# Fourth-Order Differential Equation with Deviating Argument

Authors: M. Bartušek; M. Cecchi; Z. Došlá; M. Marini
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider the fourth-order differential equation with middle-term and deviating argumentx(4)(t)+q(t)x(2)(t)+r(t)f(x(φ(t)))=0, in case when the corresponding second-order equation h″+q(t)h=0 is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.


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## Body

## 1. Introduction

The aim of this paper is to investigate the fourth-order nonlinear differential equation with middle-term and deviating argument(1.1)x(4)(t)+q(t)x(2)(t)+r(t)f(x(φ(t)))=0.The following assumptions will be made.(i)
q is a continuously differentiable bounded away from zero function, that is, q(t)≥q0>0 for large t such that
(1.2)∫0∞|q′(t)|dt<∞.(ii)
r,φ are continuous functions for t≥0, r is not identically zero for large t, φ(t)≥0,  and φ(0)=0, limt→∞φ(t)=∞.(iii)
f is a continuous function such that f(u)u>0 for u≠0.Observe that (i) implies that there exists a positive constantQ such that q(t)≤Q and the linear second-order equation(1.3)h′′(t)+q(t)h(t)=0
is oscillatory. Moreover, solutions of (1.3) are bounded together with their derivatives, see for example, [1, Theorem  2].By a solution of (1.1) we mean a function x defined on [Tx,∞), Tx≥0, which is differentiable up to the fourth order and satisfies (1.1) on [Tx,∞) and sup{|x(t)|:t≥T}>0 for T≥Tx.A solutionx of (1.1) is said to be asymptotically linear (AL-solution) if either(1.4)lim⁡t→∞x(t)=cx≠0,lim⁡t→∞x′(t)=0,
or(1.5)lim⁡t→∞|x(t)|=∞,lim⁡t→∞x′(t)=dx≠0,
for some constants cx,dx.Fourth-order nonlinear differential equations naturally appear in models concerning physical, biological, and chemical phenomena, such as, for instance, problems of elasticity, deformation of structures, or soil settlement, see, for example, [2, 3].When (1.3) is nonoscillatory and h is its eventually positive solution, it is known that (1.1) can be written as the two-term equation(1.6)(h2(t)(x′′(t)h(t))′)′+h(t)r(t)f(x(t))=0.
In this case, the question of oscillation and asymptotics of such class of equations has been investigated with sufficient thoroughness, see, for example, the papers [3–10] or the monographs [11, 12] and references therein.Nevertheless, as far we known, there are only few results concerning (1.1) when (1.3) is oscillatory. For instance, the equation without deviating argument(1.7)x(n)(t)+q(t)x(n-2)(t)+r(t)f(x(t))=0
has been investigated by Kiguradze in [13] in case q(t)≡1 and by the authors in [14, 15] when q satisfies (i). In particular, in [14] the oscillation of (1.1) in the case n=3 is studied. In [15], the existence of positive bounded and unbounded solutions as well as of oscillatory solutions for (1.7) has been considered and the case n=4 has been analyzed in detail. Other results can be found in [16] and references therein, in which the existence and uniqueness of almost periodic solutions for equations of type (1.1) with almost periodic coefficients q,r are studied.Motivated by [14, 15], here we study the existence of AL-solutions for (1.1). The approach is completely different from the one used in [15], in which an iteration process, jointly with a comparison with the linear equation y(4)+q(t)y(2)=0, is employed. Our tools are based on a topological method, certain integral inequalities, and some auxiliary functions. In particular, for proving the continuity in the Fréchet space C[t0,∞) of the fixed point operators here considered, we use a similar argument to that in the Vitali convergence theorem.Our results extend to the case with deviating argument analogues ones stated in [15] for (1.7) when n=4. We obtain sharper conditions for the existence of unbounded AL-solutions of (1.1), and, in addition, we show that under additional assumptions on q,r, these conditions become also necessary for the existence of AL-solutions, in both the bounded and unbounded cases. In the final part, we consider the particular case(1.8)f(u)=|u|λsgn⁡u(λ>0)
and we study the possible coexistence of bounded and unbounded AL-solutions. The role of deviating argument and the one of the growth of the nonlinearity are also discussed and illustrated by some examples.


