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Abstract and Applied Analysis
VolumeΒ 2012, Article IDΒ 715981, 15 pages
http://dx.doi.org/10.1155/2012/715981
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: PaulΒ Eloe

Copyright Β© 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.

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=π‘£ξ…žξ…žξ…ž(𝑑)βˆ’3π‘Ž2(𝑑)π‘£ξ…žξ…ž(𝑑)+6π‘Ž1(𝑑)π‘£ξ…ž(𝑑)+2π‘Ž0(𝑑)𝑣(𝑑)=0(1.1) with complex valued variable coefficients π‘Ž0(𝑑), π‘Ž1(𝑑), and π‘Ž2(𝑑).

A solution of (1.1) is said to be oscillatory if it has an infinite sequence of zeros in (𝑑0,∞), 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 πΆπ‘˜(𝑑0,∞) be the set of π‘˜ times differentiable functions on (𝑑0,∞). By substitution 𝑣(𝑑)=𝑒(𝑑)π‘’βˆ«π‘‘π‘‘0π‘Ž2(𝑠)𝑑𝑠 equation (1.1) with π‘Ž2(𝑑)∈𝐢2(𝑑0,∞) turns to the following equation: 𝑃𝑒=π‘’ξ…žξ…žξ…ž(𝑑)+3𝐼1(𝑑)π‘’ξ…ž(𝑑)+2𝐼2(𝑑)𝑒(𝑑)=0,(1.2) where the functions 𝐼1(𝑑) and  𝐼2(𝑑) are given by 𝐼1(𝑑)=2π‘Ž1(𝑑)+π‘Žξ…ž2(𝑑)βˆ’π‘Ž22(𝑑),𝐼2(𝑑)=π‘Ž0(𝑑)+3π‘Ž1(𝑑)π‘Ž2(𝑑)βˆ’π‘Ž32π‘Ž(𝑑)+2ξ…žξ…ž(𝑑)2.(1.3)

If 0<|π‘’βˆ«βˆžπ‘‘0π‘Ž2(𝑠)𝑑𝑠|<∞ or ξ€œβˆ’βˆž<βˆžπ‘‘0β„œξ€Ίπ‘Ž2ξ€»(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 πœ‚π‘—(𝑑)Charπ‘—ξ€·πœ‚(𝑑)=Char𝑗=π‘’βˆ’βˆ«π‘‘π‘‘0πœ‚π‘—(𝑠)π‘‘π‘ π‘ƒξ‚€π‘’βˆ«π‘‘π‘‘0πœ‚π‘—(𝑠)𝑑𝑠,𝑗=1,2,3.(1.5) By direct calculations Char𝑗(𝑑)=πœ‚π‘—ξ…žξ…ž(𝑑)+3πœ‚ξ…žπ‘—(𝑑)πœ‚π‘—(𝑑)+πœ‚3𝑗+3πœ‚π‘—(𝑑)𝐼1(𝑑)+2𝐼2(𝑑),𝑗=1,2,3.(1.6) To consider the case of complex coefficients, we are using asymptotic solutions of (1.2) in Euler form 𝑒(𝑑)=π‘’βˆ«π‘‘π‘‘0πœ‚(𝑠)𝑑𝑠 with phase functions πœ‚π‘—(𝑑), 𝑗=1,2,3, that are approximate solutions of the characteristic equation Char(πœ‚π‘—)=0.

Theorem 1.1 (see [4]). If Mammana's condition 𝑀(𝑑)=𝐼2(𝑑)βˆ’3πΌξ…ž1(𝑑)4>0π‘œπ‘Ÿπ‘€(𝑑)<0(1.7) is satisfied except at isolated points at which 𝑀(𝑑) 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(𝑑)≀0,𝐼2(𝑑)>0,𝑑>𝑑0,ξ€œ(1.8)βˆžπ‘‘0𝐼2ξ€·(𝑑)βˆ’βˆ’πΌ1ξ€Έ(𝑑)3/2𝑑𝑑=∞(1.9) are satisfied. Then (1.2) with the real coefficients is oscillatory.

The adjoint transformation 𝐼2(𝑑)β†’(3πΌξ…ž1(𝑑)/2)βˆ’πΌ2(𝑑), or 𝑀(𝑑)β†’βˆ’π‘€(𝑑) transforms (1.2) to its adjoint equation βˆ’π‘€ξ…žξ…žξ…ž(𝑑)βˆ’3𝐼1(𝑑)π‘€ξ…ž(𝑑)+(2𝐼2(𝑑)βˆ’3πΌξ…ž1(𝑑))𝑀(𝑑)=0. Note that condition (1.9) is not invariant with respect to the adjoint transformation 𝑀(𝑑)β†’βˆ’π‘€(𝑑). 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 π‘Š[𝑑,π‘Ž,𝑏]=π‘Ž(𝑑)𝑏′(𝑑)βˆ’π‘Žβ€²(𝑑)𝑏(𝑑) be the Wronskian of two differentiable functions π‘Ž(𝑑) and 𝑏(𝑑). The following asymptotic theorem is proved by using Levinson's asymptotic theorem [8].

