# Oscillation for higher order differential equations with a middle term

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## Abstract

We study the existence of bounded oscillatory solutions for a higher order differential equation, considered as a perturbation of an associated linear equation. Jointly with this, we study the nonexistence of solutions vanishing at infinity and, as an application, we obtain in the linear case an asymptotic equivalency criterion.

## Keywords

Nontrivial Solution Force Term Independent Solution Middle Term Oscillatory Solution## 1 Introduction

Hence, *q* is bounded from above and ${lim}_{t\to \mathrm{\infty}}q(t)={q}_{\mathrm{\infty}}<\mathrm{\infty}$. Note that the function *r* may change sign.

By a solution of (1) we mean a function *x* differentiable up to order *n* which satisfies (1) on $[{T}_{x},\mathrm{\infty})$, ${T}_{x}\ge 0$. A solution of (1) is said to be *proper* if $sup\{|x(t)|:t\ge T\}>0$ for any $T\ge {T}_{x}$. As usual, a solution *x* of (1) is said to be *oscillatory* if *x* changes sign for large *t*.

*i.e.*, for the equation

*r*is positive and sufficient large in some sense, then for

*n*even every proper solution of (4) is oscillatory. Moreover, for

*n*odd every proper solution of (4) is oscillatory, or is vanishing at infinity together with its derivatives, or admits the asymptotic representation

where *c*, *φ* are suitable constants and *ε* is a continuous function for $t\ge 0$ which vanishes at infinity. According to [1], in this case (4) is said to have *property A*′.

The existence of solutions vanishing at infinity, and the closely related problem on existence of bounded oscillatory solutions, has attracted the attention in many papers, see, *e.g.* the monograph [2] and references therein. In particular, nonlinear equations with middle term have been investigated in many directions, as a perturbation of an associated linear equation. For third and fourth order equations, we refer especially to [3, 4, 5], in which the property A, or its generalizations, has been studied and to [6, 7, 8, 9, 10] for oscillation problems. In particular, in [5] a good and detailed discussion of known oscillation criteria is given as well.

has been considered. In [11] sufficient conditions are obtained for the existence of solutions, which are equivalent to a polynomial. In [12] a criterion is given for existence of nonoscillatory solutions with non-zero limit at infinity. Moreover, for *n* even, an oscillation result is obtained too. In [13], (1) is studied as a perturbation of (2), under the assumption (H). Finally, the case of higher order equations with forcing term has been considered in the recent papers [14, 15, 16], see also the last section.

Observe that if *q* is a positive constant, then (2) has oscillatory and bounded solutions not vanishing at infinity. If *q* is not constant and (H) is satisfied, then these properties remain to hold for the second order equation (3), see, *e.g.* [[17], Chapter 6], or [[18], Theorem 2]. Thus, it is natural to ask under which assumptions these properties are valid also for (2) (when *q* is not constant) and for the more general case (1).

In this paper, we answer both these questions. In particular, our main result yields the existence of oscillatory solutions of (1), which are bounded and not vanishing at infinity. Our results complete recent ones in [[13], Corollary 1, Corollary 3], extend similar ones in [[1], Theorem 1.3, Theorem 1.4], which are proved for (4), and generalize ones in [3, 10], which are stated for the particular cases $n=3,4$.

In Section 3 we study the problem of the nonexistence of solutions vanishing at infinity of (1). This result will be employed in Section 4 and Section 5, namely to prove the existence of oscillatory solutions of (2) (Section 4) and the uniqueness of solutions of (1), which have the same asymptotics as solutions of (2) (Section 5).

*r*on the change of the oscillatory character, passing from (2) to the linear equation

More precisely, we will give conditions under which (2) and (5) are asymptotically equivalent. Some suggestions for further researches complete the paper.

## 2 Preliminaries

We start with some basic properties of solutions of (3), which will be useful in the sequel. Obviously, assumption $q(t)\ge {q}_{0}>0$ implies that (3) is oscillatory. Moreover, the following holds.

**Lemma 1**

*Let*(H)

*hold and let*

*u*

*be a nontrivial solution of*(3).

*Then*

*u*

*is oscillatory and*

*Proof* Since *q* is of bounded variation for $t\ge 0$, all solutions of (3) are bounded together with their derivatives, see *e.g.* [[18], Theorem 2]. Moreover, the function *q* can be represented as $q(t)=a(t)/b(t)$, where *a*, *b* are positive nonincreasing and differentiable functions such that ${lim}_{t\to \mathrm{\infty}}a(t)>0$ and ${lim}_{t\to \mathrm{\infty}}b(t)>0$, for details see [[19], Lemma 5.4.1]. Hence, taking ${t}_{0}$ such that ${u}^{\prime}({t}_{0})=0$ and applying [[13], Lemma 2], we get the assertion. □

*u*,

*v*be two linearly independent solutions of (3) with Wronskian equal to 1. Set

Then *w* and *ζ* are bounded in $[0,\mathrm{\infty})\times [0,\mathrm{\infty})$.