## 2. Unbounded Solutions

Here we study the existence of unbounded AL-solutions of (1.1). Our first main result is the following.Theorem 2.1.
For anyc,  0<c<∞, there  exists an unbounded solution x of (1.1) such that
(2.1)lim⁡t→∞x′(t)=c,lim⁡t→∞x(i)(t)=0,i=2,3,
provided
(2.2)∫0∞|r(t)|F(φ(t))dt<∞,
where for u>0(2.3)F(u)=max⁡{f(v):|v-u|≤12u}.Proof.
Without loss of generality, we prove the existence of solutions of (1.1) satisfying (2.1) for c=1.
Letu and v be two linearly independent solutions of (1.3) with Wronskian d=1. Denote
(2.4)w(s,t)=u(s)v(t)-u(t)v(s),z(s,t)=∂∂tw(s,t).
As claimed by the assumptions on q, all solutions of (1.3) and their derivatives are bounded. Thus, put
(2.5)M=sup⁡{|w(s,t)|+|z(s,t)|:s≥0,t≥0},L=2(2M+1)q0.
Let t̅≥t0 be such that φ(t)≥t0 for t≥t̅. Define
(2.6)φ̅(t)={φ(t)ift≥t̅,φ(t̅)ift0≤t≤t̅,
and choose t0≥0 large so that
(2.7)∫t0∞|r(s)|F(φ̅(s))ds≤12L,1q0∫t0∞|q′(t)|dt≤12.
Denote byC[t0,∞) the Fréchet space of all continuous functions on [t0,∞), endowed with the topology of uniform convergence on compact subintervals of [t0,∞), and consider the set Ω⊂C[t0,∞) given by
(2.8)Ω={x∈C[t0,∞):t2≤x(t)≤3t2}.
Let T>t0 and define on [t0,T] the function
(2.9)g(t)=γ′′(t)+q(t)γ(t),
where
(2.10)γ(t)=-∫tT∫τ∞r(s)f(x(φ̅(s)))w(s,τ)dsdτ
and x∈Ω. Then,
(2.11)γ′(t)=∫t∞r(s)f(x(φ̅(s)))w(s,t)ds,(2.12)γ′′(t)=∫t∞r(s)f(x(φ̅(s)))z(s,t)ds,γ′′′(t)=-r(t)f(x(φ̅(t)))-q(t)γ′(t).
Moreover, g(T)=γ′′(T), and it holds for t∈[t0,T] that
(2.13)g′(t)=γ′′′(t)+q(t)γ′(t)+q′(t)γ(t)=-r(t)f(x(φ̅(t)))+q′(t)γ(t).
Integrating, we obtain
(2.14)g(t)=g(T)-∫tTg′(s)ds=γ′′(T)+∫tTr(s)f(x(φ̅(s)))ds-∫tTq′(s)γ(s)ds.
From here and (1.3), we get
(2.15)γ(t)=1q(t)(γ′′(T)-γ′′(t)+∫tTr(s)f(x(φ̅(s)))ds-∫tTq′(s)γ(s)ds).
Thus,
(2.16)|γ(t)|≤1+2Mq0∫t∞|r(s)|F(φ̅(s))ds+1q0max⁡t≤s≤T|γ(s)|∫t0∞|q′(s)|ds,
and so
(2.17)(1-1q0∫t0∞|q′(s)|ds)max⁡t≤s≤T|γ(s)|≤1+2Mq0∫t∞|r(s)|F(φ̅(s))ds,
or, in view of (2.7),
(2.18)|γ(t)|≤L∫t∞|r(s)|F(φ̅(s))ds.
Thus, from (2.10), as T→∞, we get
(2.19)|∫t∞∫τ∞r(s)f(x(φ̅(s)))w(s,τ)dsdτ|≤L∫t∞|r(s)|F(φ̅(s))ds.
Hence, the operator   𝒯:Ω→Ω  given by
(2.20)T(x)(t)=t-∫t0t∫σ∞∫τ∞r(s)f(x(φ̅(s)))w(s,τ)dsdτdσ
is well defined for any x∈Ω. Moreover, in view of (2.19), we have
(2.21)|T′(x)(t)-1|≤L∫t∞|r(s)|F(φ̅(s))ds.
From here, in virtue of (2.7) we get
(2.22)|T(x)(t)-t|≤Lt∫t0∞|r(s)|F(φ̅(s))ds≤12t.
Hence, 𝒯(Ω)⊂Ω. From (2.5) and (2.11), we have
(2.23)|T′′(x)(t)|=|γ′(t)|≤M∫t∞|r(s)|F(φ̅(s))ds,
and so limt→∞𝒯′′(x)(t)=0. Similarly,
(2.24)|T′′′(x)(t)|=|γ′′(t)|≤M∫t∞|r(s)|F(φ̅(s))ds,
and thus,  limt→∞𝒯′′′(x)(t)=0, too. In addition,
(2.25)T(4)(x)(t)=γ′′′(t)=-q(t)T′′(x)(t)-r(t)f(x(φ̅(t))).
Hence, any fixed point of 𝒯 is a solution of (1.1) for large t.