Theorem 2.1. Assume that there exists complex-valued phase functions πœ‚π‘—(𝑑)∈𝐢2(𝑑0,∞), 𝑗=1,2,3, such that expressions β„œ[πœ‚π‘—(𝑑)βˆ’πœ‚π‘˜(𝑑)], π‘˜β‰€π‘— do not change a sign, that is, β„œξ€Ίπœ‚π‘—(𝑑)βˆ’πœ‚π‘˜ξ€»ξ€Ίπœ‚(𝑑)≀0π‘œπ‘Ÿβ„œπ‘—(𝑑)βˆ’πœ‚π‘˜ξ€»(𝑑)β‰₯0,π‘˜,𝑗=1,2,3,π‘˜β‰€π‘—,𝑑>𝑑0,ξ€œ(2.1)βˆžπ‘‘0ξ€·||πœ‚13||+||πœ‚(𝑑)23||ξ€Έ||(𝑑)Char2||(𝑑)||||𝐺(𝑑)𝑑𝑑<∞,πœ‚π‘—π‘˜(𝑑)=πœ‚π‘—(𝑑)βˆ’πœ‚π‘˜ξ€œ(𝑑),π‘˜,𝑗=1,2,3,(2.2)βˆžπ‘‘0ξ‚€||πœ‚13||(𝑑)2+||πœ‚23||(𝑑)2||Char2(𝑑)βˆ’Char3||(𝑑)||πœ‚23||(𝑑)𝐺(𝑑)𝑑𝑑<∞,(2.3) where 𝐺(𝑑)=π‘Šπ‘‘,πœ‚12,πœ‚13ξ€»βˆ’πœ‚12(𝑑)πœ‚13(𝑑)πœ‚23𝐺(𝑑),ξ…ž(𝑑)𝐺(𝑑)+πœ‚1(𝑑)+πœ‚2(𝑑)+πœ‚3(𝑑)=0.(2.4)
Then solutions of (1.2) may be represented in the form π‘’π‘˜(𝑑)=3𝑗=1πœ‘π‘—ξ€·π›Ώ(𝑑)π‘—π‘˜+πœ€π‘—π‘˜ξ€ΈπΆ(𝑑)𝑗,limπ‘‘β†’βˆžπœ€π‘—π‘˜(𝑑)=0,𝑗,π‘˜=1,2,3,(2.5) where Char𝑗(𝑑), 𝑗=1,2,3 are defined in (1.5), (1.6), and πœ‘π‘—(𝑑)=π‘’βˆ«π‘‘π‘‘0πœ‚π‘—(𝑠)𝑑𝑠,π›Ώπ‘—π‘˜=ξ‚»1,𝑗=π‘˜,0,π‘—β‰ π‘˜.(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 π‘’βˆ«π‘‘π‘‘0πœ‚π‘—(𝑠)𝑑𝑠 corresponding to the phase functions πœ‚π‘—(𝑑)∈𝐢2(𝑑0,∞), 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 πœ‚π‘˜(𝑑) is oscillatory if and only if ξ€œβˆžπ‘‘0β„‘ξ€Ίπœ‚π‘˜ξ€»(𝑑)𝑑𝑑=∞.(2.7)

The following theorem we deduce from Theorem 2.2 by choosing πœ‚1π‘Ž(𝑑)=βˆ’ξ…ž(𝑑)π‘Ž(𝑑),πœ‚2π‘Ž(𝑑)=π‘Ž(𝑑)βˆ’ξ…ž(𝑑)π‘Ž(𝑑),πœ‚3π‘Ž(𝑑)=βˆ’π‘Ž(𝑑)βˆ’ξ…ž(𝑑).π‘Ž(𝑑)(2.8)

Theorem 2.3. Assume that there exists a complex-valued function π‘Ž(𝑑)∈𝐢3(𝑑0,∞) such that π‘Žβˆ’1/2(𝑑)∈𝐢2(𝑑0,∞), and β„œ[]ξ€·π‘‘π‘Ž(𝑑)π‘‘π‘œπ‘’π‘ π‘›π‘œπ‘‘π‘β„Žπ‘Žπ‘›π‘”π‘’π‘‘β„Žπ‘’π‘ π‘–π‘”π‘›π‘œπ‘›0ξ€Έ,ξ€œ,∞(2.9)βˆžπ‘‘0|||3𝐼1(𝑑)+π‘Ž2(𝑑)+4π‘Ž1/2ξ€·π‘Ž(𝑑)βˆ’1/2ξ€Έ(𝑑)ξ…žξ…ž|||𝑑𝑑||||ξ€œπ‘Ž(𝑑)<∞,(2.10)βˆžπ‘‘0||||||2𝐼2(𝑑)+3𝐼1ξ‚΅π‘Ž(𝑑)π‘Ž(𝑑)βˆ’ξ…ž(𝑑)π‘Žξ‚Ά+ξ‚€π‘Ž(𝑑)βˆ’1(𝑑)π‘’βˆ«π‘‘π‘‘0π‘Ž(𝑠)π‘‘π‘ ξ‚ξ…žξ…žξ…žπ‘Žβˆ’1(𝑑)π‘’βˆ«π‘‘π‘‘0π‘Ž(𝑠)𝑑𝑠||||||π‘‘π‘‘π‘Ž2(𝑑)<∞.(2.11) Then (1.2) with complex coefficients has one nonoscillatory solution and two linearly independent oscillatory solutions if and only if ξ€œβˆžπ‘‘0β„‘[]ξ€œπ‘Ž(𝑑)𝑑𝑑=∞,π‘œπ‘Ÿβˆžπ‘‘0β„‘[]π‘Ž(𝑑)𝑑𝑑=βˆ’βˆž.(2.12)

By taking π‘Ž(𝑑)=πœ†/𝑑 from Theorem 2.3, we get the following corollary.

Corollary 2.4. Assume that for some complex number πœ†β‰ 0ξ€œβˆžπ‘‘0𝑑2||||2𝐼2(𝑑)+1βˆ’πœ†2𝑑3||||ξ€œπ‘‘π‘‘<∞,βˆžπ‘‘0𝑑||||3𝐼1πœ†(𝑑)+2βˆ’1𝑑2||||𝑑𝑑<∞.(2.13) Then (1.2) with complex coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if β„‘[πœ†]>0.(2.14)

By taking πœ†=1 from Corollary 2.4, we obtain well-known result [4].

