# On Asymptotic Behavior of Solutions of Generalized Emden-Fowler Differential Equations with Delay Argument

Authors: Alexander Domoshnitsky; Roman Koplatadze
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The following differential equationu(n)(t)+p(t)|u(σ(t))|μ(t)signu(σ(t))=0 is considered. Here p∈Lloc(R+;R+),  μ∈C(R+;(0,+∞)),  σ∈C(R+;R+),  σ(t)≤t, and limt→+∞⁡σ(t)=+∞. We say that the equation is almost linear if the condition  limt→+∞⁡μ(t)=1 is fulfilled, while if lim⁡supt→+∞⁡μ(t)≠1 or lim⁡inft→+∞⁡μ(t)≠1, then the equation is an essentially nonlinear differential equation. In the case of almost linear and essentially nonlinear differential equations with advanced argument, oscillatory properties have been extensively studied, but there are no results on delay equations of this sort. In this paper, new sufficient conditions implying Property A for delay Emden-Fowler equations are obtained.


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## Body

## 1. Introduction

This work deals with oscillatory properties of solutions of a functional differential equation of the form(1)u(n)(t)+p(t)|u(σ(t))|μ(t)sign⁡u(σ(t))=0,
where
(2)p∈Lloc⁡(R+;R),μ∈C(R+;(0;+∞)),σ∈C(R+;R+),σ(t)≤tfort∈R+,hhhhhhhlim⁡t→+∞σ(t)=+∞.It will always be assumed that the condition(3)p(t)≥0fort∈R+
is fulfilled.Lett0∈R+. A function u:[t0;+∞)→R is said to be a proper solution of (1) if it is locally absolutely continuous together with its derivatives up to order n-1 inclusive,
(4)sup⁡{|u(s)|:s∈[t;+∞)}>0fort≥t0,
and there exists a function u¯∈C(R+;R) such that u¯(t)≡u(t) on [t0;+∞) and the equality u¯(n)(t)+p(t)|u¯(σ(t))|μ(t)sign⁡u¯(σ(t))=0 holds for t∈[t0:+∞). A proper solution u:[t0:+∞)→R of (1) is said to be oscillatory if it has a sequence of zeros tending to +∞. Otherwise the solution u is said to be nonoscillatory.Definition 1.
We say that (1) has Property A if any of its proper solutions is oscillatory whennis even and either is oscillatory or satisfies
(5)|u(i)(t)|↓0fort↑+∞(i=0,…,n-1),
when n is odd.Definition 2.
We say that (1) is almost linear if the condition lim⁡t→+∞⁡μ(t)=1 holds, while if  lim⁡⁡sup⁡t→+∞⁡μ(t)≠1 or lim⁡⁡inf⁡t→+∞⁡μ(t)≠1, then we say that the equation is an essentially nonlinear differential equation.The Emden-Fowler equation originated from theories concerning gaseous dynamics in astrophysics in the middle of the nineteenth century. In the study of stellar structure at that time it was important to investigate the equilibrium configuration of the mass of spherical clouds of gas. Lord Kelvin in 1862 assumed that the gaseous cloud is under convective equilibrium and then Lane [1] studied the equation
(6)1t2ddt(t2dudt)+uγ=0.The Emden-Fowler equations were first considered only for second-order equations and written in the form(7)ddt(p(t)dudt)+q(t)uγ=0,t≥0,
which could be reduced in the case of positive and continuous coefficients to the equation
(8)x′′+a(t)xγ=0,t≥0.To avoid difficulties of definingxγ when x(t) is negative and γ is not an integer, the equation
(9)x′′(t)+a(t)|x(t)|γsign⁡x(t)=0,t≥0,
was usually considered. The mathematical foundation of the theory of such equations was built by Fowler [2] and the description of the results can be found in Chapter 7 of [3].We see also the Emden-Fowler equation in gas dynamics and fluid mechanics (see Sansone [4], page 431 and the paper [5]). Nonoscillation of these equations is important in various applications. Note that the zero of such solutions corresponds to an equilibrium state in a fluid with spherical distribution of density and under mutual attraction of its particles. The Emden-Fowler equations can be either oscillatory (i.e., all proper solutions have a sequence of zeros tending to zero) or nonoscillatory, if solutions are eventually positive or negative, or, in contrast with the case of linear differential equations of second order, may possess both oscillating and nonoscillating solutions. For example, for the equation
(10)x′′(t)+tμ|x(t)|γsign⁡x(t)=0,t≥0,
it was proven in [2] that for μ≥-2>-(γ+3)/2 all solutions oscillate, for μ<-(γ+3)/2—all solutions nonoscillate, and for -(γ+3)/2≤μ<-2 there are both oscillating and nonoscillating solutions.The Emden-Fowler equation presents one of the classical objects in the theory of differential equations. Tests for oscillation and nonoscillation of all solutions and existence of oscillating solutions were obtained in the works [6–8]. In [9] for the case 0<γ<1,it was obtained that all solutions of the equation
(11)x′′(t)+a(t)|xγ(t)|sign⁡x(t)=0,t≥0,
oscillate if and only if
(12)∫0∞tγa(t)dt=∞.The latest research results in this area are presented in the book [8]. Behavior of solutions to nth order Emden-Fowler equations can be essentially more complicated. Properties A and B defined by Kiguradze are studied in the abovementioned book.There are essentially less results on oscillation of delay Emden-Fowler equations. Oscillation properties of nonlinear delay differential equations, where Emden-Fowler equations were also included as a particular case, were studied in [10–20]. Results of these papers are discussed in [13, 15], where various examples demonstrating essentialities of conditions are also presented. Note that for delay differential equations there are no results on nonoscillation of all solutions and only existence of nonoscillating solutions is studied. Actually, the results on oscillation of delayed equations are based on the approaches existing for ordinary differential equations with development in the direction of preventing the obstructive influence of delay. In the paper [15] the following equation
(13)x(n)(t)+a(t)f[x(σ(t))]=0,t≥0,neven,
and its particular case
(14)x(n)(t)+a(t)|x(σ(t))|γsign⁡x(σ(t))=0,hhhhhhhhhhhhhhhhhhhhht≥0,neven,
are considered. It was obtained for the last equation under some standard assumptions on the coefficients [15] that in the case 0<γ<1,
(15)∫0∞‍σ(n-1)γ(t)a(t)dt=∞,
all solutions oscillate. We see that the integral depends on deviation of argument σ(t) and the power of the equation n. For the equation
(16)x(n)(t)+a(t)∏i=1m‍|x(σi(t))|γisign⁡x(σi(t))=0,hhhhhhhhhhhhhhhhhhhhhhhht≥0,neven,
where γi is the ratio of two positive odd integers, σ(t)≤σi(t)≤t for i=1,…,m,and σ(t)→∞ as t→∞,each of the following conditions (a), (b), and (c) ensures oscillation of all solutions:(a)
(17)∫0∞σ(n-1)γ(t)a(t)dt=∞forγ=∑i=1m‍γi<1;(b)
(18)∫0∞σ(n-1)α(t)a(t)dt=∞forγ=1,0<α<1;(c)
(19)∫0∞σn-1(t)a(t)dt=∞,σ′(t)≥0forγ>1.Most proofs of results on oscillation of all solutions to second order equations utilize the fact that if a nonoscillating solution exists, the signs of the solutionx(t) and its second derivative x′′(t) are opposite to each other for sufficiently large t. Then a growth of nonoscillating solution is estimated and the authors come to contradiction with conditions that proves oscillation of all solutions. Note that delays disturb oscillation. Instead of tγ,σγ(t) appears. The principle is clear: for oscillation of all solutions we have to achieve a corresponding smallness of the delay t-σ(t). All this is more complicated if we study nth order equations. In this case also the fact that x(t) and its nth derivative x(n)(t) have different signs for sufficiently large t is used, but the technique is more complicated.In the papers [21–28] a generalization of Emden-Fowler equations was considered. The powers in these papers can be functions and not constants. In many cases, it leads to essentially new oscillation properties of such equations. Surprisingly, oscillation behavior of equations, with the power λ and with functional power μ(t) such that lim⁡t→∞⁡μ(t)=λ, can be quite different. The main purpose of our paper is to study conditions under which the generalized (in this sense) equations preserve the known oscillation properties of Emden-Fowler equations and conditions under which these properties are not preserved. Oscillatory properties of almost linear and essentially nonlinear differential equation with advanced argument have already been studied in [21–28]. In this paper we study oscillation properties of nth order delay Emden-Fowler equations.


