Advances in Difference Equations

, 2013:220

# On power and non-power asymptotic behavior of positive solutions to Emden-Fowler type higher-order equations

Open Access
Research
Part of the following topical collections:
1. Progress in Functional Differential and Difference Equations

## Abstract

For the equation

${y}^{\left(n\right)}={y}^{k},\phantom{\rule{1em}{0ex}}k>1,n=12,13,14,$

the existence of positive solutions with non-power asymptotic behavior is proved, namely

$y={\left({x}^{\ast }-x\right)}^{-\alpha }h\left(log\left({x}^{\ast }-x\right)\right),\phantom{\rule{1em}{0ex}}\alpha =\frac{n}{k-1},x<{x}^{\ast },$

where ${x}^{\ast }$ is an arbitrary point, h is a positive periodic non-constant function on R.

To prove this result, the Hopf bifurcation theorem is used.

## Keywords

asymptotic behavior Emden-Fowler higher-order equations

## Introduction

For the equation
${y}^{\left(n\right)}=p\left(x,y,{y}^{\prime },\dots ,{y}^{\left(n-1\right)}\right){|y|}^{k}sgny,\phantom{\rule{1em}{0ex}}n\ge 2,k>1,$
(1)
Kiguradze posed the problem on the asymptotic behavior of its positive solutions such that
$\underset{x\to {x}^{\ast }-0}{lim}y\left(x\right)=\mathrm{\infty }.$
(2)
He found an asymptotic formula for these solutions to (1) with $n=2$ (see ) and supposed all such solutions to have power asymptotic behavior for other n, too. The problem was solved for $n=3$ and $n=4$ . For these n, it was proved that all such solutions behave as
$y\left(x\right)=C{\left({x}^{\ast }-x\right)}^{-\alpha }\left(1+o\left(1\right)\right),\phantom{\rule{1em}{0ex}}x\to {x}^{\ast }-0,$
(3)
with
$\alpha =\frac{n}{k-1},\phantom{\rule{2em}{0ex}}C={\left(\frac{\alpha \left(\alpha +1\right)\cdots \left(\alpha +n-1\right)}{{p}_{0}}\right)}^{\frac{1}{k-1}},$
(4)

${p}_{0}=\mathrm{const}>0$ - is a limit of $p\left(x,{y}_{0},\dots ,{y}_{n-1}\right)$ as $x\to {x}^{\ast }-0$, ${y}_{0}\to \mathrm{\infty },\dots ,{y}_{n-1}\to \mathrm{\infty }$.

So, the hypothesis of Kiguradze was confirmed in this case.

The existence of solutions satisfying (3) was proved for arbitrary $n\ge 2$. For $2\le n\le 11$, an $\left(n-1\right)$-parametric family of such solutions to equation (1) was proved to exist (see , , Ch.I(5.1)).

For the equation
${y}^{\left(n\right)}={y}^{k},\phantom{\rule{1em}{0ex}}k>1,$
(5)
a negative answer to the conjecture of Kiguradze for large n was obtained. It was proved  that for any N and $K>1$, there exist an integer $n>N$ and $k\in \mathbf{R}$, $1, such that equation (5) has a solution
$y={\left({x}^{\ast }-x\right)}^{-\alpha }h\left(log\left({x}^{\ast }-x\right)\right),$
(6)

where α is defined by (4), h is a positive periodic non-constant function on R.

Still, it was not clear how large n should be for the existence of that type of solutions.

## Preliminary results

Suppose the following conditions hold:
1. (A)
The continuous positive function $p\left(x,{y}_{0},\dots ,{y}_{n-1}\right)$ has a limit ${p}_{0}=\mathrm{const}>0$ as $x\to {x}^{\ast }-0$, ${y}_{0}\to \mathrm{\infty },\dots ,{y}_{n-1}\to \mathrm{\infty }$, and for some $\gamma >0$, it holds
$p\left(x,{y}_{0},\dots ,{y}_{n-1}\right)-{p}_{0}=O\left({\left({x}^{\ast }-x\right)}^{\gamma }+\sum _{j=0}^{n-1}{y}_{j}^{-\gamma }\right).$
(7)

2. (B)
For some ${K}_{1}>0$ and $\mu >0$ in a neighborhood of ${x}^{\ast }$ for sufficiently large ${y}_{0},\dots ,{y}_{n-1}$, ${z}_{0},\dots ,{z}_{n-1}$, it holds
$|p\left(x,{y}_{0},\dots ,{y}_{n-1}\right)-p\left(x,{z}_{0},\dots ,{z}_{n-1}\right)|\le {K}_{1}\underset{j}{max}|{y}_{j}^{-\mu }-{z}_{j}^{-\mu }|.$
(8)

Then equation (1) can be transformed (see  or , Ch.I(5.1)) by using the substitution
${x}^{\ast }-x={e}^{-t},\phantom{\rule{2em}{0ex}}y=\left(C+v\right){e}^{\alpha t},$
(9)
where C and α are defined by (4). The derivatives ${y}^{\left(j\right)}$, $j=0,1,\dots ,n-1$, become
${e}^{\left(\alpha +j\right)t}\cdot {L}_{j}\left(v,{v}^{\prime },\dots ,{v}^{\left(j\right)}\right),$
where ${v}^{\left(j\right)}=\frac{{d}^{j}v}{d{t}^{j}}$ and ${L}_{j}$ is a linear function with
${L}_{j}\left(0,0,\dots ,0\right)=C\alpha \left(\alpha +1\right)\cdots \left(\alpha +j-1\right)\ne 0$

and the coefficient of ${v}^{\left(j\right)}$ equal to 1.

Thus (1) is transformed into
${e}^{\left(\alpha +n\right)t}\cdot {L}_{n}\left(v,{v}^{\prime },\dots ,{v}^{\left(n\right)}\right)={\left(C+v\right)}^{k}{e}^{\alpha kt}\stackrel{˜}{p}\left(t,v,{v}^{\prime },\dots ,{v}^{\left(n-1\right)}\right),$
(10)

where the function $\stackrel{˜}{p}\left(t,{v}_{0},\dots ,{v}_{n-1}\right)$ is obtained from $p\left(x,{y}_{0},\dots ,{y}_{n-1}\right)$ with $x,{y}_{0},\dots ,{y}_{n-1}$ properly expressed in terms of $t,{v}_{0},\dots ,{v}_{n-1}$. This function tends to ${p}_{0}$ as $t\to \mathrm{\infty }$, $v\to 0,\dots ,{v}^{\left(n-1\right)}\to 0$.

Due to condition (8) for the function $p\left(x,{y}_{0},\dots ,{y}_{n-1}\right)$, we obtain the following inequalities for sufficiently large t and sufficiently small ${v}_{0},\dots ,{v}_{n-1}$, ${w}_{0},\dots ,{w}_{n-1}$:
$\begin{array}{c}|\stackrel{˜}{p}\left(t,{v}_{0},\dots ,{v}_{n-1}\right)-\stackrel{˜}{p}\left(t,{w}_{0},\dots ,{w}_{n-1}\right)|\hfill \\ \phantom{\rule{1em}{0ex}}\le {K}_{1}\underset{j}{max}{e}^{-\mu \left(\alpha +j\right)t}|{L}_{j}^{-\mu }\left({v}_{0},\dots ,{v}_{n-1}\right)-{L}_{j}^{-\mu }\left({w}_{0},\dots ,{w}_{n-1}\right)|.\hfill \end{array}$
Since ${L}_{j}\left(0,0,\dots ,0\right)\ne 0$, the function ${L}_{j}^{-\mu }$ is a ${C}^{\mathrm{\infty }}$ one in a neighborhood of 0 and
$|\stackrel{˜}{p}\left(t,{v}_{0},\dots ,{v}_{n-1}\right)-\stackrel{˜}{p}\left(t,{w}_{0},\dots ,{w}_{n-1}\right)|\le {K}_{2}{e}^{-\mu \alpha t}\underset{j}{max}|{v}_{j}-{w}_{j}|$

for some ${K}_{2}>0$.

