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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 715981, 15 pages
Research Article

On Oscillations of Solutions of Third-Order Dynamic Equation

Department of Mathematics, Kent State University at Stark, 6000 Frank Avenue NW, Canton, OH 44720-7599, USA

Received 10 March 2012; Accepted 25 April 2012

Academic Editor: Paul Eloe

Copyright © 2012 Gro Hovhannisyan. 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.


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 [17]. 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 𝐶𝑘(𝑡0,) be the set of 𝑘 times differentiable functions on (𝑡0,). By substitution 𝑣(𝑡)=𝑢(𝑡)𝑒𝑡𝑡0𝑎2(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)+𝑎2(𝑡)𝑎22(𝑡),𝐼2(𝑡)=𝑎0(𝑡)+3𝑎1(𝑡)𝑎2(𝑡)𝑎32𝑎(𝑡)+2(𝑡)2.(1.3)

If 0<|𝑒𝑡0𝑎2(𝑠)𝑑𝑠|< or <𝑡0𝑎2(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 𝜂𝑗(𝑡)Char𝑗𝜂(𝑡)=Char𝑗=𝑒𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠𝑃𝑒𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠,𝑗=1,2,3.(1.5) By direct calculations Char𝑗(𝑡)=𝜂𝑗(𝑡)+3𝜂𝑗(𝑡)𝜂𝑗(𝑡)+𝜂3𝑗+3𝜂𝑗(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)=𝑒𝑡𝑡0𝜂(𝑠)𝑑𝑠 with phase functions 𝜂𝑗(𝑡), 𝑗=1,2,3, that are approximate solutions of the characteristic equation Char(𝜂𝑗)=0.

Theorem 1.1 (see [4]). If Mammana's condition 𝑀(𝑡)=𝐼2(𝑡)3𝐼1(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)𝑡0𝐼2(𝑡)𝐼1(𝑡)3/2𝑑𝑡=(1.9) are satisfied. Then (1.2) with the real coefficients is oscillatory.

The adjoint transformation 𝐼2(𝑡)(3𝐼1(𝑡)/2)𝐼2(𝑡), or 𝑀(𝑡)𝑀(𝑡) transforms (1.2) to its adjoint equation 𝑤(𝑡)3𝐼1(𝑡)𝑤(𝑡)+(2𝐼2(𝑡)3𝐼1(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 𝜂𝑗(𝑡)𝐶2(𝑡0,), 𝑗=1,2,3, such that expressions [𝜂𝑗(𝑡)𝜂𝑘(𝑡)], 𝑘𝑗 do not change a sign, that is, 𝜂𝑗(𝑡)𝜂𝑘𝜂(𝑡)0𝑜𝑟𝑗(𝑡)𝜂𝑘(𝑡)0,𝑘,𝑗=1,2,3,𝑘𝑗,𝑡>𝑡0,(2.1)𝑡0||𝜂13||+||𝜂(𝑡)23||||(𝑡)Char2||(𝑡)||||𝐺(𝑡)𝑑𝑡<,𝜂𝑗𝑘(𝑡)=𝜂𝑗(𝑡)𝜂𝑘(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)=𝑊𝑡,𝜂12,𝜂13𝜂12(𝑡)𝜂13(𝑡)𝜂23𝐺(𝑡),(𝑡)𝐺(𝑡)+𝜂1(𝑡)+𝜂2(𝑡)+𝜂3(𝑡)=0.(2.4)
Then solutions of (1.2) may be represented in the form 𝑢𝑘(𝑡)=3𝑗=1𝜑𝑗𝛿(𝑡)𝑗𝑘+𝜀𝑗𝑘𝐶(𝑡)𝑗,lim𝑡𝜀𝑗𝑘(𝑡)=0,𝑗,𝑘=1,2,3,(2.5) where Char𝑗(𝑡), 𝑗=1,2,3 are defined in (1.5), (1.6), and 𝜑𝑗(𝑡)=𝑒𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠,𝛿𝑗𝑘=1,𝑗=𝑘,0,𝑗𝑘.(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 𝑒𝑡𝑡0𝜂𝑗(𝑠)𝑑𝑠 corresponding to the phase functions 𝜂𝑗(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 𝜂𝑘(𝑡) is oscillatory if and only if 𝑡0𝜂𝑘(𝑡)𝑑𝑡=.(2.7)

