- Research Article
- Open Access

# Sufficient and Necessary Conditions for Oscillation of th-Order Differential Equation with Retarded Argument

- Jin-fa Cheng
^{1}and - Yu-ming Chu
^{2}Email author

**2009**:892936

https://doi.org/10.1155/2009/892936

© J.-f. Cheng and Y.-m. Chu. 2009

**Received:**10 July 2009**Accepted:**9 December 2009**Published:**14 December 2009

## Abstract

Necessary and sufficient conditions are found for oscillation of the solutions of a class of strongly superlinear and strongly sublinear differential equations of even order with retarded argument.

## Keywords

- Taylor Expansion
- Oscillatory Behavior
- Nonlinear Differential Equation
- Functional Differential Equation
- Bounded Solution

## 1. Introduction

We consider the following th-order differential equation with retarded argument:

Firstly, we introduce several conditions as follows:

() , for and .

() , for and .

As customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise the solution is called nonoscillatory.

Definition 1.1.

The function is said to be strongly superlinear if there exists , such that is a nondecreasing function with respect to for each fixed

It is easy to see that the function is nondecreasing with respect to for if is strongly superlinear. The function is nondecreasing with respect to for if is nondecreasing with respect to .

Definition 1.2.

The function is said to be strongly sublinear if there exists , such that is a nonincreasing function with respect to for each fixed

We should indicate that there are many ways in which one can define the concept of strongly superlinearity, superlinearity, strongly sublinearity and sublinearity, to characterize functions satisfying different conditions. For example, in [1] the strongly superlinearity is used to specify functions with specific behavior at 0 and ; in [2] the superlinearity and sublinearity are defined for multivariable functions. In this paper, we adopt the definitions as in monograph [3].

In particular, if where , and is the quotient of odd positive integers, then (1.1) becomes

It is easy to see that is strongly superlinear for and is strongly sublinear for . If ; then (1.2) reduces to

Equation (1.3) is the well-known Emden-Fowler equation [4].

Recently, many remarkable results have been established for the oscillation of solutions of the second- and higher-order functional differential equations. For example, Theorem A is presented in [2].

Theorem A

For (1.3), the well-known Theorems B–D are presented in [5–7].

Theorem C [see [5]]

Theorem D [see [7]]

In [8], Waltman studied the oscillation of the solutions for the equation

Equation (1.8) is the prototype of (1.1) and (1.2). Theorems E and F were proved in [8].

Theorem E

Theorem F

Some other related results can be found in [2, 4, 9–12] and the references cited therein. Due to some problems of theoretical and technical character in handling with higher-order nonlinear differential equations, there are only a few results which concern necessary and sufficient conditions for the oscillatory behavior for (1.1). So there are a lot of things worth further consideration for (1.1). The main purpose of this paper is to establish necessary and sufficient conditions for (1.1). The obtained results extend the above theorems.

## 2. Main Results

In order to establish our main results we need introduce and establish two lemmas.

If is a positive and -times differentiable function on , and is nonpositive and not identically zero on any subinterval , then there exist and an integer such that is odd and

(i) for , ,

(ii) for

(iii) for , ,

Lemma 2.2.

for and .

Proof.

Our main result is Theorem 2.3.

Theorem 2.3.

- (a)
Suppose that is a nondecreasing function with respect to and for . If

- (b)
If is a strongly superlinear function, then every solution of (1.1) oscillates if and only if

- (c)
If is a nondecreasing function with respect to and , then every bounded solution of (1.1) oscillates if and only if

- (d)
If is a strongly sublinear function, then every solution of (1.1) oscillates if and only if

for any .

- (a)
Assume that (2.4) holds. Choose sufficiently large such that

for and some .

Observing that if satisfies the equation

then is a solution of (1.1). Therefore it suffices to show that (2.9) has a bounded nonoscillatory solution.

Consider the functional set

Clearly, we have , and therefore .

Now, we define the functions as follows:

Therefore for It follows from the Lebesgue convergence theorem that and .

- (b)
Sufficiency. Assume that for each . We will prove that every solution of (1.1) oscillates. Otherwise, assume that (1.1) has a nonoscillatory solution . Without loss of generality, assume that for . Then according to Lemma 2.1, there exists an odd integer and such that

There are two possible cases.

Case 1 ( ).

which contradicts with (2.5).

Case 2 ( ).

Using the same method as in the proof of Case 1, we get

which contradicts with (2.5).

- (c)
Sufficiency. Without loss of generality, we assume that is a bounded positive solution. We divided the proof into two cases.

Case 1 ( ).

which contradicts with (2.6).

Case 2 ( ).

which contradicts with (2.6).

- (d)
Sufficiency. Without loss of generality, we assume that is a finally positive solution, that is, for . We consider the following two cases.

Case 1 ( ).

which contradicts with (2.7).

Case 2 ( ).

It follows from (iii) of Lemma 2.1, that

which contradicts with (2.7).

Necessity.

Let be the Banach space of all real-valued continuous functions endowed with the norm

Define the mapping on by

where the integration is times.

By Lemma 2.2, for one has

Equation (2.66) and the definition of the operator imply that . On the other hand, we clearly see that for . Therefore, .

It is routine to prove that is continuous and is relatively compact in the topology of the Frechet space . Therefore, there exists such that follows from the well-known Schauder's fixed point Theorem. It is easy to see that is the solution of (1.1).

The proof of Theorem 2.3 is completed.

Remark 2.4.

If , then and . For (1.2) we can derive Corollary 2.5 from Theorem 2.3.

Corollary 2.5.

- (a)
If

- (b)
If , then every solution of (1.2) oscillates if and only if

(c) If , then every bounded solution of (1.2) oscillates if and only if

(d) If , then every solution of (1.2) oscillates if and only if

It is easy to see that Theorem A can be obtained directly from our Corollary 2.5(c).

For , we have Corollary 2.6 for (1.3).

Corollary 2.6.

- (a)
If

- (b)
If , then every solution of (1.3) oscillates if and only if

(c) If , then every bounded solution of (1.3) oscillates if and only if

(d) If , then every solution of (1.3) oscillates if and only if

We clearly see that our results in Corollary 2.6(a), (b), and (d) are exactly corresponding to the results in Theorems B, C, and D, respectively.

Remark 2.7.

We notice that if (2.21) is replaced by (2.77), then Corollary 2.8 follows from the proof of Theorem 2.3(b).

Corollary 2.8.

If , then one clearly sees that Theorem F is the special case of Corollary 2.8.

Example 2.9.

satisfies the assumptions of Theorem 2.3(a) but does not satisfy the assumptions of Theorem 2.3(b) and (c); hence there exists a bounded nonoscillatory solution. In fact is one such solution.

Example 2.10.

satisfies the assumptions of Theorem 2.3(d). Hence every solution of (1.1) is oscillatory. In fact is one such solution.

## Declarations

### Acknowledgments

The authors wish to thank the anonymous referees for the very careful reading of the manuscript and fruitful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 60850005) and the Natural Science Foundation of Zhejiang Province (Grant nos. D7080080 and Y607128).

## Authors’ Affiliations

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