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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 638271, 15 pages
http://dx.doi.org/10.1155/2011/638271
Research Article

On Constants in Nonoscillation Criteria for Half-Linear Differential Equations

Department of Mathematics, Mendel University in Brno, ZemΔ›dΔ›lskΓ‘ 1, 613 00 Brno, Czech Republic

Received 15 June 2011; Accepted 22 August 2011

Academic Editor: Victor M.Β Perez Garcia

Copyright Β© 2011 Simona FiΕ‘narovΓ‘ and Robert MaΕ™Γ­k. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the half-linear differential equation (π‘Ÿ(𝑑)Ξ¦(π‘₯ξ…ž))ξ…ž+𝑐(𝑑)Ξ¦(π‘₯)=0, where Ξ¦(π‘₯)=|π‘₯|π‘βˆ’2π‘₯, 𝑝>1. Using the modified Riccati technique, we derive new nonoscillation criteria for this equation. The results are closely related to the classical Hille-Nehari criteria and allow to replace the fixed constants in known nonoscillation criteria by a certain one-parametric expression.

1. Introduction

In this paper we consider the equation𝐿[π‘₯]ξ€·ξ€·π‘₯∢=π‘Ÿ(𝑑)Ξ¦ξ…žξ€Έξ€Έξ…ž+𝑐(𝑑)Ξ¦(π‘₯)=0,(1.1) where Ξ¦(π‘₯)=|π‘₯|π‘βˆ’2π‘₯, 𝑝>1, π‘ŸβˆˆπΆ((𝑑0,∞),ℝ+), π‘βˆˆπΆ((𝑑0,∞),ℝ) for some 𝑑0. Under a solution of this equation, we understand every continuously differentiable function π‘₯ such that π‘ŸΞ¦(π‘₯β€²) is differentiable and (1.1) holds on (𝑑0,∞). This equation is called half-linear, since a constant multiple of any solution is also a solution of (1.1).

If 𝑝=2, then (1.1) reduces to the linear equationξ€·π‘Ÿ(𝑑)π‘₯ξ…žξ€Έξ…ž+𝑐(𝑑)π‘₯=0.(1.2)

For detailed discussion related to general theory as well as applications, we refer to [1]. According to [1], the classical linear Sturmian comparison theory extends to (1.1) and hence, if a solution has infinitely many zeros in a neighborhood of infinity, then the same is true for every solution. In this case, we say that (1.1) is oscillatory. In the opposite case, we say that (1.1) is nonoscillatory, as the following definition shows. Note that due to homogeneity of the set of all solutions, we can restrict ourselves to solutions which are positive in a neighborhood of infinity.

Definition 1 (nonoscillatory equation). Equation (1.1) is said to be nonoscillatory if there exist number 𝑇β‰₯𝑑0 and solution π‘₯ of (1.1) which satisfies π‘₯(𝑑)>0 for every 𝑑β‰₯𝑇.

DoΕ‘lΓ½ and ŘezníčkovΓ‘ [2] viewed (1.1) as a perturbation of another nonoscillatory half-linear differential equation𝐿[π‘₯]ξ€·ξ€·π‘₯∢=π‘Ÿ(𝑑)Ξ¦ξ…žξ€Έξ€Έξ…ž+̃𝑐(𝑑)Ξ¦(π‘₯)=0(1.3) and proved the following result. Note that π‘ž denotes the conjugate number to 𝑝 in Theorem A and in the whole paper, that is, (1/𝑝)+(1/π‘ž)=1 holds.

Theorem A (see [2, Theorem  2]). Let β„ŽβˆˆπΆ1 be a positive function such that β„Žβ€²(𝑑)>0 for large 𝑑, say 𝑑>𝑇, βˆ«βˆžπ‘Ÿβˆ’1(𝑑)β„Žβˆ’2(𝑑)(β„Žβ€²(𝑑))2βˆ’π‘d𝑑<∞, and denote 𝐹1ξ€œ(𝑑)=βˆžπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2ξ€·β„Ž(𝑠)ξ…žξ€Έ(𝑠)π‘βˆ’2.(1.4) Suppose that limπ‘‘β†’βˆžπΉ1ξ€·β„Ž(𝑑)π‘Ÿ(𝑑)β„Ž(𝑑)Ξ¦ξ…žξ€Έ(𝑑)=∞,(1.5)limπ‘‘β†’βˆžπΉ21(𝑑)π‘Ÿ(𝑑)β„Ž3ξ€·β„Ž(𝑑)ξ…žξ€Έ(𝑑)π‘βˆ’2𝐿[β„Ž](𝑑)=0.(1.6) If limsupπ‘‘β†’βˆžπΉ1(ξ€œπ‘‘)𝑑𝑇(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(1𝑠)d𝑠<,2π‘žliminfπ‘‘β†’βˆžπΉ1ξ€œ(𝑑)𝑑𝑇(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘3(𝑠)d𝑠>βˆ’2π‘ž(1.7) for some π‘‡βˆˆβ„ sufficiently large, then (1.1) is nonoscillatory.

Theorem A is sharp in the sense that a convenient choice of the function β„Ž allows to prove explicit sharp nonoscillation criteria. As a particular example, choosing β„Ž(𝑑)=𝑑(π‘βˆ’1)/𝑝ln2/𝑝𝑑 and ̃𝑐(𝑑)=((π‘βˆ’1)/𝑝)π‘π‘‘βˆ’π‘, DoΕ‘lΓ½ and ŘezníčkovΓ‘ derived the following result for the perturbed Euler differential equationξ€·Ξ¦ξ€·π‘₯ξ…žξ€Έξ€Έξ…ž+ξ‚΅ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘π‘‘βˆ’π‘ξ‚Ά+𝛿(𝑑)Ξ¦(π‘₯)=0.(1.8)

Theorem B (see [2, Corollary  1]). If limsupπ‘‘β†’βˆž1ξ€œln𝑑𝑑𝛿(𝑠)π‘ π‘βˆ’1ln21𝑠d𝑠<2ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘βˆ’1,liminfπ‘‘β†’βˆž1ξ€œln𝑑𝑑𝛿(𝑠)π‘ π‘βˆ’1ln23𝑠d𝑠>βˆ’2ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘βˆ’1,(1.9) then (1.8) is nonoscillatory.