Solving (10) for ${v}^{\left(n\right)}$ and using formulae (4), we obtain the equation
${v}^{\left(n\right)}={\left(C+v\right)}^{k}\stackrel{˜}{p}\left(t,v,{v}^{\prime },\dots ,{v}^{\left(n-1\right)}\right)-{p}_{0}{C}^{k}-\sum _{j=0}^{n-1}{a}_{j}{v}^{\left(j\right)},$
(11)
where ${a}_{j}$ are the coefficients of the linear function ${L}_{n}$. Equation (11) can be written as
${v}^{\left(n\right)}=k{C}^{k-1}{p}_{0}v-\sum _{j=0}^{n-1}{a}_{j}{v}^{\left(j\right)}+f\left(v\right)+g\left(t,v,{v}^{\prime },\dots ,{v}^{\left(n-1\right)}\right),$
(12)
where
Besides, for sufficiently large t and sufficiently small ${v}_{0},\dots ,{v}_{n-1}$, ${w}_{0},\dots ,{w}_{n-1}$, it holds
$\begin{array}{r}|g\left(t,{v}_{0},\dots ,{v}_{n-1}\right)-g\left(t,{w}_{0},\dots ,{w}_{n-1}\right)|\\ \phantom{\rule{1em}{0ex}}\le |{\left(C+{v}_{0}\right)}^{k}-{\left(C+{w}_{0}\right)}^{k}|\cdot |\stackrel{˜}{p}\left(t,{v}_{0},\dots ,{v}_{n-1}\right)-{p}_{0}|\\ \phantom{\rule{2em}{0ex}}+{\left(C+{w}_{0}\right)}^{k}|\stackrel{˜}{p}\left(t,{v}_{0},\dots ,{v}_{n-1}\right)-\stackrel{˜}{p}\left(t,{w}_{0},\dots ,{w}_{n-1}\right)|\\ \phantom{\rule{1em}{0ex}}\le {K}_{3}\underset{j}{max}|{w}_{j}-{v}_{j}|{e}^{-min\left(\gamma ,\mu \right)\cdot min\left(\alpha ,1\right)t}.\end{array}$
Suppose that V is the vector with coordinates ${V}_{j}={v}^{\left(j\right)}$, $j=0,\dots ,n-1$. Then equation (12) can be written as
$\frac{dV}{dt}=AV+F\left(V\right)+G\left(t,V\right),$
(13)
where A is a constant $n×n$ matrix
$A=\left(\begin{array}{cccccc}0& 1& 0& 0& \cdots & 0\\ 0& 0& 1& 0& \cdots & 0\\ 0& 0& 0& 1& \cdots & 0\\ \cdot & \cdot & \cdot & \cdot & \cdots & \cdot \\ 0& 0& 0& 0& \cdots & 1\\ -{\stackrel{˜}{a}}_{0}& -{a}_{1}& -{a}_{2}& -{a}_{3}& \cdots & -{a}_{n-1}\end{array}\right),$
with
$\begin{array}{rcl}-{\stackrel{˜}{a}}_{0}& =& {a}_{0}-k{c}^{k-1}{p}_{0}={a}_{0}-k\alpha \left(\alpha +1\right)\cdots \left(\alpha +n-1\right)\\ =& {a}_{0}-\left(\alpha +1\right)\cdots \left(\alpha +n-1\right)\left(\alpha +n\right)\end{array}$
and eigenvalues satisfying the equation
$\begin{array}{rcl}0& =& det\left(A-\lambda E\right)={\left(-1\right)}^{n+1}\left(-{\stackrel{˜}{a}}_{0}-{a}_{1}\lambda -\cdots -{a}_{n-1}{\lambda }^{n-1}-{\lambda }^{n}\right)\\ =& {\left(-1\right)}^{n+1}\left(\left(\alpha +1\right)\left(\alpha +2\right)\cdots \left(\alpha +n\right)-\left(\lambda +\alpha \right)\cdots \left(\lambda +\alpha +n-1\right)\right),\end{array}$
which is equivalent to
$\prod _{j=0}^{n-1}\left(\lambda +\alpha +j\right)=\prod _{j=0}^{n-1}\left(1+\alpha +j\right).$
(14)
The mappings $F:{\mathbf{R}}^{n}\to {\mathbf{R}}^{n}$ and $G:\mathbf{R}×{\mathbf{R}}^{n}\to {\mathbf{R}}^{n}$ satisfy the following estimates as $t\to \mathrm{\infty }$:
$\left\{\begin{array}{c}\parallel F\left(V\right)\parallel =O\left({\parallel V\parallel }^{2}\right),\hfill \\ \parallel {F}_{V}^{\mathrm{\prime }}\left(V\right)\parallel =O\left(\parallel V\parallel \right),\hfill \\ \parallel G\left(t,V\right)\parallel =O\left({e}^{-2\beta t}\right),\hfill \\ \parallel G\left(t,V\right)-G\left(t,W\right)\parallel \le K\parallel V-W\parallel {e}^{-2\beta t}\hfill \end{array}$
(15)

with some constants $\beta >0$, $K>0$.

Lemma 1 

Suppose that (15) holds and A is an arbitrary constant $n×n$ matrix. Then there exists a solution $V\left(t\right)$ to equation (13) tending to zero as $t\to \mathrm{\infty }$.

Lemma 2 

Let the conditions of Lemma  1 hold. If equation (14) has m roots with negative real part, then there exists an m-parametric family of solutions $V\left(t\right)$ to equation (13) tending to zero as $t\to \mathrm{\infty }$.

If equation (13) has a solution $V\left(t\right)$ tending to 0 as $t\to \mathrm{\infty }$ and ${V}_{0}\left(t\right)$ is its first coordinate, then the function
$y\left(x\right)=\left({V}_{0}\left(-log\left({x}^{\ast }-x\right)\right)+C\right)\cdot {\left({x}^{\ast }-x\right)}^{-\alpha }$

with C and α defined by (4) is a solution to (1) such that (2) and (3) hold.

Theorem 1 [2, 3]

Suppose that conditions (A) and (B) are satisfied. Then for such ${x}^{\ast }$ there exists a solution to (1) with power asymptotic behavior (3).

Investigating the signs of the real parts of the roots of equation (14), by the Routh-Hurwitz criterion, we can prove the following theorem.

Theorem 2 [2, 3]

Suppose that $3\le n\le 11$ and conditions (A) and (B) are satisfied. Then there exists an $\left(n-1\right)$-parametric family of solutions to equation (1) with power asymptotic behavior (3).

