Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

Abstract

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

1. Introduction

Consider an ordinary nonautonomous differential equation of the 𝑛th order 𝐿(𝑢(𝑡))=^𝑛_𝑗=0𝑎_𝑗(𝑡)𝑢^(𝑗)(𝑡)=0,𝑡>𝑡₀,𝑎_𝑛=1, (1) with complex-valued continuous variable coefficients 𝑎_𝑗(𝑡),  𝑗=0,…𝑛−1.

A solution of (1) is said to be oscillatory if it has an infinite sequence of zeros in (𝑡₀,∞) and nonoscillatory otherwise. Equation (1) is said to be nonoscillatory if all nontrivial solutions are nonoscillatory.

Oscillation theorems for ordinary differential equation of the 𝑛th order in the case of real variable coefficients have been studied in many papers (see [1, 2] and references therein). To the best of the author’s knowledge the oscillations of the solutions of nonautonomous 𝑛th order equations with complex coefficients have not been studied yet (except [3]).

Let 𝐶^𝑘(𝑡₀,∞) be the set of 𝑘 times differentiable functions on (𝑡₀,∞) and 𝐿₁(𝑡₀,∞) the set of Lebesgue absolutely integrable functions on (𝑡₀,∞).

To consider the case of complex coefficients we are using asymptotic solutions of (1) in Euler form 𝑢(𝑡)=𝑒^{∫𝑡𝑡0𝑓(𝑠)𝑑𝑠}. Define the characteristic function of (1) depending on a phase function 𝑓_𝑗(𝑡)∈𝐶^𝑛−1(𝑡₀,∞)𝐶ℎ_𝑗(𝑡)=𝑒^{−∫𝑡𝑡0𝑓𝑗(𝑠)𝑑𝑠}𝐿𝑒^{∫𝑡𝑡0𝑓𝑗(𝑠)𝑑𝑠},𝑗=1,2,…,𝑛. (2)

By direct calculations for 𝑗=1,2,…,𝑛 we get 𝐶ℎ_𝑗(𝑡)=^𝑛+1_𝑘=1𝑓_𝑘𝑗(𝑡)𝑎_𝑘−1(𝑡)=𝑓_𝑛+1,𝑗(𝑡)+𝑓_𝑛𝑗𝑎_𝑛−1(𝑡)+⋯+𝑓_2𝑗𝑎₁(𝑡)+𝑓_1𝑗𝑎₀(𝑡), (3) where 𝑓_1𝑗(𝑡)=1,𝑓_𝑘+1,𝑗(𝑡)=𝑑𝑑𝑡+𝑓_𝑗(𝑡)𝑓_𝑘𝑗(𝑡),𝑘=1,2,3,…,𝑛. (4) For example, 𝑓_2𝑗(𝑡)=𝑓_𝑗(𝑡),𝑓_3𝑗(𝑡)=𝑓²_𝑗(𝑡)+𝑓^′_𝑗(𝑡),𝑓_4𝑗(𝑡)=𝑓³_𝑗(𝑡)+3𝑓^′_𝑗𝑓_𝑗(𝑡)+𝑓_𝑗^′′(𝑡). (5) Define the auxiliary square matrix 𝐹(𝑡) with the entries 𝑓_𝑘𝑗(𝑡) (see (4)) 𝐹(𝑡)=𝑓_𝑘𝑗(𝑡)^𝑛_𝑘,𝑗=1, (6) and denote by 𝐹_𝑗(𝑡),  𝑗=1,…,𝑛 the (𝑛,𝑗) minors of the matrix 𝐹(𝑡).

2. Main Theorems

The basic method of this paper is a new version of Levinson’s asymptotic theorem (see [4, 5]).

Theorem 1. Assume there exist complex-valued functions 𝑓_𝑗(𝑡)∈𝐶^𝑛−1(𝑡₀,∞),  𝑗=1,2,…,𝑛 such that for all 𝑡>𝑡₀ expressions ℜ[𝑓_𝑗(𝑡)−𝑓_𝑘(𝑡)], 1≤𝑘<𝑗≤𝑛 do not change a sign; that is, ℜ𝑓_𝑗(𝑡)−𝑓_𝑘(𝑡)≤0𝑜𝑟ℜ𝑓_𝑗(𝑡)−𝑓_𝑘(𝑡)≥0,1≤𝑘<𝑗≤𝑛,𝑡>𝑡₀,_∞𝑡0𝐹_𝑗(𝑡)𝐶ℎ_𝑘(𝑡)𝐹₀(𝑡)𝐹𝑑𝑡<∞,𝑗=1,2,…,𝑛−1,𝑘=1,2,…,𝑛,_′0(𝑡)𝐹₀(𝑡)+^𝑛_𝑗=1𝑓_𝑗(𝑡)+𝑎_𝑛−1(𝑡)=0,𝑡>𝑡₀, (7)(8)(9) where characteristic functions 𝐶ℎ_𝑘(𝑡),  𝑘=1,2,…,𝑛 are defined in (2), (3), and 𝐹₀(𝑡)≡det(𝐹(𝑡)). (10)
Then solutions of (1) may be represented in the form 𝑢(𝑡)=^𝑛_𝑗=1𝜑_𝑗(𝑡)1+𝜀_𝑗(𝑡)𝐶_𝑗,lim_𝑡→∞𝜀_𝑗(𝑡)=0,𝑗=1,2,…,𝑛, (11) where 𝜑_𝑗(𝑡)=𝑒^{∫𝑡𝑡0𝑓𝑗(𝑠)𝑑𝑠},𝑗=1,2,…,𝑛. (12)

Note that condition (9) is the well-known Abel’s identity that is satisfied if (12) are exact solutions of (1).

Theorem 1 means that the error functions 𝜀_𝑗(𝑡)→0 as 𝑡→∞ if the weighted characteristic functions 𝐹_𝑗(𝑡)𝐶ℎ_𝑘(𝑡)/𝐹₀(𝑡) are absolutely integrable in (𝑡₀,∞).

In Theorem 1 the weighted characteristic functions are the entries of the perturbation matrix from Levinson’s Theorem (see Remark 18 in Section 3).

Note that the error functions 𝜀_𝑗(𝑡) in representation (11) may be estimated via the characteristic function (see, e.g., Theorem 2.2 in [6]).

Theorem 1 may be used also for the applications mentioned in [5], the stability theory, and the Dirac equation with complex coefficients (see, e.g., [6–8]).

We will say that (1) has the asymptotic solutions 𝑒^{∫𝑡𝑡0𝑓𝑗(𝑠)𝑑𝑠} corresponding to the phase functions 𝑓_𝑗(𝑡)∈𝐶^𝑛−1(𝑡₀,∞), 𝑗=1,2,…,𝑛 if (7)–(9) are satisfied.