The constants in this criterion are optimal in some sense. Really, liminfπ‘‘β†’βˆž1ξ€œln𝑑𝑑𝛿(𝑠)π‘ π‘βˆ’1ln21𝑠d𝑠>2ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘βˆ’1(1.10) guarantees oscillation of (1.8) (see [2, Theorem  1]).

A variant of Theorem A without convergent integral of π‘Ÿβˆ’1β„Žβˆ’2(β„Žβ€²)2βˆ’π‘ is the following.

Theorem C (see [3, Theorem  2]). Let β„ŽβˆˆπΆ1 be a positive function such that β„Žβ€²(𝑑)>0 for large 𝑑, say 𝑑>𝑇, 𝐹2(ξ€œπ‘‘)=𝑑𝑇dπ‘ π‘Ÿ(𝑠)β„Ž2ξ€·β„Ž(𝑠)ξ…žξ€Έ(𝑠)π‘βˆ’2.(1.11) Suppose that (1.5) and (1.6) with 𝐹1 replaced by 𝐹2 hold. If the integral ∫∞(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠)d𝑠 is convergent, and limsupπ‘‘β†’βˆžπΉ2ξ€œ(𝑑)βˆžπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘1(𝑠)d𝑠<,2π‘žliminfπ‘‘β†’βˆžπΉ2ξ€œ(𝑑)βˆžπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘3(𝑠)d𝑠>βˆ’,2π‘ž(1.12) then (1.1) is nonoscillatory.

If we take ̃𝑐(𝑑)=((π‘βˆ’1)/𝑝)π‘π‘‘βˆ’π‘+(1/2)((π‘βˆ’1)/𝑝)π‘βˆ’1π‘‘βˆ’π‘lnβˆ’2𝑑 and β„Ž(𝑑)=𝑑(π‘βˆ’1)/𝑝ln1/𝑝𝑑, then Theorem C can be applied to the perturbed Riemann-Weber equationξ€·Ξ¦ξ€·π‘₯ξ…ž+ξƒ©ξ‚΅ξ€Έξ€Έπ‘βˆ’1π‘ξ‚Άπ‘π‘‘βˆ’π‘+12ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘βˆ’1π‘‘βˆ’π‘lnβˆ’2ξƒͺ𝑑+𝛿(𝑑)Ξ¦(π‘₯)=0(1.13) and we obtain the following statement.

Theorem D (see [3, Corollary  2]). If βˆ«βˆžπ‘‘π›Ώ(𝑠)π‘ π‘βˆ’1ln𝑠d𝑠 converges and limsupπ‘‘β†’βˆžξ€œln(ln𝑑)βˆžπ‘‘π›Ώ(𝑠)π‘ π‘βˆ’11ln𝑠d𝑠<2ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘βˆ’1,liminfπ‘‘β†’βˆžξ€œln(ln𝑑)βˆžπ‘‘π›Ώ(𝑠)π‘ π‘βˆ’13ln𝑠d𝑠>βˆ’2ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘βˆ’1,(1.14) then (1.13) is nonoscillatory.

The aim of this paper is to improve Theorems A, B, C, and D and show that the constants in the inequalities involving limsup and liminf can be replaced by a certain one-parametric expression. Roughly speaking, these theorems claim that the nonoscillation is preserved if the perturbation which is measured by the expressions 𝐹1(ξ€œπ‘‘)𝑑𝑇(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠)d𝑠,𝐹2(ξ€œπ‘‘)βˆžπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠)d𝑠(1.15) is bounded in a strip between 1/2π‘ž and βˆ’3/2π‘ž for large 𝑑. We show that there is a possibility to shift this strip down. In other words, we show that if the inequality involving limes inferior is not satisfied, it can be relaxed provided the condition involving limes superior is strengthened properly. Together with these results, we prove also similar results of a different type, where (1.1) is viewed as a standalone equation and not as a perturbation of another equation (Theorems 3.1 and 3.2).

2. Preliminary Results

The main tool used in the paper is the method based on the Riccati equation𝑅[𝑀]∢=𝑀′+𝑐(𝑑)+(π‘βˆ’1)π‘Ÿ1βˆ’π‘ž(𝑑)|𝑀|π‘ž=0(2.1) which can be obtained from (1.1) by substitution 𝑀=π‘ŸΞ¦(π‘₯β€²/π‘₯). Our results are obtained from the following necessary and sufficient condition for nonoscillation of (1.1) which can be found, for example, in [1, Theorem 2.2.1].

Lemma 2.1. Equation (1.1) is nonoscillatory if and only if there exists a differentiable function 𝑀 which satisfies the Riccati type inequality 𝑅[𝑀](𝑑)≀0(2.2) for large 𝑑.

Our results heavily depend on the following relationship between the Riccati type differential operator 𝑅[β‹…] and the so-called modified Riccati operator (the operator on the right-hand side of (2.3)).

