# On Oscillations of Solutions of Third-Order Dynamic Equation

Authors: Gro Hovhannisyan
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction

Consider an ordinary nonautonomous differential equation of the third order(1.1)Lv=v′′′(t)-3a2(t)v′′(t)+6a1(t)v′(t)+2a0(t)v(t)=0
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.LetCk(t0,∞) be the set of k times differentiable functions on (t0,∞). By substitution v(t)=u(t)e∫t0t‍a2(s)ds equation (1.1) with a2(t)∈C2(t0,∞) turns to the following equation:
(1.2)Pu=u′′′(t)+3I1(t)u′(t)+2I2(t)u(t)=0,
where the functions I1(t) and  I2(t) are given by
(1.3)I1(t)=2a1(t)+a2′(t)-a22(t),I2(t)=a0(t)+3a1(t)a2(t)-a23(t)+a2′′(t)2.If0<|e∫t0∞‍a2(s)ds|<∞ or
(1.4)-∞<∫t0∞‍R[a2(s)]ds<∞,
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)(1.5)Charj(t)=Char(ηj)=e-∫t0t‍ηj(s)dsP(e∫t0t‍ηj(s)ds),j=1,2,3.
By direct calculations
(1.6)Charj(t)=ηj′′(t)+3ηj′(t)ηj(t)+ηj3+3ηj(t)I1(t)+2I2(t),j=1,2,3.
To consider the case of complex coefficients, we are using asymptotic solutions of (1.2) in Euler form u(t)=e∫t0t‍η(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(1.7)M(t)=I2(t)-3I1′(t)4>0orM(t)<0
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(1.8)I1(t)≤0,I2(t)>0,t>t0,(1.9)∫t0∞‍(I2(t)-(-I1(t))3/2)dt=∞
are satisfied. Then (1.2) with the real coefficients is oscillatory.The adjoint transformationI2(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

LetW[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 functionsηj(t)∈C2(t0,∞), j=1,2,3, such that expressions ℜ[ηj(t)-ηk(t)], k≤j do not change a sign, that is,
(2.1)R[ηj(t)-ηk(t)]≤0orR[ηj(t)-ηk(t)]≥0,k,j=1,2,3,k≤j,t>t0,(2.2)∫t0∞‍(|η13(t)|+|η23(t)|)|Char2(t)||G(t)|dt<∞,ηjk(t)=ηj(t)-ηk(t),k,j=1,2,3,(2.3)∫t0∞‍(|η13(t)|2+|η23(t)|2)|Char2(t)-Char3(t)||η23(t)G(t)|dt<∞,
where
(2.4)G(t)=W[t,η12,η13]-η12(t)η13(t)η23(t),G′(t)G(t)+η1(t)+η2(t)+η3(t)=0.
Then solutions of (1.2) may be represented in the form
(2.5)uk(t)=∑j=13‍φj(t)(δjk+ɛjk(t))Cj,lim⁡t→∞ɛjk(t)=0,j,k=1,2,3,
where Charj(t), j=1,2,3 are defined in (1.5), (1.6), and
(2.6)φj(t)=e∫t0t‍ηj(s)ds,δjk={1,j=k,0,j≠k.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∫t0t‍ηj(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
(2.7)∫t0∞‍I[ηk(t)]dt=∞.The following theorem we deduce from Theorem2.2 by choosing
(2.8)η1(t)=-a′(t)a(t),η2(t)=a(t)-a′(t)a(t),η3(t)=-a(t)-a′(t)a(t).Theorem 2.3.
Assume that there exists a complex-valued functiona(t)∈C3(t0,∞) such that a-1/2(t)∈C2(t0,∞), and
(2.9)R[a(t)]doesnotchangethesignon(t0,∞),(2.10)∫t0∞‍|3I1(t)+a2(t)+4a1/2(t)(a-1/2(t))′′|dt|a(t)|<∞,(2.11)∫t0∞‍|2I2(t)+3I1(t)(a(t)-a′(t)a(t))+(a-1(t)e∫t0t‍a(s)ds)′′′a-1(t)e∫t0t‍a(s)ds|dta2(t)<∞.
Then (1.2) with complex coefficients has one nonoscillatory solution and two linearly independent oscillatory solutions if and only if
(2.12)∫t0∞‍I[a(t)]dt=∞,or∫t0∞‍I[a(t)]dt=-∞.By takinga(t)=λ/t from Theorem 2.3, we get the following corollary.Corollary 2.4.