*q*is not constant, it is easy to see that a basis of the space of solutions of (2) is given by

and *u*, *v* are two independent solutions of (3). The question whenever the sets (7) and (8) are, roughly speaking, close as $t\to \mathrm{\infty}$ is considered in Section 5.

*M*such that in a neighborhood of infinity

The following holds.

**Theorem A** ([[13], Theorem 1])

*Assume*$n\ge 3$.

*Let for any positive constant*

*μ*

*and for some*$j\in \{0,\dots ,n-3\}$

*Then*,

*for any solution*

*y*

*of*(2)

*such that*$y(t)=O({t}^{j})$

*as*$t\to \mathrm{\infty}$,

*there exists a solution*

*x*

*of*(1)

*such that for large*

*t*

*where* ${\epsilon}_{i}$ *are functions of bounded variation for large* *t* *and* ${lim}_{t\to \mathrm{\infty}}{\epsilon}_{i}(t)=0$, $i=0,\dots ,n-1$.

**Remark 1**If

*f*satisfies in a neighborhood

*I*of zero

for some $K>0$, then every nontrivial solution of (1) is proper. This follows from [[1], Theorem 11.5] with $h(t)={q}_{\mathrm{max}}+K|r(t)|$ and $\omega (u)=u$ for $u\in I$.

## 3 Solutions vanishing at infinity

Our main result in this section is concerned with the nonexistence of solutions vanishing at infinity.

**Theorem 1**

*Let*$n\ge 3$, $f\in {C}^{1}(\mathbb{R})$

*and*

*Then*(1)

*does not have nontrivial solutions*

*x*(

*oscillatory or nonoscillatory*)

*satisfying*

where $F\in C[0,\mathrm{\infty})$.

**Lemma 2**

*Assume*$n\ge 3$.

*Let*$F\in C[0,\mathrm{\infty})$

*satisfy*

*Then*

*where*

*w*

*is defined by*(6),

*is a solution of*(15).

*Moreover*,

*there exists*$M>0$

*such that for*$t\ge 0$

*Proof* If $F(t)\equiv 0$ for large *t*, then the assertion holds. Thus, assume that $F\not\equiv 0$ for large *t*.

*z*is well defined. Choose $\overline{t}$ large so that

*m*be a constant such that $|w(s,\tau )|\le m$, $|\zeta (s,\tau )|\le m$ for $s\ge \tau \ge \overline{t}$. From (16) we have

where $M={\ell}_{2}/(1-{\ell}_{1})(n-3)!$. If $n=3$, then (26) holds by (25).

*α*given in (19). Fix $\tilde{t}\ge \overline{t}$. In virtue of $\alpha (\tilde{t},T)=\alpha (\tilde{t},{T}_{0})+\alpha ({T}_{0},T)$ we have

Hence, *z* is well defined. A direct computation shows that *z* is a solution of (15).

*z*satisfies the estimation (17). Using (26) we have

which yields (17). □

**Remark 2** Under the stronger assumption ${\int}_{0}^{\mathrm{\infty}}{t}^{n-2}|F(t)|\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty}$, Lemma 2 follows with a standard calculation.

**Lemma 3**

*Let*$n\ge 3$

*and*$y\in {C}^{(n-1)}[T,\mathrm{\infty})$, $T\ge 0$,

*be such that*

*Then*

*Proof*Let $b\in (T,\mathrm{\infty})$. Since

*y*and ${y}^{(n-2)}$ are bounded, we can apply [[2], Lemma 5.2] with $m=n-1$ and we obtain

*b*to ∞, there exists a positive constant ${C}_{3}$ such that

which gives the assertion, because ${lim}_{t\to \mathrm{\infty}}y(t)=0$. □

The next auxiliary result is a Gronwall type lemma, which proof is elementary and so it is omitted.

**Lemma 4**

*Let*

*g*

*and*

*Ψ*

*be nonnegative continuous functions for*$t\ge {t}_{0}\ge 0$

*such that*

*Ψ*

*and*

*Ψg*

*belong to*${L}^{1}[{t}_{0},\mathrm{\infty})$.