Theorem 3 [2, 3, 5]

Suppose that $n=3$ or $n=4$ in equation (1), the continuous positive function $p\left(x,{y}_{0},\dots ,{y}_{n-1}\right)$ is Lipschitz continuous in ${y}_{0},\dots ,{y}_{n-1}$ and has a limit ${p}_{0}>0$ as $x\to {x}^{\ast }-0$, ${y}_{0}\to \mathrm{\infty },\dots ,{y}_{n-1}\to \mathrm{\infty }$. Then any positive solution to this equation with a vertical asymptote $x={x}^{\ast }$ has asymptotic behavior (3).

To prove the main results of this article, we use the Hopf bifurcation theorem .

Theorem (Hopf)

Consider the α-parameterized dynamical system $\stackrel{˙}{x}={L}_{\alpha }x+{Q}_{\alpha }\left(x\right)$ in a neighborhood of $0\in {\mathbf{R}}^{n}$ with linear operators ${L}_{\alpha }$ and smooth enough functions ${Q}_{\alpha }\left(x\right)=O\left({|x|}^{2}\right)$ as $x\to 0$. Let ${\lambda }_{\alpha }$ and ${\overline{\lambda }}_{\alpha }$ be simple complex conjugated eigenvalues of the operators ${L}_{\alpha }$. Suppose that $Re{\lambda }_{\stackrel{˜}{\alpha }}=Re{\overline{\lambda }}_{\stackrel{˜}{\alpha }}=0$ for some $\stackrel{˜}{\alpha }$ and the operator ${L}_{\stackrel{˜}{\alpha }}$ has no other eigenvalues with zero real part.

If $Re\frac{d{\lambda }_{\alpha }}{d\alpha }\left(\stackrel{˜}{\alpha }\right)\ne 0$, then there exist continuous mappings $\epsilon ↦\alpha \left(\epsilon \right)\in \mathbf{R}$, $\epsilon ↦T\left(\epsilon \right)\in \mathbf{R}$, and $\epsilon ↦b\left(\epsilon \right)\in {\mathbf{R}}^{n}$ defined in a neighborhood of 0 and such that $\alpha \left(0\right)=\stackrel{˜}{\alpha }$, $T\left(0\right)=2\pi /Im{\lambda }_{\stackrel{˜}{\alpha }}$, $b\left(0\right)=0$, $b\left(\epsilon \right)\ne 0$ for $\epsilon \ne 0$, and the solutions to the problems
$\stackrel{˙}{x}={L}_{\alpha \left(\epsilon \right)}x+{Q}_{\alpha \left(\epsilon \right)}\left(x\right),\phantom{\rule{1em}{0ex}}x\left(0\right)=b\left(\epsilon \right)$

are $T\left(\epsilon \right)$-periodic and non-constant.

## Main results

In this section, the result about the existence of solutions with non-power asymptotic behavior is proved for equation (5) with $n=12,13,14$.

Theorem 4 For $n=12,13,14$, there exists $k>1$ such that equation (5) has a solution $y\left(x\right)$ with
$\begin{array}{c}{y}^{\left(j\right)}\left(x\right)={\left({x}^{\ast }-x\right)}^{-\alpha -j}{h}_{j}\left(log\left({x}^{\ast }-x\right)\right),\hfill \\ \phantom{\rule{1em}{0ex}}j=0,1,\dots ,n-1,\hfill \end{array}$

where α is defined by (4) and ${h}_{j}$ are periodic positive non-constant functions on R.

Proof To apply the Hopf bifurcation theorem, we investigate equation (13) with $G\left(t,V\right)\equiv 0$ corresponding to the case of the constant function p and the roots of the algebraic equation (14). F is a vector function with all zero components $F\left(V\right)=\left(0,\dots ,0,{F}_{n-1}\left(V\right)\right)$, $V=\left({V}_{0},\dots ,{V}_{n-1}\right)$, and
$\begin{array}{c}{F}_{n-1}\left(V\right)=\left({\left(C+{V}_{0}\right)}^{k}-{C}^{k}-k{C}^{k-1}{V}_{0}\right)=O\left({{V}_{0}}^{2}\right),\phantom{\rule{1em}{0ex}}{V}_{0}\to 0,\hfill \\ \frac{d}{dV}{F}_{n-1}\left(V\right)=O\left(|{V}_{0}|\right),\phantom{\rule{1em}{0ex}}{V}_{0}\to 0.\hfill \end{array}$

If equation (14) has a pair of pure imaginary roots, we have to check other conditions of this theorem and then apply it.

Proposition 1 For any integer $n>11$, there exist $\alpha >0$ and $q>0$ such that
$\prod _{j=0}^{n-1}\left(qi+\alpha +j\right)=\prod _{j=0}^{n-1}\left(1+\alpha +j\right)$
(16)

with ${i}^{2}=-1$.

Remark 1 In the particular case $n=12$, this result was obtained by Vyun .

Proof Consider the positive functions ${\rho }_{n}\left(\alpha \right)$ and ${\sigma }_{n}\left(\alpha \right)$ defined for all $\alpha >0$ via the equations
$\prod _{j=0}^{n-1}\left({\rho }_{n}{\left(\alpha \right)}^{2}+{\left(\alpha +j\right)}^{2}\right)=\prod _{j=0}^{n-1}{\left(1+\alpha +j\right)}^{2}$
(17)
and
$\sum _{j=0}^{n-1}arg\left({\sigma }_{n}\left(\alpha \right)i+\alpha +j\right)=2\pi$
(18)

supposing $argz\in \left[0,2\pi \right)$ for all $z\in \mathbb{C}\setminus \left\{0\right\}$.

First, we prove the functions to be well defined for all $\alpha >0$.

The product ${\prod }_{j=0}^{n-1}\left({q}^{2}+{\left(\alpha +j\right)}^{2}\right)$ is continuous and strictly increasing as a function of $q>0$.

It tends to ${\prod }_{j=0}^{n-1}{\left(\alpha +j\right)}^{2}<{\prod }_{j=0}^{n-1}{\left(1+\alpha +j\right)}^{2}$ as $q\to 0$ and to +∞ as $q\to +\mathrm{\infty }$. Hence, for any $\alpha >0$, there exists a unique $q>0$ such that ${\prod }_{j=0}^{n-1}\left({q}^{2}+{\left(\alpha +j\right)}^{2}\right)={\prod }_{j=0}^{n-1}{\left(1+\alpha +j\right)}^{2}$.

In the same way, for any $\alpha >0$, the sum ${\sum }_{j=0}^{n-1}arg\left(qi+\alpha +j\right)$ is a continuous function of $q>0$ strictly increasing from 0 to $\frac{\pi n}{2}>2\pi$. So, there exists a unique $q>0$ such that the sum is equal to 2π.

Since both the product and the sum considered are ${C}^{1}$-functions with positive partial derivative in $q>0$, the implicit function theorem provides both ${\rho }_{n}\left(\alpha \right)$ and ${\sigma }_{n}\left(\alpha \right)$ to be ${C}^{1}$-functions, too.

Now it is sufficient to prove the existence of $\alpha >0$ such that ${\rho }_{n}\left(\alpha \right)$ and ${\sigma }_{n}\left(\alpha \right)$ are equal to the same value q, which makes the two sides of (16) be equal.

