On Oscillations of Solutions of Third-Order Dynamic Equation

Abstract

We are proving the new oscillation theorems for the solutions of third-order linear nonautonomous differential equation with complex coefficients. In the case of real coefficients we derive the oscillation criterion that is invariant with respect to the adjoint transformation. Our main tool is a new version of Levinson's asymptotic theorem.

1. Introduction

Consider an ordinary nonautonomous differential equation of the third order 𝐿𝑣=𝑣^(𝑡)−3𝑎₂(𝑡)𝑣^(𝑡)+6𝑎₁(𝑡)𝑣^(𝑡)+2𝑎₀(𝑡)𝑣(𝑡)=0(1.1) with complex valued variable coefficients 𝑎₀(𝑡), 𝑎₁(𝑡), and 𝑎₂(𝑡).

A solution of (1.1) is said to be oscillatory if it has an infinite sequence of zeros in (𝑡₀,∞), and nonoscillatory, otherwise. Equation (1.1) is said to be non-oscillatory if all solutions are non-oscillatory and is said to be oscillatory if there exists at least one oscillatory solution.

Oscillation theorems for ordinary differential equation of the third order in the case of real variable coefficients have been studied in [1–7]. To the best of the author's knowledge, the oscillations of the solutions of nonautonomous third order equations with complex coefficients have not been studied yet.

Let 𝐶^𝑘(𝑡₀,∞) be the set of 𝑘 times differentiable functions on (𝑡₀,∞). By substitution 𝑣(𝑡)=𝑢(𝑡)𝑒^{∫𝑡𝑡0𝑎2(𝑠)𝑑𝑠} equation (1.1) with 𝑎₂(𝑡)∈𝐶²(𝑡₀,∞) turns to the following equation: 𝑃𝑢=𝑢^(𝑡)+3𝐼₁(𝑡)𝑢^(𝑡)+2𝐼₂(𝑡)𝑢(𝑡)=0,(1.2) where the functions 𝐼₁(𝑡) and  𝐼₂(𝑡) are given by 𝐼₁(𝑡)=2𝑎₁(𝑡)+𝑎^2(𝑡)−𝑎²²(𝑡),𝐼₂(𝑡)=𝑎₀(𝑡)+3𝑎₁(𝑡)𝑎₂(𝑡)−𝑎³²(𝑡)+𝑎^2(𝑡)2.(1.3)

If 0<|𝑒^{∫∞𝑡0𝑎2(𝑠)𝑑𝑠}|<∞ or −∞<^∞𝑡₀ℜ𝑎₂(𝑠)𝑑𝑠<∞,(1.4) then the solutions of (1.1) and (1.2) have the same oscillation properties; that is, (1.1) is oscillatory if and only if (1.2) is oscillatory.

Define characteristic (Weierstrass) function of (1.2) depending on a phase function 𝜂_𝑗(𝑡)Char_𝑗(𝑡)=Char𝜂_𝑗=𝑒^{−∫𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠}𝑃𝑒^{∫𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠},𝑗=1,2,3.(1.5) By direct calculations Char_𝑗(𝑡)=𝜂^𝑗(𝑡)+3𝜂^𝑗(𝑡)𝜂_𝑗(𝑡)+𝜂^3𝑗+3𝜂_𝑗(𝑡)𝐼₁(𝑡)+2𝐼₂(𝑡),𝑗=1,2,3.(1.6) To consider the case of complex coefficients, we are using asymptotic solutions of (1.2) in Euler form 𝑢(𝑡)=𝑒^{∫𝑡𝑡0𝜂(𝑠)𝑑𝑠} with phase functions 𝜂_𝑗(𝑡), 𝑗=1,2,3, that are approximate solutions of the characteristic equation Char(𝜂_𝑗)=0.

Theorem 1.1 (see [4]). If Mammana's condition 𝑀(𝑡)=𝐼₂(𝑡)−3𝐼^1(𝑡)4>0𝑜𝑟𝑀(𝑡)<0(1.7) is satisfied except at isolated points at which 𝑀(𝑡) may vanish, then (1.2) is oscillatory if and only if its adjoint is oscillatory.

We will show that Mammana's condition (1.7) is connected with the dichotomy condition of Levinson, and it has a topological character (see condition (2.24) below).

Theorem 1.2 (see Lazer [5]). Assume that conditions 𝐼₁(𝑡)≤0,𝐼₂(𝑡)>0,𝑡>𝑡₀,(1.8)_∞𝑡0𝐼²(𝑡)−−𝐼₁(𝑡)^3/2𝑑𝑡=∞(1.9) are satisfied. Then (1.2) with the real coefficients is oscillatory.

The adjoint transformation 𝐼₂(𝑡)→(3𝐼^1(𝑡)/2)−𝐼₂(𝑡), or 𝑀(𝑡)→−𝑀(𝑡) transforms (1.2) to its adjoint equation −𝑤^(𝑡)−3𝐼₁(𝑡)𝑤^(𝑡)+(2𝐼₂(𝑡)−3𝐼^1(𝑡))𝑤(𝑡)=0. Note that condition (1.9) is not invariant with respect to the adjoint transformation 𝑀(𝑡)→−𝑀(𝑡). In the case of real coefficients under some restrictions, we will give the criterion of oscillations of solutions of (1.2) that is invariant with respect to the adjoint transformation (see Theorem 2.9 below).