Assume that for some complex numberλ≠0(2.13)∫t0∞‍t2|2I2(t)+1-λ2t3|dt<∞,∫t0∞‍t|3I1(t)+λ2-1t2|dt<∞.
Then (1.2) with complex coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if
(2.14)I[λ]>0.By takingλ=1 from Corollary 2.4, we obtain well-known result [4].Corollary 2.5.
Assume that conditions(2.15)∫t0∞‍t2|I2(t)|dt<∞,∫t0∞‍t|I1(t)|dt<∞
are satisfied. Then (1.2) with complex coefficients is non-oscillatory.By takinga(t)=-1/tln(t) from Theorem 2.3 we get another corollary.Corollary 2.6.
Assume that conditions(2.16)∫t0∞‍t2ln⁡2(t)|2I2(t)+3I1(t)t|dt<∞,∫t0∞‍tln⁡(t)|3I1(t)-1t2|dt<∞
are satisfied. Then (1.2) with complex coefficients is non-oscillatory.Example 2.7.
From Corollary2.6, (1.2) with
(2.17)I1(t)=13t2,I2(t)=-12t3
is non-oscillatory. Note that Corollary 2.5 is not applicable for this example since condition (2.15) fails.In the caseM(t)≡0 by taking a(t)=i3I1(t), from Theorem 2.3, we deduce the following theorem.Theorem 2.8.
Assume thatI1(t)∈C3(t0,∞), I1(t)≥β>0, and
(2.18)M(t)=I2(t)-3I1′(t)4≡0,t∈(t0,∞),∫t0∞‍|(I1-1/4(t))′′I1-1/4(t)|dt<∞,(2.19)∫t0∞‍|((I1-1/4(t))′′I1-3/4(t))′|dt<∞.
Then (1.2) with the real coefficients has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if
(2.20)∫t0∞‍I1(t)dt=∞.Another result may be proved by the different choice of the phase functions as follows:(2.21)ηj(t)=d1/3(t)e-iπ(2j+1)/3-I1(t)d1/3(t)eiπ(2j+1)/3-G′(t)3G(t),j=1,2,3,
where G(t) is defined in (2.4), and
(2.22)d(t)=M(t)+M2(t)+I13(t),M(t)=I2(t)-3I1′(t)4.Define 3 auxiliary regions on the real plane(2.23)R1={(I1,M)∈R2∣I1≤0,M<0,I13+M2<0},R2={(I1,M)∈R2∣I1<0,M≤0,I13+M2≥0},R3=R2∖(R1∪R2)=((I1,M)∈R2,M>0,orI1>0).Theorem 2.9.
Assume thatI1(t)∈C1(t0,∞), R0 is simply connected region R0⊂Rj for some j=1,2,3, and conditions (2.2) and (2.3),
(2.24)(I1(t),M(t))∈R0,t>t0
are satisfied. Then (1.2) has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if
(2.25)∫t0∞R[(M2(t)+I13(t)+M(t))1/3+(M2(t)+I13(t)-M(t))1/3]dt=∞
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 coefficientsI1(t)=I1, I2(t)=I2 from Theorem 2.9, one can deduce the obvious result that (1.2) is oscillatory if and only if I13+I22>0. Indeed in this case condition (2.25) turns to I13+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:
(2.26)M(t)≥0,orM(t)≤0,I1(t)≥0,t>t0.
If I1(t)≥0, then M(t) under condition (2.24) may change the sign.Theorem2.9 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 thatI2(t) is real, it does not change the sign, and conditions
(2.27)∫t0∞‍|(I2-1/3)′′′(t)I2-1/3|dt<∞,∫t0∞‍|(I2-1/6)′′(t)I2-1/6|dt<∞
are satisfied. Then equation
(2.28)u′′′(t)+2I2(t)u(t)=0
has one non-oscillatory solution and two linearly independent oscillatory solutions if and only if
(2.29)∫t0∞‍|I2(t)|1/3dt=∞.In the caseM(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(2.30)u′′′(t)+(1t3+1-(i+μ)2t2)u′(t)+(1t4+(i+μ)2-1t3)u(t)=0,
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]).
LetΛ(t)=diag(λ1(t),…,λn(t)) be an n×n diagonal matrix function which satisfies dichotomy condition.
For each pair of integersi and j in [1,n](i≠j) exist constants K1,K2 such that for all x and t, t0≤t≤x<∞(3.1)∫tx‍R[λi(s)-λj(s)]ds≤K1,or∫tx‍R[λi(s)-λj(s)]ds≥K2.