Theorem 2. The asymptotic solution of (1) (corresponding to the phase function 𝑓_𝑘(𝑡)) generates the oscillatory solution of (1) if there exists another asymptotic solution with the phase 𝑓_𝑗(𝑡) such that ℜ𝑓_𝑘(𝑡)−𝑓_𝑗(𝑡)≡0, (13) and one of the following conditions is satisfied: ℑ𝑓_𝑘(𝑡)+𝑓_𝑗(𝑡)≡0,_∞𝑡0ℑ𝑓_𝑘(𝑡)𝑑𝑡=∞𝑜𝑟^∞_𝑡0ℑ𝑓_𝑘(𝑡)𝑑𝑡=−∞,_∞𝑡0ℑ𝑓_𝑘(𝑡)−𝑓_𝑗(𝑡)𝑑𝑡=∞𝑜𝑟^∞_𝑡0ℑ𝑓_𝑘(𝑡)−𝑓_𝑗(𝑡)𝑑𝑡=−∞. (14)(15)

The interesting question is to obtain the converse of Theorem 2.

Conditions (13) and (14) mean that the phase functions 𝑓_𝑘(𝑡),𝑓_𝑗(𝑡) are complex congugate. For the equations with real coefficients the complex phase functions appear in complex conjugate pairs. But for the equations with complex coefficients this is not true.

Since every solution of (1) with the constant complex coefficients (without repeated characteristic roots) is the linear combination of the exponents, in view of Theorem 2, the following questions arise.

Question 1. Describe the complex numbers 𝑧₁,𝑧₂,…,𝑧_𝑛 such that 𝑒^𝑧1+𝑒^𝑧2+⋯+𝑒^𝑧𝑛=0. (16)

Question 2. For which complex constant coefficients 𝑎_𝑗,  𝑗=0,1,…,𝑛 a polynomial ∑^𝑛_𝑗=0𝑎_𝑗𝑧^𝑗,  𝑛≥2 has at least one pair of complex conjugate zeros with nonzero imaginary parts?

Question 1 is complicated in the multidimensional case (see [9]), but by the substitution 𝑒^𝑦𝑛−1=𝑒^𝑧𝑛−1+𝑒^𝑧𝑛 one can reduce 𝑛 dimensional equation to the two equations: (𝑛−1)-dimensional 𝑒^𝑧1+𝑒^𝑧2+⋯+𝑒^𝑧𝑛−2+𝑒^𝑦𝑛−1=0 and 3-dimensional 𝑒^𝑦𝑛−1=𝑒^𝑧𝑛−1+𝑒^𝑧𝑛.

For the two dimensional case there is a simple answer to Question 1: 𝑒^𝑧1+𝑒^𝑧2=0 (17) if and only if (𝐴)ℜ𝑧₁−𝑧₁ℑ𝑧=0,₁−𝑧₂=(1+2𝑘)𝜋,𝑘=0,±1,±2,…. (18)

One can answer Question 2 for quadratic, cubic, and quartic equations with the complex coefficients since the simple formulas for solutions are available for these cases (see Remark 14, Theorem 15 below).

Theorem 2 shows that the oscillations of the solutions could be produced not only by the complex conjugate phase functions, which one can see from the following example.

Example 3. The equation 𝑢^′′′′(𝑡)+3+4𝑖2𝑡²𝑢^′′(𝑡)−3+4𝑖𝑡³𝑢^′(𝑡)+45+40𝑖16𝑡⁴𝑢(𝑡)=0 (19) is oscillatory, but there is no complex conjugate pair among its phase functions with nonzero imaginary parts: 𝑓₁(𝑡)=1+2𝑖2𝑡,𝑓₂(𝑡)=5−2𝑖,𝑓2𝑡₃(𝑡)=12𝑡,𝑓₄(𝑡)=5.2𝑡 (20) Note that 𝑢_𝑗(𝑡)=𝑒^{∫𝑡𝑡0𝑓𝑗(𝑠)𝑑𝑠}, 𝑗=1,2,3,4 are exact solutions of (19), 𝑢₁(𝑡)−𝑢₃(𝑡) is an oscillatory solution of (19), and (13), (15) are satisfied.

Example 4. If 𝑟(𝑡),  𝜑(𝑡)∈[0,𝜋) are a real-valued 𝐶²(𝑡₀,∞) functions such that 𝑟^−1/2(𝑡)𝑒^{−𝜑(𝑡)/2′′}𝑟^−1/2(𝑡)𝑒^{−𝜑(𝑡)/2}∈𝐿₁𝑡₀,∞, (21) then the equation 𝑢^′′(𝑡)+𝑟²(𝑡)𝑒^{2𝑖𝜑(𝑡)}𝑢(𝑡)=0 (22) is oscillatory if and only if 𝜑(𝑡)≡0,^∞_𝑡0𝑟(𝑠)𝑑𝑠=∞,𝑡>𝑡₀. (23)

The following theorem is inspired by the asymptotic theorems for two-term differential equation from [10, 11]. We deduce it from Theorem 1 by choosing the phase function 𝑓_𝑘(𝑡) as an approximate solution of the characteristic equation 𝐶ℎ_𝑘(𝑡)=0 (see [10]): 𝑓_𝑘(𝑡)=𝑏_𝑘𝑎(𝑡)−(𝑛−1)𝑎^′(𝑡)2𝑎(𝑡),𝑘=1,2,…,𝑛. (24) Note that Theorem 1 may be used to obtain other oscillation theorems by taking different approximate solutions of the characteristic equation (see [10, 12]).

Theorem 5. Assume there exists a complex-valued function 𝑎(𝑡)∈𝐶^𝑛−1(𝑡₀,∞),  𝑎(𝑡)≠0, and complex numbers 𝑏_𝑘,  𝑘=1,2,…,𝑛 such that 𝑏₁+𝑏₂+⋯+𝑏_𝑛ℜ𝑏=0_𝑘−𝑏_𝑗𝑎(𝑡)𝑑𝑜𝑛𝑜𝑡𝑐ℎ𝑎𝑛𝑔𝑒𝑎𝑠𝑖𝑔𝑛𝑜𝑛𝑡₀,∞,1≤𝑘<𝑗≤𝑛,_∞𝑡0𝑎^(1−𝑛)/2(𝑡)𝑒^{−𝑏𝑘∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}×𝐿𝑎^(1−𝑛)/2(𝑡)𝑒^{𝑏𝑘∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}𝑑𝑡<∞,𝑘=1,…,𝑛, (25)(26)(27) the set {𝑏_𝑘𝑎(𝑡)}^𝑛_𝑘=1 contains a set 𝑆={𝑏_𝑘𝑎(𝑡)}^2𝑟_𝑘=1 of 𝑟 pairs of complex conjugate functions, and ^∞_𝑡0ℑ𝑏_𝑘𝑎(𝑡)𝑑𝑡=±∞,𝑘=1,…,2𝑚,𝑚≤𝑟,_∞𝑡0ℑ𝑏_𝑘𝑎(𝑡)𝑑𝑡<∞,𝑘=2𝑚+1,𝑚+2,…,2𝑟. (28)(29)
Then (1) with 𝑎_𝑛−1(𝑡)≡0 has 2𝑚 oscillatory and (𝑛−2𝑚) nonoscillatory linearly independent solutions.

By taking 𝑎(𝑡)=1/𝑡 from Theorem 5 we get the following corollary.