The following theorem we deduce from Theorem 2.2 by choosing 𝜂1𝑎(𝑡)=(𝑡)𝑎(𝑡),𝜂2𝑎(𝑡)=𝑎(𝑡)(𝑡)𝑎(𝑡),𝜂3𝑎(𝑡)=𝑎(𝑡)(𝑡).𝑎(𝑡)(2.8)

Theorem 2.3. Assume that there exists a complex-valued function 𝑎(𝑡)𝐶3(𝑡0,) such that 𝑎1/2(𝑡)𝐶2(𝑡0,), and []𝑡𝑎(𝑡)𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑐𝑎𝑛𝑔𝑒𝑡𝑒𝑠𝑖𝑔𝑛𝑜𝑛0,,(2.9)𝑡0|||3𝐼1(𝑡)+𝑎2(𝑡)+4𝑎1/2𝑎(𝑡)1/2(𝑡)|||𝑑𝑡||||𝑎(𝑡)<,(2.10)𝑡0||||||2𝐼2(𝑡)+3𝐼1𝑎(𝑡)𝑎(𝑡)(𝑡)𝑎+𝑎(𝑡)1(𝑡)𝑒𝑡𝑡0𝑎(𝑠)𝑑𝑠𝑎1(𝑡)𝑒𝑡𝑡0𝑎(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 𝑡0[]𝑎(𝑡)𝑑𝑡=,𝑜𝑟𝑡0[]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 𝜆0𝑡0𝑡2||||2𝐼2(𝑡)+1𝜆2𝑡3||||𝑑𝑡<,𝑡0𝑡||||3𝐼1𝜆(𝑡)+21𝑡2||||𝑑𝑡<.(2.13) Then (1.2) with complex coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if [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<,𝑡0|||𝑡ln(𝑡)3𝐼11(𝑡)𝑡2|||𝑑𝑡<(2.16) are satisfied. Then (1.2) with complex coefficients is non-oscillatory.

Example 2.7. From Corollary 2.6, (1.2) with 𝐼11(𝑡)=3𝑡2,𝐼21(𝑡)=2𝑡3(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)=𝑖3𝐼1(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)3𝐼1(𝑡)4𝑡0,𝑡0,,𝑡0|||𝐼11/4(𝑡)𝐼11/4|||(𝑡)𝑑𝑡<,(2.18)𝑡0|||𝐼11/4(𝑡)𝐼13/4(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 𝑡0𝐼1(𝑡)𝑑𝑡=.(2.20)

Another result may be proved by the different choice of the phase functions as follows: 𝜂𝑗(𝑡)=𝑑1/3(𝑡)𝑒𝑖𝜋(2𝑗+1)/3𝐼1(𝑡)𝑑1/3𝑒(𝑡)𝑖𝜋(2𝑗+1)/3𝐺(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)3𝐼1(𝑡)4.(2.22)

Define 3 auxiliary regions on the real plane 𝑅1=𝐼1,𝑀𝑅2𝐼10,𝑀<0,𝐼31+𝑀2,𝑅<02=𝐼1,𝑀𝑅2𝐼1<0,𝑀0,𝐼31+𝑀2,𝑅03=𝑅2𝑅1𝑅2=𝐼1,𝑀𝑅2,𝑀>0,or𝐼1.>0(2.23)

Theorem 2.9. Assume that 𝐼1(𝑡)𝐶1(𝑡0,), 𝑅0 is simply connected region 𝑅0𝑅𝑗 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 𝑡0𝑀2(𝑡)+𝐼31(𝑡)+𝑀(𝑡)1/3+𝑀2(𝑡)+𝐼31(𝑡)𝑀(𝑡)1/3𝑑𝑡=(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|||𝐼21/3(𝑡)𝐼21/3|||𝑑𝑡<,𝑡0|||𝐼21/6(𝑡)𝐼21/6|||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 𝑢1(𝑡)+𝑡3+1(𝑖+𝜇)2𝑡2𝑢1(𝑡)+𝑡4+(𝑖+𝜇)21𝑡3𝑢(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),,𝜆𝑛(𝑡)) be an 𝑛×𝑛 diagonal matrix function which satisfies dichotomy condition.
For each pair of integers 𝑖 and 𝑗 in [1,𝑛](𝑖𝑗) exist constants 𝐾1,𝐾2 such that for all x and t, 𝑡0𝑡𝑥<𝑥𝑡𝜆𝑖(𝑠)𝜆𝑗(𝑠)𝑑𝑠𝐾1,𝑜𝑟𝑥𝑡𝜆𝑖(𝑠)𝜆𝑗(𝑠)𝑑𝑠𝐾2.(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))𝑒𝑡𝑡0Λ(𝑠)𝑑𝑠𝐶,lim𝑡𝜀(𝑡)=0,(3.4) where 𝐸 is the identity matrix, 𝜀(𝑡) is the 𝑛×𝑛 error matrix-function, and 𝐶=(𝐶1,𝐶𝑛)tr is a constant column vector.