Lemma 2.2 ([4, Lemma  2.2]). Let β„Ž and 𝑀 be differentiable functions and 𝑣=β„Žπ‘π‘€βˆ’πΊ,𝐺=π‘Ÿβ„ŽΞ¦(β„Žβ€²), then one has the identity β„Žπ‘π‘…[𝑀][β„Ž]=𝑣′+β„ŽπΏ+(π‘βˆ’1)π‘Ÿ1βˆ’π‘žβ„Žβˆ’π‘žπ»(𝑑,𝑣),(2.3) where 𝐻(𝑑,𝑣)=|𝑣+𝐺|π‘žβˆ’π‘žΞ¦βˆ’1(𝐺)π‘£βˆ’|𝐺|π‘ž.

3. Main Results

In contrast to Theorems A and C in the first pair of theorems, we do not consider (1.1) as a perturbation of a nonoscillatory equation, but we consider this equation as a standalone problem.

Theorem 3.1. Let β„Ž be a function such that β„Ž(𝑑)>0 and β„Žβ€²(𝑑)β‰ 0, both for large 𝑑. Suppose that the following conditions hold: ξ€œβˆždπ‘‘π‘Ÿ(𝑑)β„Ž2(||β„Žπ‘‘)ξ…ž(||𝑑)π‘βˆ’2<∞,limπ‘‘β†’βˆž||Ξ¦ξ€·β„Žπ‘Ÿ(𝑑)β„Ž(𝑑)ξ…žξ€Έ||ξ€œ(𝑑)βˆžπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2=∞.(3.1) If limsupπ‘‘β†’βˆžξ€œβˆžπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξ€œπ‘‘[β„Ž](1β„Ž(𝑠)𝐿𝑠)d𝑠<π‘žξ‚€βˆšβˆ’π›Ό+,2𝛼liminfπ‘‘β†’βˆžξ€œβˆžπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξ€œπ‘‘β„Ž[β„Ž]1(𝑠)𝐿(𝑠)d𝑠>π‘žξ‚€βˆšβˆ’π›Όβˆ’ξ‚2𝛼(3.2) for some 𝛼>0, then (1.1) is nonoscillatory.

Proof. Denote 𝐺∢=π‘Ÿβ„ŽΞ¦(β„Žβ€²), π‘…βˆΆ=π‘Ÿβ„Ž2|β„Žβ€²|π‘βˆ’2 and 𝛼𝑣(𝑑)=βˆ’π‘žξ‚΅ξ€œβˆžπ‘‘π‘…βˆ’1ξ‚Ά(𝑠)dπ‘ βˆ’1βˆ’ξ€œπ‘‘[β„Ž]β„Ž(𝑠)𝐿(𝑠)d𝑠.(3.3) We have (π‘βˆ’1)π‘Ÿ1βˆ’π‘ž(𝑑)β„Žβˆ’π‘ž(𝑑)𝐻(𝑑,𝑣(𝑑))=(π‘βˆ’1)π‘Ÿ1βˆ’π‘ž(𝑑)β„Žβˆ’π‘žξ€Ί||||(𝑑)𝑣(𝑑)+𝐺(𝑑)π‘žβˆ’π‘žΞ¦βˆ’1||||(𝐺(𝑑))𝑣(𝑑)βˆ’πΊ(𝑑)π‘žξ€»||β„Ž=(π‘βˆ’1)π‘Ÿ(𝑑)ξ…ž||(𝑑)𝑝||||1+𝑣(𝑑)||||𝐺(𝑑)π‘žβˆ’π‘žπ‘£(𝑑)ξƒ­.𝐺(𝑑)βˆ’1(3.4)
Consider the function 𝐹(π‘₯)=|1+π‘₯|π‘žβˆ’π‘žπ‘₯βˆ’1. This function satisfies 𝐹(0)=0=πΉξ…ž(0) and πΉξ…žξ…ž(0)=π‘ž(π‘žβˆ’1). Hence, by the Taylor formula, the function 𝐹(π‘₯) can be approximated by (π‘ž(π‘žβˆ’1)/2)π‘₯2 in a neighborhood of π‘₯=0.
Conditions of the theorem imply that 𝑣(𝑑)=∫𝐺(𝑑)βˆ’(𝛼/π‘ž)βˆ’βˆžπ‘‘π‘…βˆ’1∫(𝑠)d𝑠𝑑[β„Ž]β„Ž(𝑠)𝐿(𝑠)dπ‘ πΊβˆ«(𝑑)βˆžπ‘‘π‘…βˆ’1(𝑠)dπ‘ βŸΆ0asπ‘‘βŸΆβˆž,(3.5) hence, for every πœ€>0, there exists π‘‡βˆˆβ„ such that 𝐹𝑣(𝑑)≀𝐺(𝑑)π‘ž(π‘žβˆ’1)2ξƒ©π‘ž1+πœ€βˆšπ›Όξƒͺ𝑣2(𝑑)𝐺2(𝑑)(3.6) holds for 𝑑>𝑇. At the same time πœ€ can be taken so small (will be specified later how) and 𝑇 so large that for 𝑑>𝑇 we have π›Όπœ€βˆ’π‘žβˆ’βˆš2π›Όπ‘ž<ξ€œβˆžπ‘‘π‘…βˆ’1ξ€œ(𝑠)d𝑠𝑑[β„Ž]π›Όβ„Ž(𝑠)𝐿(𝑠)d𝑠<βˆ’π‘ž+√2π›Όπ‘žβˆ’πœ€.(3.7) Consequently, (π‘βˆ’1)π‘Ÿ1βˆ’π‘ž(𝑑)β„Žβˆ’π‘žπ‘ž(𝑑)𝐻(𝑑,𝑣(𝑑))≀2ξƒ©π‘ž1+πœ€βˆšπ›Όξƒͺ||β„Žπ‘Ÿ(𝑑)ξ…ž||(𝑑)𝑝𝑣2(𝑑)𝐺2=π‘ž(𝑑)2ξƒ©π‘ž1+πœ€βˆšπ›Όξƒͺξ‚€βˆ«(𝛼/π‘ž)+βˆžπ‘‘π‘…βˆ’1∫(𝑠)dπ‘ π‘‘ξ‚β„Ž(𝑠)𝐿[β„Ž](𝑠)d𝑠2ξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2<ξ‚€ξ‚€βˆš(π‘ž/2)1+πœ€π‘ž/π›Όβˆšξ‚ξ‚ξ‚€ξ‚€ξ‚ξ‚2𝛼/π‘žβˆ’πœ€2ξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2.(3.8) Let 𝑀=β„Žβˆ’π‘(𝑣+𝐺). By Lemma 2.2, we have β„Žπ‘[]𝛼(𝑑)𝑅𝑀(𝑑)<βˆ’π‘žξ‚΅ξ€œβˆžπ‘‘π‘…βˆ’1ξ‚Ά(𝑠)dπ‘ βˆ’2π‘…βˆ’1ξ‚€ξ‚€βˆš(𝑑)+(π‘ž/2)1+πœ€π‘ž/π›Όβˆšξ‚ξ‚ξ‚€ξ‚€ξ‚ξ‚2𝛼/π‘žβˆ’πœ€2ξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2=ξ‚€ξ‚€βˆš(π‘ž/2)1+πœ€π‘ž/π›Όβˆšξ‚ξ‚ξ‚€ξ‚€ξ‚ξ‚2𝛼/π‘žβˆ’πœ€2βˆ’(𝛼/π‘ž)ξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2.(3.9) Consider the function in the numerator of the last fraction π‘žπ‘“(πœ€)=2ξƒ©π‘ž1+πœ€βˆšπ›Όβˆšξƒͺ2π›Όπ‘žξƒͺβˆ’πœ€2βˆ’π›Όπ‘ž.(3.10) We have 𝑓(0)=0 and by a direct computation π‘žπ‘“β€²(πœ€)=2π‘žβˆšπ›Όξƒ©βˆš2π›Όπ‘žξƒͺβˆ’πœ€2ξƒ©π‘žβˆ’π‘ž1+πœ€βˆšπ›Όβˆšξƒͺ2π›Όπ‘žξƒͺ,βˆ’πœ€(3.11) and hence βˆšπ‘“β€²(0)=(1βˆ’βˆš2)𝛼<0. This means that πœ€ can be taken so small that βˆšπ‘“(πœ€)<1βˆ’22πœ€βˆšπ›Ό.(3.12) Combining (3.9) and (3.12) we have β„Žπ‘[]<√(𝑑)𝑅𝑀(𝑑)ξ‚€ξ‚€1βˆ’2ξ‚ξ‚πœ€βˆš/2π›Όξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2<0(3.13) for 𝑑>𝑇 and (1.1) is nonoscillatory by Lemma 2.1.