Corollary 6. Assume that for some complex numbers 𝑏₁,𝑏₂,…,𝑏_𝑛, the set {𝑏_𝑗}^𝑛_𝑗=1 contains a set 𝑆={𝑏_𝑗}^2𝑟_𝑘=1 of 𝑟 pairs of complex conjugate numbers, and conditions (25), (26), ℑ𝑏_𝑗ℑ𝑏≠0,𝑗=1,…,2𝑚,_𝑗𝑎≡0,𝑗=2𝑚+1,…,2𝑟,₀(𝑡)𝑡^𝑛−1+^𝑛_𝑗=1𝑎_𝑗(𝑡)𝑡^{𝑗𝑛−1−𝑗}_𝑝=1𝑏_𝑘+𝑛+1−2𝑝2∈𝐿₁𝑡₀,∞ (30)(31) are true for all 𝑘=1,2,…,𝑛. Then (1) with 𝑎_𝑛−1(𝑡)≡0 has 2𝑚 oscillatory and (𝑛−2𝑚) nonoscillatory linearly independent solutions.

From Corollary 6 follows the well-known result.

Corollary 7 (see [1]). Assume that conditions ^∞_𝑡0𝑡^{𝑛−1−𝑗}𝑎_𝑗(𝑡)𝑑𝑡<∞,𝑗=0,1,2,…,𝑛−2 (32) are satisfied. Then (1) with 𝑎_𝑛−1(𝑡)≡0 is nonoscillatory.

For the fourth order equation we deduce the following result.

Corollary 8. Assume that for some real number 𝜀>0 and for the fixed number j from the set {0,1,2}𝑡^3−𝑗𝑎_𝑗(𝑡)+9𝑗²−15𝑗+18−2^𝑗+1𝜀²32𝑡^4−𝑗∈𝐿₁𝑡₀,𝑡,∞_3−𝑘𝑎_𝑘∈𝐿₁𝑡₀,∞,𝑘=0,1,2,𝑘≠𝑗. (33) Then equation 𝑢^′′′′(𝑡)+𝑎₂(𝑡)𝑢^′′(𝑡)+𝑎₁(𝑡)𝑢^′(𝑡)+𝑎₀(𝑡)𝑢(𝑡)=0 (34) is nonoscillatory.

Remark 9. If 𝑗=0 conditions (33) turn to ^∞_𝑡0𝑡³𝑎₀(𝑡)+9−𝜀²16𝑡⁴𝑑𝑡<∞,_∞𝑡0𝑡^3−𝑘𝑎_𝑘(𝑡)𝑑𝑡<∞,𝑘=1,2. (35)

Remark 10. To compare condition (35) with the well-known real coefficient case, note that (34) with 𝑎₁(𝑡)=𝑎₂(𝑡)≡0 and real valued 𝑎₀(𝑡) is nonoscillatory if (see [1]) ^∞_𝑡𝑠²𝑎₀(𝑠)+916𝑠⁴𝑑𝑠=^∞_𝑡𝑠²𝑎₀(𝑠)9𝑑𝑠+16𝑡>0. (36)

By taking 𝜀=𝑖𝜉, 𝑗=0 one can get another result.

Corollary 11. Assume that for some real number 𝜉 conditions ^∞_𝑡0𝑡^3−𝑘𝑎_𝑘(𝑡)𝑑𝑡<∞,^∞_𝑡0𝑡³𝑎₀(𝑡)+9+𝜉²16𝑡⁴𝑑𝑡<∞,𝑘=1,2, (37) are satisfied. Then (34) has 2 oscillatory and 2 nonoscillatory (linearly independent) solutions.

For two terms equation: 𝑢^(𝑛)(𝑡)+𝑎₀(𝑡)𝑢(𝑡)=0,𝑎₀(𝑡)=𝑎₀(𝑡)𝑒^𝑖𝜑,𝑛≥2,𝜑=0or𝜑=𝜋, (38) with real valued coefficient 𝑎₀(𝑡) by choosing 𝑎(𝑡)=𝑎₀^1/𝑛,𝑏_𝑗=𝑒^{𝑖𝜑/𝑛}𝑒^{𝑖𝜋(2𝑗+1)/𝑛},𝑗=1,…,𝑛, (39) from Theorem 5 one can deduce the following theorem.

Theorem 12. Assume 𝑎(𝑡)=|𝑎₀|^1/𝑛∈𝐶^𝑛−1(𝑡₀,∞), and for all 𝑘=1,…,𝑛^∞_𝑡0𝑎^(1−𝑛)/2(𝑡)𝑒^{−𝑏𝑘∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}𝑑^𝑛𝑑𝑡^𝑛+𝑎₀(𝑡)×𝑎^(1−𝑛)/2(𝑡)𝑒^{𝑏𝑘∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}𝑑𝑡<∞,_∞𝑡0𝑎₀(𝑡)^1/𝑛𝑑𝑡=∞. (40)(41) Then (38) has 2𝑚=2[(𝑛+𝜑/𝜋)/2]−(2𝜑/𝜋) oscillatory and 𝑛−2𝑚 nonoscillatory linearly independent solutions. Here [⋅] is the integral part of a real number.

Remark 13. If 𝑎₀(𝑡)=𝑡^𝛾, then conditions (40), (41) are simplified to 2𝛾>−.𝑛−1 (42)

Remark 14. For the quadratic equations with complex coefficients 𝑧²+𝑎₁𝑧+𝑎₀=0, it is easy to see that it has one pair of complex conjugate zeros with nonzero imaginary parts if and only if the coefficients 𝑎₀,𝑎₁ are real, and 𝑎²₁−4𝑎₀<0.

Theorem 15. The cubic equation with complex constant coefficients 𝑃(𝑧)=𝑧³+𝑎₂𝑧²+𝑎₁𝑧+𝑎₀=0 (43) has a pair of complex conjugate solutions with nonzero imaginary parts if and only if one of the following conditions is true: 𝑎₀=𝑎₀,𝑎₁=𝑎₁,𝑎₂=𝑎₂,𝐷₃(𝑡)>0 (44) or 𝑎₂≠𝑎₂,𝐷₂(𝑡)<0,𝑃𝑧₁=0, (45) where 𝐷₃𝑎=−108¹3−𝑎²₂9³𝑎−108⁰2−𝑎₁𝑎₂6+𝑎³₂27²=𝑎²₁𝑎²₂−4𝑎³₁−4𝑎³₂𝑎₀−27𝑎²₀+18𝑎₀𝑎₁𝑎₂,𝐷₂=𝑎₁−𝑎₁2(𝑎₂−𝑎₂)²−𝑎₀−𝑎₀𝑎₂−𝑎₂, (46)(47) are the discriminants of (43) and 𝑃(𝑧₁)−𝑃(𝑧₁)=0 correspondingly, 𝑧₁=𝐷₂−𝑎₁−𝑎₁2𝑎₂−𝑎₂. (48)

Example 16. For the equation 𝑧³−(5+4𝑖)𝑧²+(11+8𝑖)𝑧−15−20𝑖=0 (49) we have 𝑎₀−𝑎₀=−40𝑖,𝑎₁−𝑎₁=16𝑖,𝑎₂−𝑎₂𝐷=−8𝑖≠0,₂=𝑎₁−𝑎₁2(𝑎₂−𝑎₂)²−𝑎₀−𝑎₀𝑎₂−𝑎₂𝑧=1−5<0,₁=√1−5+1=2𝑖+1,𝑃𝑧₁=𝑃𝑧₁=0 (50) and condition (45) is satisfied.

