Abstract

We study necessary and sufficient conditions for the oscillation of the third-order nonlinear ordinary differential equation with damping term and deviating argument . Motivated by the work of Kiguradze (1992), the existence and asymptotic properties of nonoscillatory solutions are investigated in case when the differential operator is oscillatory.

1. Introduction

The aim of this paper is to investigate the third order nonlinear functional differential equation with deviating argument

The following assumptions will be made. are continuous functions for , , , , and is a continuous function, such that

In this paper we will restrict our attention to solutions of (1.1) which are defined in a neighborhood of infinity and for any of this neighborhood. As usual, a solution of (1.1) is said to be oscillatory if it has a sequence of zeros converging to infinity; otherwise it is said to be nonoscillatory.

Throughout the paper we assume that the operator is oscillatory, that is, the second-order equation

is oscillatory.

It is well known, see, for example, [1], that if (1.3) is nonoscillatory, then (1.1) can be written as a two-term equation of the form

where are continuous positive functions for

Asymptotic properties of equations of type (1.4) have been widely investigated in the literature. We refer to [17] in case when (), that is, the disconjugate differential operator is in the so-called canonical form [5, 8, 9] when this property does not occur. Some of these results extend the pioneering works [10, 11], devoted to the equation

where is the quotient of odd positive integers. Other contributions deal with the solvability of certain boundary value problems associated to equations of type (1.1) on compact or noncompact intervals see, for example, [12, 13] or [14, 15], respectively, and references therein.

Recently, oscillation criteria for (1.4) with damping term, that is, for

have been presented in [9] by using a generalized Riccati transformation and an integral averaging technique. Here oscillation means that any solution of this equation is oscillatory or satisfies . Several examples [9, Examples  1–5] concern the case when the second-order equation (1.3) is nonoscillatory and so such an equation can be reduced to a two-term equation of the form (1.4).

If the differential operator is oscillatory, then very little is known. According to Kiguradze [16], we say that (1.1) has property A if each of its solutions either is oscillatory, or satisfies the condition

If any solution of (1.1) is either oscillatory, or satisfies the condition (1.7), or admits the asymptotic representation

where and are constants, the continuous functions vanish at infinity and satisfies the inequality for large , then we say that (1.1) has weak property A.

For the results in [16] deal with the equation

Theorem 1.1 (see [16, Theorem  1.5]). Let be a nondecreasing function satisfying Then the condition is necessary and sufficient in order that (1.9) has weak property A.

Theorem 1.2 (see [16, Corollary  1.5]). Let for some and Then (1.9) has property A.

In our previous paper [1] we have investigated (1.1) without deviating argument (i.e., ), especially when (1.3) is nonoscillatory. More precisely, the nonexistence of possible types of nonoscillatory solutions is examined, independently on the oscillation of (1.3).

Motivated by [1, 16], here we continue such a study, by giving necessary and sufficient conditions in order that all solutions of (1.1) are either oscillatory or satisfy . The property A for (1.1) is also considered and an extension to (1.1) of Theorem 1.1 is presented.

The role of the deviating argument and some phenomena for (1.1), which do not occur when (1.3) is nonoscillatory, are presented. Our results depend on a a priori classification of nonoscillatory solutions which is based on the concept of phase function [17] and on a suitable energy function. A fixed point method is also employed and sharp upper and lower estimates for bounded nonoscillatory solutions of (1.1) are established by means of a suitable “cut” function. This approach enables us to assume instead of where

2. Classification of Nonoscillatory Solutions

A function defined in a neighborhood of infinity, is said to change its sign, if there exists a sequence such that .

The following theorem shows the possible types of nonoscillatory solutions for (1.1). It is worth noting that here can change sign.

Theorem 2.1. Any nonoscillatory solution of (1.1) either satisfies or

Proof. Without loss of generality suppose that there exists a solution of (1.1) and such that , , on .
O. Boruvka [17] proved that if (1.3) is oscillatory, then there exists a continuously differentiable function , called a phase function, such that and
Using this result, we can consider the change of variables for , , . Thus, and Substituting into (1.1), we obtain From here and (2.3) we obtain Because , we have for large Since , (2.4) yields and so , that is, is decreasing. If there exists such that , becomes eventually negative, which is a contradiction. Then and is nondecreasing. Let be such that on . Thus, using (2.8) we obtain Hence, , which contradicts the positivity of . Finally, the case on cannot occur, because, if on , then which is a contradiction.

Remark 2.2. Theorem 2.1 extends [1, Proposition  2] for (1.1) with and improves [11, Theorem  3.2] for (1.5).

The following lemma is similar to [16, Lemma  2.2].

Lemma 2.3. Any solution of (1.1) satisfies

Proof. In view of Theorem 2.1, it is sufficient to prove (2.10) and (2.11) for solutions of (1.1) such that for large If (2.10) does not hold, then, in view of Theorem 2.1, a positive constant exists such that for large and, hence, , a contradiction.
Now let us prove (2.11) and, without loss of generality, assume , for . Suppose, for the sake of contradiction, that . Then there exists such that either
If , then , which is a contradiction with Theorem 2.1. Thus on . From this and the Taylor theorem we obtain which gives , a contradiction to the positivity of and so (2.11) holds.