2. Main Theorems

Let 𝑊[𝑡,𝑎,𝑏]=𝑎(𝑡)𝑏′(𝑡)−𝑎′(𝑡)𝑏(𝑡) be the Wronskian of two differentiable functions 𝑎(𝑡) and 𝑏(𝑡). The following asymptotic theorem is proved by using Levinson's asymptotic theorem [8].

Theorem 2.1. Assume that there exists complex-valued phase functions 𝜂_𝑗(𝑡)∈𝐶²(𝑡₀,∞), 𝑗=1,2,3, such that expressions ℜ[𝜂_𝑗(𝑡)−𝜂_𝑘(𝑡)], 𝑘≤𝑗 do not change a sign, that is, ℜ𝜂_𝑗(𝑡)−𝜂_𝑘(𝑡)≤0𝑜𝑟ℜ𝜂_𝑗(𝑡)−𝜂_𝑘(𝑡)≥0,𝑘,𝑗=1,2,3,𝑘≤𝑗,𝑡>𝑡₀,(2.1)_∞𝑡0||𝜂₁₃(𝑡)||+||𝜂₂₃(𝑡)||||Char₂(𝑡)||||𝐺(𝑡)||𝑑𝑡<∞,𝜂^𝑗𝑘(𝑡)=𝜂_𝑗(𝑡)−𝜂_𝑘(𝑡),𝑘,𝑗=1,2,3,(2.2)_∞𝑡0||𝜂¹³(𝑡)||²+||𝜂₂₃(𝑡)||²||Char²(𝑡)−Char₃(𝑡)||||𝜂₂₃(𝑡)𝐺(𝑡)||𝑑𝑡<∞,(2.3) where 𝐺(𝑡)=𝑊𝑡,𝜂₁₂,𝜂₁₃−𝜂₁₂(𝑡)𝜂₁₃(𝑡)𝜂₂₃(𝑡),𝐺^(𝑡)𝐺(𝑡)+𝜂¹(𝑡)+𝜂₂(𝑡)+𝜂₃(𝑡)=0.(2.4)
Then solutions of (1.2) may be represented in the form 𝑢_𝑘(𝑡)=³_𝑗=1𝜑_𝑗(𝑡)𝛿_𝑗𝑘+𝜀_𝑗𝑘(𝑡)𝐶_𝑗,lim_𝑡→∞𝜀_𝑗𝑘(𝑡)=0,𝑗,𝑘=1,2,3,(2.5) where Char_𝑗(𝑡), 𝑗=1,2,3 are defined in (1.5), (1.6), and 𝜑_𝑗(𝑡)=𝑒^{∫𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠},𝛿_𝑗𝑘=10,,𝑗𝑗≠=𝑘𝑘.,(2.6)

Note that conditions (2.2) and (2.3) are given in terms of characteristic functions.

We will say that (1.2) has asymptotic solutions 𝑒^{∫𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠} corresponding to the phase functions 𝜂_𝑗(𝑡)∈𝐶²(𝑡₀,∞), 𝑗=1,2,3 if (2.1)–(2.3) are satisfied.

Theorem 2.2. The solution of (1.2) corresponding to the asymptotic solution with the phase 𝜂_𝑘(𝑡) is oscillatory if and only if ^∞𝑡₀ℑ𝜂_𝑘(𝑡)𝑑𝑡=∞.(2.7)

The following theorem we deduce from Theorem 2.2 by choosing 𝜂₁(𝑡)=−𝑎^(𝑡)𝑎(𝑡),𝜂²(𝑡)=𝑎(𝑡)−𝑎^(𝑡)𝑎(𝑡),𝜂³(𝑡)=−𝑎(𝑡)−𝑎^(𝑡)𝑎(𝑡).(2.8)

Theorem 2.3. Assume that there exists a complex-valued function 𝑎(𝑡)∈𝐶³(𝑡₀,∞) such that 𝑎^−1/2(𝑡)∈𝐶²(𝑡₀,∞), and ℜ[𝑎(𝑡)]𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑐ℎ𝑎𝑛𝑔𝑒𝑡ℎ𝑒𝑠𝑖𝑔𝑛𝑜𝑛𝑡₀,∞,(2.9)_∞𝑡0|||3𝐼₁(𝑡)+𝑎²(𝑡)+4𝑎^1/2(𝑡)𝑎^−1/2(𝑡)^|||𝑑𝑡||𝑎(𝑡)||<∞,(2.10)_∞𝑡0||||||2𝐼₂(𝑡)+3𝐼₁(𝑡)𝑎(𝑡)−𝑎^(𝑡)𝑎(𝑡)+𝑎⁻¹(𝑡)𝑒^{∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}^𝑎⁻¹(𝑡)𝑒^{∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}||||||𝑑𝑡𝑎²(𝑡)<∞.(2.11) Then (1.2) with complex coefficients has one nonoscillatory solution and two linearly independent oscillatory solutions if and only if ^∞𝑡₀ℑ[𝑎(𝑡)]𝑑𝑡=∞,𝑜𝑟^∞𝑡₀ℑ[𝑎(𝑡)]𝑑𝑡=−∞.(2.12)

By taking 𝑎(𝑡)=𝜆/𝑡 from Theorem 2.3, we get the following corollary.