3. Proofs

We are going to use Levinson’s asymptotic theorem as it appears in [5].

Theorem 17 (see [5]). Let Λ(𝑡)=diag(𝜆₁(𝑡),…,𝜆_𝑛(𝑡)) be an 𝑛×𝑛 diagonal matrix-function which satisfies dichotomy condition.
For each pair of integers 𝑖 and 𝑗 in [1,𝑛] (𝑖≠𝑗) exist constants 𝐾₁,𝐾₂ such that for all 𝑥 and 𝑡, 𝑡₀≤𝑡≤𝑥<∞^𝑥_𝑡ℜ𝜆_𝑖(𝑠)−𝜆_𝑗(𝑠)𝑑𝑠≤𝐾₁,𝑜𝑟^𝑥_𝑡ℜ𝜆_𝑖(𝑠)−𝜆_𝑗(𝑠)𝑑𝑠≥𝐾₂. (51) Let the 𝑛×𝑛 matrix 𝑃(𝑡) satisfy 𝑃(𝑡)∈𝐿₁(𝑡₀,∞) or ^𝑥_𝑡|𝑃(𝑡)|𝑑𝑠<∞, (52) by which one means that each entry in 𝑃(𝑡) has an absolutely convergent infinite integral. Then the system 𝑌^′(𝑡)=(Λ(𝑡)+𝑃(𝑡))𝑌(𝑡) (53) has a vector solution 𝑌(𝑡) with the asymptotic form 𝑌(𝑡)=(𝐸+𝜀(𝑡))𝑒^{∫𝑡𝑡0Λ(𝑠)𝑑𝑠}𝐶,lim_𝑡→∞𝜀(𝑡)=0, (54) where 𝐸 is the identity matrix, 𝜀(𝑡) is the 𝑛×𝑛 error matrix-function, and 𝐶=(𝐶₁,…𝐶_𝑛)^𝑡𝑟 is a constant column-vector.

Proof of Theorem 1. Rewrite (1) as a system 𝑦^′(𝑡)(𝑡)𝑦(𝑡),𝐴(𝑡)==𝐴0100⋯00010⋯0⋯⋯⋯0⋯1−𝑎₀(𝑡)−𝑎₁(𝑡)−𝑎₂(𝑡)−𝑎₃⋯−𝑎_𝑛−1(𝑡),𝑦(𝑡)=𝑢(𝑡)𝑢^′(𝑡)⋮𝑢^(𝑛−1). (55)(56) By transformation 𝑦(𝑡)=Φ(𝑡)𝑧(𝑡), (57) where matrix-function Φ(𝑡) is defined via phase functions 𝑓_𝑗(𝑡): Φ(𝑡)=𝐹(𝑡)𝑒^{∫𝑡𝑡0𝑓1(𝑠)𝑑𝑠}, (58) and the entries of the matrix 𝐹(𝑡) are defined in (4), (6), we get 𝑧^′(𝑡)=Φ⁻¹(𝑡)𝐴(𝑡)Φ(𝑡)−Φ^′(𝑡)𝑧(𝑡), (59) or, in view of identity, 𝐴(𝑡)Φ(𝑡)−Φ^′(𝑡)(𝑡)Λ(𝑡)+𝐵(𝑡)=Φ𝐹₀(𝑡), (60) we have 𝑧^′(𝑡)=Λ(𝑡)+𝐵(𝑡)𝐹₀(𝑡)𝑧(𝑡), (61) where Λ(𝑡)=00000𝑓₂(𝑡)−𝑓₁(𝑡)0000𝑓₃(𝑡)−𝑓₁(𝑡)00⋯⋯00⋯0𝑓_𝑛(𝑡)−𝑓₁(𝑡),𝐵(𝑡)=−𝐹₁𝐶ℎ₁−𝐹₁𝐶ℎ₂⋯−𝐹₁𝐶ℎ_𝑛𝐹₂𝐶ℎ₁𝐹₂𝐶ℎ₂⋯𝐹₂𝐶ℎ_𝑛−𝐹₃𝐶ℎ₁−𝐹₃𝐶ℎ₂⋯−𝐹₃𝐶ℎ_𝑛()⋯⋯⋯⋯−1^𝑛𝐹_𝑛𝐶ℎ₁(−1)^𝑛𝐹_𝑛𝐶ℎ₂⋯(−1)^𝑛𝐹_𝑛𝐶ℎ_𝑛. (62)(63) Here and further we often suppress the time variable 𝑡 for the simplicity.

Remark 18. Remarkable formulas (61)–(63) show that any 𝑛th order (1) may be transformed to the first order system 𝑧^′(𝑡)=𝐴₁(𝑡)𝑧(𝑡), where the matrix 𝐴₁(𝑡) is a sum of diagonal matrix Λ(𝑡) and perturbation matrix 𝑃(𝑡); that is, 𝐴₁(𝑡)=Λ(𝑡)+𝑃(𝑡) (Levinson form). Moreover, if the diagonal matrix is chosen in terms of phase functions as in (62), then the perturbation matrix 𝑃(𝑡)=𝐵(𝑡)/𝐹₀(𝑡) is the weighted characteristic function (i.e., the entries of matrix B are proportional to the characteristic functions 𝐶ℎ_𝑗(𝑡)). Our conjecture is that it is true not only for (1), but also for any first order system. For planar systems it was proved in [7].