Corollary 2.5. Assume that conditions ξ€œβˆžπ‘‘0𝑑2||𝐼2||ξ€œ(𝑑)𝑑𝑑<∞,βˆžπ‘‘0𝑑||𝐼1||(𝑑)𝑑𝑑<∞(2.15) are satisfied. Then (1.2) with complex coefficients is non-oscillatory.

By taking π‘Ž(𝑑)=βˆ’1/𝑑ln(𝑑) from Theorem 2.3 we get another corollary.

Corollary 2.6. Assume that conditions ξ€œβˆžπ‘‘0𝑑2ln2||||(𝑑)2𝐼2(𝑑)+3𝐼1(𝑑)𝑑||||ξ€œπ‘‘π‘‘<∞,βˆžπ‘‘0|||𝑑ln(𝑑)3𝐼11(𝑑)βˆ’π‘‘2|||𝑑𝑑<∞(2.16) are satisfied. Then (1.2) with complex coefficients is non-oscillatory.

Example 2.7. From Corollary 2.6, (1.2) with 𝐼11(𝑑)=3𝑑2,𝐼21(𝑑)=βˆ’2𝑑3(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(𝑑)≑0 by taking βˆšπ‘Ž(𝑑)=𝑖3𝐼1(𝑑), from Theorem 2.3, we deduce the following theorem.

Theorem 2.8. Assume that 𝐼1(𝑑)∈𝐢3(𝑑0,∞), 𝐼1(𝑑)β‰₯𝛽>0, and 𝑀(𝑑)=𝐼2(𝑑)βˆ’3πΌξ…ž1(𝑑)4𝑑≑0,π‘‘βˆˆ0ξ€Έ,ξ€œ,βˆžβˆžπ‘‘0|||𝐼1βˆ’1/4ξ€Έ(𝑑)ξ…žξ…žπΌ1βˆ’1/4|||ξ€œ(𝑑)𝑑𝑑<∞,(2.18)βˆžπ‘‘0|||𝐼1βˆ’1/4ξ€Έ(𝑑)ξ…žξ…žπΌ1βˆ’3/4(𝑑)ξ…ž|||𝑑𝑑<∞.(2.19) Then (1.2) with the real coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if ξ€œβˆžπ‘‘0√𝐼1(𝑑)𝑑𝑑=∞.(2.20)

Another result may be proved by the different choice of the phase functions as follows: πœ‚π‘—(𝑑)=𝑑1/3(𝑑)π‘’βˆ’π‘–πœ‹(2𝑗+1)/3βˆ’πΌ1(𝑑)𝑑1/3𝑒(𝑑)π‘–πœ‹(2𝑗+1)/3βˆ’πΊξ…ž(𝑑)3𝐺(𝑑),𝑗=1,2,3,(2.21) where 𝐺(𝑑) is defined in (2.4), and 𝑑(𝑑)=𝑀(𝑑)+𝑀2(𝑑)+𝐼31(𝑑),𝑀(𝑑)=𝐼2(𝑑)βˆ’3πΌξ…ž1(𝑑)4.(2.22)

Define 3 auxiliary regions on the real plane 𝑅1=𝐼1ξ€Έ,π‘€βˆˆπ‘…2∣𝐼1≀0,𝑀<0,𝐼31+𝑀2ξ€Ύ,𝑅<02=𝐼1ξ€Έ,π‘€βˆˆπ‘…2∣𝐼1<0,𝑀≀0,𝐼31+𝑀2ξ€Ύ,𝑅β‰₯03=𝑅2⧡𝑅1βˆͺ𝑅2ξ€Έ=𝐼1ξ€Έ,π‘€βˆˆπ‘…2,𝑀>0,or𝐼1ξ€Έ.>0(2.23)

Theorem 2.9. Assume that 𝐼1(𝑑)∈𝐢1(𝑑0,∞), 𝑅0 is simply connected region 𝑅0βŠ‚π‘…π‘— for some 𝑗=1,2,3, and conditions (2.2) and (2.3), 𝐼1ξ€Έ(𝑑),𝑀(𝑑)βˆˆπ‘…0,𝑑>𝑑0(2.24) are satisfied. Then (1.2) has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if ξ€œβˆžπ‘‘0β„œξƒ¬ξ‚΅ξ”π‘€2(𝑑)+𝐼31ξ‚Ά(𝑑)+𝑀(𝑑)1/3+𝑀2(𝑑)+𝐼31ξ‚Ά(𝑑)βˆ’π‘€(𝑑)1/3𝑑𝑑=∞(2.25) for at least one of cubic roots.

Note that condition (2.25) is invariant with respect to the adjoint transformation 𝑀(𝑑)β†’βˆ’π‘€(𝑑).

For the case of the real constant coefficients 𝐼1(𝑑)=𝐼1, 𝐼2(𝑑)=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(𝑑)β‰₯0,or𝑀(𝑑)≀0,𝐼1(𝑑)β‰₯0,𝑑>𝑑0.(2.26) If 𝐼1(𝑑)β‰₯0, then 𝑀(𝑑) under condition (2.24) may change the sign.

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

Theorem 2.11. Assume that 𝐼2(𝑑) is real, it does not change the sign, and conditions ξ€œβˆžπ‘‘0|||𝐼2βˆ’1/3ξ€Έξ…žξ…žξ…ž(𝑑)𝐼2βˆ’1/3|||ξ€œπ‘‘π‘‘<∞,βˆžπ‘‘0|||𝐼2βˆ’1/6ξ€Έξ…žξ…ž(𝑑)𝐼2βˆ’1/6|||𝑑𝑑<∞(2.27) are satisfied. Then equation π‘’ξ…žξ…žξ…ž(𝑑)+2𝐼2(𝑑)𝑒(𝑑)=0(2.28) has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if ξ€œβˆžπ‘‘0||𝐼2||(𝑑)1/3𝑑𝑑=∞.(2.29)

In the case 𝑀(𝑑)=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 π‘’ξ…žξ…ž(𝑑)+𝐼1(𝑑)𝑒(𝑑)=0.