## 2. Some Auxiliary Lemmas

In the sequel,C~loc⁡([t0;+∞)) will denote the set of all functions u:[t0;+∞)→R absolutely continuous on any finite subinternal of [t0;+∞) along with their derivatives of order up to including n-1.Lemma 3 (see [28]).
Letu∈C~loc⁡n-1([t0;+∞)), u(t)>0, u(n)(t)≤0  for  t≥t0, and u(n)(t)≢0 in any neighborhood of +∞. Then there exist t1≥t0 and  ℓ∈{0,…,n-1} such that  ℓ+n is odd and
(20ℓ)u(i)(t)>0fort≥t1(i=0,…,ℓ-1),(-1)i+ℓu(i)(t)>0fort≥t1(i=ℓ,…,n-1).Remark 4.
Ifn is odd and  ℓ=0, then it means that in (200) only the second inequalities are fulfilled.Lemma 5 (see [29]).
Letu∈C~loc⁡n-1([t0;+∞)) and let (20ℓ) be fulfilled for some  ℓ∈{0,…,n-1}with ℓ+n odd. Then(21)∫t0+∞‍tn-ℓ-1|u(n)(t)|dt<+∞.
If, moreover,(22)∫t0+∞‍tn-ℓ|u(n)(t)|dt=+∞,
then there exists t*>t0 such that
(23i)u(i)(t)tℓ-i↓,u(i)(t)tℓ-i-1↑(i=0,…,ℓ-1),(24)u(t)≥tℓ-1ℓ!u(ℓ-1)(t)fort≥t*,(25)u(ℓ-1)(t)≥t(n-ℓ)!∫t+∞‍sn-ℓ-1|u(n)(s)|ds+1(n-ℓ)!∫t*t‍sn-ℓ|u(n)(s)|dsfort≥t*.