To prove basic formula (60) note that by using Laplace expansion by the minors of the determinants: 𝐹₀𝑓=det¹¹𝑓₁₂⋯𝑓_1𝑛𝑓₂₁𝑓₂₂⋯𝑓_2𝑛𝑓⋮⋮⋮⋮_𝑛1𝑓_𝑛2⋯𝑓_𝑛𝑛,𝑓0=det¹¹𝑓₁₂⋯𝑓_1𝑛𝑓₂₁𝑓₂₂⋯𝑓_2𝑛𝑓⋮⋮⋮⋮_𝑘1𝑓_𝑘2⋯𝑓_𝑘𝑛,𝑘=1,…,𝑛−1, (64) we get identities: 𝐹₁𝑓_𝑛1−𝐹₂𝑓_𝑛2+⋯+(−1)^𝑛−1𝐹_𝑛𝑓_𝑛𝑛=𝐹₀(𝑡),𝐹₁𝑓_𝑘1−𝐹₂𝑓_𝑘2+⋯+(−1)^𝑛−1𝐹_𝑛𝑓_𝑘𝑛=0,𝑘=1,…,𝑛−1, (65) or (𝑓_1𝑗≡1,𝑓_2𝑗=𝑓_𝑗)𝐹₁−𝐹₂+𝐹₃+⋯+(−1)^𝑛−1𝐹_𝑛𝑓=0,₁𝐹₁−𝑓₂𝐹₂+⋯+(−1)^𝑛−1𝑓_𝑛𝐹_𝑛𝑓=0,₂₁+𝑓^′₁𝐹₁−𝑓²₂+𝑓^′₂𝐹₂+𝑓²₃+𝑓^′₃𝐹₃+(−1)^𝑛−1𝑓²_𝑛+𝑓^′_𝑛𝐹_𝑛=0. (66) By direct calculations we get 𝐴Φ−Φ^′₁₁=𝑒^{∫𝑡𝑓1𝑑𝑠}𝑓₂₁−𝑓₁Φ𝐵=0,Λ+𝐹₀₁₁=−𝐹₁+𝐹₂−𝐹₃+⋯+(−1)^𝑛𝐹_𝑛𝐶ℎ₁𝐹₀=0,𝐴Φ−Φ^′₂₁=𝑓₂−𝑓₁𝑒^{∫𝑡𝑓1𝑑𝑠}Φ𝐵Λ+𝐹₀₂₁=𝑓₂−𝑓₁𝑒^{∫𝑡𝑓1𝑑𝑠}+𝑒^{∫𝑡𝑓1𝑑𝑠}−𝐹₁+𝐹₂+⋯+(−1)^𝑛𝐹_𝑛𝐶ℎ₂𝐹₀=𝑓₂−𝑓₁𝑒^{∫𝑡𝑓1𝑑𝑠},𝐴Φ−Φ^′_𝑛1=−𝑎₀−𝑎₁𝑓₁−𝑎₂𝑓₃₁−⋯−𝑎_𝑛−1𝑓_𝑛1−𝑓_𝑛1−𝑓^′_𝑛1𝑓₁𝑒^{∫𝑡𝑓1𝑑𝑠}=−𝑒^{∫𝑡𝑓1𝑑𝑠}𝐶ℎ₁,Φ𝐵Λ+𝐹_0𝑛1=𝑒^{∫𝑡𝑓1𝑑𝑠}−𝑓_𝑛1𝐹₁+𝑓_𝑛1𝐹₂+⋯+(−1)^𝑛𝑓_𝑛𝑛𝐹_𝑛𝐶ℎ₁𝐹₀=−𝑒^{∫𝑡𝑓1𝑑𝑠}𝐶ℎ₁, (67) and the same way we get 𝐴Φ−Φ^′_𝑗𝑘=Φ𝐵Λ+𝐹_0𝑗𝑘,𝑗,𝑘=1,2,…,𝑛. (68)

To apply Theorem 17 to system (61) note that from (7) follows dichotomy condition (51) of Theorem 17. Condition (52) of Theorem 17 turns to 𝐵(𝑡)/𝐹₀(𝑡)∈𝐿₁(𝑡₀,∞): 𝐹_𝑗𝐶ℎ_𝑘(𝑡)𝐹₀(𝑡)∈𝐿₁𝑡₀,∞,𝑘,𝑗=1,2,3,…,𝑛, (69) or (8) in view of  −𝐹₁+𝐹₂−⋯+(−1)^𝑛𝐹_𝑛=0.

From Theorem 17 applied to system (61) we get 𝑧(𝑡)=𝑧₀(𝑡)(𝐸+𝜀(𝑡))𝐶,lim_𝑡→∞𝜀(𝑡)𝑧=0,₀(𝑡)=𝑒^{∫𝑡𝑡0Λ(𝑠)𝑑𝑠}=10000𝑒^{∫𝑡𝑡0(𝑓2−𝑓1)𝑑𝑠}0000𝑒^{∫𝑡𝑡0(𝑓3−𝑓1)𝑑𝑠}0000𝑒^{∫𝑡𝑡0(𝑓4−𝑓1)𝑑𝑠}. (70) From (57): 𝑦(𝑡)=Φ(𝑡)𝑧(𝑡)=Φ(𝑡)𝑧₀(𝑡)(𝐸+𝜀(𝑡))𝐶, (71) we get representation (11). If functions {𝑓_𝑗(𝑡)}^𝑛_𝑗=1 are exact solutions of 𝐶ℎ(𝑓_𝑗)=0, 𝑗=1,2,…𝑛, then 𝐹(𝑡)𝑒^{∫𝑡𝑡0𝑓1𝑑𝑠}𝑒^{∫𝑡𝑡0Λ𝑑𝑠} is the fundamental solution of (55). Abel’s identity 𝐶=det(𝐹(𝑡))𝑒^{∫𝑡𝑡0𝑓1𝑑𝑠}𝑒^{∫𝑡𝑡0Λ𝑑𝑠}𝑒^{−∫𝑡𝑡0Tr(𝐴(𝑠))𝑑𝑠}=𝐹₀(𝑡)𝑒^{∫𝑡𝑡0(𝑓1+𝑓2+⋯+𝑓𝑛+𝑎𝑛−1)(𝑠)𝑑𝑠} (72) may be written in the form (9). We always choose (approximate) phase functions 𝑓_𝑗(𝑡), 𝑗=1,2,…,𝑛 such that (9) is satisfied.

Proof of Theorem 2. From the assumptions (13), (14) of Theorem 2 one can generate an asymptotic solution as the difference of two asymptotic solutions: 𝑢₀(𝑡)=𝑒^{∫𝑡𝑡0𝑓𝑘(𝑠)𝑑𝑠}−𝑒^{∫𝑡𝑡0𝑓𝑗(𝑠)𝑑𝑠}=2𝑖𝑒^{∫𝑡𝑡0ℜ[𝑓𝑘(𝑠)]𝑑𝑠}sin^𝑡_𝑡0ℑ𝑓_𝑘(𝑠)𝑑𝑠, (73) which is oscillatory. Indeed if ∫^𝑡_𝑡0ℑ[𝑓_𝑘(𝑠)]𝑑𝑠=∞, or ∫^𝑡_𝑡0ℑ[𝑓_𝑘(𝑠)]𝑑𝑠=−∞, then the infinite sequence of zeros 𝑡_𝑝 of 𝑢₀(𝑡) is given by ^𝑡𝑝_𝑡0ℑ𝑓_𝑘(𝑠)𝑑𝑠=2𝑝𝜋,𝑝=1,2,3,…or𝑝=−1,−2,−3,…. (74)
From the assumptions (13), (15) for the solution 𝑢₀(𝑡) one can generate another infinite sequence of zeros 𝑡_𝑚 that is given by ^𝑡𝑚_𝑡0ℑ𝑓_𝑘(𝑠)−𝑓_𝑗(𝑠)𝑑𝑠=2𝑚𝜋,𝑚=1,2,3,…,or𝑚=−1,−2,−3,…. (75) Indeed we have the following: 𝑢₀𝑡_𝑚=𝑒^{∫𝑡𝑚𝑡0𝑓𝑗(𝑠)𝑑𝑠}𝑒^{∫𝑡𝑡0ℜ[𝑓𝑘(𝑠)−𝑓𝑗(𝑠)]𝑑𝑠}𝑒^{𝑖∫𝑡𝑡0ℑ[𝑓𝑘(𝑠)−𝑓𝑗(𝑠)]𝑑𝑠}−1=0. (76) Theorem 2 is followed from representation (11).