Proof of Theorem 2.1. Rewrite (1.2) as a system 𝑦(𝑡)=𝐴(𝑡)𝑦(𝑡),𝐴(𝑡)=0100012𝐼2(𝑡)3𝐼1(𝑢𝑡)0,𝑦(𝑡)=𝑢(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 𝜂𝑗(𝑡) as follows: 𝑒Φ(𝑡)=𝑡𝑡0𝜂1(𝑠)𝑑𝑠𝜇1𝑒(𝑡)𝑡𝑡0𝜂2(𝑠)𝑑𝑠𝜇2𝑒(𝑡)𝑡𝑡0𝜂3(𝑠)𝑑𝑠𝜇3𝜂(𝑡)1(𝑡)𝜇1(𝑒𝑡)𝑡𝑡0𝜂1(𝑠)𝑑𝑠𝜂2(𝑡)𝜇2(𝑒𝑡)𝑡𝑡0𝜂2(𝑠)𝑑𝑠𝜂3(𝑡)𝜇3(𝑒𝑡)𝑡𝑡0𝜂3(𝑠)𝑑𝑠𝜂1(𝑡)+𝜂21(𝑡)𝜇1𝑒(𝑡)𝑡𝑡0𝜂1(𝑠)𝑑𝑠𝜂2(𝑡)+𝜂22(𝑡)𝜇2𝑒(𝑡)𝑡𝑡0𝜂2(𝑠)𝑑𝑠𝜂3(𝑡)+𝜂23(𝑡)𝜇3𝑒(𝑡)𝑡𝑡0𝜂3𝑑𝑠,(3.7) we get the following: 𝑧(𝑡)=Φ1(𝑡)𝐴(𝑡)Φ(𝑡)Φ(𝑡)𝑧(𝑡),or𝑧(𝑡)=(𝐷(𝑡)+𝐵(𝑡))𝑧(𝑡),(3.8) where 1𝐵(𝑡)=𝜂𝐺(𝑡)23(𝑡)Char1𝜇(𝑡)1𝜂23Char2(𝑡)𝜇2𝑒(𝑡)𝑡𝑡0𝜂21𝑑𝑠𝜇1𝜂23Char3(𝑡)𝜇3𝑒(𝑡)𝑡𝑡0𝜂31𝑑𝑠𝜇2𝜂31Char1(𝑡)𝜇1𝑒(𝑡)𝑡𝑡0𝜂12𝑑𝑠𝜂31(𝑡)Char2𝜇(𝑡)2𝜂31Char3(𝑡)𝜇3𝑒(𝑡)𝑡𝑡0𝜂32𝑑𝑠𝜇3𝜂12Char1(𝑡)𝜇1𝑒(𝑡)𝑡𝑡0𝜂13𝑑𝑠𝜇3𝜂12Char2(𝑡)𝜇2𝑒(𝑡)𝑡𝑡0𝜂23𝑑𝑠𝜂12(𝑡)Char3,𝜇(𝑡)𝐷(𝑡)=1(𝑡)𝜇10𝜇(𝑡)002(𝑡)𝜇20𝜇(𝑡)003(𝑡)𝜇3(𝑡),𝜂𝑗𝑘(𝑡)=𝜂𝑗(𝑡)𝜂𝑘(𝑡).(3.9)
Choosing specific auxiliary functions 𝜇1(𝑡)=1,𝜇2(𝑡)=𝑒𝑡𝑡0𝜂21(𝑠)𝑑𝑠,𝜇3(𝑡)=𝑒𝑡𝑡0𝜂31(𝑠)𝑑𝑠,(3.10) we have 𝐷(𝑡)=0000𝜂21000𝜂311,𝐵(𝑡)=𝜂𝐺(𝑡)32Char1𝜂23Char2𝜂23Char3𝜂31Char1𝜂31Char2𝜂31Char3𝜂12Char1𝜂12Char2𝜂12Char3.(3.11) Here and further we suppress the time variable 𝑡 for the simplicity.
From Liouville's formula detΦ(𝑡)=𝐶𝑒𝑡𝑡0Tr(𝐴(𝑠))𝑑𝑠=𝐶 applied to (3.7) with the assumption that 𝜂𝑗(𝑡) are solutions of Char𝑗(𝑡)=0,𝜇1=𝜇2=𝜇3=1, we get the following 𝐶=det(Φ(𝑡))=𝐺(𝑡)𝑒𝑡𝑡0(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 𝜂𝑗(𝑡) such that (3.13) is satisfied (see (2.4)). From (3.13), we get that 𝜂1𝜂3Char12𝜂(𝑡)1𝜂2Char13(𝑡)𝐺(𝑡)=𝜂1(𝑡)𝜂2(𝑡)𝜂3𝐺(𝑡)(𝑡),𝐺(𝑡)(3.14)Char1(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: 𝑥𝑡𝜂𝑘𝑗𝑑𝑠0,or𝑥𝑡𝜂𝑘𝑗𝑑𝑠0,𝑥𝑡,𝑗𝑘,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)Char𝑗(𝑡),𝜂𝐺(𝑡)23(𝑡)Char𝑗(𝑡)𝐺(𝑡)𝐿1𝑡0,,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)=𝜂𝐺(𝑡)13(𝑡)Char2(𝑡)𝜂𝐺(𝑡)12Char3𝐺.(3.18) Assuming that 𝜂213(𝑡)Char2(𝑡)Char3(𝑡)𝜂23(𝑡)𝐺(𝑡)𝐿1𝑡0,,(3.19) condition 𝜂13Char1/𝐺𝐿1(𝑡0,) may be dropped as well since 𝜂13Char1𝐺=𝜂13𝜂13Char2𝜂12Char3𝐺𝜂23=𝜂13𝜂13Char2Char3+𝜂23Char3𝜂23𝐺.