Compare the functions ${\rho }_{n}\left(\alpha \right)$ and ${\sigma }_{n}\left(\alpha \right)$ near the boundaries of their common domain.

Equation (17) defining the function ${\rho }_{n}\left(\alpha \right)$ may be written as
$\prod _{j=0}^{n-1}\left(1+\frac{2j}{\alpha }+\frac{{j}^{2}}{{\alpha }^{2}}+{\left(\frac{{\rho }_{n}\left(\alpha \right)}{\alpha }\right)}^{2}\right)=\prod _{j=0}^{n-1}{\left(1+\frac{j+1}{\alpha }\right)}^{2}.$

This shows that $\frac{{\rho }_{n}\left(\alpha \right)}{\alpha }\to 0$ as $\alpha \to +\mathrm{\infty }$.

Equation (18) defining the function ${\sigma }_{n}\left(\alpha \right)$ may be written as
$\sum _{j=0}^{n-1}arctan\frac{\frac{{\sigma }_{n}\left(\alpha \right)}{\alpha }}{1+\frac{j}{\alpha }}=2\pi .$

This shows that $\frac{{\sigma }_{n}\left(\alpha \right)}{\alpha }\to tan\frac{2\pi }{n}>0$ as $\alpha \to +\mathrm{\infty }$. Thus, ${\rho }_{n}\left(\alpha \right)<{\sigma }_{n}\left(\alpha \right)$ for sufficiently large α.

Now, to prove Proposition 1, it is sufficient to show that ${\rho }_{n}\left(\alpha \right)>{\sigma }_{n}\left(\alpha \right)$ for sufficiently small α. To compare the functions ${\rho }_{n}\left(\alpha \right)$ and ${\sigma }_{n}\left(\alpha \right)$ for small $\alpha >0$, we need some lemmas.

Lemma 3 For all $\alpha >0$, it holds ${\rho }_{n}{\left(\alpha \right)}^{2}<2\left(\alpha +n\right)-1$.

Proof Suppose that ${\rho }_{n}{\left(\alpha \right)}^{2}\ge 2\left(\alpha +n\right)-1$ for some $\alpha >0$. Then
$\begin{array}{rcl}\prod _{j=0}^{n-1}\left({\rho }_{n}{\left(\alpha \right)}^{2}+{\left(\alpha +j\right)}^{2}\right)& \ge & \prod _{j=0}^{n-1}\left(2\left(\alpha +n\right)-1+{\left(\alpha +j\right)}^{2}\right)\\ >& \prod _{j=0}^{n-1}\left(2\left(\alpha +j+1\right)-1+{\left(\alpha +j\right)}^{2}\right)=\prod _{j=0}^{n-1}{\left(1+\left(\alpha +j\right)\right)}^{2}.\end{array}$

This contradiction with the definition of ${\rho }_{n}\left(\alpha \right)$ completes the proof of Lemma 3. □

Lemma 4 For all $\alpha >0$, it holds ${\rho }_{n+1}\left(\alpha \right)>{\rho }_{n}\left(\alpha \right)$.

Proof According to the definition of ${\rho }_{n}\left(\alpha \right)$ by (17) and Lemma 3, we have
$\begin{array}{rcl}\prod _{j=0}^{n}\left({\rho }_{n}{\left(\alpha \right)}^{2}+{\left(\alpha +j\right)}^{2}\right)& =& \prod _{j=0}^{n-1}{\left(1+\alpha +j\right)}^{2}\cdot \left({\rho }_{n}{\left(\alpha \right)}^{2}+{\left(\alpha +n\right)}^{2}\right)\\ <& \prod _{j=0}^{n-1}{\left(1+\alpha +j\right)}^{2}\cdot \left(2\left(\alpha +n\right)-1+{\left(\alpha +n\right)}^{2}\right)<\prod _{j=0}^{n}{\left(1+\alpha +j\right)}^{2}.\end{array}$

In order to make the first and the last products be equal, we have to replace ${\rho }_{n}\left(\alpha \right)$ in the first one by a greater value. This means that ${\rho }_{n+1}\left(\alpha \right)>{\rho }_{n}\left(\alpha \right)$ and Lemma 4 is proved. □

Lemma 5 For all $\alpha >0$, it holds ${\sigma }_{n+1}\left(\alpha \right)<{\sigma }_{n}\left(\alpha \right)$.

Proof According to the definition of ${\sigma }_{n}\left(\alpha \right)$ by (18), we have
$\sum _{j=0}^{n}arg\left({\sigma }_{n}\left(\alpha \right)i+\alpha +j\right)=2\pi +arg\left({\sigma }_{n}\left(\alpha \right)i+\alpha +n\right)>2\pi .$

In order to make the sum equal 2π, we have to replace ${\sigma }_{n}\left(\alpha \right)$ by a smaller value. So, ${\sigma }_{n+1}\left(\alpha \right)<{\sigma }_{n}\left(\alpha \right)$ and Lemma 5 is proved. □

Due to Lemmas 3, 4, 5 proved, it is sufficient now for the proof of Proposition 1 to show that ${\rho }_{12}\left(\alpha \right)>{\sigma }_{12}\left(\alpha \right)$ for sufficiently small $\alpha >0$.

Lemma 6 It holds ${\rho }_{12}\left(\alpha \right)>2$ for all sufficiently small $\alpha >0$.

Proof Straightforward exact calculations show that
$\underset{\alpha \to 0}{lim}\prod _{j=0}^{11}\left({2}^{2}+{\left(\alpha +j\right)}^{2}\right)=\prod _{j=0}^{11}\left(4+{j}^{2}\right)=192\text{,}175\text{,}659\text{,}520\text{,}000\text{,}000<2\cdot {10}^{17}$
and
$\underset{\alpha \to 0}{lim}\prod _{j=0}^{11}{\left(1+\alpha +j\right)}^{2}={\left(12!\right)}^{2}=229\text{,}442\text{,}532\text{,}802\text{,}560\text{,}000>2\cdot {10}^{17}.$
So, for sufficiently small $\alpha >0$, we have
$\prod _{j=0}^{11}\left({2}^{2}+{\left(\alpha +j\right)}^{2}\right)<2\cdot {10}^{17}<\prod _{j=0}^{11}{\left(1+\alpha +j\right)}^{2}.$

Hence, for these α, in order to avoid contradiction with the definition of ${\rho }_{12}\left(\alpha \right)$, the inequality ${\rho }_{12}{\left(\alpha \right)}^{2}>{2}^{2}$ is necessary. Lemma 6 is proved. □

Lemma 7 It holds ${\sigma }_{12}\left(\alpha \right)<2$ for sufficiently small $\alpha >0$.