*If*

*for some nonnegative constant*

*A*,

*then*

*In particular*, *if* $A=0$, *then* $g(t)=0$ *for each* $t\in [{t}_{0},\mathrm{\infty})$.

*Proof of Theorem 1* Assume that there exists a nontrivial solution *x* of (1) defined on $[{T}_{x},\mathrm{\infty})$ and satisfying (14).

*u*,

*v*be linearly independent solutions of (3) with Wronskian 1. By Lemma 1,

*h*is a solution of the second order equation

where ${C}_{1}$, ${C}_{2}$ are suitable constants.

We claim that ${C}_{1}={C}_{2}=0$. Indeed, if ${C}_{1}^{2}+{C}^{2}>0$, then ${C}_{1}u+{C}_{2}v$ is nontrivial solution of (3) and applying Lemma 1 we get a contradiction.

*t*

By Lemma 4, we find that *x* is identically zero for large *t*. Since $f\in {C}^{1}$, (34) and (35) imply (12), by Remark 1 every nontrivial solution *x* of (1) is proper and so *x* is identically zero for $t\ge {T}_{x}$. The proof is complete. □

**Remark 3** If $f(u)={|u|}^{\lambda}sgnu$, $\lambda \ge 1$, then the assumption $f\in {C}^{1}(\mathbb{R})$ in Theorem 1 is satisfied. If $\lambda <1$, then $f\notin {C}^{1}(\mathbb{R})$ and Theorem 1 is not valid. The following example illustrates that assumptions of Theorem 1 are optimal.

**Example 1**Consider the third order equation

where $r(t)=\alpha (\alpha +1)(\alpha +2){(t+1)}^{\alpha \lambda -\alpha -3}+\alpha {(t+1)}^{\alpha \lambda -\alpha -1}$ and $\alpha >0$.

If $\lambda <1$, then ${\int}_{0}^{\mathrm{\infty}}r(t)\phantom{\rule{0.2em}{0ex}}dt<\mathrm{\infty}$, *i.e.* condition (13) is satisfied, however, we have $f\notin {C}^{1}(\mathbb{R})$. If $\lambda \ge 1$, condition (13) is not satisfied.

One can check that in both cases $x(t)={(t+1)}^{-\alpha}$ is a solution vanishing at infinity of (36). This shows the strictness of both assumptions of Theorem 1, that is, $f\in {C}^{1}(\mathbb{R})$ and (13).

## 4 Oscillation in the linear case

In this section we prove the existence of oscillatory solutions of (2), which are bounded and not vanishing at infinity.

**Theorem 2**

*Let*$n\ge 3$,

*u*

*be a nontrivial solution of*(3)

*and*

*Then*(2)

*has an oscillatory solution*

*ϕ*

*such that*

*where*

*ε*

*is a continuous function on*$[0,\mathrm{\infty})$

*and*${lim}_{t\to \mathrm{\infty}}\epsilon (t)=0$.

*In particular*,

To prove this theorem, we give asymptotic expressions of the integrals in (8).

**Lemma 5**

*Let*$n\ge 3$

*and*(37)

*hold*.

*If*

*u*

*is a nontrivial*(

*oscillatory*)

*solution of*(3),

*then there exist constants*${c}_{i}$, $i=0,1,\dots ,n-2$, ${c}_{n-2}\ne 0$,

*and a function*

*ε*

*such that*

*where* *ε* *is a continuous function on* $[0,\mathrm{\infty})$ *and* ${lim}_{t\to \mathrm{\infty}}\epsilon (t)=0$.

*Proof*In view of (H), we have ${lim}_{t\to \mathrm{\infty}}q(t)={q}_{\mathrm{\infty}}$, $0<{q}_{\mathrm{\infty}}<\mathrm{\infty}$. Let

*μ*be an integer, $\mu \ge 1$. Using (3) we get

and ${lim}_{t\to \mathrm{\infty}}{\epsilon}_{0}(t)=0$.

*ν*be an integer, $\nu \ge 1$. Using (3) we have

and ${lim}_{t\to \mathrm{\infty}}{\epsilon}_{1}(t)=0$.

When $n=3$, the assertion follows from (38) with $\mu =1$. Similarly, when $n=4$, the assertion follows from (39) with $\nu =1$.

*j*be an integer and $n-3\ge j\ge 2$. Fixed $k>0$ and integrating by parts, we obtain

in virtue of Lemma 1 and (37), we get ${lim}_{t\to \mathrm{\infty}}{H}_{j}(t)=0$.

*n*be odd, $n\ge 5$. By using recursively (40) with

where ${c}_{i}$ are suitable constants and ${K}_{1}$ is a continuous function on $[0,\mathrm{\infty})$ and ${lim}_{t\to \mathrm{\infty}}{K}_{1}(t)=0$.