## 3. Necessary Conditions for the Existence of a Solution of Type(20ℓ)

The following notation will be used throughout the work:(26)α=inf⁡{μ(t):t∈R+},β=sup⁡{μ(t):t∈R+},(27)σ(-1)(t)=sup⁡{s≥0,σ(s)≤t},σ(-k)=σ(-1)∘σ(-(k-1)),k=2,3,….Clearlyσ(-1)(t)≥t, and σ(-1) is nondecreasing and coincides with the inverse of σ when the latter exists.Definition 6.
Lett0∈R+. By Uℓ,t0 one denotes the set of all proper solutions u:[t0,+∞)→R of (1) satisfying the condition (20ℓ) with some t1≥t0.Lemma 7.
Let the conditions (2),(3) be fulfilled, letℓ∈{1,…,n-1}with  ℓ+n odd, and let u∈Uℓ,t0 be a positive proper solution of (1). If, moreover, α≥1, β<+∞,
(28ℓ)∫0+∞‍tn-ℓ(σ(t))(ℓ-1)μ(t)p(t)dt=+∞,
then for any M∈(1;+∞) there exists  t*>t0 such that for any k∈N(29)u(ℓ-1)(t)≥ρk,ℓ,t*(α)(t)fort≥σ(-k)(t*),
where α is given by the first equality of (26) and
(30)ρ1,ℓ,t*(α)(t)=ℓ!exp⁡⁡{Mℓ(α)∫σ(-1)(t*)t‍∫s+∞‍ξn-ℓ-2hhhh×(σ(ξ))1+(ℓ-1)μ(ξ)hhhh×p(ξ)dξds∫σ(-1)(t*)t‍},(31)ρi,ℓ,t*(α)(t)=ℓ!+1(n-ℓ)!×∫σ(-i)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)×p(ξ)hh×(1ℓ!ρi-1,ℓ,t*(σ(ξ)))μ(ξ)dξdshhh(i=2,…,k),(32)Mℓ(α)={1ℓ!(n-ℓ)!ifα=1,Mifα>1.Proof.
Lett0∈R+,  ℓ∈{1,…,n-1} with  ℓ+n odd and u∈Uℓ,t0 (see Definition 6) is solution of (1). Since β<+∞, according to (1), (20ℓ), and (28ℓ), it is clear that condition (22) holds. Thus, by Lemma 5 there exists t2>t1 such that the conditions (23i)–(25) with t*=t2 are fulfilled and
(33)u(ℓ-1)(t)≥t(n-ℓ)!∫t+∞‍sn-ℓ-1p(s)(u(σ(s)))μ(s)ds+1(n-ℓ)!∫t2t‍sn-ℓp(s)(u(σ(s)))μ(s)dshhhhhhhhhhhhhhhhhhhhhfort≥t2.
Observe that there existst3>t2 such that σ(t)≥t2 for t≥t3. Thus, by (24), for any t≥t3 we get
(34)u(ℓ-1)(t)≥t(n-ℓ)!∫t+∞‍sn-ℓ-1p(s)(u(σ(s)))μ(s)ds-1(n-ℓ)!∫t2t‍sd∫s+∞‍ξn-ℓ-1p(ξ)hhh×(u(σ(ξ)))μ(ξ)dξds≥1(n-ℓ)!∫t3t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)hhh×(1ℓ!u(ℓ-1)(σ(ξ)))μ(ξ)p(ξ)dξds.
According to(28ℓ) and (23ℓ-1), choose t*>t3 such that
(35)1(n-ℓ)!∫t3t*‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)hhh×(u(ℓ-1)(σ(ξ))ℓ!)μ(ξ)p(ξ)dξds>ℓ!.
By (34) and (35) we have
(36)u(ℓ-1)(t)≥ℓ!+1(n-ℓ)!∫t*t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)hhh×(u(ℓ-1)(σ(ξ))ℓ!)μ(ξ)dξdshhhhhhhhhhhhhhhhhhhhhhhhhhfort≥t*.
Letα=1. Since u(ℓ-1)(t)→+∞ as t→+∞, without loss of generally we can assume that u(ℓ-1)(σ(ξ))≥ℓ!  for ξ≥t3. Then by (23ℓ) from (36) we get
(37)u(ℓ-1)(t)≥ℓ!+1ℓ!(n-ℓ)!∫t*t‍∫s+∞‍ξn-ℓ-2hhh×(σ(ξ))1+(ℓ-1)μ(ξ)u(ℓ-1)(ξ)p(ξ)dξdshhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhfort≥t*.
It is obvious that(38)x′(t)≥u(ℓ-1)(t)ℓ!(n-ℓ)!∫t+∞‍ξn-ℓ-2(σ(ξ))1+(ℓ-1)μ(ξ)p(ξ)dξ,
where
(39)x(t)=ℓ!+1ℓ!(n-ℓ)!∫t*t‍∫s+∞‍ξn-ℓ-2×(σ(ξ))1+(ℓ-1)μ(ξ)p(ξ)u(ℓ-1)(ξ)dξds.
Thus, according to(23ℓ-1) (37), and (39) from (38) we get
(40)x′(t)≥x(t)ℓ!(n-ℓ)!∫t+∞‍ξn-ℓ-2(σ(ξ))1+(ℓ-1)μ(ξ)p(ξ)dξhhhfort≥t*.
Therefore(41)x(t)≥ℓ!exp⁡{∫t*t‍∫s+∞‍1ℓ!(n-ℓ)!×∫t*t‍∫s+∞‍ξn-ℓ-2(σ(ξ))1+(ℓ-1)μ(ξ)p(ξ)dξds}hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhfort≥t*.
Hence, according to (37) and (39)
(42)u(ℓ-1)(t)≥ρ1,ℓ,t*(1)(t)fort≥σ(-1)(t*),
where
(43)ρ1,ℓ,t*(1)(t)=ℓ!exp⁡{1ℓ!(n-ℓ)!×∫σ(-1)(t*)t‍∫s+∞‍ξn-ℓ-2×(σ(ξ))1+(ℓ-1)μ(ξ)p(ξ)dξds∫σ(-1)(t*)t‍}.
Thus, according to (36) and (42)
(44)u(ℓ-1)(t)≥ρi,ℓ,t*(1)(t)fort≥σ(-i)(t*)(i=1,…,k),
where
(45)ρi,ℓ,t*(1)(t)=ℓ!+1(n-ℓ)!×∫σ(-i)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)hhh×(1ℓ!ρi-1,ℓ,t*(1)(σ(ξ)))μ(ξ)dξdshhhhhhhhhhhhhhhhhhhhhhhhh(i=2,…,k).
Now assume thatα>1 and M∈(1,+∞). Since u(ℓ-1)(t)↑+∞ for t↑+∞, without loss of generality we can assume that (u(ℓ-1)(σ(ξ))/ℓ!)α-1≥ℓ!(n-ℓ)!M for ξ≥t*. Therefore, from (36) we have
(46)u(ℓ-1)(t)≥ℓ!+M∫t*t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)hhh×p(ξ)u(ℓ-1)(σ(ξ))dξdshhhhhhhhhhhhhhhhhhhhhhhhfort≥t*.
Taking into account (46), as above we can find that if α>1, then
(47)u(ℓ-1)(t)≥ρk,ℓ,t*(α)(t)fort≥σ(-k)(t*),
where
(48)ρ1,ℓ,t*(α)(t)=ℓ!exp⁡{M∫σ(-1)(t*)t‍∫s+∞‍ξn-ℓ-2(σ(ξ))1+(ℓ-1)μ(ξ)hhhhh×p(ξ)dξds∫σ(-1)(t*)t‍}hhhhfort≥σ(-1)(t*),(49)ρi,ℓ,t*(α)(t)=ℓ!+1(n-ℓ)!×∫σ(-i)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)×(1ℓ!ρi-1,ℓ,t*(α)(σ(ξ)))μ(ξ)dξdsfort≥σ(-i)(t*)(i=2,…,k).
According to (43)–(45) and (47)–(49), it is obvious that for any α≥1, k∈N, and M>1 there exists t*∈R+ such that (29)–(31) hold, where Mℓ(α) is defined by (32). This proves the validity of the lemma.Analogously we can prove.Lemma 8.
Let conditions (2), (3), (28ℓ) be fulfilled, letℓ∈{1,…,n-1} withℓ+n odd, 1≤β<+∞, and let u∈Uℓ,t0 be a positive proper solution of (1). Then for any M∈(1;+∞) there exists t*>t0 such that for any k∈N(50)u(ℓ-1)(t)≥ρ~k,ℓ,t*(β)(t)fort≥σ(-k)(t*),
where β is defined by the second equality of (26) and
(51)ρ~1,ℓ,t*(β)(t)=ℓ!exp⁡×{∫σ(-1)(t*)t‍M(β)×∫σ(-1)(t*)t‍∫s+∞‍ξn-ℓ-2×(σ(ξ))1+ℓμ(ξ)-βhhh×p(ξ)dξds∫σ(-1)(t*)t‍},(52)ρ~i,ℓ,t*(β)(t)=ℓ!+1(n-ℓ)!×∫σ(-i)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)hhh×(1ℓ!ρ~i-1,ℓ,t*(σ(ξ)))μ(ξ)p(ξ)dξdshhhhhhhhhhhhhhhhhhhhhhhhhhhhh(i=2,…,k),(53)M(β)={1ℓ!(n-ℓ)!ifβ=1,Mifβ>1.Remark 9.
