here, .

Lemma 1.1. Assume , and , let is an eventually bounded positive solution of inequality (1.2), and set If is a positive odd integer, then eventually, If is a positive even integer, then eventually,

Proof. We have the following cases.
Case 1 (If is a positive odd integer). Because is bounded, and , thus is bounded. From (1.2), we have . Assume , since , decreases, set , then .
Thus, that is, . Simile, , this is a contradiction and is bounded, therefore, .
Again, since it decreases, set , thus . Next, we proof . Assume , then, then, . Simile, , this is a contradiction and is bounded, therefore, . That is, . Similarly, we obtain
Case 2 (If is a positive even integer). The proof of Case 2 is similar to that of part Case 1, therefore, it is omitted. We obtain
The proof is complete.

Lemma 1.2 (see [3, page 21]). Let , and suppose that a function satisfies the inequality: Then, cannot be a nonnegative function.

Lemma 1.3. Suppose that , and . Let be an eventually positive solution of (1.2) and set then eventually

Proof. From (1.2) and (1.7), eventually, we have and the hypotheses on , and yield that is not eventually zero. Thus, is eventually nonzero. Hence, if does not hold, then eventually , or .Case 1. , then there exists such that for . Then, . In view of (1.3) and , we obtain According to Lemma 1.2, we obtain . This is a contradiction and so is eventually positive.Case 2. , one argues that and repeats the argument to obtain that which contradicts the offset after .
The proof is complete.

2. Comparison Theory of Existence for Eventually Positive Solution

Theorem 2.1. Assume all conditions of Lemma 1.1 hold, is a positive odd integer. And, for sufficiently large .Then (1.1) has an eventually bounded positive solution if and only if inequality (1.2) has an eventually bounded positive solution.

Proof. It is clear that an eventually bounded positive solution of (1.1) is also an eventually bounded positive solution of (1.2). So, it suffices to prove that if (1.2) has an eventually bounded positive solution , for , then so does (1.1). Set It follows from Lemma 1.1 and (1.2) that eventually, By using (1.8) and integrating (1.2) from to , we obtain By repeating the same procedure times and by using (1.8), we are led to the inequality: Using Tonelli′s theorem, we reverse the order of integration and obtain That is, Let be such that (2.6) hold, and . Now, we consider the set of functions and define an operator on as follows: Then, it follows from Lebesgue′s dominated convergence theorem that is continuous. By using (2.6), it is easy to see that maps into itself, and for any , we have , for . Next, we define the sequence Then, by using (2.6) and a simple induction, we can easily see that Set then satisfies Again, set , then satisfies , and Thus, is a positive solution of (1.1) for .
The following: Assume that there exists , such that , for , and . Then, which implies which contradicts . Thus, is an eventually bounded positive solution of (1.1).
The proof is complete.

Theorem 2.2. Assume all conditions of Lemma 1.3 hold, is a positive odd integer. And, for sufficiently large .Then, (1.1) has an eventually positive solution if and only if inequality (1.2) has an eventually positive solution.

Proof. It is clear that an eventually positive solution of (1.1) is also an eventually positive solution of (1.2). So, it suffices to prove that if (1.2) has an eventually positive solution , for , then so does (1.1). Set
It follows from Lemma 1.3 and (1.2) that eventually, , which implies that there exists a nonnegative even integer , such that eventually We consider the following possible cases.
Case 1 (). Since is a positive integer, is an even integer, we can easily see that there exists a , such , and .
By using (2.17) and integrating (1.2) from to , we obtain By repeating the same procedure times and by using (2.31), we are led to the inequality: Using Tonelli′s theorem, we reverse the order of integration and obtain That is, Let be such that (2.21) hold, and . Now, we consider the set of functions and define an operator on as follows: Then, it follows from Lebesgue′s dominated convergence theorem that is continuous. By using (2.21), it is easy to see that maps into itself, and for any , we have , for . Next, we define the sequence Then, by using (2.21) and a simple induction, we can easily see that Set , then satisfies: Again, set , then satisfies , and Thus, is a positive solution of (1.1) for .
Consider the following: Assume that there exists , such that , for , and . Then, which implies which contradicts . Thus, is an eventually positive solution of equation.
Case 2 (). By using (2.17) and integrating (1.2) from to , we obtain Let be such that (2.17) and hold. Integrating (2.31) from to and using (2.17), we have Using a method similar to the proof of Case 1 yields that (1.1) also has an eventually positive solution.
The proof is complete.

Theorem 2.3. Assume all conditions of Lemma 1.1 hold, is a positive even integer, and, for sufficiently large .Then, (1.1) has an eventually positive solution if and only if inequality (1.2) has an eventually positive solution.

Proof. It is clear that an eventually bounded positive solution of (1.1) is also an eventually bounded positive solution of (1.2). So, it suffices to prove that if (1.2) has an eventually bounded positive solution , for , then so does (1.1). Set It follows from Lemma 1.1 and (1.2) that eventually, By using (1.9) and integrating (1.2) from to , we obtain By repeating the same procedure times and by using (1.9), we are led to the inequality: Using Tonelli′s theorem, we reverse the order of integration and obtain That is, Let be such that (2.38) hold, and . Now, we consider the set of functions: and define an operator on as follows: Then, it follows from Lebesgue′s dominated convergence theorem that is continuous. By using (2.38), it is easy to see that maps into itself and, for any , we have , for . Next, we define the sequence : Then, by using (2.38) and a simple induction, we can easily see that Set , then satisfies Again, set , then satisfies , and Thus, is a positive solution of (1.1) for .
Consider the following Assume that there exists , such that , for , and . Then, which implies which contradicts . Thus, is an eventually positive solution of (1.1).
The proof is complete.

Example 2.4. Consider high-order neutral differential equation with distributed delay Here, . It is easy to see that From Theorem 2.2, we have that (1.1) has an eventually positive solution if and only if inequality (1.2) has an eventually positive solution. In fact, is a positive solution of (1.1).

Acknowledgments

This research is supported by the Natural Sciences Foundation of Shanxi Province and Scientific Research Project of Shanxi Datong University. The authors are greatly indebted to the referees for their valuable suggestions and comments.

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