Let the n×n matrix N(t) satisfy N(t)∈L1(t0,∞) or
(3.2)∫tx‍|N(t)|ds<∞,
by which we mean that each entry in N(t) has an absolutely convergent infinite integral. Then the system
(3.3)Y′(t)=(Λ(t)+N(t))Y(t)
has a vector solution Y(t) with the asymptotic form
(3.4)Y(t)=(E+ɛ(t))e∫t0t‍Λ(s)dsC,lim⁡t→∞ɛ(t)=0,
where E is the identity matrix, ɛ(t) is the n×n error matrix-function, and C=(C1,…Cn)tr is a constant column vector.Proof of Theorem2.1.
Rewrite (1.2) as a system
(3.5)y′(t)=A(t)y(t),A(t)=(010001-2I2(t)-3I1(t)0),y(t)=(u(t)u′(t)u′′(t)).
By transformation
(3.6)y(t)=Φ(t)z(t),
where matrix function Φ(t) is defined via phase functions ηj(t) as follows:
(3.7)Φ(t)=(e∫t0t‍η1(s)dsμ1(t)e∫t0t‍η2(s)dsμ2(t)e∫t0t‍η3(s)dsμ3(t)η1(t)μ1(t)e∫t0t‍η1(s)dsη2(t)μ2(t)e∫t0t‍η2(s)dsη3(t)μ3(t)e∫t0t‍η3(s)ds(η1′(t)+η12(t))μ1(t)e∫t0t‍η1(s)ds(η2′(t)+η22(t))μ2(t)e∫t0t‍η2(s)ds(η3′(t)+η32(t))μ3(t)e∫t0t‍η3ds),
we get the following:
(3.8)z′(t)=Φ-1(t)(A(t)Φ(t)-Φ′(t))z(t),orz′(t)=(D(t)+B(t))z(t),
where
(3.9)B(t)=1G(t)(η23(t)Char1(t)μ1η23Char2(t)μ2(t)e∫t0t‍η21dsμ1η23Char3(t)μ3(t)e∫t0t‍η31dsμ2η31Char1(t)μ1(t)e∫t0t‍η12dsη31(t)Char2(t)μ2η31Char3(t)μ3(t)e∫t0t‍η32dsμ3η12Char1(t)μ1(t)e∫t0t‍η13dsμ3η12Char2(t)μ2(t)e∫t0t‍η23dsη12(t)Char3(t)),D(t)=(μ1′(t)μ1(t)000μ2′(t)μ2(t)000μ3′(t)μ3(t)),ηjk(t)=ηj(t)-ηk(t).
Choosing specific auxiliary functions(3.10)μ1(t)=1,μ2(t)=e∫t0t‍η21(s)ds,μ3(t)=e∫t0t‍η31(s)ds,
we have
(3.11)D(t)=(0000η21000η31),B(t)=1G(t)(η32Char1η23Char2η23Char3η31Char1η31Char2η31Char3η12Char1η12Char2η12Char3).
Here and further we suppress the time variable t for the simplicity.
From Liouville's formuladetΦ(t)=Ce∫t0t‍Tr(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
(3.12)C=det⁡(Φ(t))=G(t)e∫t0t‍(η1+η2+η3)(s)ds.
The Liouville's formula may be written in the form
(3.13)G′(t)G(t)+η1(t)+η2(t)+η3(t)=0.
We always are choosing the phase functions ηj(t) such that (3.13) is satisfied (see (2.4)). From (3.13), we get that
(3.14)(η1-η3)Char12(t)-(η1-η2)Char13(t)G(t)=-η1(t)-η2(t)-η3(t)-G′(t)G(t),(3.15)Char1(t)-Char2(t)η1(t)-η2(t)=Char1(t)-Char3(t)η1(t)-η3(t)=Char2(t)-Char3(t)η2(t)-η3(t).
To apply Theorem 3.1 to system (3.8) note that from (2.1) it follows dichotomy condition (3.1) of Theorem 3.1:
(3.16)∫tx‍R[ηkj]ds≥0,or∫tx‍R[ηkj]ds≤0,x≥t,j≠k,k,j=1,2,3.
Condition (3.2) of Theorem 3.1 turns to B(t)∈L1(t0,∞), and it is followed from
(3.17)η13(t)Charj(t)G(t),η23(t)Charj(t)G(t)∈L1(t0,∞),j=1,2,3.
One can drop conditionη23(t)Char1(t)/G(t)∈L1(t0,∞) since from (3.15) we have
(3.18)η23(t)Char1(t)G(t)=η13(t)Char2(t)G(t)-η12Char3G.