Theorem 3.2. Let β„Ž be a function such that β„Ž(𝑑)>0 and β„Žβ€²(𝑑)β‰ 0, both for large 𝑑. Suppose that ξ€œβˆž[β„Ž]β„Ž(𝑑)𝐿(𝑑)d𝑑isconvergent,limπ‘‘β†’βˆž||Ξ¦ξ€·β„Žπ‘Ÿ(𝑑)β„Ž(𝑑)ξ…žξ€Έ||ξ€œ(𝑑)𝑑dπ‘ π‘Ÿ(𝑠)β„Ž2(||β„Žπ‘ )ξ…ž(||𝑠)π‘βˆ’2=∞.(3.14) If limsupπ‘‘β†’βˆžξ€œπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξ€œβˆžπ‘‘[β„Ž](1β„Ž(𝑠)𝐿𝑠)d𝑠<π‘žξ‚€βˆšβˆ’π›Ό+,2𝛼liminfπ‘‘β†’βˆžξ€œπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2(𝑠)|β„Žξ…ž(𝑠)|π‘βˆ’2ξ€œβˆžπ‘‘β„Ž[β„Ž]1(𝑠)𝐿(𝑠)d𝑠>π‘žξ‚€βˆšβˆ’π›Όβˆ’ξ‚2𝛼(3.15) for some 𝛼>0, then (1.1) is nonoscillatory.

Proof. With π‘…βˆΆ=π‘Ÿβ„Ž2|β„Žβ€²|π‘βˆ’2 we take 𝑣𝛼(𝑑)=π‘žξ‚΅ξ€œπ‘‘π‘…βˆ’1ξ‚Ά(𝑠)dπ‘ βˆ’1+ξ€œβˆžπ‘‘β„Ž[β„Ž](𝑠)𝐿(𝑠)d𝑠(3.16) and the proof is the same as the proof of Theorem 3.1.

The following theorems are variants of Theorems 3.1 and 3.2. In these theorems we view (1.1) as a perturbation of another (nonoscillatory) equation (1.3).

Theorem 3.3. Let β„Ž be a function such that β„Ž(𝑑)>0 and β„Žβ€²(𝑑)β‰ 0, both for large 𝑑. Suppose that (3.1) and limsupπ‘‘β†’βˆžπ‘Ÿ(𝑑)β„Ž3||β„Ž(𝑑)ξ…ž||π‘βˆ’2𝐿[]ξƒ©ξ€œ(𝑑)β„Ž(𝑑)βˆžπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξƒͺ2=0(3.17) hold. If limsupπ‘‘β†’βˆžξ€œβˆžπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξ€œπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(1𝑠)d𝑠<π‘žξ‚€βˆšβˆ’π›Ό+,2𝛼liminfπ‘‘β†’βˆžξ€œβˆžπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξ€œπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘1(𝑠)d𝑠>π‘žξ‚€βˆšβˆ’π›Όβˆ’ξ‚2𝛼(3.18) for some 𝛼>0, then (1.1) is nonoscillatory.