Example 2.12. Equation π‘’ξ…žξ…žξ…žξ‚΅1(𝑑)+𝑑3+1βˆ’(𝑖+πœ‡)2𝑑2ξ‚Άπ‘’ξ…žξ‚΅1(𝑑)+𝑑4+(𝑖+πœ‡)2βˆ’1𝑑3𝑒(𝑑)=0,(2.30) where πœ‡ is a real number and is oscillatory by Corollary 2.4 since conditions (2.13) and (2.14) are satisfied with πœ†=𝑖+πœ‡. Note that for this example Theorem 1.2 is not applicable since both conditions (1.8) and (1.9) fail even when πœ‡=0.

3. Proofs

Our main tool is Levinson's asymptotic theorem.

Theorem 3.1 (see [8]). Let Ξ›(𝑑)=diag(πœ†1(𝑑),…,πœ†π‘›(𝑑)) 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, 𝑑0≀𝑑≀π‘₯<βˆžξ€œπ‘₯π‘‘β„œξ€Ίπœ†π‘–(𝑠)βˆ’πœ†π‘—ξ€»(𝑠)𝑑𝑠≀𝐾1ξ€œ,π‘œπ‘Ÿπ‘₯π‘‘β„œξ€Ίπœ†π‘–(𝑠)βˆ’πœ†π‘—ξ€»(𝑠)𝑑𝑠β‰₯𝐾2.(3.1) Let the 𝑛×𝑛 matrix 𝑁(𝑑) satisfy 𝑁(𝑑)∈𝐿1(𝑑0,∞) or ξ€œπ‘₯𝑑||||𝑁(𝑑)𝑑𝑠<∞,(3.2) by which we mean that each entry in 𝑁(𝑑) has an absolutely convergent infinite integral. Then the system π‘Œξ…ž(𝑑)=(Ξ›(𝑑)+𝑁(𝑑))π‘Œ(𝑑)(3.3) has a vector solution π‘Œ(𝑑) with the asymptotic form π‘Œ(𝑑)=(𝐸+πœ€(𝑑))π‘’βˆ«π‘‘π‘‘0Ξ›(𝑠)𝑑𝑠𝐢,limπ‘‘β†’βˆžπœ€(𝑑)=0,(3.4) where 𝐸 is the identity matrix, πœ€(𝑑) is the 𝑛×𝑛 error matrix-function, and 𝐢=(𝐢1,…𝐢𝑛)tr is a constant column vector.