Assuming that
(3.19)η132(t)(Char2(t)-Char3(t))η23(t)G(t)∈L1(t0,∞),
condition η13Char1/G∈L1(t0,∞) may be dropped as well since
(3.20)η13Char1G=η13(η13Char2-η12Char3)Gη23=η13(η13(Char2-Char3)+η23Char3)η23G.
So condition (3.2) of Theorem 3.1 turns to
(3.21)(|η23(t)|+|η13(t)|)|Charj(t)||G(t)|,|η13(t)|2|Char2-Char3(t)||η23(t)G(t)|∈L1(t0,∞),j=2,3,
or (2.2) and (2.3). From Theorem 3.1 applied to system (3.8) and we get that
(3.22)z(t)=z0(t)(E+ɛ(t))C,z0(t)=e∫t0t‍D(s)ds=(μ1(t)000μ2(t)000μ3(t))C1,y(t)=Φ(t)z(t)=Φ(t)z0(t)(E+ɛ(t))C,
or representation (2.5).Proof of Theorem2.2.
Theorem2.2 is followed from Theorem 2.1 since in representation (2.5) one may choose asymptotic solutions as follows:
(3.23)φ2(t)=e∫t0t‍R[η2(s)]dssin⁡∫t0t‍I[η2(s)]dsds,φ3(t)=e∫t0t‍R[η2(s)]dscos⁡∫t0t‍I[η2(s)]ds
which are oscillating if and only if condition (2.7) is satisfied.Proof of Theorem2.3.
Theorem2.3 is deduced from Theorem 2.2 by choosing phase functions as in (2.8). From
(3.24)2R[η21]=2R[η13]=R[η23]=2R[a(t)],
condition (2.1) turns to condition (2.9). From (2.4), we get that
(3.25)G(t)=2a3(t),η1+η2+η3+G′(t)G(t)=0.
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
(3.26)2η13Char2G=1a2(2I2+3I1(a-a′a)+(a-1(t)e∫t0t‍a(s)ds)′′′a-1(t)e∫t0t‍a(s)ds),η13(Char2-Char3)G=3I1(t)a(t)+a(t)+4(a-1/2)′′a-1/2.
In view of
(3.27)2η13G=η23G=4η132η23G=1a2,
conditions (2.2) and (2.3) of Theorem 2.2 turn to (2.10) and (2.11).
Further the asymptotic solution(3.28)e∫t0t‍η2(s)ds+e∫t0t‍η3(s)ds=2a(t)e∫t0t‍R[a(s)]dscos⁡∫t0t‍I[a(s)]ds
is oscillating if and only if (2.12) is satisfied. Indeed, the solution corresponding to the asymptotic solution e∫t0t‍η1ds=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 Corollaries2.4 and 2.6.
We deduce Corollaries2.4 and 2.6 from Theorem 2.3 by the special choice of function a(t) as follows:
(3.29)a(t)=λtln⁡γ(t),λ≠0.
From (2.10) and (2.11) we get that
(3.30)tln⁡γ(t)|3I1+γ(γ-2)t2ln⁡2(t)+λ2t2ln⁡2γ(t)-1t2|∈L1(t0,∞),t2ln⁡2γ(t)|2I2+3I1t(1+γln⁡(t)+λln⁡γ(t))+γ(γ-1)(γ-2)t3ln⁡3(t)+λγ(γ-2)t3ln⁡2+γ(t)-γt3ln⁡(t)+λ3t3ln⁡3γ(t)-λt3ln⁡γ(t)|∈L1(t0,∞),
or in the case γ=0(3.31)∫t0∞‍t(3I1(t)+λ2-1t2)dt<∞,∫t0∞‍t2|2I2(t)+3(1+λ)I1(t)t+λ(λ2-1)t3|dt<∞,
which is equivalent to (2.13).
Further from (2.12) we get condition (2.14) in the case γ=0:
(3.32)∫t0∞‍I[λ]dttln⁡γ(t)=∞,orI[λ]>0,γ=0.
The proof of Corollary 2.6 is followed from (3.30) to (3.27) by choosing γ=-λ=1.Proof of Theorem2.8.