Proof Consider the limit
$\begin{array}{c}\underset{\alpha \to 0}{lim}\sum _{j=0}^{11}arg\left(2i+\alpha +j\right)\hfill \\ \phantom{\rule{1em}{0ex}}=arg2i+arctan2+arctan1+arctan\frac{2}{3}+arctan\frac{1}{2}+\sum _{j=5}^{11}arctan\frac{2}{j}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{5\pi }{4}+arctan\frac{2}{3}+\sum _{j=5}^{11}arctan\frac{2}{j}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{5\pi }{4}+arctan\frac{\frac{2}{3}+\frac{2}{5}}{1-\frac{2}{3}\cdot \frac{2}{5}}+arctan\frac{\frac{2}{6}+\frac{2}{7}}{1-\frac{2}{6}\cdot \frac{2}{7}}+arctan\frac{\frac{2}{8}+\frac{2}{9}}{1-\frac{2}{8}\cdot \frac{2}{9}}+arctan\frac{\frac{2}{10}+\frac{2}{11}}{1-\frac{2}{10}\cdot \frac{2}{11}}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{5\pi }{4}+arctan\frac{16}{11}+arctan\frac{13}{19}+arctan\frac{1}{2}+arctan\frac{21}{53}\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{5\pi }{4}+arctan\frac{\frac{16}{11}+\frac{13}{19}}{1-\frac{16}{11}\cdot \frac{13}{19}}+arctan\frac{\frac{1}{2}+\frac{21}{53}}{1-\frac{1}{2}\cdot \frac{21}{53}}=\frac{5\pi }{4}+arctan447+arctan\frac{19}{17}.\hfill \end{array}$
Note that
$tan\left(arctan447+arctan\frac{19}{17}\right)=\frac{447+\frac{19}{17}}{1-447\cdot \frac{19}{17}}=-\frac{3\text{,}809}{4\text{,}238}.$

Hence, $arctan447+arctan\frac{19}{17}>\frac{3\pi }{4}$ and ${\sum }_{j=0}^{11}arg\left(2i+\alpha +j\right)>2\pi$ for sufficiently small $\alpha >0$. Thus, for these α, we have ${\sigma }_{12}\left(\alpha \right)<2$, which completes the proof of Lemma 7. □

Now Proposition 1 is also proved.  □

Proposition 2 For any $\alpha >0$ and any integer $n>1$, all roots $\lambda \in \mathbb{C}$ to equation (14) are simple.

Proof Since we consider a polynomial equation of degree n, it is sufficient to prove the existence of n different roots to (14). We will show that for any integer m such that $-n, there exists ${\mu }_{m}\in \mathbb{C}$ satisfying
$\prod _{j=0}^{n-1}|{\mu }_{m}+j|=\prod _{j=0}^{n-1}\left(1+\alpha +j\right)$
(19)
and
$\sum _{j=0}^{n-1}arg\left({\mu }_{m}+j\right)=m\pi$
(20)

with argz denoting the principal value of the argument lying in the open-closed interval $\left(-\pi ,\pi \right]$. Surely, all these 2n complex numbers ${\mu }_{m}$ are different. Those with even m generate, via the relation ${\lambda }_{m}+\alpha ={\mu }_{m}$, just n different roots ${\lambda }_{m}$ to (14).

We begin to accomplish this plan by noting that the set of μ satisfying equation (20) with $m=0$ is the real semi-axis $\left(0,+\mathrm{\infty }\right)$ containing a single point satisfying (19), namely ${\mu }_{0}=1+\alpha$.

Similarly, the set of μ satisfying equation (20) with $m=n$ is the real unbounded interval $\left(-\mathrm{\infty },1-n\right)$ containing a single point satisfying (19), namely ${\mu }_{n}=\alpha -n$.

Now consider the cases $0 and the upper complex half-plane. For any $\omega >0$, the smooth function
${\varphi }_{\omega }\left(r\right)=\sum _{j=0}^{n-1}arg\left(r+\omega i+j\right)=\sum _{j=0}^{n-1}arccot\frac{r+j}{\omega }$

monotonically decreases from to 0 as r increases from −∞ to +∞. So, for any $\omega >0$ and $b\in \left(0,n\pi \right)$, there exists a unique value r such that ${\varphi }_{\omega }\left(r\right)=b$. Due to the inequality $\frac{d{\varphi }_{\omega }}{dr}\left(r\right)<0$, the implicit function theorem provides the existence of the smooth functions ${r}_{m}\left(\omega \right)$ satisfying ${\varphi }_{\omega }\left({r}_{m}\left(\omega \right)\right)=m\pi$.

Note that if $r\le -m$, then $r+j<0$ for all $j and $r+m\le 0$. Hence,
$\underset{\omega \to +0}{\underset{_}{lim}}\sum _{j=0}^{n-1}arccot\frac{r+j}{\omega }\ge \underset{\omega \to +0}{\underset{_}{lim}}\sum _{j=0}^{m-1}arccot\frac{r+j}{\omega }+\underset{\omega \to +0}{\underset{_}{lim}}arccot\frac{0}{\omega }=m\pi +\frac{\pi }{2}>m\pi$

and such r cannot be the value of ${r}_{m}\left(\omega \right)$ for sufficiently small $\omega >0$.

Similarly, if $r\ge 1-m$, then $r+j>0$ for all $j>m-1$ and $r+m-1\ge 0$. Hence,
$\begin{array}{rcl}\underset{\omega \to +0}{\stackrel{‾}{lim}}\sum _{j=0}^{n-1}arccot\frac{r+j}{\omega }& \le & \underset{\omega \to +0}{\stackrel{‾}{lim}}\sum _{j=0}^{m-2}arccot\frac{r+j}{\omega }+\frac{\pi }{2}+\underset{\omega \to +0}{\stackrel{‾}{lim}}\sum _{j=m}^{n-1}arccot\frac{r+j}{\omega }\\ \le & \left(m-1\right)\pi +\frac{\pi }{2}+0

and such r cannot be the value of ${r}_{m}\left(\omega \right)$ for sufficiently small $\omega >0$.

So, if $\omega >0$ is sufficiently small, then ${r}_{m}\left(\omega \right)$ satisfies the inequality $-m<{r}_{m}\left(\omega \right)<1-m$ and thereby is negative.

Consider the product ${\prod }_{j=0}^{n-1}|{r}_{m}\left(\omega \right)+\omega i+j|$ with $0 and investigate its behavior for small $\omega >0$.

If $j\ge m$, then for sufficiently small $\omega >0$, we have $|{r}_{m}\left(\omega \right)+j|={r}_{m}\left(\omega \right)+j and
$\prod _{j=m}^{n-1}|{r}_{m}\left(\omega \right)+j|\le \prod _{j=m}^{n-1}j<\prod _{j=m}^{n-1}\left(1+j\right).$
(21)
If $j\le m-1$, then for sufficiently small $\omega >0$, we have $|{r}_{m}\left(\omega \right)+j|=-{r}_{m}\left(\omega \right)-j
$\prod _{j=0}^{m-1}|{r}_{m}\left(\omega \right)+j|\le \prod _{j=0}^{m-1}|1+\left(m-1-j\right)|=\prod _{J=0}^{m-1}\left(1+J\right),\phantom{\rule{1em}{0ex}}J=m-1-j.$
(22)
Combining (21) and (22), we obtain, for sufficiently small $\omega >0$,
$\prod _{j=0}^{n-1}|{r}_{m}\left(\omega \right)+j|<\prod _{j=0}^{n-1}\left(1+j\right),$
and
$\prod _{j=0}^{n-1}|{r}_{m}\left(\omega \right)+\omega i+j|<\prod _{j=0}^{n-1}\left(1+\alpha +j\right).$
As for large ω, the left-hand side of the above inequality evidently tends to +∞ as $\omega \to +\mathrm{\infty }$ and hence is greater than its right-hand side for sufficiently large ω. By continuity there exists ${\omega }_{m}>0$ such that
$\prod _{j=0}^{n-1}|{r}_{m}\left({\omega }_{m}\right)+{\omega }_{m}i+j|=\prod _{j=0}^{n-1}\left(1+\alpha +j\right).$

Thus, we can take ${\mu }_{m}={r}_{m}\left({\omega }_{m}\right)+{\omega }_{m}i\in \mathbb{C}$ to satisfy (19) and (20) for $0. For $-n, we can take the conjugates ${\mu }_{m}=\overline{{\mu }_{-m}}$. Thus, the existence of all ${\mu }_{m}$ needed is proved. This completes the proof of Proposition 2. □

Lemma 8 If $12\le n\le 14$, $\alpha >0$, and $q>0$ satisfy the polynomial equation
$\prod _{j=0}^{n-1}\left({\left(\alpha +j\right)}^{2}+{q}^{2}\right)=\prod _{j=0}^{n-1}{\left(\alpha +j+1\right)}^{2},$

then $2\alpha +4<{q}^{2}<3\alpha +5$.