Corollary 2.4. Assume that for some complex number 𝜆≠0^∞𝑡₀𝑡²||||2𝐼₂(𝑡)+1−𝜆²𝑡³||||𝑑𝑡<∞,^∞𝑡₀𝑡||||3𝐼₁(𝑡)+𝜆²−1𝑡²||||𝑑𝑡<∞.(2.13) Then (1.2) with complex coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if ℑ[𝜆]>0.(2.14)

By taking 𝜆=1 from Corollary 2.4, we obtain well-known result [4].

Corollary 2.5. Assume that conditions ^∞𝑡₀𝑡²||𝐼₂(𝑡)||𝑑𝑡<∞,^∞𝑡₀𝑡||𝐼₁(𝑡)||𝑑𝑡<∞(2.15) are satisfied. Then (1.2) with complex coefficients is non-oscillatory.

By taking 𝑎(𝑡)=−1/𝑡ln(𝑡) from Theorem 2.3 we get another corollary.

Corollary 2.6. Assume that conditions ^∞𝑡₀𝑡²ln²(𝑡)||||2𝐼₂(𝑡)+3𝐼¹(𝑡)𝑡||||𝑑𝑡<∞,^∞𝑡₀𝑡ln(𝑡)|||3𝐼₁(𝑡)−1𝑡²|||𝑑𝑡<∞(2.16) are satisfied. Then (1.2) with complex coefficients is non-oscillatory.

Example 2.7. From Corollary 2.6, (1.2) with 𝐼₁(𝑡)=13𝑡²,𝐼₂(𝑡)=−12𝑡³(2.17) is non-oscillatory. Note that Corollary 2.5 is not applicable for this example since condition (2.15) fails.

In the case 𝑀(𝑡)≡0 by taking 𝑎(𝑡)=𝑖√3𝐼₁(𝑡), from Theorem 2.3, we deduce the following theorem.

Theorem 2.8. Assume that 𝐼₁(𝑡)∈𝐶³(𝑡₀,∞), 𝐼₁(𝑡)≥𝛽>0, and 𝑀(𝑡)=𝐼₂(𝑡)−3𝐼^1(𝑡)4≡0,𝑡∈𝑡⁰,∞,^∞𝑡₀|||𝐼_1−1/4(𝑡)^𝐼^1−1/4(𝑡)|||𝑑𝑡<∞,(2.18)_∞𝑡0|||𝐼_1−1/4(𝑡)^𝐼^1−3/4(𝑡)^|||𝑑𝑡<∞.(2.19) Then (1.2) with the real coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if ^∞𝑡₀√𝐼₁(𝑡)𝑑𝑡=∞.(2.20)

Another result may be proved by the different choice of the phase functions as follows: 𝜂_𝑗(𝑡)=𝑑^1/3(𝑡)𝑒^{−𝑖𝜋(2𝑗+1)/3}−𝐼¹(𝑡)𝑑^1/3(𝑡)𝑒^{𝑖𝜋(2𝑗+1)/3}−𝐺^(𝑡)3𝐺(𝑡),𝑗=1,2,3,(2.21) where 𝐺(𝑡) is defined in (2.4), and 𝑑(𝑡)=𝑀(𝑡)+𝑀²(𝑡)+𝐼³¹(𝑡),𝑀(𝑡)=𝐼₂(𝑡)−3𝐼^1(𝑡)4.(2.22)

Define 3 auxiliary regions on the real plane 𝑅𝑅₁=𝐼₁,𝑀∈𝑅²∣𝐼₁≤0,𝑀<0,𝐼³¹+𝑀²<0₂,𝑅=𝐼₁,𝑀∈𝑅²∣𝐼₁<0,𝑀≤0,𝐼³¹+𝑀²≥0₃,=𝑅²⧵𝑅₁∪𝑅₂=𝐼₁,𝑀∈𝑅²,𝑀>0,or𝐼₁>0.(2.23)

Theorem 2.9. Assume that 𝐼₁(𝑡)∈𝐶¹(𝑡₀,∞), 𝑅₀ is simply connected region 𝑅₀⊂𝑅_𝑗 for some 𝑗=1,2,3, and conditions (2.2) and (2.3), 𝐼₁(𝑡),𝑀(𝑡)∈𝑅₀,𝑡>𝑡₀(2.24) are satisfied. Then (1.2) has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if ^∞𝑡₀ℜ𝑀²(𝑡)+𝐼³¹(𝑡)+𝑀(𝑡)^1/3+𝑀²(𝑡)+𝐼³¹(𝑡)−𝑀(𝑡)^1/3𝑑𝑡=∞(2.25) for at least one of cubic roots.

Note that condition (2.25) is invariant with respect to the adjoint transformation 𝑀(𝑡)→−𝑀(𝑡).

For the case of the real constant coefficients 𝐼₁(𝑡)=𝐼₁, 𝐼₂(𝑡)=𝐼₂ from Theorem 2.9, one can deduce the obvious result that (1.2) is oscillatory if and only if 𝐼³¹+𝐼²²>0. Indeed in this case condition (2.25) turns to 𝐼³¹+𝐼²²>0, and conditions (2.2), (2.3), and (2.24) could be dropped.