Proof. Denote 𝐺∢=π‘Ÿβ„ŽΞ¦(β„Žβ€²), π‘…βˆΆ=π‘Ÿβ„Ž2|β„Žβ€²|π‘βˆ’2 as in Theorem 3.1. Further 𝛼𝑣(𝑑)=βˆ’π‘žξ‚΅ξ€œβˆžπ‘‘π‘…βˆ’1ξ‚Ά(𝑠)dπ‘ βˆ’1βˆ’ξ€œπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠)d𝑠.(3.19) Similarly as in the proof of Theorem 3.1 we get (3.4), 𝑣(𝑑)=∫𝐺(𝑑)βˆ’(𝛼/π‘ž)βˆ’βˆžπ‘‘π‘…βˆ’1∫(𝑠)d𝑠𝑑(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠)dπ‘ πΊβˆ«(𝑑)βˆžπ‘‘π‘…βˆ’1(𝑠)dπ‘ βŸΆ0asπ‘‘βŸΆβˆž,(3.20) and for sufficiently small πœ€>0, there exists π‘‡βˆˆβ„ such that (π‘βˆ’1)π‘Ÿ1βˆ’π‘ž(𝑑)β„Žβˆ’π‘žξ‚€ξ‚€βˆš(𝑑)𝐻(𝑑,𝑣(𝑑))<(π‘ž/2)1+πœ€π‘ž/π›Όβˆšξ‚ξ‚ξ‚€ξ‚€ξ‚ξ‚2𝛼/π‘žβˆ’πœ€2ξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2(3.21) holds for 𝑑>𝑇. Using this estimate and Lemma 2.2, we see that the function 𝑀=β„Žβˆ’π‘(𝑣+𝐺) satisfies β„Žπ‘[]𝐿[]+ξ‚€ξ‚€βˆš(𝑑)𝑅𝑀(𝑑)<β„Ž(𝑑)β„Ž(𝑑)(π‘ž/2)1+πœ€π‘ž/π›Όβˆšξ‚ξ‚ξ‚€ξ‚€ξ‚ξ‚2𝛼/π‘žβˆ’πœ€2βˆ’(𝛼/π‘ž)ξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2(3.22) for 𝑑>𝑇. From (3.17) it follows that 𝐿[]ξ‚΅ξ€œβ„Ž(𝑑)β„Ž(𝑑)𝑅(𝑑)βˆžπ‘‘π‘…βˆ’1ξ‚Ά(𝑠)d𝑠2<√2βˆ’12πœ€βˆšπ›Ό(3.23) for large 𝑑. Using this and (3.12) we get β„Žπ‘[]𝐿[]+√(𝑑)𝑅𝑀(𝑑)<β„Ž(𝑑)β„Ž(𝑑)ξ‚€ξ‚€1βˆ’2ξ‚ξ‚πœ€βˆš/2π›Όξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2=1π‘…ξ€·βˆ«(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2ξƒ¬βˆš1βˆ’22πœ€βˆšξ‚πΏ[]ξ‚΅ξ€œπ›Ό+β„Ž(𝑑)β„Ž(𝑑)𝑅(𝑑)βˆžπ‘…βˆ’1ξ‚Ά(𝑑)d𝑑2ξƒ­<1ξ€·βˆ«π‘…(𝑑)βˆžπ‘‘π‘…βˆ’1ξ€Έ(𝑠)d𝑠2ξƒ¬βˆš1βˆ’22πœ€βˆšβˆšπ›Ό+2βˆ’12πœ€βˆšπ›Όξƒ­=0(3.24) for large 𝑑. Hence (1.1) is nonoscillatory by Lemma 2.1.

Corollary 3.4. If there exists 𝛼>0 such that limsupπ‘‘β†’βˆž1ξ€œln𝑑𝑑𝛿(𝑠)π‘ π‘βˆ’1ln2𝑠d𝑠<π‘βˆ’1π‘ξ‚Άπ‘βˆ’1ξ‚€βˆšβˆ’π›Ό+,2𝛼liminfπ‘‘β†’βˆž1ξ€œln𝑑𝑑𝛿(𝑠)π‘ π‘βˆ’1ln2𝑠d𝑠>π‘βˆ’1π‘ξ‚Άπ‘βˆ’1ξ‚€βˆšβˆ’π›Όβˆ’ξ‚,2𝛼(3.25) then (1.8) is nonoscillatory.

Proof. Choose β„Ž(𝑑)=𝑑(π‘βˆ’1)/𝑝ln2/𝑝𝑑 and ̃𝑐(𝑑)=((π‘βˆ’1)/𝑝)π‘π‘‘βˆ’π‘. The fact that (3.1) and (3.17) hold has been proved in [2, Corollary  1]. Further, ξ€œβˆžπ‘‘1π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2𝑝dπ‘ β‰ˆξ‚Άπ‘βˆ’1π‘βˆ’21ln𝑑(3.26) as shown also in [2] and the statement follows from Theorem 3.3.