Proof It can be proved in the same way for all n mentioned. We show this for $n=12$.

First, compute the right-hand side of the equation:
$\begin{array}{c}\prod _{j=0}^{11}{\left(\alpha +j+1\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}={\alpha }^{24}+156{\alpha }^{23}+11\text{,}518{\alpha }^{22}+535\text{,}392{\alpha }^{21}+17\text{,}581\text{,}135{\alpha }^{20}\hfill \\ \phantom{\rule{2em}{0ex}}+433\text{,}823\text{,}676{\alpha }^{19}+8\text{,}353\text{,}410\text{,}208{\alpha }^{18}+128\text{,}665\text{,}048\text{,}512{\alpha }^{17}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}612\text{,}229\text{,}817\text{,}055{\alpha }^{16}+16\text{,}625\text{,}859\text{,}652\text{,}116{\alpha }^{15}+142\text{,}196\text{,}061\text{,}481\text{,}318{\alpha }^{14}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}013\text{,}438\text{,}536\text{,}648\text{,}512{\alpha }^{13}+6\text{,}032\text{,}418\text{,}472\text{,}347\text{,}265{\alpha }^{12}\hfill \\ \phantom{\rule{2em}{0ex}}+29\text{,}989\text{,}851\text{,}619\text{,}249\text{,}236{\alpha }^{11}\hfill \\ \phantom{\rule{2em}{0ex}}+124\text{,}253\text{,}074\text{,}219\text{,}885\text{,}468{\alpha }^{10}+427\text{,}135\text{,}043\text{,}298\text{,}835\text{,}872{\alpha }^{9}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}209\text{,}806\text{,}045\text{,}835\text{,}003\text{,}760{\alpha }^{8}+2\text{,}795\text{,}060\text{,}589\text{,}044\text{,}133\text{,}696{\alpha }^{7}\hfill \\ \phantom{\rule{2em}{0ex}}+5\text{,}194\text{,}030\text{,}186\text{,}679\text{,}450\text{,}688{\alpha }^{6}+7\text{,}613\text{,}724\text{,}634\text{,}416\text{,}755\text{,}712{\alpha }^{5}\hfill \\ \phantom{\rule{2em}{0ex}}+8\text{,}564\text{,}233\text{,}279\text{,}835\text{,}510\text{,}784{\alpha }^{4}+7\text{,}096\text{,}936\text{,}674\text{,}284\text{,}421\text{,}120{\alpha }^{3}\hfill \\ \phantom{\rule{2em}{0ex}}+4\text{,}059\text{,}952\text{,}667\text{,}309\text{,}260\text{,}800{\alpha }^{2}+1\text{,}424\text{,}017\text{,}035\text{,}657\text{,}216\text{,}000\alpha \hfill \\ \phantom{\rule{2em}{0ex}}+229\text{,}442\text{,}532\text{,}802\text{,}560\text{,}000.\hfill \end{array}$
Now, estimate the left-hand side supposing ${q}^{2}\ge 3\alpha +5>0$:
$\begin{array}{c}\prod _{j=0}^{11}\left({\left(\alpha +j\right)}^{2}+{q}^{2}\right)\hfill \\ \phantom{\rule{1em}{0ex}}\ge \prod _{j=0}^{11}\left({\left(\alpha +j\right)}^{2}+3\alpha +5\right)\hfill \\ \phantom{\rule{1em}{0ex}}\ge {\alpha }^{24}+168{\alpha }^{23}+13\text{,}216{\alpha }^{22}+647\text{,}658{\alpha }^{21}+22\text{,}191\text{,}136{\alpha }^{20}\hfill \\ \phantom{\rule{2em}{0ex}}+565\text{,}650\text{,}624{\alpha }^{19}+11\text{,}143\text{,}609\text{,}279{\alpha }^{18}+174\text{,}022\text{,}752\text{,}156{\alpha }^{17}\hfill \\ \phantom{\rule{2em}{0ex}}+2\text{,}192\text{,}303\text{,}359\text{,}180{\alpha }^{16}+22\text{,}557\text{,}120\text{,}652\text{,}044{\alpha }^{15}+191\text{,}221\text{,}185\text{,}335\text{,}728{\alpha }^{14}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}343\text{,}463\text{,}278\text{,}373\text{,}840{\alpha }^{13}+7\text{,}851\text{,}135\text{,}965\text{,}424\text{,}751{\alpha }^{12}\hfill \\ \phantom{\rule{2em}{0ex}}+38\text{,}226\text{,}775\text{,}470\text{,}470\text{,}448{\alpha }^{11}\hfill \\ \phantom{\rule{2em}{0ex}}+155\text{,}030\text{,}143\text{,}411\text{,}290\text{,}136{\alpha }^{10}+522\text{,}520\text{,}458\text{,}095\text{,}057\text{,}994{\alpha }^{9}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}457\text{,}064\text{,}439\text{,}886\text{,}002\text{,}624{\alpha }^{8}+3\text{,}337\text{,}255\text{,}633\text{,}900\text{,}992\text{,}816{\alpha }^{7}\hfill \\ \phantom{\rule{2em}{0ex}}+6\text{,}209\text{,}925\text{,}089\text{,}367\text{,}687\text{,}345{\alpha }^{6}+9\text{,}237\text{,}499\text{,}888\text{,}429\text{,}090\text{,}764{\alpha }^{5}\hfill \\ \phantom{\rule{2em}{0ex}}+10\text{,}723\text{,}421\text{,}856\text{,}201\text{,}549\text{,}372{\alpha }^{4}+9\text{,}360\text{,}016\text{,}963\text{,}404\text{,}522\text{,}912{\alpha }^{3}\hfill \\ \phantom{\rule{2em}{0ex}}+5\text{,}777\text{,}193\text{,}048\text{,}791\text{,}013\text{,}360{\alpha }^{2}+2\text{,}247\text{,}088\text{,}906\text{,}508\text{,}241\text{,}600\alpha \hfill \\ \phantom{\rule{2em}{0ex}}+413\text{,}920\text{,}896\text{,}501\text{,}672\text{,}000.\hfill \end{array}$
The difference of this polynomial and the previous one is equal to
$\begin{array}{c}\prod _{j=0}^{11}\left({\left(\alpha +j\right)}^{2}+3\alpha +5\right)-\prod _{j=0}^{11}{\left(\alpha +j+1\right)}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}=12{\alpha }^{23}+1\text{,}698{\alpha }^{22}+112\text{,}266{\alpha }^{21}+4\text{,}610\text{,}001{\alpha }^{20}+131\text{,}826\text{,}948{\alpha }^{19}\hfill \\ \phantom{\rule{2em}{0ex}}+2\text{,}790\text{,}199\text{,}071{\alpha }^{18}+45\text{,}357\text{,}703\text{,}644{\alpha }^{17}+580\text{,}073\text{,}542\text{,}125{\alpha }^{16}\hfill \\ \phantom{\rule{2em}{0ex}}+5\text{,}931\text{,}260\text{,}999\text{,}928{\alpha }^{15}+49\text{,}025\text{,}123\text{,}854\text{,}410{\alpha }^{14}+330\text{,}024\text{,}741\text{,}725\text{,}328{\alpha }^{13}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}818\text{,}717\text{,}493\text{,}077\text{,}486{\alpha }^{12}+8\text{,}236\text{,}923\text{,}851\text{,}221\text{,}212{\alpha }^{11}\hfill \\ \phantom{\rule{2em}{0ex}}+30\text{,}777\text{,}069\text{,}191\text{,}404\text{,}668{\alpha }^{10}+95\text{,}385\text{,}414\text{,}796\text{,}222\text{,}122{\alpha }^{9}\hfill \\ \phantom{\rule{2em}{0ex}}+247\text{,}258\text{,}394\text{,}050\text{,}998\text{,}864{\alpha }^{8}+542\text{,}195\text{,}044\text{,}856\text{,}859\text{,}120{\alpha }^{7}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}015\text{,}894\text{,}902\text{,}688\text{,}236\text{,}657{\alpha }^{6}+1\text{,}623\text{,}775\text{,}254\text{,}012\text{,}335\text{,}052{\alpha }^{5}\hfill \\ \phantom{\rule{2em}{0ex}}+2\text{,}159\text{,}188\text{,}576\text{,}366\text{,}038\text{,}588{\alpha }^{4}+2\text{,}263\text{,}080\text{,}289\text{,}120\text{,}101\text{,}792{\alpha }^{3}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}717\text{,}240\text{,}381\text{,}481\text{,}752\text{,}560{\alpha }^{2}+823\text{,}071\text{,}870\text{,}851\text{,}025\text{,}600\alpha \hfill \\ \phantom{\rule{2em}{0ex}}+184\text{,}478\text{,}363\text{,}699\text{,}112\text{,}000,\hfill \end{array}$