(3.20) So condition (3.2) of Theorem 3.1 turns to ||𝜂23(||+||𝜂𝑡)13(||||𝑡)Char𝑗(||𝑡)||||,||𝜂𝐺(𝑡)13||(𝑡)2||Char2Char3||(𝑡)||𝜂23||(𝑡)𝐺(𝑡)𝐿1𝑡0,,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)=𝑒𝑡𝑡0𝐷(𝑠)𝑑𝑠=𝜇1(𝑡)000𝜇2(𝑡)000𝜇3𝐶(𝑡)1,𝑦(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)=𝑒𝑡𝑡0[𝜂2(𝑠)]𝑑𝑠sin𝑡𝑡0𝜂2(𝑠)𝑑𝑠𝑑𝑠,𝜑3(𝑡)=𝑒𝑡𝑡0[𝜂2(𝑠)]𝑑𝑠cos𝑡𝑡0𝜂2(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 𝜂221𝜂=213𝜂=23[𝑎]=2(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=1𝑎22𝐼2+3𝐼1𝑎𝑎𝑎+𝑎1(𝑡)𝑒𝑡𝑡0𝑎(𝑠)𝑑𝑠𝑎1(𝑡)𝑒𝑡𝑡0𝑎(𝑠)𝑑𝑠,𝜂13Char2Char3𝐺=3𝐼1(𝑡)𝑎𝑎(𝑡)+𝑎(𝑡)+41/2𝑎1/2.(3.26) In view of 2𝜂13𝐺=𝜂23𝐺=4𝜂213𝜂23𝐺=1𝑎2,(3.27) conditions (2.2) and (2.3) of Theorem 2.2 turn to (2.10) and (2.11).
Further the asymptotic solution 𝑒𝑡𝑡0𝜂2(𝑠)𝑑𝑠+𝑒𝑡𝑡0𝜂3(𝑠)𝑑𝑠=2𝑒𝑎(𝑡)𝑡𝑡0[𝑎(𝑠)]𝑑𝑠cos𝑡𝑡0[]𝑎(𝑠)𝑑𝑠(3.28) is oscillating if and only if (2.12) is satisfied. Indeed, the solution corresponding to the asymptotic solution 𝑒𝑡𝑡0𝜂1𝑑𝑠=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: 𝜆𝑎(𝑡)=𝑡ln𝛾(𝑡),𝜆0.(3.29) From (2.10) and (2.11) we get that 𝑡ln𝛾||||(𝑡)3𝐼1+𝛾(𝛾2)𝑡2ln2+𝜆(𝑡)2𝑡2ln2𝛾1(𝑡)𝑡2||||𝐿1𝑡0,𝑡,2ln2𝛾(||||𝑡)2𝐼2+3𝐼1𝑡𝛾1++𝜆ln(𝑡)ln𝛾+(𝑡)𝛾(𝛾1)(𝛾2)𝑡3ln3+(𝑡)𝜆𝛾(𝛾2)𝑡3ln2+𝛾𝛾(𝑡)𝑡3+𝜆ln(𝑡)3𝑡3ln3𝛾𝜆(𝑡)𝑡3ln𝛾||||(𝑡)𝐿1𝑡0,,(3.30) or in the case 𝛾=0𝑡0𝑡3𝐼1𝜆(𝑡)+21𝑡2𝑑𝑡<,𝑡0𝑡2||||2𝐼2(𝑡)+3(1+𝜆)𝐼1(𝑡)𝑡+𝜆𝜆21𝑡3||||𝑑𝑡<,(3.31) which is equivalent to (2.13).
Further from (2.12) we get condition (2.14) in the case 𝛾=0: 𝑡0[𝜆]𝑑𝑡𝑡ln𝛾[𝜆](𝑡)=,or>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)=𝑖3𝐼1(𝑡), we have 2𝐼2(𝑡)=𝑎(𝑡)𝑎(𝑡). Condition (2.10) turns to (2.18) as follows: 3𝐼1+𝑎2𝑎𝑎+41/2𝑎1/2=𝑄(𝑡)𝐿1𝑡0𝑎,,𝑄(𝑡)=41/2𝑎1/2,(3.33) and condition (2.11) turns to (2.19) since 1a22𝐼2+3𝐼1𝑎𝑎(𝑡)𝑎+𝑣(𝑡)𝑣=𝑣(𝑡)𝑎2𝑣𝑎=𝑄(𝑡)2𝑎(𝑡)+𝑄(𝑡),(3.34) where 𝑣(𝑡)=𝑎1(𝑡)𝑒𝑡𝑡0𝑎(𝑠)𝑑𝑠𝑎,𝑄(𝑡)=41/2𝑎1/2𝐼=411/4𝐼11/4.(3.35)