Theorem 3.5. Let β„Ž be a function such that β„Ž(𝑑)>0 and β„Žβ€²(𝑑)β‰ 0, both for large 𝑑. Suppose that the following conditions hold: ξ€œβˆž(𝑐(𝑑)βˆ’Μƒπ‘(𝑑))β„Žπ‘(𝑑)d𝑑isconvergent,limπ‘‘β†’βˆžπ‘Ÿ||Ξ¦ξ€·β„Ž(𝑑)β„Ž(𝑑)ξ…žξ€Έ||ξ€œ(𝑑)𝑑dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2=∞,limsupπ‘‘β†’βˆžπ‘Ÿ(𝑑)β„Ž3||β„Ž(𝑑)ξ…ž||(𝑑)π‘βˆ’2𝐿[]ξƒ©ξ€œβ„Ž(𝑑)𝑑dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξƒͺ2=0.(3.27) If limsupπ‘‘β†’βˆžξ€œπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξ€œβˆžπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(1𝑠)d𝑠<π‘žξ‚€βˆšβˆ’π›Ό+,2𝛼liminfπ‘‘β†’βˆžξ€œπ‘‘dπ‘ π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2ξ€œβˆžπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘1(𝑠)d𝑠>π‘žξ‚€βˆšβˆ’π›Όβˆ’ξ‚2𝛼(3.28) for some 𝛼>0, then (1.1) is nonoscillatory.

Proof. Denote π‘…βˆΆ=π‘Ÿβ„Ž2|β„Žβ€²|π‘βˆ’2 as in the proof of Theorem 3.1. We take 𝑣𝛼(𝑑)=π‘žξ‚΅ξ€œπ‘‘π‘…βˆ’1ξ‚Ά(𝑠)dπ‘ βˆ’1+ξ€œβˆžπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠)d𝑠,(3.29) and the proof is the same as the proof of Theorem 3.3.

Corollary 3.6. If βˆ«βˆžπ‘‘π›Ώ(𝑠)π‘ π‘βˆ’1ln𝑠d𝑠 converges and there exists 𝛼>0 such that limsupπ‘‘β†’βˆžξ€œln(ln𝑑)βˆžπ‘‘π›Ώ(𝑠)π‘ π‘βˆ’1ξ‚΅ln𝑠d𝑠<π‘βˆ’1π‘ξ‚Άπ‘βˆ’1ξ‚€βˆšβˆ’π›Ό+,2𝛼liminfπ‘‘β†’βˆžξ€œln(ln𝑑)βˆžπ‘‘π›Ώ(𝑠)π‘ π‘βˆ’1ξ‚΅ln𝑠d𝑠>π‘βˆ’1π‘ξ‚Άπ‘βˆ’1ξ‚€βˆšβˆ’π›Όβˆ’ξ‚,2𝛼(3.30) then (1.13) is nonoscillatory.

Proof. We take ̃𝑐(𝑑)=((π‘βˆ’1)/𝑝)π‘π‘‘βˆ’π‘+1/2((π‘βˆ’1)/𝑝)π‘βˆ’1π‘‘βˆ’π‘lnβˆ’2𝑑 and β„Ž(𝑑)=𝑑(π‘βˆ’1)/𝑝ln1/𝑝𝑑. Then, as shown in the proof of [3, Corollary  2], all assumptions of Theorem 3.5 hold and ξ€œβˆžπ‘‘1π‘Ÿ(𝑠)β„Ž2||β„Ž(𝑠)ξ…ž||(𝑠)π‘βˆ’2𝑝dπ‘ β‰ˆξ‚Άπ‘βˆ’1π‘βˆ’2ln(ln𝑑).(3.31) Hence, the statement follows from Theorem 3.5.

In the following theorem we employ the technique used in previous results directly for Riccati operator from (2.1) rather than for modified Riccati operator from (2.3). This method yields a result which is (as far as we know) new even in the linear case (Theorem 3.7) and offers also a simple and alternative proof of known results (see Remark 6).

Theorem 3.7. Let the following conditions hold: ξ€œβˆžπ‘Ÿ1βˆ’π‘ž(𝑑)d𝑑<∞(3.32) and for some 𝛼>0limsupπ‘‘β†’βˆžξ‚΅ξ€œβˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ π‘βˆ’1ξ€œπ‘‘π‘(𝑠)d𝑠<βˆ’π›Ό+𝛼1/π‘ž,liminfπ‘‘β†’βˆžξ‚΅ξ€œβˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ π‘βˆ’1ξ€œπ‘‘π‘(𝑠)d𝑠>βˆ’π›Όβˆ’π›Ό1/π‘ž.(3.33) Then (1.1) is nonoscillatory.

Proof. From the assumptions of the theorem it follows that there exist πœ€ and 𝑇 such that 0<πœ€<𝛼1/π‘ž and βˆ’π›Όβˆ’π›Ό1/π‘žξ‚΅ξ€œ+πœ€<βˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ π‘βˆ’1ξ€œπ‘‘π‘(𝑠)d𝑠<βˆ’π›Ό+𝛼1/π‘žβˆ’πœ€(3.34) for every 𝑑>𝑇. Define βˆ«π‘€(𝑑)=βˆ’π›Ό(βˆžπ‘‘π‘Ÿ1βˆ’π‘ž(𝑠)d𝑠)1βˆ’π‘βˆ’βˆ«π‘‘π‘(𝑠)d𝑠. Direct computation shows π‘€ξ…ž(𝑑)+𝑐(𝑑)+(π‘βˆ’1)π‘Ÿ1βˆ’π‘ž||||(𝑑)𝑀(𝑑)π‘ž=βˆ’π›Ό(π‘βˆ’1)π‘Ÿ1βˆ’π‘žξ‚΅ξ€œ(𝑑)βˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ βˆ’π‘+(π‘βˆ’1)π‘Ÿ1βˆ’π‘ž(||||π›Όξ‚΅ξ€œπ‘‘)βˆžπ‘‘π‘Ÿ1βˆ’π‘ξ‚Ά(𝑠)d𝑠1βˆ’π‘+ξ€œπ‘‘||||𝑐(𝑠)dπ‘ π‘ž=(π‘βˆ’1)π‘Ÿ1βˆ’π‘žξ‚΅ξ€œ(𝑑)βˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ βˆ’π‘ξƒ©||||ξ‚΅ξ€œβˆ’π›Ό+𝛼+βˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ π‘βˆ’1ξ€œπ‘‘||||𝑐(𝑠)dπ‘ π‘žξƒͺ<(π‘βˆ’1)π‘Ÿ1βˆ’π‘žξ‚΅ξ€œ(𝑑)βˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ βˆ’π‘ξ€·||π›Όβˆ’π›Ό+1/π‘ž||βˆ’πœ€π‘žξ€Έ<0(3.35) for every 𝑑>𝑇. The nonoscillation of (1.1) follows from Lemma 2.1.