Let us show that𝒯(Ω) is relatively compact, that is, 𝒯(Ω) consists of functions equibounded and equicontinuous on every compact interval of [t0,∞). Because 𝒯(Ω)⊂Ω, the equiboundedness follows. Moreover, in view of (2.7), 𝒯′(u)(t) is bounded for any u∈Ω, which yields the equicontinuity of the elements in  𝒯(Ω).
Now we prove the continuity of𝒯 in Ω. Let {xn}, n∈ℕ, be a sequence in Ω, which uniformly converges to x̅∈Ω on every compact interval of [t0,∞). Fixing T>t0, in virtue of (2.23), the dominated convergence Lebesgue theorem gives
(2.26)lim⁡n→∞∫σT∫τ∞r(s)(f(xn(φ̅(s))))w(s,τ)dsdτ=∫σT∫τ∞r(s)(f(x(φ̅(s))))w(s,τ)dsdτ.
Moreover,
(2.27)|∫σ∞∫τ∞r(s)(f(xn(φ̅(s)))-f(x̅(φ̅(s))))w(s,τ)dsdτ|≤|∫σT∫τ∞r(s)(f(xn(φ̅(s)))-f(x̅(φ̅(s))))w(s,τ)dsdτ|+∫T∞∫τ∞|r(s)|(f(xn(φ̅(s)))+f(x̅(φ̅(s))))|w(s,τ)|dsdτ.
In view of (2.19), we have
(2.28)∫T∞∫τ∞|r(s)|(f(xn(φ̅(s)))+f(x̅(φ̅(s))))|w(s,τ)|dsdτ≤2M∫T∞|r(s)|F(φ̅(s))ds.
Thus, choosing  T sufficiently large, we get from (2.27)
(2.29)lim⁡n→∞∫σ∞∫τ∞r(s)f(xn(φ̅(s)))w(s,τ)dsdτ=∫σ∞∫τ∞r(s)f(x̅(φ̅(s)))w(s,τ)dsdτ,
and so the continuity of 𝒯 in Ω follows. By the Tychonoff fixed point theorem, the operator 𝒯 has a fixed point x, which is an unbounded solution of (1.1) satisfying (2.1).Remark 2.2.
With minor modifications, Theorem2.1 gives also the existence of eventually negative unbounded AL-solutions. The details are omitted.Remark 2.3.
Whenφ(t)≡t, Theorem 2.1 is related with Theorem  1 in [15], from which the existence of unbounded AL-solutions of (1.1) can be obtained under stronger assumptions. A comparison between Theorem  1 in [15] and Theorem 2.1 is given in Section 4.Our next result gives a necessary condition for the existence of unbounded solutionsx of (1.1) satisfying for large t and some α and β(2.30)0<α≤x′(t)≤β.Theorem 2.4.
Assume eitherr(t)≥0 or r(t)≤0.
Equation (1.1) does not have eventually positive solutions x satisfying (2.30) for large t and some α and β provided
(2.31)∫0∞|r(t)|F̅(φ(t))dt=∞,
where for u>0(2.32)F̅(u)=min⁡{f(v):α2u≤v≤2βu}.Proof.
Assumer(t)≥0, and let x be an eventually positive solution of (1.1) satisfying (2.30). Then, there exists τ such that
(2.33)α2t≤x(t)≤2tβfort≥τ.
Consequently, in view of (2.31), we have
(2.34)lim⁡t→∞∫τtr(s)f(x(φ(s)))ds=∞.
Thus, integrating (1.1), we get
(2.35)lim⁡t→∞(x′′′(t)+∫τtq(s)x′′(s)ds)=-∞.
Furthermore,
(2.36)|∫τtq(s)x′′(s)ds|=|q(t)x′(t)-q(τ)x′(τ)-∫τtq′(s)x′(s)ds|≤2βQ+β∫τ∞|q′(s)|ds<∞,
where Q=sups≥0q(s). Hence limt→∞x′′′(t)=-∞, which gives a contradiction with the boundedness of x′. Finally, if r(t)≤0, the argument is similar and the details are left to the reader.