Proof of Example 4. Asymptotic solutions of (22) we choose in the form 𝑢_𝑗(𝑡)=𝑒^{∫𝑡𝑡0𝑓𝑗(𝑠)𝑑𝑠},𝑓_𝑗=𝑥_𝑗−𝑥^′_𝑗(𝑡)2𝑥_𝑗(𝑡),𝑥₁(𝑡)=𝑖𝑟(𝑡)𝑒^{𝑖𝜑(𝑡)},𝑥₂(𝑡)=−𝑥₁(𝑡) (77) or 𝑢_𝑗(𝑡)=𝐶𝑥_𝑗^−1/2(𝑡)𝑒^{∫𝑡𝑡0𝑥𝑗(𝑠)𝑑𝑠},𝑗=1,2. (78) From (4) we have the following: 𝐹(𝑡)=𝑓11₁(𝑡)𝑓₂(𝑡),𝐹₀(𝑡)=det(𝐹)=𝑓₂(𝑡)−𝑓₁(𝑡)=−2𝑥₁(𝑡),𝐹₁=𝐹₂=1,𝐶ℎ_𝑗=𝑓^′_𝑗(𝑡)+𝑓²(𝑡)−𝑥²₁(𝑡)=𝑥₁^−1/2′′(𝑡)𝑥₁^1/2(𝑡). (79)
So if (8) is satisfied, that is, 𝑥₁^−1/2′′(𝑡)𝑥₁^1/2(𝑡)∈𝐿₁𝑡₀,∞, (80) then solutions of 𝑢^′′(𝑡)+𝑟²(𝑡)𝑒^{2𝑖𝜑(𝑡)}𝑢(𝑡) may be written in the form following: 𝑢(𝑡)=𝐶₁1+𝜀₁(𝑡)𝑢₁(𝑡)+𝐶₂1+𝜀₂(𝑡)𝑢₂(𝑡),lim_𝑡→∞𝜀_𝑗(𝑡)=0,𝑗=1,2. (81) If 𝑢(𝑡) has a zero for some complex numbers 𝐶_1,2,|𝐶₁|+|𝐶₂|≠0, then 𝑥₁^−1/2(𝑡)𝐶₁𝑒^{∫𝑡𝑡0𝑥1(𝑠)𝑑𝑠}+𝐶₂𝑒^{∫𝑡𝑡0𝑥2(𝑠)𝑑𝑠}=0, (82) or 𝐶₁𝑒^{𝑖𝑦(𝑡)}+𝐶₂𝑒^{−𝑖𝑦(𝑡)}𝑦(𝑡)==0,_𝑡𝑡0𝑟(𝑠)cos𝜑(𝑠)+𝑖sin𝜑(𝑠)𝑑𝑠, (83) or −𝐶₂𝐶₁=𝑒^{2𝑖𝑦(𝑡)}. (84) By taking the derivative by 𝑡 of both sides we get 0=2𝑖𝑦^′(𝑡)𝑒^{2𝑖𝑦(𝑡)} (85) and since 𝑦^′(𝑡)=𝑟(𝑡)cos𝜑(𝑡)+𝑖sin𝜑(𝑡)≠0, (86) we get 0=𝑒^{2𝑖𝑦(𝑡)}=𝑒^{2𝑖∫𝑡𝑡0𝑟(𝑠)cos(𝜑(𝑠))𝑑𝑠}𝑒^{−2∫𝑡𝑡0𝑟sin(𝜑(𝑠))𝑑𝑠}, (87) or 0=𝑒^{−2∫𝑡𝑡0𝑟sin(𝜑(𝑠))𝑑𝑠}. (88) By taking the derivative by 𝑡 of the both sides we get 0=−2𝑟(𝑡)sin𝜑(𝑡)𝑒^{−2∫𝑡𝑡0𝑟sin(𝜑(𝑠))𝑑𝑠} (89) or 𝜑(𝑡)≡0. (90) So condition (8) turns to (21).

Proof of Theorem 5. We deduce Theorem 5 from Theorem 1 by choosing 𝑓_𝑗(𝑡) as in (24), (25). By calculations we get 𝐹₀(𝑡)=_{1≤𝑘<𝑚≤𝑛}𝑏_𝑘−𝑏_𝑚𝑎^{1+2+⋯+𝑛−1}=_{1≤𝑘<𝑚≤𝑛}𝑏_𝑘−𝑏_𝑚𝑎^{(𝑛−1)𝑛/2}. (91) Note that (9) is satisfied in view of 𝑏₁+⋯+𝑏_𝑛=0: 𝑓₁+𝑓₂+⋯+𝑓_𝑛+𝐹^′₀𝐹₀=𝑏₁+⋯+𝑏_𝑛−𝑛(𝑛−1)𝑎^′2𝑎²𝑎(𝑡)+𝑛(𝑛−1)𝑎^′2𝑎=0. (92) Further we have the following: 𝐹_𝑗=_{𝑘<𝑚<𝑛;𝑘,𝑚≠𝑗}𝑏_𝑘−𝑏_𝑚𝑎^{1+2+⋯+𝑛−2}=_{𝑘<𝑚<𝑛;𝑘,𝑚≠𝑗}𝑏_𝑘−𝑏_𝑚𝑎^{(𝑛−1)(𝑛−2)/2}, (93) and condition (8), in view of Definition (2), turns to (27): 𝐹_𝑗(𝑡)𝐶ℎ_𝑘(𝑡)𝐹₀(𝑡)=𝐶_𝑗𝐶ℎ_𝑘(𝑡)𝑎^1−𝑛(𝑡)=𝐶_𝑗𝑎^1−𝑛𝑎^(𝑛−1)/2𝑒^{−𝑏𝑘∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}×𝐿𝑎^(1−𝑛)/2𝑒^{𝑏𝑘∫𝑡𝑡0𝑎(𝑠)𝑑𝑠},𝐶_𝑗=∏_{𝑘<𝑚<𝑛;𝑘,𝑚≠𝑗}𝑏_𝑘−𝑏_𝑚∏_{1≤𝑘<𝑚≤𝑛}𝑏_𝑘−𝑏_𝑚. (94) Further (7) turns to (26). So under conditions of Theorem 5 conditions (7)–(9) are satisfied and Theorem 1 is applicable. Further from (28) we get (13), (14) is satisfied for 𝑘=1,2,…,2𝑚, and Theorem 5 is followed from Theorem 1 and Theorem 2.

Proof of Corollary 6. Corollary 6 is followed from Theorem 5 by taking 𝑎(𝑡)=1/𝑡. From (27) we have 𝑡^{(𝑛−1)/2−𝑏𝑘𝑛}_𝑗=0𝑎_𝑗(𝑡)𝑡^{𝑏𝑘+(𝑛−1)/2(𝑗)}∈𝐿₁𝑡₀,∞,𝑘=1,…,𝑛, (95) or, since (𝑡^𝛽)^(𝑗)=𝛽(𝛽−1)⋯(𝛽−𝑗+1)𝑡^𝛽−𝑗,  𝑗=1,2,…,𝑛, we get ^𝑛_𝑗=1𝑎_𝑗(𝑡)𝑡^{𝑛−1−𝑗}𝑏_𝑘+𝑛−12×𝑏_𝑘+𝑛−32⋯𝑏_𝑘+𝑛+1−2𝑗2+𝑎₀(𝑡)𝑡^𝑛−1∈𝐿₁𝑡₀,∞, (96) or condition (27) turns to (31). Further conditions (28),(29) turn to (30).