Corollary 3.8. For πœ†<0 denote by πœ‡(πœ†) the positive root of the equation 𝑧1/π‘ž+𝑧+πœ†=0. Denote π΄βˆ—=liminfπ‘‘β†’βˆžξ€·βˆ«βˆžπ‘‘π‘Ÿ1βˆ’π‘žξ€Έ(𝑠)dπ‘ π‘βˆ’1βˆ«π‘‘π΄π‘(𝑠)d𝑠,βˆ—=limsupπ‘‘β†’βˆžξ‚΅ξ€œβˆžπ‘‘π‘Ÿ1βˆ’π‘žξ‚Ά(𝑠)dπ‘ π‘βˆ’1ξ€œπ‘‘π‘(𝑠)d𝑠.(3.36) If (3.32) holds and π΄βˆ—β‰€βˆ’2π‘βˆ’1π‘ξ‚΅π‘βˆ’1π‘ξ‚Άπ‘βˆ’1,π΄βˆ—<ξ€·πœ‡ξ€·π΄βˆ—ξ€Έξ€Έ1/π‘žξ€·π΄βˆ’πœ‡βˆ—ξ€Έ,(3.37) then (1.1) is nonoscillatory.

Proof. It follows from the fact that 𝑦=πœ‡1/π‘ž(π‘₯)βˆ’πœ‡(π‘₯) is for π‘₯β‰€βˆ’((2π‘βˆ’1)/𝑝)((π‘βˆ’1)/𝑝)π‘βˆ’1 explicit formula for the curve given parametrically by π‘₯=βˆ’π›Όβˆ’π›Ό1/π‘ž, 𝑦=βˆ’π›Ό+𝛼1/π‘ž for 𝛼β‰₯π‘žβˆ’π‘. This curve is increasing for 𝛼>π‘žβˆ’π‘ and (3.33) means that the point (π΄βˆ—,π΄βˆ—) is below this curve. The same is ensured by inequalities (3.37).

4. Concluding Remarks and Comments

Remark 1. If we put 𝛼=1/2 in Theorems 3.3 and 3.5, we get Theorems A and C. The constant βˆšβˆ’π›Ό+2𝛼 from the condition with limes superior is maximal with this choice. As far as we know, Theorems 3.3 and 3.5 are new if 𝛼≠1/2 and Theorems 3.1 and 3.2 are new for every 𝛼>0. Similarly, if we put 𝛼=π‘žβˆ’π‘ in Theorem 3.7, then we get [1, Theorem  3.1.5].

Remark 2. If 𝛼>1/2, then both βˆšβˆ’π›ΌΒ±2𝛼 are decreasing functions of the variable 𝛼 and thus the bounds for 𝐹1∫(𝑑)𝑑𝑇(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠) and 𝐹2∫(𝑑)βˆžπ‘‘(𝑐(𝑠)βˆ’Μƒπ‘(𝑠))β„Žπ‘(𝑠)d𝑠 which guarantee nonoscillation in Theorems 3.3 and 3.5 also decrease. Closer investigation shows that the bound for limes inferior decreases faster and thus the maximal allowed difference between limes superior and inferior is allowed to be bigger if both are small. A similar remark applies also to the results from Theorems 3.1 and 3.2.

Remark 3. The nonoscillation criteria in the previous theorems are written in the form limsupπ‘‘β†’βˆž1𝑓(𝑑)<π‘žξ‚€βˆšβˆ’π›Ό+2𝛼,liminfπ‘‘β†’βˆž1𝑓(𝑑)>π‘žξ‚€βˆšβˆ’π›Όβˆ’ξ‚,2𝛼(4.1) where the definition of 𝑓(𝑑) varies for each particular theorem. Note that if inequalities (4.1) hold for some 𝛼 which satisfies 0<𝛼<1/2, then they hold also for 𝛼=1/2. In view of this fact it is reasonable to suppose 𝛼β‰₯1/2.

Remark 4. Denote π‘“βˆ—=liminfπ‘‘β†’βˆžπ‘“(𝑑),π‘“βˆ—=limsupπ‘‘β†’βˆžπ‘“(𝑑).(4.2) It is easy to show that the parametric curve √π‘₯=(1/π‘ž)(βˆ’π›Όβˆ’2𝛼),β€‰β€‰βˆšπ‘¦=(1/π‘ž)(βˆ’π›Ό+2𝛼),  𝛼β‰₯1/2, is an increasing function with nonparametric equation βˆšπ‘¦=π‘₯βˆ’(2/π‘ž)+(2/π‘ž)1βˆ’2π‘žπ‘₯. Since (4.1) expresses the fact that the point (π‘“βˆ—,π‘“βˆ—) is below this curve, inequalities (4.1) are satisfied if π‘“βˆ—3β‰€βˆ’2π‘ž,π‘“βˆ—<π‘“βˆ—βˆ’2π‘ž+2π‘žβˆš1βˆ’2π‘žπ‘“βˆ—.(4.3)