Proof of Theorem 2.9. Let choose the phase functions 𝜂𝑗(𝑡) as in (2.21). We deduce Theorem 2.9 from Theorem 2.2. By calculations 𝐺(𝑡)=33𝐼31+3𝑑2+3𝑑𝐼12𝑑𝐼1𝑖𝑑=𝑖3𝐼3𝑑1+31𝑑2+𝐼1𝐼3𝑑ln31𝑑2,𝜂(3.36)12=32𝐼1𝑑1/3𝑑1/3𝑖32𝐼1𝑑1/3+𝑑1/3,𝜂23=𝑖3𝐼1𝑑1/3+𝑑1/3,(3.37) to deduce dichotomy conditions (2.1) from (2.24), it is enough to show that 𝜂2𝜂3𝐼>0,if1,𝑀𝑅1,𝜂2𝜂3𝐼0,if1,𝑀𝑅2𝑅1,𝜂1𝜂2𝐼>0,if1,𝑀𝑅2,𝜂1𝜂2𝐼0,if1,𝑀𝑅2𝑅2.(3.38)

Case 1 (𝐼31(𝑡)+𝑀2(𝑡)0,𝑡>𝑡0). In this case 𝑑 and 𝜂1 are real, 𝜂2,3 are complex conjugate and from (3.37) 𝜂1𝜂2𝜂=1𝜂3=32𝐼1𝑑1/3𝑑1/3𝜂,2𝜂3=0.(3.39) If 𝐼1<0,𝑀0, then 𝑑=𝑀+𝑀2+𝐼31<𝑀+|𝑀|=0, [𝜂1𝜂2]>0.
If 𝐼10,𝑀>0, then 𝑑=𝑀+𝑀2+𝐼31𝑀>0, [𝜂1𝜂2]<0.
Otherwise if 𝐼1>0, then 𝑑>𝑀+𝑀20, 𝑑1/3>0 and 𝑑2=𝑀+𝑀2+𝐼312=𝐼31+2𝑀21+𝐼1+31𝑀2𝐼31,𝑑2/3𝐼1𝜂>0,1𝜂2=32𝐼1𝑑1/3𝑑1/30.(3.40)
Further condition (2.7) turns to (2.25) since [𝜂2]=(1/2)[𝜂2𝜂3].

