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

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.

1. Introduction

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

(1.1)

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

(1.2)

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

(1.3)

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

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

(1.4)

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

Theorem B [see [5, 7]]

If , then (1.3) has a bounded nonoscillatory solution if and only if

(1.5)

Theorem C [see [5]]

If , then all solutions of (1.3) are oscillatory if and only if

(1.6)

Theorem D [see [7]]

If , then (1.3) is oscillatory if and only if

(1.7)

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

(1.8)

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

Theorem E

If satisfies (i) and for and (ii) is continuous and non-negative, then (1.8) has a bounded and eventually monotonic solution if and only if

(1.9)

Theorem F

Suppose that the conditions (i) and (ii) in Theorem E are satisfied. If

(1.10)

for some , then all solutions of (1.8) are oscillatory if and only if

(1.11)

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.

Lemma 2.1 (see [13–15]).

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.

If is a strongly sublinear function, then

(2.1)

for and .

Proof.

From and together with Definition 1.2 we clearly see that

(2.2)

where . From we know that , and therefore

(2.3)

Our main result is Theorem 2.3.

Theorem 2.3.

The following statements are true.

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

(2.4)

for some constants , then (1.1) has a bounded nonoscillatory solution.

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

(2.5)

for any .

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

(2.6)

for each .

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

(2.7)

for any .

Proof.

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

(2.8)

for and some .

Observing that if satisfies the equation

(2.9)

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

Consider the functional set

(2.10)

Define the operator as follows:

(2.11)

Then we have

(2.12)

Clearly, we have , and therefore .

Now, we define the functions as follows:

(2.13)

where

(2.14)

Since the function is nondecreasing with respect to and , a straightforward verification shows the validity of the inequalities

(2.15)

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

It is easy to see that is the desired bounded and nonoscillatory solution of  (2.9)

• (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

(2.16)

There are two possible cases.

Case 1 ().

In this case we see that

(2.17)

Since is an increasing function, hence for and some constants , one has

(2.18)

Making use of the Taylor expansion we get

(2.19)

From (1.1) and (2.17) together with (2.19) we get

(2.20)

The strong superlinearity of leads to

(2.21)

which implies

(2.22)

From (2.22) we have

(2.23)

By using the elementary inequality for , we have

(2.24)

Therefore, we get

(2.25)

(2.26)

or

(2.27)

which contradicts with (2.5).

Case 2 ().

Making use of (2.21) we have

(2.28)

For , it follows from (iii) of Lemma 2.1 that

(2.29)

For sufficiently large , one has

(2.30)

Let , then

(2.31)

and therefore

(2.32)

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

(2.33)

that is

(2.34)

which contradicts with (2.5).

Conversely, if every solution of (1.1) oscillates, then (2.5) holds. Otherwise (2.4) holds. Theorem 2.3(a) implies that (1.1) has a nonoscillatory solution.

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

Case 1 ().

The same argument as in the proof of Theorem 2.3(b) implies that inequality (2.26) holds for , that is,

(2.35)

which contradicts with (2.6).

Case 2 ().

From the proof of Theorem 2.3(b) we also clearly see that

(2.36)

which contradicts with (2.6).

Conversely, if every bounded solution of (1.1) oscillates, and then (2.6) holds. Otherwise (2.4) holds, then Theorem 2.3(a) implies that (1.1) has a nonoscillatory bounded solution.

• (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 ().

In this case we see that

(2.37)

then we know that

(2.38)

and there exist constants and such that and for The strong sublinearity of implies that

(2.39)

The same argument as in the proof of Case  1 of Theorem 2.3(b) yields

(2.40)

Integrating from to leads to

(2.41)

That is

(2.42)

Let

(2.43)

then , and

(2.44)

and for , one has

(2.45)

Therefore

(2.46)

or

(2.47)

By condition (H2), we can choose such that and for . Then making use of Lemma 2.2, we have

(2.48)

From (2.47) and (2.48) together with we get

(2.49)

which contradicts with (2.7).

Case 2 ().

That is,

(2.50)

From and for we know that

(2.51)

and there exist constants and such that and for The strong sublinearity of leads to

(2.52)

It follows from (iii) of Lemma 2.1, that

(2.53)

and thus

(2.54)

Let , then , , and

(2.55)

where is also even. According to the same process as the one used in the proof of Case  1 of Theorem 2.3(d) we conclude that

(2.56)

By condition (H2), we can choose such that and for . Now making use of Lemma 2.2, we have

(2.57)

From (2.56) and (2.57) together with we clearly see that

(2.58)

which contradicts with (2.7).

Necessity.

If every solution of (1.1) oscillates, then (2.7) holds. Otherwise, assuming that

(2.59)

for some constants , we should prove that (1.1) has a nonoscillatory solution. From (2.59) we know that there exist and some such that

(2.60)

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

(2.61)

and let be the subset of defined by

(2.62)

Define the mapping on by

(2.63)

where the integration is times.

By Lemma 2.2, for one has

(2.64)

for sufficient large , that is,

(2.65)

From (2.60) and (2.65) we get

(2.66)

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.

If is even, then the following statements are true.

• (a) If

(2.67)

then (1.2) has a bounded nonoscillatory solution.

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

(2.68)

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

(2.69)

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

(2.70)

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.

If , then the following statements are true.

• (a) If

(2.71)

then (1.3) has a bounded nonoscillatory solution.

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

(2.72)

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

(2.73)

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

(2.74)

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.

If , then (1.1) becomes

(2.75)

From the proof of Theorem 2.3(b) we indicate that the strongly superlinearity of can be replaced by the condition

(2.76)

In fact, if is a nonoscillatory solution of (2.75), then from Theorem 2.3(a) we may assume that is unbounded, and (2.76) implies that , and there exists such that and for . Then we get

(2.77)

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 all solutions of (2.75) oscillate if and only if

(2.78)

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

Example 2.9.

The equation

(2.79)

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.

The equation

(2.80)

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