Choosing $\mu =(n+1)/2$ in (38), from (41) the assertion follows for *n* odd, $n\ge 5$.

*n*be even, $n\ge 6$. By using a similar argument to the one above given and applying recursively (40) with

where ${\overline{c}}_{i}$ are suitable constants and ${K}_{2}$ is a continuous function on $[0,\mathrm{\infty})$ and ${lim}_{t\to \mathrm{\infty}}{K}_{2}(t)=0$.

Choosing $\nu =(n-2)/2$ in (39), from (42) the assertion follows. □

*Proof of Theorem 2* By Lemma 5, (2) has a solution ${u}^{\prime}(t)+\epsilon (t)$ if *n* is odd, and $u(t)+\epsilon (t)$ if *n* is even, where ${lim}_{t\to \mathrm{\infty}}\epsilon (t)=0$. According to Lemma 1 this solution is oscillatory, bounded, and not vanishing at infinity. □

## 5 Oscillation in the nonlinear case

Our main results are given by the following.

**Theorem 3** *Let* $n\ge 3$ *and* *u* *be a nontrivial solution of* (3).

*Assume*(37)

*and*

*Then for any real numbers*${c}_{0}$, ${c}_{1}$, (1)

*has a solution*

*x*,

*defined on*$[{T}_{x},\mathrm{\infty})$, ${T}_{x}\ge 0$,

*such that*

*where* *ε* *is a continuous function on* $[{T}_{x},\mathrm{\infty})$ *and* ${lim}_{t\to \mathrm{\infty}}\epsilon (t)=0$.

*Consequently*, (1)

*has oscillatory solutions*

*x*

*such that*

*Proof* By Theorem 2, (2) has an oscillatory solution *ϕ*, which is bounded and not vanishing at infinity. Applying Theorem A with $j=0$, (1) has a solution with the same asymptotic properties as that one of (2). □

**Theorem 4**

*Let*$n\ge 3$

*and*

*u*,

*v*

*be two linearly independent solutions of*(3).

*Assume*(37)

*and for any positive constant*

*μ*

*where the function* $\overline{f}$ *is defined in the Preliminaries*.

*Then for any vector*$({c}_{0},{c}_{1},\dots ,{c}_{n-1})\in {\mathbb{R}}^{n}$

*there exists a solution*

*x*

*of*(1),

*defined on*$[{T}_{x},\mathrm{\infty})$, ${T}_{x}\ge 0$,

*such that*

*where* *ε* *is a continuous function on* $[{T}_{x},\mathrm{\infty})$ *and* ${lim}_{t\to \mathrm{\infty}}\epsilon (t)=0$.

*If*,

*in addition*, $f\in {C}^{1}(\mathbb{R})$

*and there exists*$M>0$

*such that*

*then the solution* *x* *given by* (45) *is unique*.

*Proof* *Existence.* As noticed above, the set (8) is a basis for the space of solutions of (2). Applying Theorem A with $j=n-3$ and Theorem 3, (1) has a solution *x* satisfying (45).

*Uniqueness.*For the sake of simplicity, let

*n*be even. The case

*n*odd follows in a similar way. Suppose, by contradiction, that for $({c}_{0},{c}_{1},\dots ,{c}_{n-1})\in {\mathbb{R}}^{n}$ there exist two different solutions

*x*and $\overline{x}$ of (1) satisfying (45). Then for $\overline{z}(t)=x(t)-\overline{x}(t)$ we have

where $R(t)=r(t){f}^{\prime}(\xi (t))$, for $t\ge 1$ and a suitable constant ${M}_{1}>0$.

where ${M}_{2}={max}_{|v|\le {u}_{0}}|{f}^{\prime}(v)|$. Hence we can apply Theorem 1 to (48), which gives a contradiction with (47). □

**Remark 4** Theorem 3 extends [[1], Theorem 1.4] and Theorem 4 [[1], Theorem 1.3] stated for (1) with $r(t)>0$.

Thus, according to Theorem 4, for a fixed vector $({c}_{0},{c}_{1},\dots ,{c}_{n-1})$ there exists a unique solution of (49) which has the asymptotic representation (45).

Now, consider (49), where *n* is even, $r(t)>0$ for $t\ge 0$ and $\lambda \ne 1$. By [[1], Corollary 1.6], if ${\int}^{\mathrm{\infty}}{t}^{n-3}r(t)\phantom{\rule{0.2em}{0ex}}dt=\mathrm{\infty}$, then every solution of (49) is oscillatory. Obviously, condition (37) is satisfied. Therefore, the condition (43) in Theorem 3 is optimal.