Lemma 2.4. Let for large . If is a solution of (1.1) such that for large and the function is nonincreasing for large , then

Proof. By contradiction, assume Then for large If there exists such that , then for large , which is a contradiction. Thus for large , which is again a contradiction to Theorem 2.1.

In view of Theorem 2.1, any nonoscillatory solution of (1.1) is one of the following types.

Type I: satisfies for large

Type II: satisfies for large

Type III: satisfies for large

Remark 2.5. Nonoscillatory solution of (1.1), such that changes its sign, is usually called weakly oscillatory solution. Note that weakly oscillatory solutions can be either of Type II or Type III. When in [1, Theorem  6] conditions are given under which (1.1) does not have weakly oscillatory solutions, especially solutions of Type III. This result will be used later for proving that the only nonoscillatory solutions of (1.1) are of Type I.

3. Necessary Condition for Oscillation

Our main result here deals with the existence of solutions of Type II.

Theorem 3.1. Assume is continuously differentiable and is bounded away from zero, that is, If then for any there exists a solution of (1.1) satisfying

Proof. We prove the existence of solutions of (1.1) satisfying (3.3) for .
Let and be two linearly independent solutions of (1.3) with Wronskian . By assumptions on , all solutions of (1.3) and their derivatives are bounded; see, for example, [18, Theorem  2]. Put
and denote Thus, there exists such that for any Hence Let be large so that Let be such that for . Define Denote by the Fréchet space of all continuous functions on endowed with the topology of uniform convergence on compact subintervals of . Consider the set given by Let be fixed and let . For any consider the “cut” function Then Consider the function Then and Integrating from to we have From here and (3.11) we get Thus, in view of (3.5), we obtain or, in view of (3.6), that is, Hence that is, In view of (3.19), using the Cauchy criterion, the limit exists finitely for any fixed This fact means that the operator is well defined for any . Moreover, from (3.19) we have for and so, in view of (3.6), maps into itself.
Let us show that is relatively compact, that is, consists of functions which are equibounded and equicontinuous on every compact interval of Because the equiboundedness follows. Moreover, for any we have
and so which proves the equicontinuity of the elements of .
Now we prove the continuity of on Let be a sequence in which converges uniformly on every compact interval of to . Because is relatively compact, the sequence admits a subsequence, denoted again by for sake of simplicity, which is convergent to From (3.4) we obtain
where is defined in (1.13). Hence, in virtue of the Lebesgue dominated convergence theorem, converges pointwise to on that is, Choosing a sufficiently large and using (3.22), we obtain Then, from (3.26), converges point-wise to and, in view of the uniqueness of the limit, is the only cluster point of the compact sequence , that is, the continuity of in . Applying the Tychonov fixed point theorem, there exists a solution of the integral equation which is a solution of (1.1) with the required properties.

Remark 3.2. Theorem 3.1 partially extends [16, Theorem  1.4] for .

4. Sufficient Condition for Oscillation

In this section we give a sufficient condition for oscillation of (1.1) in the sense that any solution is either oscillatory or satisfies .

Theorem 4.1. Assume and continuously differentiable for large satisfying If then any (nonoscillatory) solution of (1.1) satisfies Moreover, any nonoscillatory solution with for large satisfies for

Proof. To prove the first assertion, it is sufficient to show that (1.1) does not have solutions of Type II. By contradiction, let be a solution of (1.1) such that for .
Consider the function given by (2.14). Then
By Lemma 2.4, we have . Since we get Consequently, we have from (4.5) and, as , we get a contradiction. Hence, any nonoscillatory solution satisfies (4.4).
Now let be a solution of (1.1) such that    for . Hence is of Type I and
Because is bounded and is nonincreasing, (2.11) and (4.9) yield and so . Consider the function defined by Then we have for Since is decreasing and is bounded, in view of (2.10) we obtain and so .

Remark 4.2. Theorem 4.1generalizes Theorem  1.1. In [11, Theorem  3.8] the first part of Theorem 4.1 is proved, by a different method, for the particular (1.5) under assumptions (4.2) and (4.3).

Applying Theorems 3.1 and 4.1 we get the following result which extendsTheorem  1.1for (1.9).

Corollary 4.3. Assume (4.1) and that is continuously differentiable for large satisfying (4.2) then, condition (4.3) is necessary and sufficient in order to every solution of (1.1) is either oscillatory or satisfies (4.4).

From Theorem 4.1 and its proof, we have the following results.

Corollary 4.4. Assume that is continuously differentiable for large satisfying (4.2). If (4.1) and (4.3) are satisfied, then for any nonoscillatory solution of (1.1) one has where is defined by (4.11).
In addition, if , then is a nonoscillatory solution of (1.1) if and only if for large .