In Lemma7, the condition β<+∞ cannot be replaced by the condition β=+∞. Indeed, let c∈(0,1). Consider (1), where n is even and
(54)σ(t)≡t,p(t)=n!tlog⁡1/cttn+1(ct-1)log⁡1/ct,β(t)=log⁡1/ct,t≥2c.
It is obvious that the functionu(t)=c-(1/t) is the solution of (1) and it satisfies the condition (201) for t≥(2/c). On the other hand, the condition (281) holds, but the condition (22) is not fulfilled.Theorem 10.
Letℓ∈{1,…,n-1} withℓ+n be odd, let β<+∞ and the conditions (2), (3), (28ℓ), and let
(55ℓ)∫0+∞‍tn-ℓ-1(σ(t))ℓμ(t)p(t)dt=+∞
be fulfilled, and for some t0∈R+, Uℓ,t0≠∅. Then for any M>1 there exists t*>t0 such that if α=1,(56)lim⁡t→+∞1t∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)hhhh×(1ℓ!ρk,ℓ,t*(1)(σ(ξ)))μ(ξ)p(ξ)dξds=0
and if α>1, then for any k∈N and δ∈(1;α],
(57)∫σ(-k)(t*)+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))(ℓ-1)μ(ξ)+δhhh×(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δp(ξ)dξds<+∞,
where α is defined by first equality of (26) and ρk,ℓ,t*(α) is given by (30)–(32).Proof.
LetM>1 and t0∈R+ such that Uℓ,t0≠∅. By definition, (1) has a proper solution u∈Uℓ,t0 satisfying the condition (20ℓ) with some tℓ≥t0. Due to (1), (20ℓ), and (28ℓ), it is obvious that condition (22) holds. Thus by Lemma 5 there exists t2>t1 such that conditions (23i)-(24) with t*=t2 are fulfilled. On the other hand, according to Lemma 7 (and its proof), we see that
(58)u(ℓ-1)(t)≥1(n-ℓ)!∫t2t‍∫s+∞‍ξn-ℓ-1p(ξ)(u(σ(ξ)))μ(ξ)dξdshhhhfort≥t2,
and there exists t*>t2 such that relation (30) is fulfilled. Without loss of generality we can assume that σ(t)≥t2 for t≥t*. Therefore, by (24), from (58) we have
(59)u(ℓ-1)(t)≥1(n-ℓ)!∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)hhhh×(1ℓ!u(ℓ-1)(σ(ξ)))μ(ξ)dξds.
Assume thatα=1. Then by (44) and (59), we have
(60)u(ℓ-1)(t)≥1(n-ℓ)!∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)×(1ℓ!ρk-1,ℓ,t*(1)(σ(ξ)))μ(ξ)dξdsfort≥σ(-k)(t*).
On the other hand, according to(23ℓ-1) and (55ℓ) it is obvious that
(61)u(ℓ-1)(t)t↓0fort↑+∞.
Therefore, from (60) we get
(62)lim⁡t→+∞1t∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)hhhh×(1ℓ!ρk,ℓ,t*(1)(σ(ξ)))μ(ξ)dξds=0.
Now assume thatα>1 and δ∈(1,α]. Then according to (47), (23ℓ-1), and (61), from (59) we have
(63)u(ℓ-1)(t)≥1(n-ℓ)!∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)hhh×(1ℓ!u(ℓ-1)(σ(ξ)))μ(ξ)-δhhh×(1ℓ!u(ℓ-1)(σ(ξ)))δdξds≥1(n-ℓ)!∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-δ(σ(ξ))δ+(ℓ-1)μ(ξ)p(ξ)hhh×(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δhhh×(1ℓ!u(ℓ-1)(ξ))δdξds≥1(n-ℓ)!∫σ(-k)(t*)t‍(uℓ-1(s)ℓ!)δhhhh×∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)p(ξ)hhhhh×(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δdξds.
Thus, we obtain(64)(v(t))δ≥1(ℓ!(n-ℓ)!)δ×(∫σ(-k)(t*)t‍vδ(s)∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)×p(ξ)(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δdξds)δ,
where v(t)=(1/ℓ!)u(ℓ-1)(t).
It is obvious that there existt1>σ(-k)(t*) such that
(65)∫σ(-k)(t*)t‍vδ(s)∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)p(ξ)hh×(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δdξds>0hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhfort≥t1.
Therefore, from (64)
(66)∫t1t‍φ′(s)(φ(s))δds≥1(ℓ!(n-ℓ)!)δ×∫t1t‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)hh×p(ξ)(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δdξds,
where
(67)φ(t)=∫σ(-k)(t*)t‍(v(s))δ∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)hhh×p(ξ)(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δdξds.
From the last inequality we get(68)∫t1t‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)p(ξ)×(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δdξds≤(ℓ!(n-ℓ)!)δδ-1[φ1-δ(t1)-φ1-δ(t)]≤(ℓ!(n-ℓ)!)δδ-1φ1-δ(t1)fort≥t1.
Passing to the limit in the latter inequality, we get(69)∫t1+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)p(ξ)hh×(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δdξds<+∞;
that is, according to (62) and (69), (56) and (57) hold, which proves the validity of the theorem.Using Lemma8 in a similar manner one can prove the following.Theorem 11.
Letℓ∈{1,…,n-1} withℓ+n be odd, let (2), (3), (28ℓ), and (55ℓ) be fulfilled, and for some t0∈R+, Uℓ,t0=∅. Then there exists t*>t0 such that if β=1, for any k∈N(70)limsup⁡t→+∞1t×∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)×(ρ~k,ℓ,t*(1)(σ(ξ)))μ(ξ)p(ξ)dξds=0,
and if 1<β<+∞, then for any k∈N and δ∈(1,β](71)∫σ(-k)(t*)+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))ℓμ(ξ)+δ-βhh×(ρ~k,ℓ,t*(β)(σ(ξ)))β-δp(ξ)dξds<+∞,
where β is defined by the second equality of (26) and ρ~k,ℓ,t*(β) is given by (51)–(53).