which is positive for any $\alpha \ge 0$. This shows that the polynomial equation cannot be satisfied by $\alpha >0$ and $q>0$ with ${q}^{2}\ge 3\alpha +5$.

In the same way, compute
$\begin{array}{c}\prod _{j=0}^{11}{\left(\alpha +j+1\right)}^{2}-\prod _{j=0}^{11}\left({\left(\alpha +j\right)}^{2}+2\alpha +4\right)\hfill \\ \phantom{\rule{1em}{0ex}}=96{\alpha }^{22}+13\text{,}156{\alpha }^{21}+844\text{,}624{\alpha }^{20}+33\text{,}778\text{,}316{\alpha }^{19}+943\text{,}838\text{,}852{\alpha }^{18}\hfill \\ \phantom{\rule{2em}{0ex}}+19\text{,}590\text{,}096\text{,}240{\alpha }^{17}+313\text{,}464\text{,}915\text{,}984{\alpha }^{16}+3\text{,}960\text{,}996\text{,}926\text{,}744{\alpha }^{15}\hfill \\ \phantom{\rule{2em}{0ex}}+40\text{,}162\text{,}617\text{,}066\text{,}616{\alpha }^{14}+330\text{,}203\text{,}929\text{,}721\text{,}796{\alpha }^{13}\hfill \\ \phantom{\rule{2em}{0ex}}+2\text{,}215\text{,}299\text{,}128\text{,}334\text{,}800{\alpha }^{12}\hfill \\ \phantom{\rule{2em}{0ex}}+12\text{,}163\text{,}303\text{,}361\text{,}220\text{,}828{\alpha }^{11}+54\text{,}651\text{,}209\text{,}110\text{,}677\text{,}476{\alpha }^{10}\hfill \\ \phantom{\rule{2em}{0ex}}+200\text{,}323\text{,}721\text{,}839\text{,}107\text{,}240{\alpha }^{9}+595\text{,}229\text{,}721\text{,}350\text{,}941\text{,}648{\alpha }^{8}\hfill \\ \phantom{\rule{2em}{0ex}}+1\text{,}419\text{,}051\text{,}246\text{,}703\text{,}474\text{,}880{\alpha }^{7}+2\text{,}673\text{,}079\text{,}829\text{,}956\text{,}829\text{,}568{\alpha }^{6}\hfill \\ \phantom{\rule{2em}{0ex}}+3\text{,}889\text{,}993\text{,}689\text{,}940\text{,}050\text{,}432{\alpha }^{5}+4\text{,}228\text{,}750\text{,}706\text{,}659\text{,}177\text{,}984{\alpha }^{4}\hfill \\ \phantom{\rule{2em}{0ex}}+3\text{,}257\text{,}831\text{,}645\text{,}648\text{,}401\text{,}920{\alpha }^{3}+1\text{,}625\text{,}109\text{,}784\text{,}526\text{,}284\text{,}800{\alpha }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+437\text{,}271\text{,}322\text{,}981\text{,}376\text{,}000\alpha +37\text{,}266\text{,}873\text{,}282\text{,}560\text{,}000.\hfill \end{array}$

Hence, ${\prod }_{j=0}^{11}{\left(\alpha +j+1\right)}^{2}>{\prod }_{j=0}^{11}\left({\left(\alpha +j\right)}^{2}+{q}^{2}\right)$ if $2\alpha +4\ge {q}^{2}$.

This contradiction yields $2\alpha +4<{q}^{2}<3\alpha +5$. So, Lemma 8 is proved. □

The condition $Re\frac{d{\lambda }_{\alpha }}{d\alpha }\left(\stackrel{˜}{\alpha }\right)\ne 0$ needed for the Hopf theorem, expressed explicitly by means of the implicit function theorem, looks like
$\begin{array}{c}{\left[\sum _{j=0}^{n-1}\frac{\alpha +j}{{q}^{2}+{\left(\alpha +j\right)}^{2}}\right]}^{2}+{\left[\sum _{j=0}^{n-1}\frac{q}{{q}^{2}+{\left(\alpha +j\right)}^{2}}\right]}^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\ne \sum _{j=0}^{n-1}\frac{\alpha +j}{{q}^{2}+{\left(\alpha +j\right)}^{2}}\sum _{j=0}^{n-1}\frac{1}{1+\alpha +j}.\hfill \end{array}$
Lemma 9 If $12\le n\le 14$, $\alpha >0$ and $0<{q}^{2}<3\alpha +5$, then
$\begin{array}{r}{\left[\sum _{j=0}^{n-1}\frac{\alpha +j}{{q}^{2}+{\left(\alpha +j\right)}^{2}}\right]}^{2}+{\left[\sum _{j=0}^{n-1}\frac{q}{{q}^{2}+{\left(\alpha +j\right)}^{2}}\right]}^{2}\\ \phantom{\rule{1em}{0ex}}>\sum _{j=0}^{n-1}\frac{\alpha +j}{{q}^{2}+{\left(\alpha +j\right)}^{2}}\sum _{j=0}^{n-1}\frac{1}{1+\alpha +j}.\end{array}$
(23)

Proof Hereafter all sums and products with no limits indicated are over $j=0,1,\dots ,n-1$.