Proof of Theorem 2.1. Rewrite (1.2) as a system π‘¦ξ…žβŽ›βŽœβŽœβŽœβŽœβŽ(𝑑)=𝐴(𝑑)𝑦(𝑑),𝐴(𝑑)=010001βˆ’2𝐼2(𝑑)βˆ’3𝐼1(βŽžβŽŸβŽŸβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽœβŽœβŽπ‘’π‘‘)0,𝑦(𝑑)=𝑒(𝑑)ξ…žπ‘’(𝑑)ξ…žξ…ž(⎞⎟⎟⎟⎟⎠.𝑑)(3.5) By transformation 𝑦(𝑑)=Ξ¦(𝑑)𝑧(𝑑),(3.6) where matrix function Ξ¦(𝑑) is defined via phase functions πœ‚π‘—(𝑑) as follows: βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπ‘’Ξ¦(𝑑)=βˆ«π‘‘π‘‘0πœ‚1(𝑠)π‘‘π‘ πœ‡1𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚2(𝑠)π‘‘π‘ πœ‡2𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚3(𝑠)π‘‘π‘ πœ‡3πœ‚(𝑑)1(𝑑)πœ‡1(𝑒𝑑)βˆ«π‘‘π‘‘0πœ‚1(𝑠)π‘‘π‘ πœ‚2(𝑑)πœ‡2(𝑒𝑑)βˆ«π‘‘π‘‘0πœ‚2(𝑠)π‘‘π‘ πœ‚3(𝑑)πœ‡3(𝑒𝑑)βˆ«π‘‘π‘‘0πœ‚3(𝑠)π‘‘π‘ ξ€·πœ‚ξ…ž1(𝑑)+πœ‚21ξ€Έ(𝑑)πœ‡1𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚1(𝑠)π‘‘π‘ ξ€·πœ‚ξ…ž2(𝑑)+πœ‚22ξ€Έ(𝑑)πœ‡2𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚2(𝑠)π‘‘π‘ ξ€·πœ‚ξ…ž3(𝑑)+πœ‚23ξ€Έ(𝑑)πœ‡3𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚3π‘‘π‘ βŽžβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽŸβŽ ,(3.7) we get the following: π‘§ξ…ž(𝑑)=Ξ¦βˆ’1ξ€·(𝑑)𝐴(𝑑)Ξ¦(𝑑)βˆ’Ξ¦ξ…žξ€Έ(𝑑)𝑧(𝑑),orπ‘§ξ…ž(𝑑)=(𝐷(𝑑)+𝐡(𝑑))𝑧(𝑑),(3.8) where 1𝐡(𝑑)=βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽœβŽπœ‚πΊ(𝑑)23(𝑑)Char1πœ‡(𝑑)1πœ‚23Char2(𝑑)πœ‡2𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚21π‘‘π‘ πœ‡1πœ‚23Char3(𝑑)πœ‡3𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚31π‘‘π‘ πœ‡2πœ‚31Char1(𝑑)πœ‡1𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚12π‘‘π‘ πœ‚31(𝑑)Char2πœ‡(𝑑)2πœ‚31Char3(𝑑)πœ‡3𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚32π‘‘π‘ πœ‡3πœ‚12Char1(𝑑)πœ‡1𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚13π‘‘π‘ πœ‡3πœ‚12Char2(𝑑)πœ‡2𝑒(𝑑)βˆ«π‘‘π‘‘0πœ‚23π‘‘π‘ πœ‚12(𝑑)Char3⎞⎟⎟⎟⎟⎟⎟⎟⎠,βŽ›βŽœβŽœβŽœβŽœβŽœβŽœβŽπœ‡(𝑑)𝐷(𝑑)=ξ…ž1(𝑑)πœ‡10πœ‡(𝑑)00ξ…ž2(𝑑)πœ‡20πœ‡(𝑑)00ξ…ž3(𝑑)πœ‡3⎞⎟⎟⎟⎟⎟⎟⎠(𝑑),πœ‚π‘—π‘˜(𝑑)=πœ‚π‘—(𝑑)βˆ’πœ‚π‘˜(𝑑).(3.9)
Choosing specific auxiliary functions πœ‡1(𝑑)=1,πœ‡2(𝑑)=π‘’βˆ«π‘‘π‘‘0πœ‚21(𝑠)𝑑𝑠,πœ‡3(𝑑)=π‘’βˆ«π‘‘π‘‘0πœ‚31(𝑠)𝑑𝑠,(3.10) we have βŽ›βŽœβŽœβŽœβŽœβŽπ·(𝑑)=0000πœ‚21000πœ‚31⎞⎟⎟⎟⎟⎠1,𝐡(𝑑)=βŽ›βŽœβŽœβŽœβŽœβŽπœ‚πΊ(𝑑)32Char1πœ‚23Char2πœ‚23Char3πœ‚31Char1πœ‚31Char2πœ‚31Char3πœ‚12Char1πœ‚12Char2πœ‚12Char3⎞⎟⎟⎟⎟⎠.(3.11) Here and further we suppress the time variable 𝑑 for the simplicity.
From Liouville's formula detΞ¦(𝑑)=πΆπ‘’βˆ«π‘‘π‘‘0Tr(𝐴(𝑠))𝑑𝑠=𝐢 applied to (3.7) with the assumption that πœ‚π‘—(𝑑) are solutions of Char𝑗(𝑑)=0,πœ‡1=πœ‡2=πœ‡3=1, we get the following 𝐢=det(Ξ¦(𝑑))=𝐺(𝑑)π‘’βˆ«π‘‘π‘‘0(πœ‚1+πœ‚2+πœ‚3)(𝑠)𝑑𝑠.(3.12) The Liouville's formula may be written in the form πΊξ…ž(𝑑)𝐺(𝑑)+πœ‚1(𝑑)+πœ‚2(𝑑)+πœ‚3(𝑑)=0.(3.13) We always are choosing the phase functions πœ‚π‘—(𝑑) such that (3.13) is satisfied (see (2.4)). From (3.13), we get that ξ€·πœ‚1βˆ’πœ‚3ξ€ΈChar12ξ€·πœ‚(𝑑)βˆ’1βˆ’πœ‚2ξ€ΈChar13(𝑑)𝐺(𝑑)=βˆ’πœ‚1(𝑑)βˆ’πœ‚2(𝑑)βˆ’πœ‚3𝐺(𝑑)βˆ’ξ…ž(𝑑),𝐺(𝑑)(3.14)Char1(𝑑)βˆ’Char2(𝑑)πœ‚1(𝑑)βˆ’πœ‚2=(𝑑)Char1(𝑑)βˆ’Char3(𝑑)πœ‚1(𝑑)βˆ’πœ‚3=(𝑑)Char2(𝑑)βˆ’Char3(𝑑)πœ‚2(𝑑)βˆ’πœ‚3.(𝑑)(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: ξ€œπ‘₯π‘‘β„œξ€Ίπœ‚π‘˜π‘—ξ€»ξ€œπ‘‘π‘ β‰₯0,orπ‘₯π‘‘β„œξ€Ίπœ‚π‘˜π‘—ξ€»π‘‘π‘ β‰€0,π‘₯β‰₯𝑑,π‘—β‰ π‘˜,π‘˜,𝑗=1,2,3.(3.16) Condition (3.2) of Theorem 3.1 turns to 𝐡(𝑑)∈𝐿1(𝑑0,∞), and it is followed from πœ‚13(𝑑)Char𝑗(𝑑),πœ‚πΊ(𝑑)23(𝑑)Char𝑗(𝑑)𝐺(𝑑)∈𝐿1𝑑0ξ€Έ,∞,𝑗=1,2,3.(3.17)
One can drop condition πœ‚23(𝑑)Char1(𝑑)/𝐺(𝑑)∈𝐿1(𝑑0,∞) since from (3.15) we have πœ‚23(𝑑)Char1(𝑑)=πœ‚πΊ(𝑑)13(𝑑)Char2(𝑑)βˆ’πœ‚πΊ(𝑑)12Char3𝐺.(3.18) Assuming that πœ‚213ξ€·(𝑑)Char2(𝑑)βˆ’Char3ξ€Έ(𝑑)πœ‚23(𝑑)𝐺(𝑑)∈𝐿1𝑑0ξ€Έ,,∞(3.19) condition πœ‚13Char1/𝐺∈𝐿1(𝑑0,∞) may be dropped as well since πœ‚13Char1𝐺=πœ‚13ξ€·πœ‚13Char2βˆ’πœ‚12Char3ξ€ΈπΊπœ‚23=πœ‚13ξ€·πœ‚13ξ€·Char2βˆ’Char3ξ€Έ+πœ‚23Char3ξ€Έπœ‚23𝐺.(3.20) So condition (3.2) of Theorem 3.1 turns to ξ€·||πœ‚23(||+||πœ‚π‘‘)13(||ξ€Έ||𝑑)Char𝑗(||𝑑)||||,||πœ‚πΊ(𝑑)13||(𝑑)2||Char2βˆ’Char3||(𝑑)||πœ‚23||(𝑑)𝐺(𝑑)∈𝐿1𝑑0ξ€Έ,∞,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(𝑑)=𝑧0(𝑑)(𝐸+πœ€(𝑑))𝐢,𝑧0(𝑑)=π‘’βˆ«π‘‘π‘‘0𝐷(𝑠)𝑑𝑠=βŽ›βŽœβŽœβŽœβŽœβŽπœ‡1(𝑑)000πœ‡2(𝑑)000πœ‡3⎞⎟⎟⎟⎟⎠𝐢(𝑑)1,𝑦(𝑑)=Ξ¦(𝑑)𝑧(𝑑)=Ξ¦(𝑑)𝑧0(𝑑)(𝐸+πœ€(𝑑))𝐢,(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: πœ‘2(𝑑)=π‘’βˆ«π‘‘π‘‘0β„œ[πœ‚2(𝑠)]π‘‘π‘ ξ€œsin𝑑𝑑0β„‘ξ€Ίπœ‚2(𝑠)𝑑𝑠𝑑𝑠,πœ‘3(𝑑)=π‘’βˆ«π‘‘π‘‘0β„œ[πœ‚2(𝑠)]π‘‘π‘ ξ€œcos𝑑𝑑0β„‘ξ€Ίπœ‚2(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 ξ€Ίπœ‚2β„œ21ξ€»ξ€Ίπœ‚=2β„œ13ξ€»ξ€Ίπœ‚=β„œ23ξ€»[π‘Ž]=2β„œ(𝑑),(3.24) condition (2.1) turns to condition (2.9). From (2.4), we get that 𝐺(𝑑)=2π‘Ž3(𝑑),πœ‚1+πœ‚2+πœ‚3+πΊξ…ž(𝑑)𝐺(𝑑)=0.(3.25) Since in conditions (2.2)-(2.3) the function 𝐺(𝑑) appears in denominator we should assume that 𝐺(𝑑)β‰ 0, or π‘Ž(𝑑)β‰ 0. By direct calculations, we get that 2πœ‚13Char2𝐺=1π‘Ž2βŽ›βŽœβŽœβŽœβŽ2𝐼2+3𝐼1ξ‚€π‘Žβˆ’π‘Žβ€²π‘Žξ‚+ξ‚€π‘Žβˆ’1(𝑑)π‘’βˆ«π‘‘π‘‘0π‘Ž(𝑠)π‘‘π‘ ξ‚ξ…žξ…žξ…žπ‘Žβˆ’1(𝑑)π‘’βˆ«π‘‘π‘‘0π‘Ž(𝑠)π‘‘π‘ βŽžβŽŸβŽŸβŽŸβŽ ,πœ‚13ξ€·Char2βˆ’Char3𝐺=3𝐼1(𝑑)ξ€·π‘Žπ‘Ž(𝑑)+π‘Ž(𝑑)+4βˆ’1/2ξ€Έξ…žξ…žπ‘Žβˆ’1/2.(3.26) In view of 2πœ‚13𝐺=πœ‚23𝐺=4πœ‚213πœ‚23𝐺=1π‘Ž2,(3.27) conditions (2.2) and (2.3) of Theorem 2.2 turn to (2.10) and (2.11).
Further the asymptotic solution π‘’βˆ«π‘‘π‘‘0πœ‚2(𝑠)𝑑𝑠+π‘’βˆ«π‘‘π‘‘0πœ‚3(𝑠)𝑑𝑠=2π‘’π‘Ž(𝑑)βˆ«π‘‘π‘‘0β„œ[π‘Ž(𝑠)]π‘‘π‘ ξ€œcos𝑑𝑑0β„‘[]π‘Ž(𝑠)𝑑𝑠(3.28) is oscillating if and only if (2.12) is satisfied. Indeed, the solution corresponding to the asymptotic solution π‘’βˆ«π‘‘π‘‘0πœ‚1𝑑𝑠=𝐢/π‘Ž(𝑑) is non-oscillatory (1/π‘Ž(𝑑) does not have zeros; otherwise π‘Ž(𝑑)∈𝐢3(𝑑0,∞) 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 π‘Ž(𝑑) as follows: πœ†π‘Ž(𝑑)=𝑑ln𝛾(𝑑),πœ†β‰ 0.(3.29) From (2.10) and (2.11) we get that 𝑑ln𝛾||||(𝑑)3𝐼1+𝛾(π›Ύβˆ’2)𝑑2ln2+πœ†(𝑑)2𝑑2ln2π›Ύβˆ’1(𝑑)𝑑2||||∈𝐿1𝑑0ξ€Έ,𝑑,∞2ln2𝛾(||||𝑑)2𝐼2+3𝐼1𝑑𝛾1++πœ†ln(𝑑)ln𝛾+(𝑑)𝛾(π›Ύβˆ’1)(π›Ύβˆ’2)𝑑3ln3+(𝑑)πœ†π›Ύ(π›Ύβˆ’2)𝑑3ln2+π›Ύβˆ’π›Ύ(𝑑)𝑑3+πœ†ln(𝑑)3𝑑3ln3π›Ύβˆ’πœ†(𝑑)𝑑3ln𝛾||||(𝑑)∈𝐿1𝑑0ξ€Έ,,∞(3.30) or in the case 𝛾=0ξ€œβˆžπ‘‘0𝑑3𝐼1πœ†(𝑑)+2βˆ’1𝑑2ξ‚Άξ€œπ‘‘π‘‘<∞,βˆžπ‘‘0𝑑2||||2𝐼2(𝑑)+3(1+πœ†)𝐼1(𝑑)𝑑+πœ†ξ€·πœ†2ξ€Έβˆ’1𝑑3||||𝑑𝑑<∞,(3.31) which is equivalent to (2.13).
Further from (2.12) we get condition (2.14) in the case 𝛾=0: ξ€œβˆžπ‘‘0β„‘[πœ†]𝑑𝑑𝑑ln𝛾[πœ†](𝑑)=∞,orβ„‘>0,𝛾=0.(3.32) The proof of Corollary 2.6 is followed from (3.30) to (3.27) by choosing 𝛾=βˆ’πœ†=1.