Theorem2.8 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:
(3.33)3I1+a2a+4(a-1/2)′′a-1/2=Q(t)∈L1(t0,∞),Q(t)=4(a-1/2)′′a-1/2,
and condition (2.11) turns to (2.19) since
(3.34)1a2(2I2+3I1(a-a′(t)a)+v′′′(t)v)=v′′′(t)a2v-a=(Q(t)2a(t))′+Q(t),
where
(3.35)v(t)=a-1(t)e∫t0t‍a(s)ds,Q(t)=4(a-1/2)′′a-1/2=4(I1-1/4)′′I1-1/4.Proof of Theorem2.9.
Let choose the phase functionsηj(t) as in (2.21). We deduce Theorem 2.9 from Theorem 2.2. By calculations
(3.36)G(t)=3(3I13+3d2+3dI1′-2d′I1)id=-i33d[1+I13d2+I13d(ln⁡I13d2)′],(3.37)η12=32(I1d1/3-d1/3)-i32(I1d1/3+d1/3),η23=i3(I1d1/3+d1/3),
to deduce dichotomy conditions (2.1) from (2.24), it is enough to show that
(3.38)R[η2-η3]>0,if(I1,M)∈R1,R[η2-η3]≤0,if(I1,M)∈R2∖R1,R[η1-η2]>0,if(I1,M)∈R2,R[η1-η2]≤0,if(I1,M)∈R2∖R2.Case 1 (I13(t)+M2(t)≥0,t>t0).
In this cased and η1 are real, η2,3 are complex conjugate and from (3.37)
(3.39)R[η1-η2]=R[η1-η3]=32(I1d1/3-d1/3),R[η2-η3]=0.
If I1<0,M≤0, then d=M+M2+I13<M+|M|=0, ℜ[η1-η2]>0.
IfI1≤0,M>0, then d=M+M2+I13≥M>0, ℜ[η1-η2]<0.
Otherwise ifI1>0, then d>M+M2≥0, d1/3>0 and
(3.40)d2=(M+M2+I13)2=I13+2M2(1+1+I13M2)≥I13,d2/3≥I1>0,R[η1-η2]=32(I1d1/3-d1/3)≤0.
Further condition (2.7) turns to (2.25) since ℑ[η2]=(1/2)ℑ[η2-η3].Case 2 (I13(t)+M2(t)<0,t>t0).
In this cased=M+i-I13-M2 is complex valued, η1,2,3,G are real. In view of
(3.41)dd¯=(M+M2+I13)(M-M2+I13)=-I13,d1/3(d¯)1/3=-I1,d¯1/3=-I1d1/3,(re-ia)1/3=r1/3eia/3¯,d¯1/3=d1/3¯,
we get that
(3.42)η2=eiπ/3d1/3-e-iπ/3I1d1/3-G′(t)3G(t)=2R[eiπ/3d1/3]-G′(t)3G(t),I[η2]=0,η1=-2R[d1/3]-G′(t)3G(t),η3=2R[e-iπ/3d1/3]-G′(t)3G(t).
Condition (2.25) fails in this case: ℜ[d1/3-d¯1/3]=0 and (1.2) is non-oscillatory. Further
(3.43)d=M+i-M2-I13=reia,tan(a)=-M2-I13M,a∈(-π2,π2),η1-η2=3(I[d1/3]-3R[d1/3])=3I[d1/3](13-cot(a3)),η1-η3=-3(I[d1/3]+3R[d1/3])=-3I[d1/3](13+cot(a3)),η2-η3=-23I[d1/3]=-23r1/3sin⁡(a3).
If M>0 then tan(a)>0, 0<a<π/2, a=tan-1-1-(I13/M2), and we get the following:
(3.44)0<a3<π6,sin⁡(a3)>0,cos⁡(a3)>0,-∞<1+I13M2≤0,I[d1/3]=r1/3sin⁡(a3)>0,cot(a3)>13,η1-η2<0,η1-η3<0,η2-η3<0.
If M<0 then tan(a)<0, -π/2<a<0, a=-tan-1-1-(I13/M2), and we get the following:
(3.45)-π6<a3<0,sin⁡(a3)<0,cos⁡(a3)>0,I[d1/3]<0,cot(a3)<-13,η1-η2<0,η1-η3<0,η2-η3>0.Proof of Theorem2.11.
By takingI1(t)≡0, we get the following:
(3.46)d(t)=2I2(t),G(t)=-6i3I2(t),η1(t)=-d1/3-G′3G=-d1/3-d′3dη2,3(t)=e±iπ/3d1/3-d′3d=1±i32d1/3-d′3d,I[η2,3]=±32d1/3,
and Theorem 2.9 turns to Theorem 2.11.



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*Source: 715981-2012-07-11.xml*