## 4. Sufficient Conditions for Nonexistence of Solutions of the Type(20ℓ)

Theorem 12.
Letβ<+∞,ℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), (28ℓ), and (55ℓ) be fulfilled, and if α=1, for any large t*∈R+ and for some k∈N(72ℓ)limsup⁡t→+∞1t∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)hhhh×(1ℓ!ρk,ℓ,t*(1)(σ(ξ)))μ(ξ)p(ξ)dξds>0
or if α>1, for same k∈N and δ∈(1,α](73ℓ)∫σ(-k)(t*)+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)hh×(1ℓ!ρk,ℓ,t*(α)(σ(ξ)))μ(ξ)-δp(ξ)dξds=+∞.
Then for anyt0∈R+ one has Uℓ,t0=∅, where α and β are defined by (26) and ρk,ℓ,t*(α) is given by (30)–(32).Proof.
Assume the contrary. Let there existt0∈R+ such that Uℓ,t0≠∅ (see Definition 6). Then (1) has a proper solution u:[t0,+∞)→R satisfying the condition (20ℓ). Since the condition of Theorem 10 is fulfilled, there exists t*>t0 such that if α=1 (if α>1), the condition (56) (the condition (57)) holds, which contradicts (72ℓ) and (73ℓ). The obtained contradiction proves the validity of the theorem.Using Theorem11 analogously we can prove the following.Theorem 13.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), (28ℓ), and (55ℓ) be fulfilled, and if β=1, for any large t*∈R+ and for some k∈N(74ℓ)limsup⁡t→+∞1t∫σ(-k)(t*)t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)hhh×(1ℓ!ρ~k,ℓ,t*(1)(σ(ξ)))μ(ξ)p(ξ)dξds>0
or if 1<β<+∞ for same k∈N and δ∈(1,β](75ℓ)∫σ(-k)(t*)+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))ℓμ(ξ)+δ-βhh×(ρ~k,ℓ,t*(β)(σ(ξ)))β-δp(ξ)dξds=+∞.
Then for any t0∈R+ we have Uℓ,t0=∅, where β is defined by the second equality of (26) and ρ~k,ℓ,t* is given by (51)–(53).Corollary 14.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), and (55ℓ) be fulfilled, α=1, β<+∞, and
(76ℓ)limsup⁡t→+∞1t∫0t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)dξds>0.
ThenUℓ,t0=∅ for any t0∈R+.Proof.
Since(77)ρ1,ℓ,t(α)(σ(t*))≥ℓ!fort≥σ(-1)(t*),
it suffices to note that by (76ℓ) the conditions (72ℓ) and (28ℓ) are fulfilled for k=1.Corollary 15.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2) and (3) be fulfilled, α=1, β<+∞, and
(78ℓ)liminf⁡t→+∞t∫t+∞‍sn-ℓ-2(σ(s))1+(ℓ-1)μ(s)p(s)ds=γ>0.
If, moreover, for some ε∈(0,γ)(79ℓ)limsup⁡t→+∞1t×∫0t‍∫s+∞‍ξn-ℓ-1(σ(ξ))μ(ξ)(ℓ-1+((γ-ε)/(ℓ!(n-ℓ)!)))p(ξ)dξds>0,
then for any t0∈R+, Uℓ,t0=∅.Proof.
Clearly by virtue of(78ℓ) conditions (28ℓ) and (55ℓ) are fulfilled. Let ε∈(0,γ). According to (78ℓ) and (79ℓ) it is obvious that, for large t, ρ1,ℓ,t*(1)(t)≥ℓ!t(γ-ε)/(ℓ!(n-ℓ)!). Therefore, by (79ℓ), for k=1,  (72ℓ) holds, which proves the validity of the corollary.Corollary 16.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), (28ℓ), and (55ℓ) be fulfilled, α>1, β<+∞, and for some δ∈(1,α](80ℓ)∫0+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)p(ξ)dξds=+∞.
Then for anyt0∈R+, Uℓ,t0=∅, where α is defined by the first condition of (3).Proof.
By(80ℓ) and (77), for k=1, the condition (73ℓ) holds, which proves the validity of the corollary.Corollary 17.
Letℓ∈{1,…,n-1} withℓ+n odd and let the conditions (2), (3), and (78ℓ) be fulfilled. Moreover, if α>1, β<+∞, and there exists m∈N such that(81)liminf⁡t→+∞σm(t)t>0,
then for any t0∈R+, Uℓ,t0=∅, where α is defined by the first condition of (3).Proof.
By(78ℓ) there exist c>0 and t1∈R+ such that
(82ℓ)t∫t+∞‍ξn-ℓ-2(σ(ξ))1+(ℓ-1)μ(ξ)p(ξ)dξ≥cfort≥t1.