Proof of Theorem 2.8. Theorem 2.8 is followed from Theorem 2.3. Indeed from 𝑀(𝑑)≑0,𝑑>𝑑0√,π‘Ž(𝑑)=𝑖3𝐼1(𝑑), we have 2𝐼2(𝑑)=βˆ’π‘Žβ€²(𝑑)π‘Ž(𝑑). Condition (2.10) turns to (2.18) as follows: 3𝐼1+π‘Ž2π‘Žξ€·π‘Ž+4βˆ’1/2ξ€Έξ…žξ…žπ‘Žβˆ’1/2=𝑄(𝑑)∈𝐿1𝑑0ξ€Έξ€·π‘Ž,∞,𝑄(𝑑)=4βˆ’1/2ξ€Έξ…žξ…žπ‘Žβˆ’1/2,(3.33) and condition (2.11) turns to (2.19) since 1a2ξ‚΅2𝐼2+3𝐼1ξ‚΅π‘Žπ‘Žβˆ’ξ…ž(𝑑)π‘Žξ‚Ά+π‘£ξ…žξ…žξ…ž(𝑑)𝑣=π‘£ξ…žξ…žξ…ž(𝑑)π‘Ž2π‘£ξ‚΅βˆ’π‘Ž=𝑄(𝑑)ξ‚Ά2π‘Ž(𝑑)ξ…ž+𝑄(𝑑),(3.34) where 𝑣(𝑑)=π‘Žβˆ’1(𝑑)π‘’βˆ«π‘‘π‘‘0π‘Ž(𝑠)π‘‘π‘ ξ€·π‘Ž,𝑄(𝑑)=4βˆ’1/2ξ€Έξ…žξ…žπ‘Žβˆ’1/2𝐼=41βˆ’1/4ξ€Έξ…žξ…žπΌ1βˆ’1/4.(3.35)