Proof of Corollary 7. Corollary 7 is followed from Corollary 6 by taking 𝑏_𝑘=𝑘−1−((𝑛−1)/2),  𝑘=1,2,…,𝑛, 𝑚=0.

Proof of Corollary 8. We deduce Corollary 8 from Corollary 6.
In the case 𝑛=4,  𝑎₃≡0 condition (29) turns to 𝑡⁻¹𝑏_𝑘+32𝑏_𝑘+12𝑏_𝑘−12𝑏_𝑘−32+𝑡𝑎₂𝑏_𝑘+32𝑏_𝑘+12+𝑡²𝑎₁𝑏_𝑘+32+𝑡³𝑎₀∈𝐿₁𝑡₀,∞,𝑘=1,2,3,4. (97)
Considering the case 𝑗=0 of Corollary 8 assume 𝑡^3−𝑘𝑎_𝑘∈𝐿₁(𝑡₀,∞),  𝑘=1,2. Then condition (97) turns to 𝑡³𝑎₀+𝑏_𝑘+3/2𝑏_𝑘+1/2𝑏_𝑘−1/2𝑏_𝑘−3/2𝑡⁴∈𝐿₁𝑡₀,∞. (98) For any real 𝜀 by choosing 𝑏₁=125−√25−𝜀²,𝑏₂=−𝑏₁,𝑏₃=125+√25−𝜀²,𝑏₄=−𝑏₃, (99) we get (𝑏²_𝑘−9/4)(𝑏²_𝑘−1/4)=(9−𝜀²)/16 and (97) turns to (33) with 𝑗=0.
In the case 𝑗=1, assuming 𝑡^3−𝑘𝑎_𝑘∈𝐿₁(𝑡₀,∞), 𝑘=0,2 (97) turns to 𝑡²𝑏_𝑘+3/2𝑎₁+𝑏²_𝑘−1/4𝑏_𝑘−3/2𝑡³∈𝐿₁𝑡₀,∞. (100) By choosing 𝑏₄=−3/2, and 𝑏₁, 𝑏₂, and 𝑏₃ as real solutions of 𝑏²_𝑘−14𝑏_𝑘−32=3−𝜀²8,𝑏₁+𝑏₂+𝑏₃=32 (101) with the negative discriminant (see (46)), we get conditions (33) with 𝑗=1: 𝑡²𝑎₁+3−𝜀²8𝑡³,𝑡^3−𝑘𝑎_𝑘∈𝐿₁𝑡₀,∞,𝑘=0,2. (102) In the case 𝑗=2, assume 𝑡^3−𝑘𝑎_𝑘∈𝐿₁(𝑡₀,∞),𝑘=0,1. Condition (97) turns to 𝑡𝑏_𝑘+3/2𝑏_𝑘+1/2𝑎₂+𝑏_𝑘−3/2𝑏_𝑘−1/2𝑡²∈𝐿₁𝑡₀,∞. (103)
By choosing 𝑏₃=−3/2, 𝑏₄=−1/2, and 𝑏_1,2 as real solutions of 𝑏_𝑘−32𝑏_𝑘−12=3−𝜀²4,𝑏₁+𝑏₂+𝑏₃+𝑏₄=0, (104) or 𝑏₁=1−1−𝜀²4,𝑏₂=1+1−𝜀²4, (105) we get nonoscillation conditions (33) with 𝑗=2: 𝑡𝑎₂+3−𝜀²8𝑡²,𝑡^3−𝑘𝑎_𝑘∈𝐿₁𝑡₀,∞,𝑘=0,1. (106)

Proof of Corollary 11. Proof of Corollary 11 is similar to proof of Corollary 8 in case 𝑗=0 with the choice 𝑏₁=(1/2)5−25+𝜉², 𝑏₂=−𝑏₁, 𝑏₃=(1/2)5+25+𝜉², and 𝑏₄=−𝑏₃. Note since Im[𝑏₁]>0,  Im[𝑏₂]<0,  and Im[𝑏₃]=Im[𝑏₄]=0 we get 2𝑚=2.

Proof of Theorem 12. We deduce Theorem 12 from Theorem 5.
From the choice (39) we get 𝑏_𝑗=𝑒^{𝑖𝜑/𝑛}𝑒^{𝑖𝜋(2𝑗+1)/𝑛},−𝑎₀(𝑡)^1/𝑛=𝑏_𝑗𝑎(𝑡),𝑗=1,…,𝑛. (107)
It is easy to see that condition (25) of Theorem 5 is satisfied. Condition (27) turns to (40). To check condition (26) consider ℜ𝑏_𝑘−𝑏_𝑗=cos𝜑+𝜋𝑛+2𝜋𝑘𝑛−cos𝜑+𝜋𝑛+2𝜋𝑗𝑛=2sin𝜑+𝜋1+𝑗+𝑘𝑛𝜋sin𝑗−𝑘𝑛. (108) Assuming 1≤𝑘<𝑗≤𝑛, we have 𝜋sin𝑗−𝑘𝑛>0, (109) and since 𝜑≡0 or 𝜑≡𝜋, the sign of ℜ[𝑏_𝑘−𝑏_𝑗] does not depend on 𝑡 and (26) is true. Let 𝑚 be the number of indices 𝑗∈{1,…,𝑛} for which ℑ𝑏_𝑗𝑎(𝑡)=𝑎₀(𝑡)^1/𝑛sin𝜑+𝜋2𝑗+1𝑛≠0. (110) Then (28) is true in view of (41).
If for some 𝑗∈{1,…,𝑛}sin𝜑+𝜋2𝑗+1𝑛=0, (111) then a nonoscillatory solution exists. This condition 𝛼+2𝑗+1𝑛𝜑∈𝑍,𝛼=𝜋, (112) that is, 𝛼=0,1 since 𝜑=0 or 𝜑=𝜋.
Consider the case 𝜑=0,𝛼=0. (113) If 𝑛 is odd, then (2𝑗+1)/𝑛 is integer if and only if 𝑗=(𝑛−1)/2 which means that the solution with 𝑗=𝑛−12,𝑛≥2 (114) is nonoscillatory, so other 2𝑚=𝑛−1=2[𝑛/2] solutions are oscillatory.
If 𝜑=0 and 𝑛 is even, then and all 2𝑚=𝑛=2[𝑛/2] solutions are oscillatory.
Further in the case 𝜑=𝜋,𝛼=1, (115) and 𝑛 is even we have the solutions with 𝑛𝑗=2−1,𝑗=𝑛−1 (116) are nonoscillatory, and the other 𝑛−2 solutions are oscillatory.
In the case 𝜑=𝜋 and 𝑛 is odd we have only one nonoscillatory solution with 𝑗=𝑛−1 and 2𝑚=𝑛−1.
So 2𝑛2𝑚=2,𝜑≡0,𝑛−2,𝜑≡𝜋,𝑛iseven,𝑛−1,𝜑≡𝜋,𝑛isodd, (117) or 2𝑚=2𝑛+𝜑/𝜋2−2𝜑𝜋. (118)