## 3. Bounded Solutions

In this section we study the existence of bounded AL-solutions of (1.1). The following holds.Theorem 3.1.
If(3.1)∫0∞|r(t)|tdt<∞,
then, for any c∈ℝ∖{0}, there exists a solution x of (1.1) satisfying
(3.2)lim⁡t→∞x(t)=c,lim⁡t→∞x(i)(t)=0,i=1,2.Proof.
Without loss of generality, we prove the existence of solutions of (1.1) satisfying (3.2) for c=1.
We proceed by a similar way to that in the proof of Theorem2.1, and we sketch the proof.
LetM be the constant given in (2.5), and let
(3.3)K=max⁡{f(u):12≤u≤32},L1=2K(2M+1)q0.
Choose t0≥0 large so that
(3.4)∫t0∞t|r(t)|dt≤12L1,1q0∫t0∞|q′(s)|ds≤12,
and define φ̅ as in (2.6). Denote by C[t0,∞) the Fréchet space of all continuous functions on [t0,∞), endowed with the topology of uniform convergence on compact subintervals of [t0,∞), and consider the set Ω⊂C[t0,∞) given by
(3.5)Ω={x∈C[t0,∞):12≤x(t)≤32}.
Let T>t0, and, for any x∈Ω, consider again the function γ given in (2.10). Reasoning as in the proof of Theorem 2.1, with minor changes, we obtain
(3.6)|∫t∞∫τ∞r(s)f(x(φ̅(s)))w(s,τ)dsdτ|≤L1∫t∞|r(s)|ds.
Hence, in virtue of (3.1), the operator  ℋ:Ω→Ω  given by
(3.7)H(x)(t)=1+∫t∞∫σ∞∫τ∞r(s)f(x(φ̅(s)))w(s,τ)dsdτdσ
is well defined and limt→∞ℋ(x)(t)=1. In view of (3.6), we get
(3.8)|H′(x)(t)|≤L1∫t∞|r(s)|ds.
A similar estimation holds for |ℋ′′(x)|. Thus, limt→∞ℋ(i)(x)(t)=0,i=1,2. In view of (3.4), from (3.8), we obtain
(3.9)|H(x)(t)-1|≤L1∫t∞s|r(s)|ds≤12,
that is, ℋ(Ω)⊂Ω. Moreover, a standard calculation gives
(3.10)H(4)(x)(t)=-q(t)H(2)(x)(t)-r(t)f(x(φ̅(s))),
and so any fixed point of ℋ is, for large t, a solution of (1.1). Proceeding by a similar way to that in the proof of Theorem 2.1, we obtain that ℋ(Ω) is relatively compact.
Now we prove the continuity ofℋ in Ω. Let {xn}, n∈ℕ, be a sequence in Ω, which uniformly converges to x̅∈Ω on every compact interval of [t0,∞). Since
(3.11)|∫τ∞r(s)(f(xn(φ̅(s))))w(s,τ)ds|≤KM∫τ∞|r(s)|ds,
in virtue of (3.1), the dominated convergence Lebesgue theorem gives
(3.12)lim⁡n→∞∫σ∞∫τ∞r(s)(f(xn(φ̅(s))))w(s,τ)dsdτ=∫σ∞∫τ∞r(s)(f(x(φ̅(s))))w(s,τ)dsdτ.
Moreover, fixing  T>t0, we have
(3.13)|∫t∞∫σ∞∫τ∞r(s)(f(xn(φ̅(s)))-f(x̅(φ̅(s))))w(s,τ)dsdτdσ|≤|∫tT∫σ∞∫τ∞r(s)(f(xn(φ̅(s)))-f(x̅(φ̅(s))))w(s,τ)dsdτdσ|+∫T∞∫σ∞∫τ∞|r(s)|(f(xn(φ̅(s)))+f(x̅(φ̅(s))))|w(s,τ)|dsdτdσ.
In view of (3.9), we have