Proof of Theorem 2.9. Let choose the phase functions πœ‚π‘—(𝑑) as in (2.21). We deduce Theorem 2.9 from Theorem 2.2. By calculations √𝐺(𝑑)=3ξ€·3𝐼31+3𝑑2+3π‘‘πΌξ…ž1βˆ’2π‘‘ξ…žπΌ1ξ€Έβˆšπ‘–π‘‘=βˆ’π‘–3𝐼3𝑑1+31𝑑2+𝐼1𝐼3𝑑ln31𝑑2ξƒͺξ…žξƒ­,πœ‚(3.36)12=32𝐼1𝑑1/3βˆ’π‘‘1/3ξ‚Άβˆ’π‘–βˆš32𝐼1𝑑1/3+𝑑1/3ξ‚Ά,πœ‚23√=𝑖3𝐼1𝑑1/3+𝑑1/3ξ‚Ά,(3.37) to deduce dichotomy conditions (2.1) from (2.24), it is enough to show that β„œξ€Ίπœ‚2βˆ’πœ‚3𝐼>0,if1ξ€Έ,π‘€βˆˆπ‘…1,β„œξ€Ίπœ‚2βˆ’πœ‚3𝐼≀0,if1ξ€Έ,π‘€βˆˆπ‘…2⧡𝑅1,β„œξ€Ίπœ‚1βˆ’πœ‚2𝐼>0,if1ξ€Έ,π‘€βˆˆπ‘…2,β„œξ€Ίπœ‚1βˆ’πœ‚2𝐼≀0,if1ξ€Έ,π‘€βˆˆπ‘…2⧡𝑅2.(3.38)

Case 1 (𝐼31(𝑑)+𝑀2(𝑑)β‰₯0,𝑑>𝑑0). In this case 𝑑 and πœ‚1 are real, πœ‚2,3 are complex conjugate and from (3.37) β„œξ€Ίπœ‚1βˆ’πœ‚2ξ€»ξ€Ίπœ‚=β„œ1βˆ’πœ‚3ξ€»=32𝐼1𝑑1/3βˆ’π‘‘1/3ξ‚Άξ€Ίπœ‚,β„œ2βˆ’πœ‚3ξ€»=0.(3.39) If 𝐼1<0,𝑀≀0, then 𝑑=𝑀+𝑀2+𝐼31<𝑀+|𝑀|=0, β„œ[πœ‚1βˆ’πœ‚2]>0.
If 𝐼1≀0,𝑀>0, then 𝑑=𝑀+𝑀2+𝐼31β‰₯𝑀>0, β„œ[πœ‚1βˆ’πœ‚2]<0.
Otherwise if 𝐼1>0, then βˆšπ‘‘>𝑀+𝑀2β‰₯0, 𝑑1/3>0 and 𝑑2=𝑀+𝑀2+𝐼31ξ‚Ά2=𝐼31+2𝑀2βŽ›βŽœβŽœβŽξƒŽ1+𝐼1+31𝑀2⎞⎟⎟⎠β‰₯𝐼31,𝑑2/3β‰₯𝐼1β„œξ€Ίπœ‚>0,1βˆ’πœ‚2ξ€»=32𝐼1𝑑1/3βˆ’π‘‘1/3≀0.(3.40)
Further condition (2.7) turns to (2.25) since β„‘[πœ‚2]=(1/2)β„‘[πœ‚2βˆ’πœ‚3].