Letδ=(1+α)/2 and M=m(1+α)/c(α-δ). Then by (82ℓ) and (30), for large t*>t1,(83)ρ1,ℓ,t*(α)(t)≥tMct≥t*.
Therefore(84)(σ(t)t)δ(1ℓ!ρ1,ℓ,t*(α)(σ(t)))μ(t)-δ≥(σ(t)t)δ(1ℓ!σMc(t))α-δ=1(ℓ!)α-δ((σ(t))1+(Mc(α-δ)/δ)t)δ>1(ℓ!)α-δ((σ(t))mt)δ.
Thus, by (81) and (78ℓ), it is obvious that (73ℓ) holds, which proves the corollary.Quite similarly one can prove the following.Corollary 18.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), (28ℓ), and (55ℓ) be fulfilled, α>1, and β<+∞. Moreover, if
(85ℓ)liminf⁡t→+∞tln⁡t∫t+∞‍ξn-ℓ-2(σ(ξ))1+(ℓ-1)μ(ξ)p(ξ)dξ>0
and for some δ∈(1,α] and m∈N(86ℓ)∫0+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))δ+(ℓ-1)μ(ξ)(ln⁡σ(ξ))mdξds=+∞,
then for any t0∈R+ one has Uℓ,t0=∅, where α and β are defined by the condition of (26).Corollary 19.
Letℓ∈{1,…,n-1} withℓ+n be odd, let the conditions (2), (3), and (28ℓ) be fulfilled, α>1, and β<+∞. Moreover, let there exist γ∈(0,1) and r∈(0,1] such that
(87ℓ)liminf⁡t→+∞tγ∫t+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)dξ>0,(88)lim⁡inf⁡t→+∞σ(t)tr>0,
and let at least one of the conditions
(89)rα≥1
or rα<1 and for some ε>0 and δ∈(1,α](90ℓ)∫0+∞‍∫s+∞‍ξn-ℓ-1-δ+(αr(1-γ)/(1-αr))-εhh×(σ(ξ))(ℓ-1)μ(ξ)p(ξ)dξds=+∞
be fulfilled. Then for any t0∈R+ one has Uℓ,t0=∅, where α is defined by (26).Proof.
It suffices to show that the condition(73ℓ) is satisfied for some k∈N and δ=(1+α)/2. Indeed, according to (87ℓ) and (88), there exist γ∈(0,1), r∈(0,1], c>0, and t1∈R+ such that(91)tγ∫t+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)dξ>cfort≥t1,(92)σ(t)≥ctrfort≥t1.
By (77), (31), and (91), from (31) we get
(93)ρ2,ℓ,t*(α)(t)≥c(n-ℓ)!∫σ(-1)(t*)t‍s-γdshhhhhhhhhhhh=c(t1-γ-σ(-1)1-γ(t*))(n-ℓ)!(1-γ)fort≥σ(-1)(t*).
Letγ1∈(γ,1). Choose t2>σ(-1)(t*) such that
(94)ρ2,ℓ,t*(α)(t)≥t1-γ1fort≥t2.
Therefore, by (91) from (31) we can find t3>t2 such that
(95)ρ3,ℓ,t*(α)(t)≥t(1-γ1)(1+αr)fort≥t3.
Hence for anyk0∈N there exists tk0 such that
(96)ρk0,ℓ,t*(α)(t)≥t(1-γ1)(1+αr+⋯+(αr)kα-2)fort≥tk0.
Assume that (89) is fulfilled. Choose k0∈N such that k0-1≥((1-r)(1+α))/((1-γ1)(α-1)). Then according to (92), (96), and (28ℓ), the condition (73ℓ) holds for k=k0 and δ=(1+α)/2. In this case, the validity of the corollary has already been proven.
Assume now that(90ℓ) is fulfilled. Let ε>0 and choose k0∈N and γ1∈(γ,1) such that
(97)(1-γ1)(1+αr+⋯+(αr)k0-2)>(1-γ)αr1-αr-ε.
Then according to (96), (92), and (90ℓ), it is obvious that (73ℓ) holds for k=k0. The proof of the corollary is complete.Using Theorem13, in a manner similar to above we can prove the following.Corollary 20.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), and (28ℓ) be fulfilled, β=1, and
(98ℓ)limsup⁡t→+∞1t∫0t‍∫s+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)dξds>0.
Then for anyt0∈R+, Uℓ,t0=∅, where β is given by (26).Corollary 21.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), and (28ℓ) be fulfilled, β=1, and
(99ℓ)liminf⁡t→+∞t∫t+∞‍ξn-ℓ-1(σ(ξ))(ℓ-1)μ(ξ)p(ξ)dξ=γ>0.
Moreover, let for someε∈(0,γ)(100ℓ)limsup⁡t→+∞1t∫0t‍∫s+∞‍ξn-ℓ-1(σ(ξ))μ(ξ)(ℓ-1+((γ-ε)/ℓ!(n-ℓ)!))hhh×p(ξ)dξds>0.
Then for anyt0∈R+ one has Uℓ,t0=∅, where β is given by (26).Corollary 22.
Letℓ∈{1,…,n-1} withℓ+n odd, let the conditions (2), (3), (28ℓ), and (55ℓ) be fulfilled, 1<β<+∞, and for some δ∈(1,β](101ℓ)∫0+∞‍∫s+∞‍ξn-ℓ-1-δ(σ(ξ))ℓμ(ξ)+δ-βp(ξ)dξds=+∞.
Then for anyt0∈R+, Uℓ,t0=∅.