Remark 2.10. Levinson's dichotomy condition (2.24) is satisfied if the modified Mammana's condition is satisfied as follows: 𝑀(𝑡)≥0,or𝑀(𝑡)≤0,𝐼₁(𝑡)≥0,𝑡>𝑡₀.(2.26) If 𝐼₁(𝑡)≥0, then 𝑀(𝑡) under condition (2.24) may change the sign.

Theorem 2.9 does not exclude the case 𝐼₂(𝑡)<0, but Theorem 1.2 does. In the case 𝐼₁(𝑡)≡0 conditions of Theorem 2.9 are simplified.

Theorem 2.11. Assume that 𝐼₂(𝑡) is real, it does not change the sign, and conditions ^∞𝑡₀|||𝐼_2−1/3^(𝑡)𝐼^2−1/3|||𝑑𝑡<∞,^∞𝑡₀|||𝐼_2−1/6^(𝑡)𝐼^2−1/6|||𝑑𝑡<∞(2.27) are satisfied. Then equation 𝑢^(𝑡)+2𝐼₂(𝑡)𝑢(𝑡)=0(2.28) has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if ^∞𝑡₀||𝐼₂(𝑡)||^1/3𝑑𝑡=∞.(2.29)

In the case 𝑀(𝑡)=0 (self-adjoint equation (1.2)), condition (2.25) turns to (2.20) (see Theorem 2.8 above), which is Leighton's (see [9]) necessary condition of oscillations for solutions of the second-order equation 𝑢^(𝑡)+𝐼₁(𝑡)𝑢(𝑡)=0.

Example 2.12. Equation 𝑢^(𝑡)+1𝑡³+1−(𝑖+𝜇)²𝑡²𝑢^(𝑡)+1𝑡⁴+(𝑖+𝜇)²−1𝑡³𝑢(𝑡)=0,(2.30) where 𝜇 is a real number and is oscillatory by Corollary 2.4 since conditions (2.13) and (2.14) are satisfied with 𝜆=𝑖+𝜇. Note that for this example Theorem 1.2 is not applicable since both conditions (1.8) and (1.9) fail even when 𝜇=0.

3. Proofs

Our main tool is Levinson's asymptotic theorem.