Case 2 (𝐼31(𝑡)+𝑀2(𝑡)<0,𝑡>𝑡0). In this case 𝑑=𝑀+𝑖𝐼31𝑀2 is complex valued, 𝜂1,2,3,𝐺 are real. In view of 𝑑𝑑=𝑀+𝑀2+𝐼31𝑀𝑀2+𝐼31=𝐼31,𝑑1/3𝑑1/3=𝐼1,𝑑1/3𝐼=1𝑑1/3,𝑟𝑒𝑖𝑎1/3=𝑟1/3𝑒𝑖𝑎/3,𝑑1/3=𝑑1/3,(3.41) we get that 𝜂2=𝑒𝑖𝜋/3𝑑1/3𝑒𝑖𝜋/3𝐼1𝑑1/3𝐺(𝑡)𝑒3𝐺(𝑡)=2𝑖𝜋/3𝑑1/3𝐺(𝑡)𝜂3G(𝑡),2𝜂=0,1𝑑=21/3𝐺(𝑡)3𝐺(𝑡),𝜂3𝑒=2𝑖𝜋/3𝑑1/3𝐺(𝑡).3𝐺(𝑡)(3.42) Condition (2.25) fails in this case: [𝑑1/3𝑑1/3]=0 and (1.2) is non-oscillatory. Further 𝑑=𝑀+𝑖𝑀2𝐼31=𝑟𝑒𝑖𝑎,tan(𝑎)=𝑀2𝐼31𝑀𝜋,𝑎2,𝜋2,𝜂1𝜂2=3𝑑1/3𝑑31/3𝑑=31/313𝑎cot3,𝜂1𝜂3=3𝑑1/3+𝑑31/3𝑑=31/313𝑎+cot3,𝜂2𝜂3=2𝑑31/3=23𝑟1/3𝑎sin3.(3.43) If 𝑀>0 then tan(𝑎)>0, 0<𝑎<𝜋/2, 𝑎=tan11(𝐼31/𝑀2), and we get the following: 𝑎0<3<𝜋6𝑎,sin3𝑎>0,cos3𝐼>0,<1+31𝑀2𝑑0,1/3=𝑟1/3𝑎sin3𝑎>0,cot3>13,𝜂1𝜂2<0,𝜂1𝜂3<0,𝜂2𝜂3<0.(3.44) If 𝑀<0 then tan(𝑎)<0, 𝜋/2<𝑎<0, 𝑎=tan11(𝐼31/𝑀2), and we get the following: 𝜋6<𝑎3𝑎<0,sin3𝑎<0,cos3𝑑>0,1/3𝑎<0,cot31<3,𝜂1𝜂2<0,𝜂1𝜂3<0,𝜂2𝜂3>0.(3.45)

Proof of Theorem 2.11. By taking 𝐼1(𝑡)0, we get the following: 𝑑(𝑡)=2𝐼2(𝑡),𝐺(𝑡)=6𝑖3𝐼2(𝑡),𝜂1(𝑡)=𝑑1/3𝐺3𝐺=𝑑1/3𝑑𝜂3𝑑2,3(𝑡)=𝑒±𝑖𝜋/3𝑑1/3𝑑=3𝑑1±𝑖32𝑑1/3𝑑𝜂3𝑑,2,3=±32𝑑1/3,(3.46) and Theorem 2.9 turns to Theorem 2.11.


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