(3.14)∫T∞∫σ∞∫τ∞|r(s)|(f(xn(φ̅(s)))+f(x̅(φ̅(s))))|w(s,τ)|dsdτdσ≤2L1∫T∞s|r(s)|ds,
and so, choosing T sufficiently large, from (3.13) we obtain the continuity of ℋ in Ω. Hence, by the Tychonoff fixed point theorem, the operator ℋ has a fixed point x, which is a bounded solution of (1.1) satisfying (3.2).Remark 3.2.
Whenn=4, Theorem 3.1 extends to equations with deviating argument of a similar result stated in [15] for (1.7). Observe that our approach used here is completely different from that in [15].The next result shows that, under additional assumptions, condition (3.1) can be also necessary for the existence of bounded AL-solutions of (1.1).Theorem 3.3.
Assume either(3.15)r(t)≥0,q′′(t)≥0forlarget
or
(3.16)r(t)≤0,q′′(t)≤0forlarget.
If
(3.17)∫0∞|r(t)|tdt=∞,
then (1.1) does not have solutions x satisfying
(3.18)0<α≤x(t)≤β,
for large t and some α and β. Consequently, every bounded solution x of (1.1) satisfies
(3.19)liminf⁡t→∞|x(t)|=0.The following lemmas are needed for proving Theorem3.3.Lemma 3.4.
Assumeq′′(t)≥0 for t≥T≥0, and let x be a solution of (1.1) satisfying (3.18) for t≥T. Then, there exist two constants M1,M2 such that for t≥T(3.20)-∫Ttsq(s)x′′(s)ds<tq′(t)x(t)-tq(t)x′(t)+M1,(3.21)∫Tt(s-T)q′(s)x′(s)ds<M2.
If q′′(t)≤0 for t≥T≥0, inequalities (3.20), (3.21) hold in the opposite order.Proof.
Supposeq′′(t)≥0on [T,∞). We have
(3.22)∫Ttsq(s)x′′(s)ds=tq(t)x′(t)-Tq(T)x′(T)-∫Ttq(s)x′(s)ds-∫Ttsq′(s)x′(s)ds.
Since
(3.23)∫Ttsq′(s)x′(s)ds=tq′(t)x(t)-Tq′(T)x(T)-∫Ttq′(s)x(s)ds-∫Ttsq′′(s)x(s)ds,∫Ttq(s)x′(s)ds=q(t)x(t)-q(T)x(T)-∫Ttq′(s)x(s)ds,
from (3.22), we get
(3.24)-∫Ttsq(s)x′′(s)ds=tq′(t)x(t)-tq(t)x′(t)+q(t)x(t)-2∫Ttq′(s)x(s)ds-∫Ttsq′′(s)x(s)ds+K1,
where K1 is a suitable constant. Since q,x are bounded, q′′(t)≥0, in view of (1.1), inequality (3.20) follows.
Moreover,q′ is nondecreasing for t≥T. Because q is a positive bounded function, then q′(t)≤0 on [T,∞). Thus, inequality (3.21) follows integrating by parts and using (1.1). Finally, if q′′(t)≤0 on [T,∞), the argument is similar.Lemma 3.5.
Letx be a solution of (1.1) satisfying (3.18) for large t. If
(3.25)∫0∞|r(t)|dt<∞,
then x′′ is bounded. If, in addition, r(t)≥0,q′′(t)≥0 for t≥T≥0 and (3.17) holds, then for large t(3.26)x′′′(t)+q(t)x′(t)<q′(t)x(t).