Theorem 3.1 (see [8]). 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<∞^𝑥𝑡ℜ𝜆_𝑖(𝑠)−𝜆_𝑗(𝑠)𝑑𝑠≤𝐾₁,𝑜𝑟^𝑥𝑡ℜ𝜆_𝑖(𝑠)−𝜆_𝑗(𝑠)𝑑𝑠≥𝐾₂.(3.1) Let the 𝑛×𝑛 matrix 𝑁(𝑡) satisfy 𝑁(𝑡)∈𝐿₁(𝑡₀,∞) or ^𝑥𝑡||𝑁(𝑡)||𝑑𝑠<∞,(3.2) by which we mean that each entry in 𝑁(𝑡) has an absolutely convergent infinite integral. Then the system 𝑌^(𝑡)=(Λ(𝑡)+𝑁(𝑡))𝑌(𝑡)(3.3) 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,(3.4) 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 2.1. Rewrite (1.2) as a system 𝐴(𝑡)=⎛⎜⎜⎜⎜⎝−2𝐼00₂𝑦^(𝑡)=10𝐴(𝑡)01𝑦(𝑡),(𝑡)−3𝐼₁(𝑡)0⎞⎟⎟⎟⎟⎠,𝑦(𝑡)=⎛⎜⎜⎜⎜⎝𝑢𝑢(𝑡)^𝑢(𝑡)_(𝑡)⎞⎟⎟⎟⎟⎠.(3.5) By transformation 𝑦(𝑡)=Φ(𝑡)𝑧(𝑡),(3.6) where matrix function Φ(𝑡) is defined via phase functions 𝜂_𝑗(𝑡) as follows: Φ(𝑡)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑒^{∫𝑡𝑡0𝜂1(𝑠)𝑑𝑠}𝜂𝜇₁(𝑡)𝑒𝜇₂(𝑡)^{∫𝑡𝑡0𝜂2(𝑠)𝑑𝑠}𝑒𝜇₃(𝑡)^{∫𝑡𝑡0𝜂3(𝑠)𝑑𝑠}₁(𝑡)𝜇₁(𝑡)𝑒^{∫𝑡𝑡0𝜂1(𝑠)𝑑𝑠}𝜂₂(𝑡)𝜇₂(𝑡)𝑒^{∫𝑡𝑡0𝜂2(𝑠)𝑑𝑠}𝜂₃(𝑡)𝜇₃(𝑡)𝑒^{∫𝑡𝑡0𝜂3(𝑠)𝑑𝑠}𝜂^1(𝑡)+𝜂²¹(𝑡)𝜇₁(𝑡)𝑒^{∫𝑡𝑡0𝜂1(𝑠)𝑑𝑠}𝜂^2(𝑡)+𝜂²²(𝑡)𝜇₂(𝑡)𝑒^{∫𝑡𝑡0𝜂2(𝑠)𝑑𝑠}𝜂^3(𝑡)+𝜂²³(𝑡)𝜇₃(𝑡)𝑒^{∫𝑡𝑡0𝜂3𝑑𝑠}⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.7) we get the following: 𝑧^(𝑡)=Φ⁻¹(𝑡)𝐴(𝑡)Φ(𝑡)−Φ^(𝑡)𝑧(𝑡),or𝑧^(𝑡)=(𝐷(𝑡)+𝐵(𝑡))𝑧(𝑡),(3.8) where 𝐵(𝑡)=1𝐺(𝑡)⎛⎜⎜⎜⎜⎜⎜⎜⎝𝜂²³(𝑡)Char₁(𝑡)𝜇¹𝜂₂₃Char₂(𝑡)𝜇₂(𝑡)𝑒^{∫𝑡𝑡0𝜂21𝑑𝑠}𝜇₁𝜂₂₃Char₃(𝑡)𝜇₃(𝑡)𝑒^{∫𝑡𝑡0𝜂31𝑑𝑠}𝜇₂𝜂₃₁Char₁(𝑡)𝜇₁(𝑡)𝑒^{∫𝑡𝑡0𝜂12𝑑𝑠}𝜂₃₁(𝑡)Char₂(𝑡)𝜇²𝜂₃₁Char₃(𝑡)𝜇₃(𝑡)𝑒^{∫𝑡𝑡0𝜂32𝑑𝑠}𝜇₃𝜂₁₂Char₁(𝑡)𝜇₁(𝑡)𝑒^{∫𝑡𝑡0𝜂13𝑑𝑠}𝜇₃𝜂₁₂Char₂(𝑡)𝜇₂(𝑡)𝑒^{∫𝑡𝑡0𝜂23𝑑𝑠}𝐷(𝑡)=⎛⎜⎜⎜⎜⎜⎜⎝𝜇_1𝜂₁₂(𝑡)Char₃(𝑡)⎞⎟⎟⎟⎟⎟⎟⎟⎠,(𝑡)𝜇₁0(𝑡)𝜇00_2(𝑡)0𝜇₂0(𝑡)𝜇_30(𝑡)𝜇₃(𝑡)⎞⎟⎟⎟⎟⎟⎟⎠,𝜂^𝑗𝑘(𝑡)=𝜂_𝑗(𝑡)−𝜂_𝑘(𝑡).(3.9)
Choosing specific auxiliary functions 𝜇₁(𝑡)=1,𝜇₂(𝑡)=𝑒^{∫𝑡𝑡0𝜂21(𝑠)𝑑𝑠},𝜇₃(𝑡)=𝑒^{∫𝑡𝑡0𝜂31(𝑠)𝑑𝑠},(3.10) we have 𝐷(𝑡)=⎛⎜⎜⎜⎜⎝000𝜂00₂₁𝜂00₃₁⎞⎟⎟⎟⎟⎠,𝐵(𝑡)=1𝐺(𝑡)⎛⎜⎜⎜⎜⎝𝜂³²Char₁𝜂₂₃Char₂𝜂₂₃Char₃𝜂₃₁Char₁𝜂₃₁Char₂𝜂₃₁Char₃𝜂₁₂Char₁𝜂₁₂Char₂𝜂₁₂Char₃⎞⎟⎟⎟⎟⎠.(3.11) Here and further we suppress the time variable 𝑡 for the simplicity.
From Liouville's formula detΦ(𝑡)=𝐶𝑒^{∫𝑡𝑡0Tr(𝐴(𝑠))𝑑𝑠}=𝐶 applied to (3.7) with the assumption that 𝜂_𝑗(𝑡) are solutions of Char_𝑗(𝑡)=0,𝜇₁=𝜇₂=𝜇₃=1, we get the following 𝐶=det(Φ(𝑡))=𝐺(𝑡)𝑒^{∫𝑡𝑡0(𝜂1+𝜂2+𝜂3)(𝑠)𝑑𝑠}.(3.12) The Liouville's formula may be written in the form 𝐺^(𝑡)𝐺(𝑡)+𝜂¹(𝑡)+𝜂₂(𝑡)+𝜂₃(𝑡)=0.(3.13) We always are choosing the phase functions 𝜂_𝑗(𝑡) such that (3.13) is satisfied (see (2.4)). From (3.13), we get that 𝜂₁−𝜂₃Char₁₂(𝑡)−𝜂₁−𝜂₂Char₁₃(𝑡)𝐺(𝑡)=−𝜂¹(𝑡)−𝜂₂(𝑡)−𝜂₃(𝑡)−𝐺^(𝑡)Char₁𝐺(𝑡),(3.14)(𝑡)−Char₂(𝑡)𝜂₁(𝑡)−𝜂₂(𝑡)=Char¹(𝑡)−Char₃(𝑡)𝜂₁(𝑡)−𝜂₃(𝑡)=Char²(𝑡)−Char₃(𝑡)𝜂₂(𝑡)−𝜂₃(𝑡).(3.15) To apply Theorem 3.1 to system (3.8) note that from (2.1) it follows dichotomy condition (3.1) of Theorem 3.1: ^𝑥𝑡ℜ𝜂_𝑘𝑗𝑑𝑠≥0,or^𝑥𝑡ℜ𝜂_𝑘𝑗𝑑𝑠≤0,𝑥≥𝑡,𝑗≠𝑘,𝑘,𝑗=1,2,3.(3.16) Condition (3.2) of Theorem 3.1 turns to 𝐵(𝑡)∈𝐿₁(𝑡₀,∞), and it is followed from 𝜂₁₃(𝑡)Char_𝑗(𝑡)𝐺(𝑡),𝜂²³(𝑡)Char_𝑗(𝑡)𝐺(𝑡)∈𝐿¹𝑡₀,∞,𝑗=1,2,3.(3.17)
One can drop condition 𝜂₂₃(𝑡)Char₁(𝑡)/𝐺(𝑡)∈𝐿₁(𝑡₀,∞) since from (3.15) we have 𝜂₂₃(𝑡)Char₁(𝑡)𝐺(𝑡)=𝜂¹³(𝑡)Char₂(𝑡)𝐺(𝑡)−𝜂¹²Char₃𝐺.(3.18) Assuming that 𝜂²¹³(𝑡)Char₂(𝑡)−Char₃(𝑡)𝜂₂₃(𝑡)𝐺(𝑡)∈𝐿¹𝑡₀,∞,(3.19) condition 𝜂₁₃Char₁/𝐺∈𝐿₁(𝑡₀,∞) may be dropped as well since 𝜂₁₃Char₁𝐺=𝜂¹³𝜂₁₃Char₂−𝜂₁₂Char₃𝐺𝜂₂₃=𝜂¹³𝜂₁₃Char₂−Char₃+𝜂₂₃Char₃𝜂₂₃𝐺.(3.20) So condition (3.2) of Theorem 3.1 turns to ||𝜂₂₃(𝑡)||+||𝜂₁₃(𝑡)||||Char_𝑗(𝑡)||||𝐺(𝑡)||,||𝜂¹³(𝑡)||²||Char₂−Char₃(𝑡)||||𝜂₂₃(𝑡)𝐺(𝑡)||∈𝐿¹𝑡₀,∞,𝑗=2,3,(3.21) or (2.2) and (2.3). From Theorem 3.1 applied to system (3.8) and we get that 𝑧(𝑡)=𝑧₀(𝑡)(𝐸+𝜀(𝑡))𝐶,𝑧₀(𝑡)=𝑒^{∫𝑡𝑡0𝐷(𝑠)𝑑𝑠}=⎛⎜⎜⎜⎜⎝𝜇¹0(𝑡)𝜇₂0000(𝑡)𝜇₃0(𝑡)⎞⎟⎟⎟⎟⎠𝐶¹𝑦(𝑡)=Φ(𝑡)𝑧(𝑡)=Φ(𝑡)𝑧₀,(𝑡)(𝐸+𝜀(𝑡))𝐶,(3.22) or representation (2.5).