## 5. Differential Equation with Property A

Theorem 23.
Let the conditions (2), (3) be fulfilled and for anyℓ∈{1,…,n-1} withℓ+n odd, the conditions (74ℓ) and (75ℓ) hold. Moreover for any large t*∈R+, if α=1 and β<+∞ for some k∈N let  (72ℓ) be fulfilled or if α>1 and β<+∞, for some k∈N, M∈(1,+∞), and δ∈(1,α], let  (72ℓ) be fulfilled. Then, if for odd n(102)∫0+∞‍tn-1p(t)dt=+∞,
then (1) has Property A, where α and β are defined by (26) and ρk,ℓ,t*(α) is given by (30)–(32).Proof.
Let (1) have a proper nonoscillatory solution u:[t0,+∞)→(0,+∞) (the case u(t)<0 is similar). Then by (2), (3), and Lemma 3 there existsℓ∈{0,…,n-1} such thatℓ+n is odd and conditions (20ℓ) hold. Since, for anyℓ∈{1,…,n-1} withℓ+n odd, the conditions of Theorem 12 are fulfilled we haveℓ∉{1,…,n-1}. Now assume thatℓ=0, n is odd, and there exists c∈(0,1) such that u(t)≥c for sufficiently large t. According to (200) since β<+∞, from (1) we have
(103)∑i=0n-1‍(n-i-1)!t1i|u(i)(t1)|≥∫t1t‍sn-1p(s)cμ(s)ds≥cβ∫t1t‍sn-1p(s)dsfort≥t1,
where t1 is a sufficiently large number. The last inequality contradicts the condition (102). The obtained contradiction proves that (1) has Property A.Theorem 24.
Let the conditions (2), (3) be fulfilled and for anyℓ∈{1,…,n-1} withℓ+n odd the conditions (28ℓ) and (55ℓ) hold. Let moreover, if β=1, for some k∈N  (74ℓ) hold or if 1<β<+∞ and for some k∈N, M∈(1,+∞), and δ∈(1,β], (75ℓ) hold. Then, if for odd n (102) is fulfilled, then (1) has Property A, where β is defined by the second equality of (26) and ρ~k,ℓ,t*(β) is given by (51)–(53).Proof.
The proof of the theorem is analogous to that of Theorem23. We simply use Theorem 13 instead of Theorem 12.Theorem 25.
Letα>1, β<+∞, let the conditions (2), (3), (281), and (551) be fulfilled, and
(104)lim⁡⁡inf⁡t→+∞(σ(t))μ(t)t>0.
If, moreover, for some k∈N, M∈(1,+∞), and δ∈(1,α], (731) holds and for odd n (102) is fulfilled, then (1) has Property A, where α and β are defined by (26) and ρk,1,t*(α) is given by (30)–(32).Proof.
It suffices to note that by (104) and (721), for anyℓ∈{2,…,n-1} there exist M>1, k∈N, and δ∈(1,α) such that condition (72ℓ) is fulfilled.Theorem 26.
Let1<β<+∞, and conditions (2), (3), (281), (29), and (104) be fulfilled. If, moreover, for some k∈N, M∈(1,+∞), and δ∈(1,β], (751) holds and for odd n (102) is fulfilled, then (1) has Property A, where β is defined by the second condition of (26) and ρ~k,1,t*(β) is given by (51)–(55).Proof.
The theorem is proved similarly to Theorem25 if we replace the condition (731) by the condition (751).Corollary 27.
Letα=1, β<+∞, and conditions (2), (3), (281), (761), and (104) be fulfilled. Then (1) has Property A, where α and β are given by (26).Proof.
By(281), (761), and (104), condition (102), and for anyℓ∈{1,…,n-1}  (76ℓ) holds. Now assume that (1) has a proper nonoscillatory solution u:[t0,+∞)→(0,+∞). Then, by (2), (3), and Lemma 3, there existsℓ∈{0,…,n-1} such thatℓ+n is odd and the condition (20ℓ) holds. Since for anyℓ∈{1,…,n-1} withℓ+n odd the conditions of Corollary 14 are fulfilled, we haveℓ∉{1,…,n-1}. Therefore n is odd andℓ=0. According to (102) and (200) it is obvious that the condition (5) holds. Therefore, (1) has Property A.Using Corollaries15–19, the validity of Corollaries 28–32 can be proven similarly to Corollary 27.Corollary 28.
Letα=1, β<+∞, and conditions (2), (3), (781), (791), and (104) be fulfilled. Then (1) has Property A, where α and β are given by (26).Corollary 29.
Letα>1, β<+∞, conditions (2), (3), (104) be fulfilled, and (801) for some δ∈(1,α] hold and if n is odd, condition (102) holds. Then (1) has Property A.Corollary 30.
Letα>1, β<+∞, the conditions (2), (3), (281), (781), (104) be fulfilled, and (81) for some m∈N holds. Then (1) has Property A.Corollary 31.
Letα>1, β<+∞, the conditions (2), (3), (281), (104), and (851) and for some δ∈(1,α] and m∈N(861) be fulfilled. Then (1) has Property A, where α and β are given by (26).Corollary 32.
Letα>1, β<+∞, and the conditions (2), (3), (281), and (104) be fulfilled. Let moreover, there exist γ∈(0,1) and r∈(0,1] such that (871) and (88) hold. Then either condition (89) or condition (901) is sufficient for (1) to have Property A.Corollary 33.
Letβ=1 and the conditions (2), (3), (281), (104), and (981) be fulfilled. Then (1) has Property A, where β is defined by the second condition of (26).Proof.
By(281), (104), and (981), the condition (102), and for anyℓ∈{1,…,n-1}  (98ℓ) holds. Therefore, by Corollary 20, for any t0∈R+ andℓ∈{1,…,n-1} withℓ+n is odd Uℓ,t0=∅. On the other hand, if n is odd andℓ=0, according to (102) it is obvious that the condition (5) holds, which proves that (1) has Property A.Using Corollaries21 and 22, we can analogously prove the following corollaries.Corollary 34.
Letβ=1 and the conditions (2), (3), (104), (991), and (1001) be fulfilled. Then (1) has Property A, where β is given by (26).Corollary 35.
Let1<β<+∞, the conditions (2), (3), (104), (281), and (291) be fulfilled, and if n is odd (102) holds. Moreover, if for some δ∈(1,β)(1011) holds, then (1) has Property A.Theorem 36.
Letα>1, β<+∞, the conditions (2), (3), (28n-1), and (29n-1) be fulfilled, and
(105)lim⁡⁡sup⁡t→+∞(σ(t))μ(t)t<+∞.
Moreover, for somek∈N, M∈(1,+∞), and δ∈(1,α], let (73n-1) be fulfilled. Then (1) has Property A, where α and β are defined by (26).Proof.
It suffices to note that by(28n-1), (29n-1), (105), and (73n-1) there exist M>1, k∈N, and δ∈(1,α] such that (28ℓ), (29ℓ), and (73ℓ) hold for anyℓ∈{1,…,n-2}.Theorem 37.
Let1<β<+∞ and the conditions (2), (3), (28n-1), (29n-1), and (105) be fulfilled, and for some k∈N, M∈(1,+∞), and δ∈(1,β], (75n-1) holds. Then (1) has Property A, where β is given by the second condition of (26).Proof.
The proof is similar to that of Theorem36 we replace the condition (73n-1) by the condition (75n-1).Corollary 38.
Letα=1, β<+∞, and the conditions (2), (3), (28n-1), (76n-1), and (105) be fulfilled. Then (1) has Property A, where α and β are given by (26).Proof.
By(28n-1), (76n-1), and (105), the condition (102), and for any ℓ∈{1,…, n-1} the condition (76ℓ) holds; it is obvious that (1) has PropertyA.Using Corollaries15–19, the validity of Corollaries 39–43 below can be proven similarly to Corollary 38.Corollary 39.
Letα=1, β<+∞, and the conditions (2), (3), (78n-1), (79n-1), and (105) be fulfilled. Then (1) has Property A, where α and β are given by (26).Corollary 40.
Letα>1, β<+∞, and the conditions (2), (3), (105), and, for some δ∈(1,α], (80n-1) be fulfilled. Then (1) has Property A, where α is given by (26).Corollary 41.
Letα>1, β<+∞, and the conditions (2), (3), (28n-1), (78n-1), (105), and for some m∈N (81) be fulfilled. Then (1) has Property A, whereαand β are given by (26).Corollary 42.
Letα>1, β<+∞, and the conditions (2), (3), (28n-1), (85n-1) and (105) be fulfilled and for some δ∈(1,α] and m∈N(86n-1) holds. Then (1) has Property A, where α and β are given by (26).Corollary 43.
Letα>1, β<+∞, and the conditions (2), (3), (28n-1), and (105) be fulfilled. Let, moreover, there exist γ∈(0,1) and r∈(0,1) such that (87n-1) and (88) hold. Then either condition (89) or condition (90n-1) is sufficient for (1) to have Property A.Corollary 44.
Letβ=1 and the conditions (2), (3), (28n-1), (105), and (98n-1) be fulfilled. Then (1) has Property A.Proof.
By(28n-1), (105), and (98n-1), the conditions (102), and for anyℓ∈{1,…,n-1}  (98ℓ) holds. Therefore, by Corollary 20, it is clear that (1) has Property A.Using Corollaries21 and 22, analogously we can prove Corollaries 45 and 5.18.Corollary 45.
Letβ=1 and the conditions (2), (3), (105), (99n-1), and (100n-1) be fulfilled. Then (1) has Property A, where β is given by (26).Corollary 46.
Let1<β<+∞ and the conditions (2), (3), (105), and (29n-1) be fulfilled. If for some δ∈(1,β], (101n-1) holds, then (1) has Property A.