If r(t)≤0,q′′(t)≤0 for t≥T≥0, inequality (3.26) holds in the opposite order.Proof.
Sincelimt→∞φ(t)=∞, there exists τ such that for t≥τ(3.27)0<α≤x(φ(t))≤β.
Without loss of generality, let τ=T. Thus, inft≥Tf(x(φ(t)))>0.
Letu and v be two linearly independent solutions of (1.3) with Wronskian d=1. By assumptions on q, all solutions of (1.3) and their derivatives are bounded. Thus, by the variation constant formula, there exist constants c1 and c2 such that
(3.28)x′′(t)=c1u(t)+c2v(t)-∫Tt(u(s)v(t)-u(t)v(s))r(s)f(x(φ(s)))ds,
and, in view of (3.25), x′′ is bounded.
Let us prove (3.26), and suppose r(t)≥0,q′′(t)≥0 on [T,∞). Multiplying (1.1) by t and integrating from T to t, we get
(3.29)tx′′′(t)-x′′(t)+∫Ttsq(s)x′′(s)ds=Tx′′′(T)-x′′(T)-∫Ttsr(s)f(x(φ(s)))ds,
or, in view of Lemma 3.4,
(3.30)tx′′′(t)≤x′′(t)+tq′(t)x(t)-tq(t)x′(t)-∫Ttsr(s)f(x(φ(s)))ds+K2,
where K2 is a suitable constant. Since x′′ is bounded and
(3.31)∫Ttsr(s)f(x(φ(s)))ds≥inf⁡t≥Tf(x(φ(t)))∫Ttsr(s)ds,
from (3.17) and (3.30), we have
(3.32)lim⁡t→∞t(x′′′(t)-q′(t)x(t)+q(t)x′(t))=-∞,
which gives the assertion. The case r(t)≤0,q′′(t)≤0 on [T,∞) can be treated in a similar way.Proof of Theorem3.3.
Supposer(t)≥0, q′′(t)≥0 for t≥T≥0. Without loss of generality, assume also that (3.27) holds for t≥T. Define
(3.33)v(t)=x′′(t)+q(t)x(t),(3.34)z(t)=x′′′(t)+q(t)x′(t)-∫Ttq′(s)x′(s)ds.
Then, z′(t)=-r(t)f(x(φ(t)))≤0 and
(3.35)z(t)=z(T)-∫Ttr(s)f(x(φ(s)))ds.
Since q′(t)≤0 for t≥T, we have
(3.36)v′(t)≤z(t)+∫Ttq′(s)x′(s)ds=z(T)-∫Ttr(s)f(x(φ(s)))ds+∫Ttq′(s)x′(s)ds.
Case I.
Assume(3.37)∫0∞r(t)dt=∞.
Since for t≥T we have q′′(t)≥0 and, as claimed, q′(t)≤0, we get
(3.38)∫Ttq′(s)x′(s)ds=q′(t)x(t)-q′(T)x(T)-∫Ttq′′(s)x(s)ds≤-q′(T)x(T).
Thus, from (3.36), we obtain limt→∞v′(t)=-∞, that is, v is unbounded. Hence, in view of (3.33), we obtain a contradiction with the boundedness of x.Case II.
Now assume (3.17) and (3.25). In view of Lemma 3.5, without loss of generality, we can suppose that (3.26) holds for t≥T. Then,
(3.39)z(T)=x′′′(T)+q(T)x′(T)<q′(T)x(T).
Hence, z(T)<0. Integrating (3.36), we get
(3.40)v(t)≤v(T)+z(T)(t-T)-∫Tt(s-T)r(s)f(x(φ(s)))ds+∫Tt(s-T)q′(s)x′(s)ds,
and, in view of Lemma 3.4, we have
(3.41)v(t)≤v(T)+z(T)(t-T)+M2.
Thus, limt→∞v(t)=-∞, that is, as before, a contradiction. Finally, the case r(t)≤0, q′′(t)≤0 for large t follows in a similar way.