Proof of Remark 14. If 𝑧²+𝑎₁𝑧+𝑎₀=0 has 1 pair of complex conjugate zeros 𝑧₁, 𝑧₂=𝑧₁, then 𝑎₀, 𝑎₁ are real numbers since 𝑎₀=𝑧₁𝑧₂=|𝑧₁|²≥0 and 𝑎₁=−𝑧₁−𝑧₂=−𝑧₁−𝑧₁, and quadratic equation 𝑧²+𝑎₁𝑧+𝑎₀=0 has two complex conjugate roots if 𝑎²₁−4𝑎₀<0. Assuming that 𝑎₀, 𝑎₁ are real numbers and 𝑎²₁−4𝑎₀<0 from quadratic formula the equation 𝑧²+𝑎₁𝑧+𝑎₀=0 has two complex conjugate roots.

Proof of Theorem 15. Assume there exists a pair of complex conjugate solutions 𝑧₁, 𝑧₁ of 𝑃(𝑧)=𝑧³+𝑎₂𝑧²+𝑎₁𝑧+𝑎₀=0; that is, 𝑃𝑧₁=0,𝑃𝑧₁=0, (119) then denoting 𝑃(𝑧)=𝑧³+𝑎₂𝑧²+𝑎₁𝑧+𝑎₀, (120) we have 𝑃(𝑧₁)=0 as well, and 𝑃𝑧₁−𝑃𝑧₁=𝑎₂−𝑎₂𝑧²₁+𝑎₁−𝑎₁𝑧₁+𝑎₀−𝑎₀=0. (121) Consider 3 cases.
First case: 𝑎₂=𝑎₂, 𝑎₁=𝑎₁, 𝑎₀=𝑎₀, 𝐷₃>0.
In this case 𝑃(𝑧₁)−𝑃(𝑧₁)=0 is satisfied. The coefficients of 𝑃(𝑧)=0 are real and (43) has two complex conjugate solutions since discriminant 𝐷₃ of 𝑃(𝑧)=0 is positive. We drop the case 𝑎₂=𝑎₂, 𝑎₁=𝑎₁, 𝑎₀=𝑎₀, 𝐷₃<0 since in that case all solutions are real.
Second case: 𝑎₂≠𝑎₂.
In this case by solving quadratic equation 𝑃(𝑧₁)−𝑃(𝑧₁)=0 we get 𝑧₁𝑎=−¹−𝑎₁2𝑎₂−𝑎₂+𝐷₂,𝐷₂=𝑎₁−𝑎₁2(𝑎₂−𝑎₂)²−𝑎₀−𝑎₀𝑎₂−𝑎₂, (122) and 𝑃(𝑧₁)=𝑃(𝑧₁)=0 implies another solution 𝑧₂ can be found by division: 𝑃(𝑧)=𝑧−𝑧₁𝑄(𝑧),𝑄(𝑧)=𝑧²+𝑧₁+𝑎₂𝑧+𝑎₁+𝑎₂𝑧₁+𝑧²₁,𝑄𝑧₂=0. (123) Note that if discriminant 𝐷₂ of the quadratic equation 𝑃(𝑧₁)−𝑃(𝑧₁)=0 is negative, then by solving this equation we get two complex conjugate roots with nonzero imaginary parts: 𝑧₂=𝑧₁𝑎=−¹−𝑎₁2𝑎₂−𝑎₂−𝐷₂. (124)
If 𝐷₂≥0, (125) then 𝑧₁ is real, and to find another (complex) solution 𝑧₃ one should solve 𝑄𝑧₃=𝑧²₃+𝑧₁+𝑎₂𝑧₃+𝑎₁+𝑎₂𝑧₁+𝑧²₁=0. (126) Since 𝑧₃ must be a solution as well, we have 𝑄𝑧₃−𝑄𝑧₃=𝑎₂−𝑎₂𝑧₃+𝑧₁+𝑎₁−𝑎₁=0, (127) and 𝑧₃ is a real number given by 𝑧₃=−𝑧₁−𝑎₁−𝑎₁𝑎₂−𝑎₂. (128) Further since 𝑧₁, 𝑧₃ are solutions by division 𝑄(𝑧)=𝑧−𝑧₃𝑅(𝑧),𝑅(𝑧)=𝑧+𝑧₁+𝑧₃+𝑎₂, (129) and one can find the third solution 𝑧₄: 𝑧₄=−𝑧₁−𝑧₃−𝑎₂. (130) This case does not work since we have two real solutions 𝑧₁, 𝑧₃ and one complex solution 𝑧₄ of 𝑃(𝑧)=0.
Third case: 𝑎₂=𝑎₂, 𝑎₁≠𝑎₁.
In this case assuming 𝑧₅, 𝑧₅ are complex conjugate solutions of 𝑃(𝑧)=0 from 𝑃(𝑧₅)−𝑃(𝑧₅)=(𝑎₁−𝑎₁)𝑧₅+𝑎₀−𝑎₀=0 we get 𝑧₅ which is a real number given by 𝑧₅𝑎=−⁰−𝑎₀𝑎₁−𝑎₁ (131) and 𝑃(𝑧₅)=0.
If 𝑧₆ is another solution of 𝑃(𝑧)=0, by division we get 𝑃(𝑧)=𝑧−𝑧₅𝑄(𝑧),𝑄(𝑧)=𝑧²+𝑧₅+𝑎₂𝑧+𝑎₁+𝑎₂𝑧₅+𝑧²₅,𝑄𝑧₆=0. (132) Since 𝑧₆=𝑧₅ is also solution of 𝑄(𝑧)=0, we get 𝑄𝑧₅−𝑄𝑧₅=𝑎₁−𝑎₁=0, (133) and we come back to the case 1.
To prove that (45) is sufficient condition in view of 𝑃(𝑧₁)=𝑃(𝑧₁), it is enough to show that 𝑧₁𝑎=−¹−𝑎₁2𝑎₂−𝑎₂−𝑎₁−𝑎₁2𝑎₂−𝑎₂²−𝑎₀−𝑎₀𝑎₂−𝑎₂, (134) but it is true since ((𝑎₁−𝑎₁)/2(𝑎₂−𝑎₂))²−(𝑎₀−𝑎₀)/(𝑎₂−𝑎₂)<0.
It is easy to see that (44) is a sufficient condition as well.

Acknowledgment

The author wish to thank Stefan Born, Oktay Duman, Roberto Gianni, Herbert Homeier, Boris Kazarnovskii, Kus Krisnawan, Christopher Landauer, Octav Olteanu, Stepan Tersian, and Gail Wolkowicz for the fruitful discussion of Question 1 and Question 2 and the anonymous referees for the careful reading of the paper and fruitful comments and suggestions.