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On power and non-power asymptotic behavior of positive solutions to Emden-Fowler type higher-order equations
Advances in Difference Equations volume 2013, Article number: 220 (2013)
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.
For the equation
Kiguradze posed the problem on the asymptotic behavior of its positive solutions such that
He found an asymptotic formula for these solutions to (1) with (see ) and supposed all such solutions to have power asymptotic behavior for other n, too. The problem was solved for and . For these n, it was proved that all such solutions behave as
- is a limit of as , .
So, the hypothesis of Kiguradze was confirmed in this case.
For the equation
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.
Suppose the following conditions hold:
The continuous positive function has a limit as , , and for some , it holds(7)
For some and in a neighborhood of for sufficiently large , , it holds(8)
where C and α are defined by (4). The derivatives , , become
where and is a linear function with
and the coefficient of equal to 1.
Thus (1) is transformed into
where the function is obtained from with properly expressed in terms of . This function tends to as , .
Due to condition (8) for the function , we obtain the following inequalities for sufficiently large t and sufficiently small , :
Since , the function is a one in a neighborhood of 0 and
for some .
Solving (10) for and using formulae (4), we obtain the equation
where are the coefficients of the linear function . Equation (11) can be written as
Besides, for sufficiently large t and sufficiently small , , it holds
Suppose that V is the vector with coordinates , . Then equation (12) can be written as
where A is a constant matrix
and eigenvalues satisfying the equation
which is equivalent to
The mappings and satisfy the following estimates as :
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 .
If equation (13) has a solution tending to 0 as and is its first coordinate, then the function
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.
If , then there exist continuous mappings , , and defined in a neighborhood of 0 and such that , , , for , and the solutions to the problems
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 .
Theorem 4 For , there exists such that equation (5) has a solution with
where α is defined by (4) and are periodic positive non-constant functions on R.
Proof To apply the Hopf bifurcation theorem, we investigate equation (13) with 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 , , and
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 , there exist and such that
Remark 1 In the particular case , this result was obtained by Vyun .
Proof Consider the positive functions and defined for all via the equations
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.
Equation (17) defining the function may be written as
This shows that as .
Equation (18) defining the function may be written 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 .
Proof Suppose that for some . Then
This contradiction with the definition of completes the proof of Lemma 3. □
Lemma 4 For all , it holds .
Proof According to the definition of by (17) and Lemma 3, we have
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 .
Proof According to the definition of by (18), we have
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 .
Proof Straightforward exact calculations show that
So, for sufficiently small , we have
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 .
Proof Consider the limit
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.
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 , there exists satisfying
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 .
Now consider the cases and the upper complex half-plane. For any , the smooth function
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 .
Note that if , then for all and . Hence,
and such r cannot be the value of for sufficiently small .
Similarly, if , then for all and . Hence,
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 .
If , then for sufficiently small , we have and
If , then for sufficiently small , we have
Combining (21) and (22), we obtain, for sufficiently small ,
As for large ω, the left-hand side of the above inequality evidently tends to +∞ as and hence is greater than its right-hand side for sufficiently large ω. By continuity there exists such that
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. □
Lemma 8 If , , and satisfy the polynomial equation
Proof It can be proved in the same way for all n mentioned. We show this for .
First, compute the right-hand side of the equation:
Now, estimate the left-hand side supposing :
The difference of this polynomial and the previous one is equal to
which is positive for any . This shows that the polynomial equation cannot be satisfied by and with .
In the same way, compute
Hence, if .
This contradiction yields . So, Lemma 8 is proved. □
The condition needed for the Hopf theorem, expressed explicitly by means of the implicit function theorem, looks like
Lemma 9 If , and , then
Proof Hereafter all sums and products with no limits indicated are over .
Multiplying inequality (23) by and then twice by , we obtain the following equivalent inequality provided :
with the polynomials and .
Put , . Substituting this into inequality (24) and multiplying the result by , we obtain another equivalent one:
with and .
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
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. □
Now, the Hopf bifurcation theorem and the lemmas proved provide, for , the existence of a family such that equation (14) with has imaginary roots and for sufficiently small ε, system (13) with has a periodic solution with period as . In particular, the coordinate of the vector is also a periodic function with the same period. Then, taking into account (9), we obtain
Put , which is a non-constant continuous periodic and positive for sufficiently small ε function and obtain the required equality
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 , ;
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.
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The research was supported by RFBR (grant 11-01-00989).
The author declares that she has no competing interests.