Proof. By Theorem 4.1, any nonoscillatory solution of (1.1) is either of Type I or Type III. If is of Type I, then for and so Hence, by using the argument in the proof of Theorem 4.1, we obtain for large
If is of Type III, then there exists a sequence of zeros of tending to such that and so . Similarly, reasoning as in the proof of Theorem 4.1, one has that for this solution. Thus, in both cases, the monotonicity of gives (4.13).
Finally, if and is an oscillatory solution, then for any and so for large . Since for some sequence , we get for large .

Theorem 4.5. Let and is continuously differentiable for satisfying (4.2). If (4.1) and (4.3) are satisfied, for any nonoscillatory solution of (1.1) defined for , one has In particular, any continuable solution with zero is oscillatory.

Proof. Assume that for and . Since , we have for . For the function defined by (4.11), from (4.12) we obtain for Because we obtain for This is a contradiction with Corollary 4.4 and the assertion follows.

We conclude this section with the following result on the continuability of solutions of (1.1).

Proposition 4.6. Assume is continuously differentiable for satisfying (4.2). Then any solution of (1.1) which is not continuable at infinity has an infinite number of zeros.

Proof. Let be a solution of (1.1) defined on . If has a finite number of zeros on , then there exists such that , on . Suppose on . Consider the function given by (2.14). Then .
Now consider these two cases. (a) Let in the left neighborhood of or changes sign in this neighborhood. Then for . Integrating this inequality on we obtain
Thus and is bounded on . From here and the boundedness of we have that is bounded on . Since () are bounded, the solution can be extended beyond , which is a contradiction.
(b) Let in the left neighborhood of , say . Then is decreasing and is bounded on . We claim that does not tend to as . Indeed, if , then there exists a sequence such that , which is a contradiction with the boundedness of . Thus is bounded and so is bounded on and is continuable, which is again a contradiction.

Remark 4.7. Theorem 4.5 and Proposition 4.6 improve [11, Corollary  3.4] and [11, Theorem  1.2] for (1.5), respectively.

5. Property A

In this section we study Property A for (1.1), that is, any solution either is oscillatory or tends to zero as .

We start with the following result on the boundedness of nonoscillatory solutions, which extends [11, Theorem  3.12] for (1.5).

Theorem 5.1. Assume that is continuously differentiable for large and there exists a constant such that Then every nonoscillatory solution of (1.1) satisfies

Proof. Without loss of generality, assume that is a solution of (1.1) such that , for . Consider the function defined by (2.14). Then that is, is nonincreasing. By Lemma 2.4, . Moreover, from (5.3) we have or Let be two linearly independent solutions of (1.3) with Wronskian . Because all solutions of (1.3) together with their derivatives are bounded, see, for example, [19, Ch.  XIV, Theorem  3.1], from (5.5) we get Moreover, from the equality using the variation of constants formula, we have where and are real constants. Thus is bounded. Moreover, Since is bounded, are bounded, and (5.5) holds, is bounded, too. From here and (2.14) the boundedness of follows.

The next result describes the asymptotic properties of nonoscillatory solutions and will be used later.

Theorem 5.2. Assume (4.1), (4.3), and is continuously differentiable satisfying (5.1). If then any nonoscillatory solution of (1.1) defined on satisfies If, moreover, then satisfies

Proof. Theorem 4.1 yields that is not of Type II. Then there exists such that for either or
By Theorem 5.1, and are bounded, that is, there exists such that for . By Corollary 4.4, the function given by (4.11), is positive for large say and has a finite limit as Moreover
From the boundedness of we get and so Hence, in view of (5.11) and (5.17), we obtain (5.12).
In order to complete the proof, define for
Then for and so is increasing. If satisfies either (5.15) or (5.16), there exists a sequence tending to such that Moreover, Now, in view of (5.20) and (5.21), we obtain and so (5.14) is satisfied.

Using the previous results, we obtain a sufficient condition for property A.

Theorem 5.3. Assume (4.1), (5.10), (5.11), and is continuously differentiable satisfying (5.1). If there exists such that then (1.1) has property A, that is, any nonoscillatory solution of (1.1) satisfies

Proof. Let be a nonoscillatory solution of (1.1) defined for . Because (5.24) implies (4.3), by Theorem 4.1, is of Type I or Type III. If is of Type I, the assertion follows applying again Theorem 4.1. Now let be of Type III and assume for By applying Theorem 5.2, in view of (5.24), we obtain According to the inequality of Nagy, see, for example, [20, V, 2, Theorem  1], we get Now consider the function given by (2.14). Then for . For a sequence of zeros of tending to we have Hence, and By Corollary 4.4, the function given by (4.11), is positive decreasing for large From here, (2.10), (5.27), and (5.28) we obtain .

Corollary 5.4. Suppose assumptions of Theorem 5.3, and for some then any nonoscillatory solution of (1.1) satisfies for large .

Proof. The assertion follows from Theorem 5.3 and [1, Theorem  6].

Acknowledgments

The first and third authors are supported by the Research Project 0021622409 of the Ministry of Education of the Czech Republic and Grant 201/08/0469 of the Czech Grant Agency. The fourth author is supported by the Research Project PRIN07-Area 01, no. 37 of the Italian Ministry of Education.