## 4. Applications

Here we present some applications of our results to a particular case of (1.1), namely, the equation(4.1)x(4)(t)+q(t)x′′(t)+r(t)|x(φ(t))|λsgn⁡x(φ(t))=0(λ>0),
jointly with some suggestions for future research.


### 4.1. Coexistence of Both Types of AL-Solutions

Applying Theorems2.1–3.3 to this equation, we obtain the following.Corollary 4.1.
(a) Letr(t)≠0 for large t.  Equation (4.1) has unbounded AL-solutions if and only if
(4.2)∫0∞|r(t)|φλ(t)dt<∞.
(b) Assume either (3.15) or (3.16). Equation (4.1) has bounded AL-solutions if and only if (3.1) holds.Corollary4.1 shows also that the deviating argument can produce a different situation concerning the unboundedness of solutions with respect to the corresponding equation without delay, as the following example illustrates.Example 4.2.
In view of Corollary4.1(a), the equation
(4.3)x(4)(t)+q(t)x(2)(t)+1(t+1)2|x(t)|3/2sgn⁡x(t)=0,
where q satisfies (i), has unbounded AL-solutions, while the corresponding ordinary equation
(4.4)x(4)(t)+q(t)x(2)(t)+1(t+1)2|x(t)|3/2sgn⁡x(t)=0,
in view of Theorem 2.4, does not have unbounded AL-solutions. Moreover, if in addition q′′(t)>0 for large t, then from Corollary 4.1(b) (4.3) does not have bounded AL-solutions.The following example shows that the opposite situation to the one described in Example4.2 can occur.Example 4.3.
Consider the equation(4.5)x(4)(t)+q(t)x(2)(t)+1(t+1)3x(t2)=0,
where q satisfies (i). From Theorem 3.1, (4.5) has bounded AL-solutions and the same occurs for the corresponding ordinary equation. Nevertheless, in view of Corollary 4.1(a), (4.5) has no unbounded AL-solutions.Examples4.2 and 4.3 illustrate also that the coexistence of both AL-solutions for (4.1) can fail. Sufficient conditions for the coexistence of these solutions immediately follow from Corollary 4.1.Corollary 4.4.
Letr(t)≠0 for large t.
(a) Assume for larget(4.6)φ(t)≥t1/λ.
If (4.1) has unbounded AL-solutions, then (4.1) also has AS bounded solutions.
(b) Assume for larget(4.7)φ(t)≤t1/λ,sgn⁡r(t)=sgn⁡q′′(t).
If (4.1) has bounded AL-solutions, then (4.1) also has unbounded AL-solutions.
For the equation without deviating argument(4.8)x(4)(t)+q(t)x′′(t)+r(t)|x(t)|λsgn⁡x(t)=0(λ>0),
from Corollary 4.4 we get the following.Corollary 4.5.
Letr(t)≠0 for large t.
(a) Assumeλ≥1. If (4.8) has unbounded AL-solutions, then (4.8) has also bounded AL-solutions.
(b) Assume0<λ≤1 and sgnr(t)=sgnq′′(t) for large t. If (4.8) has bounded AL-solutions, then (4.8) has also unbounded AL-solutions.


### 4.2. Comparison with Some Results in [15]

As claimed, the existence of unbounded AL-solutions for (4.8) follows also from Theorem  1 in [15]. For n=4 this result reads as follows.Theorem A.
If(4.9)∫0∞|r(t)|tλ+1dt<∞,
then there exists a solution x of (4.8) such that
(4.10)x(i)(t)=t(i)+εi(t),i=0,…,3,
where εi are functions of bounded variation for large t and limt→∞εi(t)=0.Therefore, whenφ(t)≡t, Theorem 2.1 ensures the existence of unbounded AL-solutions of (4.8) under a weaker condition than (4.9), namely,(4.11)∫0∞|r(t)|tλdt<∞.
On the other hand, Theorem A gives an asymptotic formula for such solutions.


### 4.3. An Open Problem

Equation (1.1) can admit also other types of nonoscillatory solutions, as the following examples show.Example 4.6.
Consider the equation(4.12)x(4)(t)+x(2)(t)-2t2+4t+26(t+1)7/2|x(t)|3/2sgn⁡x(t)=0.
In virtue of Corollary 4.1(b), (4.12) has no bounded AL-solutions. Nevertheless, this equation admits nonoscillatory bounded solutions because x(t)=(1+t)-1 is a solution of (4.12).Example 4.7.
Consider the equation(4.13)x(4)(t)+x(2)(t)+t2+4t+10(t+2)4(log⁡(t+2))3x3(t)=0.
Thus, (3.1) holds, while ∫0∞t3r(t)dt=∞. Hence, in virtue of Corollary 4.1, (4.13) has bounded AL-solutions, but no unbounded AL-solutions. Nevertheless, this equation admits nonoscillatory unbounded solutions because x(t)=log(t+2) is a solution of (4.13).The existence of nonoscillatory solutionsx satisfying either limt→∞x(t)=0 or lim|x(t)|=∞,limt→∞x′(t)=0 will be a subject of our next research.


