On power and non-power asymptotic behavior of positive solutions to Emden-Fowler type higher-order equations
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For the equation
the existence of positive solutions with non-power asymptotic behavior is proved, namely
where is an arbitrary point, h is a positive periodic non-constant function on R.
To prove this result, the Hopf bifurcation theorem is used.
Keywordsasymptotic behavior Emden-Fowler higher-order equations
- is a limit of as , .
So, the hypothesis of Kiguradze was confirmed in this case.
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.
- (A)The continuous positive function has a limit as , , and for some , it holds(7)
- (B)For some and in a neighborhood of for sufficiently large , , it holds(8)
and the coefficient of equal to 1.
where the function is obtained from with properly expressed in terms of . This function tends to as , .
for some .
with some constants , .
Lemma 1 
Suppose that (15) holds and A is an arbitrary constant matrix. Then there exists a solution to equation (13) tending to zero as .
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 to equation (13) tending to zero as .
with C and α defined by (4) is a solution to (1) such that (2) and (3) hold.
Suppose that conditions (A) and (B) are satisfied. Then for such 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.
Suppose that and conditions (A) and (B) are satisfied. Then there exists an -parametric family of solutions to equation (1) with power asymptotic behavior (3).
Suppose that or in equation (1), the continuous positive function is Lipschitz continuous in and has a limit as , . Then any positive solution to this equation with a vertical asymptote has asymptotic behavior (3).
To prove the main results of this article, we use the Hopf bifurcation theorem .
Consider the α-parameterized dynamical system in a neighborhood of with linear operators and smooth enough functions as . Let and be simple complex conjugated eigenvalues of the operators . Suppose that for some and the operator has no other eigenvalues with zero real part.
are -periodic and non-constant.
In this section, the result about the existence of solutions with non-power asymptotic behavior is proved for equation (5) with .
where α is defined by (4) and are periodic positive non-constant functions on R.
If equation (14) has a pair of pure imaginary roots, we have to check other conditions of this theorem and then apply it.
Remark 1 In the particular case , this result was obtained by Vyun .
supposing for all .
First, we prove the functions to be well defined for all .
The product is continuous and strictly increasing as a function of .
It tends to as and to +∞ as . Hence, for any , there exists a unique such that .
In the same way, for any , the sum is a continuous function of strictly increasing from 0 to . So, there exists a unique such that the sum is equal to 2π.
Since both the product and the sum considered are -functions with positive partial derivative in , the implicit function theorem provides both and to be -functions, too.
Now it is sufficient to prove the existence of such that and are equal to the same value q, which makes the two sides of (16) be equal.
Compare the functions and near the boundaries of their common domain.
This shows that as .
This shows that as . Thus, for sufficiently large α.
Now, to prove Proposition 1, it is sufficient to show that for sufficiently small α. To compare the functions and for small , we need some lemmas.
Lemma 3 For all , it holds .
This contradiction with the definition of completes the proof of Lemma 3. □
Lemma 4 For all , it holds .
In order to make the first and the last products be equal, we have to replace in the first one by a greater value. This means that and Lemma 4 is proved. □
Lemma 5 For all , it holds .
In order to make the sum equal 2π, we have to replace by a smaller value. So, 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 for sufficiently small .
Lemma 6 It holds for all sufficiently small .
Hence, for these α, in order to avoid contradiction with the definition of , the inequality is necessary. Lemma 6 is proved. □
Lemma 7 It holds for sufficiently small .
Hence, and for sufficiently small . Thus, for these α, we have , which completes the proof of Lemma 7. □
Now Proposition 1 is also proved. □
Proposition 2 For any and any integer , all roots to equation (14) are simple.
with argz denoting the principal value of the argument lying in the open-closed interval . Surely, all these 2n complex numbers are different. Those with even m generate, via the relation , just n different roots to (14).
We begin to accomplish this plan by noting that the set of μ satisfying equation (20) with is the real semi-axis containing a single point satisfying (19), namely .
Similarly, the set of μ satisfying equation (20) with is the real unbounded interval containing a single point satisfying (19), namely .
monotonically decreases from nπ to 0 as r increases from −∞ to +∞. So, for any and , there exists a unique value r such that . Due to the inequality , the implicit function theorem provides the existence of the smooth functions satisfying .
and such r cannot be the value of for sufficiently small .
and such r cannot be the value of for sufficiently small .
So, if is sufficiently small, then satisfies the inequality and thereby is negative.
Consider the product with and investigate its behavior for small .
Thus, we can take to satisfy (19) and (20) for . For , we can take the conjugates . Thus, the existence of all needed is proved. This completes the proof of Proposition 2. □
Proof It can be proved in the same way for all n mentioned. We show this for .
which is positive for any . This shows that the polynomial equation cannot be satisfied by and with .
Hence, if .
This contradiction yields . So, Lemma 8 is proved. □
Proof Hereafter all sums and products with no limits indicated are over .
with the polynomials and .
with and .
with polynomials . Straightforward though very cumbersome calculations show that , and all other in (26) are polynomials with positive coefficients.
This completes the proof of Lemma 9. □
In the similar way, we obtain the related expressions for , .
Theorem 4 is proved. □
Conclusions, concluding remarks and open problems
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 , then , ;
if , then , ;
Note that equation (14) has no pure imaginary roots if . So, the Hopf bifurcation theorem cannot be applied, but it does not follow that Theorem 4 cannot be proved for some .
If , then the inequality needed for the Hopf bifurcation theorem cannot be proved in the same way because the estimate does not hold.
The research was supported by RFBR (grant 11-01-00989).
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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.