## 6. Necessary and Sufficient Conditions

Theorem 47.
Letα>1 and β<+∞, let the conditions (2) and (3) be fulfilled and
(106)lim⁡⁡inf⁡t→+∞σ(t)t>0.
Then the condition (102) is necessary and sufficient for (1) to have Property A, where α and β are given by (26).Proof.
Necessity. Assume that (1) has Property A and
(107)∫0+∞‍tn-1p(t)dt<+∞.
Therefore, by Lemma 4.1 from [28], there exists c≠0 such that (1) has a proper solution u:[0,+∞)→R satisfying the condition lim⁡t→+∞⁡u(t)=c. But this contradicts the fact that (1) has Property A.
Sufficiency. By (106) and (102) it is obvious that the condition (801) holds. Therefore the sufficiency follows from Corollary 29.Remark 48.
In Theorem47 the condition β<+∞ cannot be replaced by the condition β=+∞. Indeed, let c∈(0,1/2), α=1/2c, and
(108)p(t)=n!tlgαtt1+n(1+ct)lgαtt≥1.
It is obvious that the condition (102) is fulfilled, but equation
(109)u(n)(t)+p(t)|u(t)|lgαtsign⁡u(t)=0
has solution u(t)=(1/t+c). Therefore, (109) does not have Property A.



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*Source: 168425-2014-01-22.xml*