Proof of Theorem 2.2. Theorem 2.2 is followed from Theorem 2.1 since in representation (2.5) one may choose asymptotic solutions as follows: 𝜑₂(𝑡)=𝑒^{∫𝑡𝑡0ℜ[𝜂2(𝑠)]𝑑𝑠}sin^𝑡𝑡₀ℑ𝜂₂(𝑠)𝑑𝑠𝑑𝑠,𝜑₃(𝑡)=𝑒^{∫𝑡𝑡0ℜ[𝜂2(𝑠)]𝑑𝑠}cos^𝑡𝑡₀ℑ𝜂₂(𝑠)𝑑𝑠(3.23) which are oscillating if and only if condition (2.7) is satisfied.

Proof of Theorem 2.3. Theorem 2.3 is deduced from Theorem 2.2 by choosing phase functions as in (2.8). From 2ℜ𝜂₂₁=2ℜ𝜂₁₃=ℜ𝜂₂₃=2ℜ[𝑎(𝑡)],(3.24) condition (2.1) turns to condition (2.9). From (2.4), we get that 𝐺(𝑡)=2𝑎³(𝑡),𝜂₁+𝜂₂+𝜂₃+𝐺^(𝑡)𝐺(𝑡)=0.(3.25) Since in conditions (2.2)-(2.3) the function 𝐺(𝑡) appears in denominator we should assume that 𝐺(𝑡)≠0, or 𝑎(𝑡)≠0. By direct calculations, we get that 2𝜂₁₃Char₂𝐺=𝑎1²⎛⎜⎜⎜⎝2𝐼₂+3𝐼₁𝑎−𝑎′𝑎+𝑎⁻¹(𝑡)𝑒^{∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}^𝑎⁻¹(𝑡)𝑒^{∫𝑡𝑡0𝑎(𝑠)𝑑𝑠}𝜂₁₃⎞⎟⎟⎟⎠,Char₂−Char₃𝐺=3𝐼¹(𝑡)𝑎(𝑡)+𝑎(𝑡)+4𝑎^−1/2^𝑎^−1/2.(3.26) In view of 2𝜂₁₃𝐺=𝜂𝐺²³=𝜂4𝜂₂₃𝐺²¹³=𝑎1²,(3.27) conditions (2.2) and (2.3) of Theorem 2.2 turn to (2.10) and (2.11).
Further the asymptotic solution 𝑒^{∫𝑡𝑡0𝜂2(𝑠)𝑑𝑠}+𝑒^{∫𝑡𝑡0𝜂3(𝑠)𝑑𝑠}=2𝑎(𝑡)𝑒^{∫𝑡𝑡0ℜ[𝑎(𝑠)]𝑑𝑠}cos^𝑡𝑡₀ℑ[𝑎(𝑠)]𝑑𝑠(3.28) is oscillating if and only if (2.12) is satisfied. Indeed, the solution corresponding to the asymptotic solution 𝑒^{∫𝑡𝑡0𝜂1𝑑𝑠}=𝐶/𝑎(𝑡) is non-oscillatory (1/𝑎(𝑡) does not have zeros; otherwise 𝑎(𝑡)∈𝐶³(𝑡₀,∞) is undefined at some points).