## 4.1. Coexistence of Both Types of AL-Solutions

Applying Theorems2.1–3.3 to this equation, we obtain the following.Corollary 4.1.
(a) Letr(t)≠0 for large t.  Equation (4.1) has unbounded AL-solutions if and only if
(4.2)∫0∞|r(t)|φλ(t)dt<∞.
(b) Assume either (3.15) or (3.16). Equation (4.1) has bounded AL-solutions if and only if (3.1) holds.Corollary4.1 shows also that the deviating argument can produce a different situation concerning the unboundedness of solutions with respect to the corresponding equation without delay, as the following example illustrates.Example 4.2.
In view of Corollary4.1(a), the equation
(4.3)x(4)(t)+q(t)x(2)(t)+1(t+1)2|x(t)|3/2sgn⁡x(t)=0,
where q satisfies (i), has unbounded AL-solutions, while the corresponding ordinary equation
(4.4)x(4)(t)+q(t)x(2)(t)+1(t+1)2|x(t)|3/2sgn⁡x(t)=0,
in view of Theorem 2.4, does not have unbounded AL-solutions. Moreover, if in addition q′′(t)>0 for large t, then from Corollary 4.1(b) (4.3) does not have bounded AL-solutions.The following example shows that the opposite situation to the one described in Example4.2 can occur.Example 4.3.
Consider the equation(4.5)x(4)(t)+q(t)x(2)(t)+1(t+1)3x(t2)=0,
where q satisfies (i). From Theorem 3.1, (4.5) has bounded AL-solutions and the same occurs for the corresponding ordinary equation. Nevertheless, in view of Corollary 4.1(a), (4.5) has no unbounded AL-solutions.Examples4.2 and 4.3 illustrate also that the coexistence of both AL-solutions for (4.1) can fail. Sufficient conditions for the coexistence of these solutions immediately follow from Corollary 4.1.Corollary 4.4.
Letr(t)≠0 for large t.
(a) Assume for larget(4.6)φ(t)≥t1/λ.
If (4.1) has unbounded AL-solutions, then (4.1) also has AS bounded solutions.
(b) Assume for larget(4.7)φ(t)≤t1/λ,sgn⁡r(t)=sgn⁡q′′(t).
If (4.1) has bounded AL-solutions, then (4.1) also has unbounded AL-solutions.
For the equation without deviating argument(4.8)x(4)(t)+q(t)x′′(t)+r(t)|x(t)|λsgn⁡x(t)=0(λ>0),
from Corollary 4.4 we get the following.Corollary 4.5.
Letr(t)≠0 for large t.
(a) Assumeλ≥1. If (4.8) has unbounded AL-solutions, then (4.8) has also bounded AL-solutions.
(b) Assume0<λ≤1 and sgnr(t)=sgnq′′(t) for large t. If (4.8) has bounded AL-solutions, then (4.8) has also unbounded AL-solutions.


## 4.2. Comparison with Some Results in [15]

As claimed, the existence of unbounded AL-solutions for (4.8) follows also from Theorem  1 in [15]. For n=4 this result reads as follows.Theorem A.
If(4.9)∫0∞|r(t)|tλ+1dt<∞,
then there exists a solution x of (4.8) such that
(4.10)x(i)(t)=t(i)+εi(t),i=0,…,3,
where εi are functions of bounded variation for large t and limt→∞εi(t)=0.Therefore, whenφ(t)≡t, Theorem 2.1 ensures the existence of unbounded AL-solutions of (4.8) under a weaker condition than (4.9), namely,(4.11)∫0∞|r(t)|tλdt<∞.
On the other hand, Theorem A gives an asymptotic formula for such solutions.


## 4.3. An Open Problem

Equation (1.1) can admit also other types of nonoscillatory solutions, as the following examples show.Example 4.6.
Consider the equation(4.12)x(4)(t)+x(2)(t)-2t2+4t+26(t+1)7/2|x(t)|3/2sgn⁡x(t)=0.
In virtue of Corollary 4.1(b), (4.12) has no bounded AL-solutions. Nevertheless, this equation admits nonoscillatory bounded solutions because x(t)=(1+t)-1 is a solution of (4.12).Example 4.7.
Consider the equation(4.13)x(4)(t)+x(2)(t)+t2+4t+10(t+2)4(log⁡(t+2))3x3(t)=0.
Thus, (3.1) holds, while ∫0∞t3r(t)dt=∞. Hence, in virtue of Corollary 4.1, (4.13) has bounded AL-solutions, but no unbounded AL-solutions. Nevertheless, this equation admits nonoscillatory unbounded solutions because x(t)=log(t+2) is a solution of (4.13).The existence of nonoscillatory solutionsx satisfying either limt→∞x(t)=0 or lim|x(t)|=∞,limt→∞x′(t)=0 will be a subject of our next research.



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*Source: 185242-2012-03-18.xml*