Multiplying inequality (23) by ${U}_{\ast }=\prod \left(1+\alpha +j\right)$ and then twice by ${V}_{\ast }=\prod \left[{q}^{2}+{\left(\alpha +j\right)}^{2}\right]$, we obtain the following equivalent inequality provided $\alpha >0$:
${U}_{\ast }\left[{\left(\sum \left(\alpha +j\right){V}_{j}\right)}^{2}+{q}^{2}{\left(\sum {V}_{j}\right)}^{2}\right]>{V}_{\ast }\sum \left(\alpha +j\right){V}_{j}\sum {U}_{j}$
(24)

with the polynomials ${U}_{j}=\frac{{U}_{\ast }}{1+\alpha +j}$ and ${V}_{j}=\frac{{V}_{\ast }}{{q}^{2}+{\left(\alpha +j\right)}^{2}}$.

Put ${q}^{2}=\frac{3\alpha +5}{1+w}$, $w>0$. Substituting this into inequality (24) and multiplying the result by ${\left(1+w\right)}^{2n-1}$, we obtain another equivalent one:
$\begin{array}{r}{U}_{\ast }\left[\left(1+w\right){\left(\sum \left(\alpha +j\right){P}_{j}\right)}^{2}+\left(3\alpha +5\right){\left(\sum {P}_{j}\right)}^{2}\right]\\ \phantom{\rule{1em}{0ex}}>{P}_{\ast }\cdot \sum \left(\alpha +j\right){P}_{j}\cdot \sum {U}_{j}\end{array}$
(25)

with ${P}_{\ast }=\prod \left[3\alpha +5+\left(1+w\right){\left(\alpha +j\right)}^{2}\right]$ and ${P}_{j}=\frac{{P}_{\ast }}{3\alpha +5+\left(1+w\right){\left(\alpha +j\right)}^{2}}$.

Both sides of inequality (25) are polynomials of α and w with non-negative integer coefficients. So, they can be computed exactly, with no rounding. This rather cumbersome computation gives the following result for the difference of the left- and right-hand sides of (25) expressed as
$\begin{array}{r}{U}_{\ast }\left[\left(1+w\right){\left(\sum \left(\alpha +j\right){P}_{j}\right)}^{2}+\left(3\alpha +5\right){\left(\sum {P}_{j}\right)}^{2}\right]\\ \phantom{\rule{1em}{0ex}}-{P}_{\ast }\sum \left(\alpha +j\right){P}_{j}\sum {U}_{j}=\sum _{j=0}^{5n-2}{\mathrm{\Delta }}_{j}{\alpha }^{j}\end{array}$
(26)

with polynomials ${\mathrm{\Delta }}_{j}\in \mathbb{R}\left[w\right]$. Straightforward though very cumbersome calculations show that ${\mathrm{\Delta }}_{5n-2}=0$, and all other ${\mathrm{\Delta }}_{j}$ in (26) are polynomials with positive coefficients.

This completes the proof of Lemma 9. □

To apply the Hopf bifurcation theorem, we need to check that equation (14) cannot have more than a single pair of imaginary conjugated roots. It can be easily obtained by considering equation (16).

Now, the Hopf bifurcation theorem and the lemmas proved provide, for $n=12,13,14$, the existence of a family ${\alpha }_{\epsilon }>0$ such that equation (14) with $\alpha ={\alpha }_{0}$ has imaginary roots $\lambda =±qi$ and for sufficiently small ε, system (13) with $\alpha ={\alpha }_{\epsilon }$ has a periodic solution ${V}_{\epsilon }\left(t\right)$ with period ${T}_{\epsilon }\to T=\frac{2\pi }{q}$ as $\epsilon \to 0$. In particular, the coordinate ${V}_{\epsilon ,0}\left(t\right)=v\left(t\right)$ of the vector ${V}_{\epsilon }\left(t\right)$ is also a periodic function with the same period. Then, taking into account (9), we obtain
$y\left(x\right)=\left(C+v\left(-ln\left({x}^{\ast }-x\right)\right)\right){\left({x}^{\ast }-x\right)}^{-\alpha }.$
Put $h\left(s\right)=C+v\left(-s\right)$, which is a non-constant continuous periodic and positive for sufficiently small ε function and obtain the required equality
$y\left(x\right)={\left({x}^{\ast }-x\right)}^{-\alpha }h\left(ln\left({x}^{\ast }-x\right)\right).$

In the similar way, we obtain the related expressions for ${y}^{\left(j\right)}\left(x\right)$, $j=0,1,\dots ,n-1$.

Theorem 4 is proved. □

## Conclusions, concluding remarks and open problems

1. 1.

Computer calculations give approximate values of α providing equation (14) to have a pure imaginary root λ. They are, with corresponding values of k, as follows:

if $n=12$, then $\alpha \approx 0.56$, $k\approx 22.4$;

if $n=13$, then $\alpha \approx 1.44$, $k\approx 10.0$;

if $n=14$, then $\alpha \approx 2.37$, $k\approx 6.9$.
1. 2.

Note that equation (14) has no pure imaginary roots if $n\le 11$. So, the Hopf bifurcation theorem cannot be applied, but it does not follow that Theorem 4 cannot be proved for some $n<12$.

2. 3.

Equation (5) with $n=3$ has solutions of type (6) with oscillatory h (see [3, 5]).

3. 4.

If $n\ge 15$, then the inequality needed for the Hopf bifurcation theorem $Re\frac{d{\lambda }_{\alpha }}{d\alpha }\left(\stackrel{˜}{\alpha }\right)\ne 0$ cannot be proved in the same way because the estimate ${q}^{2}<3\alpha +5$ does not hold.

## References

1. 1.
Kiguradze IT, Chanturia TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic, Dordrecht; 1993.
2. 2.
Astashova IV: Asymptotic behavior of solutions of certain nonlinear differential equations. 1(3). In Reports of Extended Session of a Seminar of the I. N. Vekua Institute of Applied Mathematics. Tbilis. Gos. Univ., Tbilisi; 1985:9-11. (Russian)Google Scholar
3. 3.
Astashova IV: Qualitative properties of solutions to quasilinear ordinary differential equations. In Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis. Edited by: Astashova IV. UNITY-DANA, Moscow; 2012:22-290. (Russian)Google Scholar
4. 4.
Kozlov VA: On Kneser solutions of higher order nonlinear ordinary differential equations. Ark. Mat. 1999, 37(2):305-322. 10.1007/BF02412217
5. 5.
Astashova IV: Application of dynamical systems to the study of asymptotic properties of solutions to nonlinear higher-order differential equations. J. Math. Sci. 2005, 126(5):1361-1391.
6. 6.
Marsden JE, McCracken M: The Hopf Bifurcation and Its Applications. Springer, Berlin; 1976. XIII
7. 7.
Astashova IV, Vyun SA: On positive solutions with non-power asymptotic behavior to Emden-Fowler type twelfth order differential equation. Differ. Equ. 2012, 48(11):1568-1569. (Russian)Google Scholar 