From this and Theorem 3 we get the following.

**Corollary 1** *Let* $\lambda >1$, *n* *even*, $n\ge 4$ *and* $r(t)>0$ *for* $t\ge 0$. *Then* (49) *has oscillatory solutions*.

## 6 Asymptotic equivalence of linear equations

In this section we present another consequence of our results.

and denote by ${S}_{x}$ and ${S}_{y}$ the solution space of (5) and (2), respectively.

*asymptotically equivalent*, if there exists a map $T:{S}_{y}\to {S}_{x}$ such that for every $y\in {S}_{y}$ there exists a unique $x\in {S}_{x}$ such that $T(y)=x$ and

Applying Theorem A and Theorem 1 we get the following.

**Theorem 5**

*Assume*$n\ge 3$

*and*

*Then* (2) *and* (5) *are asymptotically equivalent*.

*Proof* As we noticed above, functions ${t}^{j}$, ${\Gamma}_{u}$ and ${\Gamma}_{v}$ are linearly independent solutions of (2).

are solutions of (5). Then ${\overline{x}}_{0}-{x}_{0}$ is also a solution of (5) and tends to zero as $t\to \mathrm{\infty}$. This is a contradiction with Theorem 1. For $j\in \{1,\dots ,n-1\}$ we proceed by the same way.

and ${lim}_{t\to \mathrm{\infty}}{\eta}_{n-3}^{(n-3)}(t)=0$, we have ${c}_{n-3}=0$. Proceeding in the same way we get ${c}_{i}=0$ for $i=0,\dots ,n-2$.

Since ${\eta}_{j}$ are uniquely determined, there exists a 1-1 map between the sets ${\overline{S}}_{x}$ and ${\overline{S}}_{y}$ and so the assertion follows. □

**Remark 5** The assumption (37) is not needed in Theorem 5.

## 7 Open problems

- (1)
Does Corollary 1 hold for

*n*even and $\lambda \le 1$? - (2)
Let

*q*be bounded, but not of bounded variation on $[0,\mathrm{\infty})$. Then (3) can have unbounded oscillatory solutions, see,*e.g.*, [[17], Chapter VI-5, Theorem 3]. Similarly, if (3) is oscillatory and ${lim}_{t\to \mathrm{\infty}}q(t)=0$, then (3) can have again unbounded oscillatory solutions, as the Euler equation illustrates. In these cases, it should be interesting to find conditions under which (1) has unbounded oscillatory solutions too. - (3)
Let

*q*be unbounded. Thus, as the Armellini-Tonelli-Sansone theorem shows, (3) can have (oscillatory) solutions which tend to zero as $t\to \mathrm{\infty}$, see [20] for a detailed survey on this topic. In particular, in [20, 21] the boundedness and the existence of vanishing at infinity solutions are investigated for second order linear equations with advanced arguments. As before, also in this situation, it should be an interesting problem studying asymptotic properties of possible oscillatory solutions of (1). Note that for (1) with $n=4$, conditions which ensure that all solutions are oscillatory, can be found in [6, 7] or in [[8], Theorem 4]. - (4)Let
*g*be a continuous function on ℝ such that$\underset{|u|\to \mathrm{\infty}}{lim}{\int}_{0}^{u}g(\sigma )\phantom{\rule{0.2em}{0ex}}d\sigma =\mathrm{\infty}.$Under assumptions (H), any solution of the nonlinear equation${y}^{\u2033}(t)+q(t)g(y(t))=0$is bounded together with its first derivative, see,*e.g.*, [[18], Theorem 4]. Thus, motivated by the results here obtained, we can ask under which conditions the nonlinear equation${x}^{(n)}(t)+q(t)g({x}^{(n-2)}(t))+r(t)f(x(t))=0$has oscillatory bounded solutions.

- (5)(5)Recently, oscillation of equations with a forcing term have been studied. For example, the boundedness of any solutions is studied in [15] and the periodic case in [14]. Moreover, in [16] a two-term equation with forcing term
*e*has been considered and the oscillation is studied under additional conditions on the function*r*. Thus, it seems interesting to extend our study to the existence of oscillatory solutions and solutions vanishing at infinity for the equation with the forcing term${x}^{(n)}(t)+q(t){x}^{(n-2)}(t)+r(t)f(x(t))=e(t).$

## Notes

### Acknowledgements

The first and second authors were supported by the grant GAP201/11/0768 of the Czech Grant Agency. The authors thank both referees for their valuable comments to the paper.

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