Case 2 (𝐼31(𝑑)+𝑀2(𝑑)<0,𝑑>𝑑0). In this case 𝑑=𝑀+π‘–βˆ’πΌ31βˆ’π‘€2 is complex valued, πœ‚1,2,3,𝐺 are real. In view of 𝑑𝑑=𝑀+𝑀2+𝐼31ξ”ξ‚Άξ‚΅π‘€βˆ’π‘€2+𝐼31ξ‚Ά=βˆ’πΌ31,𝑑1/3𝑑1/3=βˆ’πΌ1,𝑑1/3𝐼=βˆ’1𝑑1/3,ξ€·π‘Ÿπ‘’βˆ’π‘–π‘Žξ€Έ1/3=π‘Ÿ1/3π‘’π‘–π‘Ž/3,𝑑1/3=𝑑1/3,(3.41) we get that πœ‚2=π‘’π‘–πœ‹/3𝑑1/3βˆ’π‘’βˆ’π‘–πœ‹/3𝐼1𝑑1/3βˆ’πΊξ…ž(𝑑)𝑒3𝐺(𝑑)=2β„œπ‘–πœ‹/3𝑑1/3ξ€»βˆ’πΊξ…ž(𝑑)ξ€Ίπœ‚3G(𝑑),β„‘2ξ€»πœ‚=0,1𝑑=βˆ’2β„œ1/3ξ€»βˆ’πΊξ…ž(𝑑)3𝐺(𝑑),πœ‚3𝑒=2β„œβˆ’π‘–πœ‹/3𝑑1/3ξ€»βˆ’πΊξ…ž(𝑑).3𝐺(𝑑)(3.42) Condition (2.25) fails in this case: β„œ[𝑑1/3βˆ’π‘‘1/3]=0 and (1.2) is non-oscillatory. Further 𝑑=𝑀+π‘–βˆ’π‘€2βˆ’πΌ31=π‘Ÿπ‘’π‘–π‘Žξ”,tan(π‘Ž)=βˆ’π‘€2βˆ’πΌ31π‘€ξ‚€βˆ’πœ‹,π‘Žβˆˆ2,πœ‹2,πœ‚1βˆ’πœ‚2=√3ℑ𝑑1/3ξ€»βˆ’βˆšξ€Ίπ‘‘3β„œ1/3𝑑=3β„‘1/31√3ξ‚€π‘Žβˆ’cot3ξƒͺ,πœ‚1βˆ’πœ‚3√=βˆ’3ℑ𝑑1/3ξ€»+βˆšξ€Ίπ‘‘3β„œ1/3𝑑=βˆ’3β„‘1/31√3ξ‚€π‘Ž+cot3ξƒͺ,πœ‚2βˆ’πœ‚3√=βˆ’2𝑑3β„‘1/3ξ€»βˆš=βˆ’23π‘Ÿ1/3ξ‚€π‘Žsin3.(3.43) If 𝑀>0 then tan(π‘Ž)>0, 0<π‘Ž<πœ‹/2, π‘Ž=tanβˆ’1ξ”βˆ’1βˆ’(𝐼31/𝑀2), and we get the following: π‘Ž0<3<πœ‹6ξ‚€π‘Ž,sin3ξ‚ξ‚€π‘Ž>0,cos3𝐼>0,βˆ’βˆž<1+31𝑀2ℑ𝑑≀0,1/3ξ€»=π‘Ÿ1/3ξ‚€π‘Žsin3ξ‚ξ‚€π‘Ž>0,cot3>1√3,πœ‚1βˆ’πœ‚2<0,πœ‚1βˆ’πœ‚3<0,πœ‚2βˆ’πœ‚3<0.(3.44) If 𝑀<0 then tan(π‘Ž)<0, βˆ’πœ‹/2<π‘Ž<0, π‘Ž=βˆ’tanβˆ’1ξ”βˆ’1βˆ’(𝐼31/𝑀2), and we get the following: βˆ’πœ‹6<π‘Ž3ξ‚€π‘Ž<0,sin3ξ‚ξ‚€π‘Ž<0,cos3ℑ𝑑>0,1/3ξ€»ξ‚€π‘Ž<0,cot31<βˆ’βˆš3,πœ‚1βˆ’πœ‚2<0,πœ‚1βˆ’πœ‚3<0,πœ‚2βˆ’πœ‚3>0.(3.45)

Proof of Theorem 2.11. By taking 𝐼1(𝑑)≑0, we get the following: 𝑑(𝑑)=2𝐼2√(𝑑),𝐺(𝑑)=βˆ’6𝑖3𝐼2(𝑑),πœ‚1(𝑑)=βˆ’π‘‘1/3βˆ’πΊβ€²3𝐺=βˆ’π‘‘1/3βˆ’π‘‘β€²πœ‚3𝑑2,3(𝑑)=π‘’Β±π‘–πœ‹/3𝑑1/3βˆ’π‘‘ξ…ž=√3𝑑1±𝑖32𝑑1/3βˆ’π‘‘ξ…žξ€Ίπœ‚3𝑑,β„‘2,3ξ€»βˆš=Β±32𝑑1/3,(3.46) and Theorem 2.9 turns to Theorem 2.11.

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