Proof of Corollaries 2.4 and 2.6. We deduce Corollaries 2.4 and 2.6 from Theorem 2.3 by the special choice of function 𝑎(𝑡) as follows: 𝑎(𝑡)=𝜆𝑡ln^𝛾(𝑡),𝜆≠0.(3.29) From (2.10) and (2.11) we get that 𝑡ln^𝛾(𝑡)||||3𝐼₁+𝛾(𝛾−2)𝑡²ln²(𝑡)+𝑡²ln𝜆^22𝛾(𝑡)−𝑡1²||||∈𝐿₁𝑡𝑡₀,∞₂,ln^2𝛾(𝑡)||||2𝐼₂+3𝐼¹+𝜆𝛾(𝛾𝑡−2)1+ln𝛾(𝑡)+ln^𝛾𝜆(𝑡)+𝛾(𝛾𝑡−³l1n)³((𝛾𝑡)−2)𝑡³ln^2+𝛾(𝑡)−𝑡³l𝛾n(𝑡)+𝑡³ln𝜆^33𝛾(𝑡)−𝑡³ln𝜆^𝛾(𝑡)||||∈𝐿¹𝑡₀,∞,(3.30) or in the case 𝛾=0^∞𝑡₀𝑡3𝐼₁(𝑡)+𝜆²−1𝑡²𝑑𝑡<∞,_∞𝑡0𝑡²||||2𝐼₂(𝑡)+3(1+𝜆)𝐼¹(𝑡)𝑡+𝜆𝜆²−1𝑡³||||𝑑𝑡<∞,(3.31) which is equivalent to (2.13).
Further from (2.12) we get condition (2.14) in the case 𝛾=0: ^∞𝑡₀ℑ[𝜆]𝑑𝑡𝑡ln^𝛾(𝑡)=∞,orℑ[𝜆]>0,𝛾=0.(3.32) The proof of Corollary 2.6 is followed from (3.30) to (3.27) by choosing 𝛾=−𝜆=1.

Proof of Theorem 2.8. Theorem 2.8 is followed from Theorem 2.3. Indeed from 𝑀(𝑡)≡0,𝑡>𝑡₀,𝑎(𝑡)=𝑖√3𝐼₁(𝑡), we have 2𝐼₂(𝑡)=−𝑎′(𝑡)𝑎(𝑡). Condition (2.10) turns to (2.18) as follows: 3𝐼₁+𝑎²𝑎+4𝑎^−1/2^𝑎^−1/2=𝑄(𝑡)∈𝐿₁𝑡₀,∞,𝑄(𝑡)=4𝑎^−1/2^𝑎^−1/2,(3.33) and condition (2.11) turns to (2.19) since 1a²2𝐼²+3𝐼₁𝑎−𝑎^(𝑡)𝑎+𝑣^(𝑡)𝑣=𝑣^(𝑡)𝑎²𝑣−𝑎=2𝑄𝑎((𝑡𝑡))^+𝑄(𝑡),(3.34) where 𝑣(𝑡)=𝑎⁻¹(𝑡)𝑒^{∫𝑡𝑡0𝑎(𝑠)𝑑𝑠},𝑄(𝑡)=4𝑎^−1/2^𝑎^−1/2=4𝐼^1−1/4^𝐼^1−1/4.(3.35)

Proof of Theorem 2.9. Let choose the phase functions 𝜂_𝑗(𝑡) as in (2.21). We deduce Theorem 2.9 from Theorem 2.2. By calculations 𝐺(𝑡)=√33𝐼³¹+3𝑑²+3𝑑𝐼^1−2𝑑^𝐼₁𝑖𝑑=−𝑖3√3𝑑1+𝐼³¹𝑑²+𝐼¹𝜂3𝑑ln𝑑𝐼³¹²^,(3.36)₁₂=32𝑑𝐼^11/3−𝑑^1/3−𝑖√32𝑑𝐼^11/3+𝑑^1/3,𝜂²³=𝑖√3𝐼¹𝑑^1/3+𝑑^1/3,(3.37) to deduce dichotomy conditions (2.1) from (2.24), it is enough to show that ℜℜ𝜂₂𝜂₂−𝜂₃>0,if𝐼₁,𝑀∈𝑅₁,ℜ𝜂−₁𝜂₃≤0,if𝐼₁,𝑀∈𝑅²⧵𝑅₁,ℜ𝜂₁−𝜂₂>0,if𝐼₁,𝑀∈𝑅₂,−𝜂₂≤0,if𝐼₁,𝑀∈𝑅²⧵𝑅₂.(3.38)

