All solutions of a fourth-order nonlinear delay differential equation are shown to converge to zero or to oscillate. Novel Riccati type techniques involving third-order linear differential equations are employed. Implications in the deflection of elastic horizontal beams are also indicated.

1. Introduction

It has been observed that the solutions of quite a few higher order functional differential equations oscillate or converge to zero (see, e.g., the recent paper [1]). Such a dichotomy may sometimes yield useful information in real problems (see the remark at the end of this note). In this paper, we report that the same phenomena occurs for fourth-order nonlinear delay differential equations of the form


(i) is a nonnegative constant, is a nonnegative function in such that it does not vanish identically on any (ii) is a positive function in (iii) and for and (iv) and for .

As is well known, fourth-order differential equations can be used to model the deflection of a beam (in particular, the function in (1.1) may be interpreted as the deflection from the equilibrium position of a horizontal beam at the spatial coordinate ). Therefore, a short note reporting our observation may be of interest in the qualitative theory of fourth-order differential equations. Indeed, in this note, we will distinct ourselves by emphasizing the role played by the function in (1.1), since the term acts as a control of the slope of the beam under consideration at the coordinate .

We will restrict our attention to those solutions of (1.1) which exist on and satisfy the condition for any . Such a solution is called oscillatory if it has arbitrarily large zeros, and nonoscillatory otherwise.

Lemma 1.1. Assume that such that for and is nonpositive for and does not vanish identically on any . If is even (or odd), then there exists (resp., ) such that for all sufficiently large for and for Furthermore, for all sufficiently large where satisfies and

Proof. The existence of in the above result is due to Kiguradze and is well known (see, e.g., [2]). Next, we will prove the inequality (1.2) for even (the odd case being similar) as follows (cf. [3]). First we may assume that there is a such that for By Taylor's formula, we have where Furthermore, since for Next, we pick such that for Then and By Taylor's formula again, we get here Hence Combining (1.7) and (1.5), we see that for all large

Lemma 1.2. Suppose the linear third-order differential equation has an eventually positive increasing solution on and is a nonoscillatory solution of (1.1) on Then there exists such that or for .

Proof. Without loss of generality, we may assume that and for . Then is a solution of the third-order delay differential equation We claim that, all solutions of (1.10) are nonoscillatory (so that is eventually positive or eventually negative). To see this, let be the solution of (1.9) such that and for Then from Lemma 1.1, we have for greater than or equal to a positive number
Suppose to the contrary that is an oscillatory solution of (1.10). We assert that oscillates. Indeed, since is oscillatory, for any there are and such that , and . Hence and . If or then either or is a zero of in . Otherwise, and so that there is such that These show that has arbitrarily large zeros.
Now that is oscillatory, either for all or else there are strictly increasing and divergent sequences and such that for and
Suppose the former case holds. Then there are such that for and and From (1.9) and (1.10), we have or, Hence we have which is a contradiction.
Suppose the latter case holds. From (1.9) and (1.10), we have from which by we obtain Since we see that which is a contradiction.

2. Asymptotic Dichotomy

We are now ready to state and prove our main result.

Theorem 2.1. Suppose that(i)equation (1.9) has an eventually positive increasing solution on (ii)there is a differentiable function such that for every (iii) Then every solution of (1.1) either oscillates or converges to .

Proof. Let be a solution of (1.1) which does not converge to Suppose to the contrary that is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that and for Furthermore, by Lemma 1.2, there is such that or for
Suppose for Then by (1.1), By Lemma 1.1, we may suppose further that and (1.2) holds for as well. Let Then by (1.1) and (1.2), we see that Hence for all which is contrary to our assumption (2.1).
Suppose for Since is eventually positive, eventually decreasing, and does not converge to , we easily see that (i) cannot be eventually nonpositive, (ii) (iii) and (iv) there is such that for
In view of (i) above, either is oscillatory or is eventually nonnegative. If is oscillatory, then is also oscillatory. Hence there exists some number such that and for From (1.1) and (2.2), we have It follows that Since the right-hand side tends to as tends to is eventually negative which is contrary to our assumption on .
As a consequence, we must have ( and) say for There are three cases to consider:
Case 1. If and does not vanish identically for , then for all large , a contradiction. If for , then is a linear function. Since , we get for , which is contrary to our assumption that for Case 2. If for , then from (1.1), we get which is contrary to (2.2).Case 3. If oscillates, then (2.8) holds so that is eventually negative, which is contrary to our assumption on This completes the proof.

3. Remarks and Example

We remark that the condition (2.2) in the above result can, in case is differentiable, be replaced by the alternate condition

for every Indeed, we only need to note that

where or and then we may follow the rest of the above arguments.

Example 3.1. Consider the fourth-order delay differential equation Here, and with . It is clear that (2.2) is satisfied. The equation has an eventually positive and eventually increasing solution . By choosing and any large we have Consequently, condition (2.1) is satisfied. Hence by Theorem 2.1, any solution of (3.3) is oscillatory or satisfies .

As our final remark, suppose we have a (sufficiently) long elastic horizontal beam modeled by (1.1). Our Theorem says that (under appropriate conditions) if the right end of the beam is fixed at some nonzero vertical position, then there must be places where there are positive as well as negative displacements of the beam from its equilibrium position. This shows that the dichotomy in our Theorem has implications in the elastic beam deflection problem.