Remark 5. Let us compare our results with known results on a particular example of perturbed Euler equation (1.8). Let us denote 𝑝𝑔(𝑑)=ξ‚Άπ‘βˆ’1π‘βˆ’11ξ€œln𝑑𝑑𝛿(𝑠)π‘ π‘βˆ’1ln2𝑔𝑠d𝑠,βˆ—=liminfπ‘‘β†’βˆžπ‘”(𝑑),π‘”βˆ—=limsupπ‘‘β†’βˆžπ‘”(𝑑),(4.4) and rewrite the conditions from Corollary 3.4 into π‘”βˆ—βˆš>βˆ’π›Όβˆ’2𝛼,π‘”βˆ—βˆš<βˆ’π›Ό+2𝛼.(4.5) It is easy to see that every equation in the form (1.8) can be associated with some point in the plane π‘”βˆ— and π‘”βˆ—. On the other hand, for every point in this plane which satisfies π‘”βˆ—β‰₯π‘”βˆ—, we can construct an equation in the form (1.8) which is associated with this point.
As mentioned before, DoΕ‘lΓ½ and ŘezníčkovΓ‘ [2] proved that (1.8) is oscillatory if π‘”βˆ—>1/2 and nonoscillatory if π‘”βˆ—>βˆ’3/2 and π‘”βˆ—<1/2. This gives two regions in π‘”βˆ—π‘”βˆ— plane with resolved oscillation properties of (1.8): the unbounded region of oscillation has the form of the angle with vertex [1/2,1/2], rays π‘”βˆ—=1/2 and π‘”βˆ—=π‘”βˆ—, open up and the bounded region of nonoscillation has the form of triangle with vertexes [1/2,1/2], [βˆ’3/2,1/2], and [βˆ’3/2,βˆ’3/2]. Corollary 3.4 allows us to extend the region of nonoscillation by the unbounded region which is between the line π‘”βˆ—=π‘”βˆ— and the curve given for 𝛼β‰₯1/2 parametrically by π‘”βˆ—βˆš=βˆ’π›Όβˆ’2𝛼, π‘”βˆ—βˆš=βˆ’π›Ό+2𝛼 or, equivalently, given by π‘”βˆ—=π‘”βˆ—βˆšβˆ’2+21βˆ’2π‘”βˆ— for π‘”βˆ—β‰€βˆ’3/2. All these regions are shown on Figure 1.
In particular, if 𝛿(𝑑)=π‘˜((π‘βˆ’1)/𝑝)π‘βˆ’1π‘‘βˆ’π‘lnβˆ’2(𝑑), then π‘”βˆ—=π‘”βˆ—=π‘˜ and (1.8) is oscillatory if π‘˜>1/2 and nonoscillatory if π‘˜β‰€1/2. This observation has been made already by Elbert and Schneider in [5] and has a close connection with the so-called conditional oscillation, see [6] for more details related to conditional oscillation.

638271.fig.001
Figure 1: Regions of oscillation and nonoscillation of perturbed Euler equation.

Remark 6. If the integral βˆ«βˆžπ‘(𝑠)d𝑠 is convergent and if we use the method from Theorem 3.7 and Corollary 3.8 with βˆ«π‘€(𝑑)=𝛼(π‘‘π‘Ÿ1βˆ’π‘ž(𝑠)d𝑠)βˆ’1+βˆ«βˆžπ‘‘π‘(𝑠)d𝑠, we get the second part of [1, Theorem  3.3.6], which is originally due for 𝑝≀2 to Kandelaki et al. ([7, Theorem  1.6]).

Remark 7. If 𝛼=1/4 and 𝑝=2, then Theorem 3.7 reduces to well-known Hille-Nehari nonoscillation criteria. In this case the constants from (3.33) reduce to 1/4 and βˆ’3/4.

Remark 8. If we use the additional condition liminfπ‘‘β†’βˆžπ‘Ÿξ€·β„Ž(𝑑)β„Ž(𝑑)Ξ¦ξ…žξ€Έ(𝑑)>0,(4.6) then the conclusion related to that of Theorems 3.2 and 3.5 can be derived from known results. Really, denote 𝑅=π‘Ÿβ„Ž2|β„Žβ€²|π‘βˆ’2 and 𝐢=β„ŽπΏ[β„Ž], where β„Ž is a positive function such that β„Žβ€²β‰ 0 and suppose that βˆ«βˆžπ‘…βˆ’1(𝑑)d𝑑=∞ and ∫∞𝐢(𝑑)d𝑑 is convergent. Under these conditions, an alternative version of Theorem 3.2 can be derived using the so-called linearization technique. This technique is based on comparison of the (non)oscillation of (1.1) with that of a certain linear equation. The relation between these equations is hidden in identity (2.3) which (after the quadratization of the last term on the right-hand side) relates the associated Riccati operators. More precisely, nonoscillation of (1.1) is implied by nonoscillation of the linear equation (𝑅(𝑑)𝑦′)+π‘ž+πœ€2𝐢(𝑑)𝑦=0,(4.7) where πœ€>0 is arbitrary. Applying the linear version (𝑝=2) of the criteria discussed in Remark 6 to the above linear equation, we obtain the conditions for limes inferior and superior from Theorem 3.2. Similarly if 𝐢=(π‘βˆ’Μƒπ‘)β„Žπ‘, where β„Ž is a positive solution of (1.3), then we get the conditions for limes inferior and superior from Theorem 3.5. Note that if 𝑝=2, then 𝑅(𝑑) and 𝐢(𝑑) are the coefficients of the equation which results from (1.2) (the special case of (1.1) for 𝑝=2) upon the transformation π‘₯=β„Žπ‘¦. We refer to [8–10] for results concerning the linearization technique and for the half-linear Hille-Nehari type criteria derived using this technique from the classical criteria mentioned in Remark 7.

Acknowledgment

Research supported by the Grant P201/10/1032 of the Czech Science Foundation.

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