Case 1 (𝐼³¹(𝑡)+𝑀²(𝑡)≥0,𝑡>𝑡₀). In this case 𝑑 and 𝜂₁ are real, 𝜂_2,3 are complex conjugate and from (3.37) ℜ𝜂₁−𝜂₂=ℜ𝜂₁−𝜂₃=32𝑑𝐼^11/3−𝑑^1/3,ℜ𝜂²−𝜂₃=0.(3.39) If 𝐼₁<0,𝑀≤0, then 𝑑=𝑀+𝑀²+𝐼³¹<𝑀+|𝑀|=0, ℜ[𝜂₁−𝜂₂]>0.
If 𝐼₁≤0,𝑀>0, then 𝑑=𝑀+𝑀²+𝐼³¹≥𝑀>0, ℜ[𝜂₁−𝜂₂]<0.
Otherwise if 𝐼₁>0, then 𝑑>𝑀+√𝑀²≥0, 𝑑^1/3>0 and 𝑑²=𝑀+𝑀²+𝐼³¹²=𝐼³¹+2𝑀²⎛⎜⎜⎝1+1+𝐼³¹𝑀²⎞⎟⎟⎠≥𝐼₃₁ℜ𝜂,𝑑^2/3≥𝐼₁>0,₁−𝜂₂=32𝑑𝐼^11/3−𝑑^1/3≤0.(3.40)
Further condition (2.7) turns to (2.25) since ℑ[𝜂₂]=(1/2)ℑ[𝜂₂−𝜂₃].

Case 2 (𝐼³¹(𝑡)+𝑀²(𝑡)<0,𝑡>𝑡₀). In this case 𝑑=𝑀+𝑖−𝐼³¹−𝑀² is complex valued, 𝜂_1,2,3,𝐺 are real. In view of 𝑑𝑑=𝑀+𝑀²+𝐼³¹𝑀−𝑀²+𝐼³¹=−𝐼³¹𝑑_1/3,𝑑^1/3=−𝐼₁,𝑑^1/3=−𝑑𝐼¹_1/3,𝑟𝑒^−𝑖𝑎^1/3=𝑟^1/3𝑒^𝑖𝑎/3,𝑑^1/3=𝑑^1/3,(3.41) we get that 𝜂₂=𝑒^𝑖𝜋/3𝑑^1/3−𝑒^{−𝑖𝜋/3}𝐼₁𝑑^1/3−𝐺^(𝑡)3𝐺(𝑡)=2ℜ𝑒^𝑖𝜋/3𝑑^1/3−𝐺^(𝑡)𝜂3G(𝑡),ℑ𝜂²=0,₁=−2ℜ𝑑^1/3−𝐺^(𝑡)3𝐺(𝑡),𝜂³=2ℜ𝑒^{−𝑖𝜋/3}𝑑^1/3−𝐺^(𝑡)3𝐺(𝑡).(3.42) Condition (2.25) fails in this case: ℜ[𝑑^1/3−𝑑^1/3]=0 and (1.2) is non-oscillatory. Further 𝑑=𝑀+𝑖−𝑀²−𝐼³¹=𝑟𝑒^𝑖𝑎,tan(𝑎)=−𝑀²−𝐼³¹𝜂₁𝑀,𝑎∈−𝜋2,𝜋2,−𝜂₂=√3ℑ𝑑^1/3−√3ℜ𝑑^1/3=3ℑ𝑑^1/31𝜂₁√3−cot𝑎3,−𝜂₃=−√3ℑ𝑑^1/3+√3ℜ𝑑^1/3=−3ℑ𝑑^1/31𝜂₂√3+cot𝑎3,−𝜂₃=−2√3ℑ𝑑^1/3=−2√3𝑟^1/3sin𝑎3.(3.43) If 𝑀>0 then tan(𝑎)>0, 0<𝑎<𝜋/2, 𝑎=tan⁻¹−1−(𝐼³¹/𝑀²), and we get the following: 0<𝑎3<𝜋6,sin𝑎3>0,cos𝑎3>0,−∞<1+𝑀𝐼³¹²ℑ𝑑≤0,_1/3=𝑟^1/3sin𝑎3>0,cot𝑎3>1𝜂₁√3,−𝜂₂<0,𝜂₁−𝜂₃<0,𝜂₂−𝜂₃<0.(3.44) If 𝑀<0 then tan(𝑎)<0, −𝜋/2<𝑎<0, 𝑎=−tan⁻¹−1−(𝐼³¹/𝑀²), and we get the following: −𝜋6<ℑ𝑎3<𝑑0,sin𝑎3<0,cos𝑎3>0,_1/3<0,cot𝑎3<−1𝜂₁√3,−𝜂₂<0,𝜂₁−𝜂₃<0,𝜂₂−𝜂₃>0.(3.45)

Proof of Theorem 2.11. By taking 𝐼₁(𝑡)≡0, we get the following: 𝑑(𝑡)=2𝐼₂(𝑡),𝐺(𝑡)=−6𝑖√3𝐼₂(𝑡),𝜂₁(𝑡)=−𝑑^1/3−𝐺′3𝐺=−𝑑^1/3−𝑑′𝜂3𝑑_2,3(𝑡)=𝑒^±𝑖𝜋/3𝑑^1/3−𝑑^3𝑑=1±𝑖√32𝑑^1/3−𝑑^3𝑑,ℑ𝜂^2,3=±√32𝑑^1/3,(3.46) and Theorem 2.9 turns to Theorem 2.11.