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On quasi-periodic solutions to a higher-order Emden-Fowler type differential equation
Boundary Value Problems volume 2014, Article number: 174 (2014)
The paper is devoted to the existence of oscillatory and non-oscillatory quasi-periodic, in some sense, solutions to a higher-order Emden-Fowler type differential equation.
The paper is devoted to the existence of oscillatory and non-oscillatory quasi-periodic, in some sense, solutions to the higher-order Emden-Fowler type differential equation
The fact of the existence of such solutions answers the two questions posed by IT Kiguradze:
Can we describe more precisely qualitative properties of oscillatory solutions to (1)?
Do all blow-up solutions to this equation (and similarly all Kneser solutions) have the power asymptotic behavior?
A lot of results on the asymptotic behavior of solutions to (1) are described in detail in . In particular (see Ch. IV, §15), the existence of oscillatory solutions to a generalization of this equation was proved (see also  Ch. I, §6.1). In  a result was formulated on non-extensibility of oscillatory solutions to (1) with odd n and . In the cases and the asymptotic behavior of all oscillatory solutions is described in –. Some results on the existence of blow-up solutions are in  (Ch. IV, §16),  (Ch. I, §5), , . Some results on the existence of some special solutions to this equation are in , , , , –.
2 On existence of quasi-periodic oscillatory solutions
In this section some results will be obtained on the existence of special oscillatory solutions. The main results of this section were formulated in .
For any integer and real there exists a periodic oscillatory function h such that for any and the function
For any let be the maximally extended solution to the equation
satisfying the initial conditions with .
For put and .
Consider the function defined by the formula
and the mapping defined by the formula
and satisfying the equality for all .
Next, consider the subset consisting of all satisfying the following conditions:
for all ,
The restriction of the projection to the set Q is a homeomorphism of Q onto the convex compact subset
For anythere existssuch thatandfor all.
Put . This J exists and is positive due to the definition of Q. On some interval all derivatives with are positive. Those with , due to (3), are negative on the same interval.
While keeping this sign combination, the function and its derivatives are bounded, which provides extensibility of as the solution to (3) outside the interval .
On the other hand, this sign combination cannot take place up to +∞. Indeed, in that case would increase providing for all , which is impossible for any positive function on the unbounded interval .
So, must change the sign combination of its derivatives. The only possible combination to be the next one corresponds to the positive derivatives with and the negative ones with .
The same arguments show that the new sign combination must also change and finally, after J changes, we arrive at the case with and with . Now, contrary to the previous cases, the function does not increase, but its first derivative is negative and decreases (recall that ). Hence this sign combination also cannot take place on an unbounded interval and therefore it must change to the case with all negative , . By the way, the function must vanish at some point , which completes the proof of Lemma 1. □
Note that is not only the first positive zero of , but the only positive one. Indeed, all with are negative at , whence, according to (3), all with decrease and are negative for all in the domain of .
To continue the proof of Theorem 1, consider the function taking each to the first positive zero of the function . To prove its continuity, we apply the implicit function theorem. The function can be considered as a local solution to the equation , where
is the ‘solution’ mapping defined on a domain including . The necessary for the implicit function theorem condition is satisfied since the left-hand side of the last inequality is equal to . Besides, any function implicitly defined near a point must be positive in some its neighborhood. Hence locally must be equal to , but neither to a non-positive zero of nor to a non-first positive one, which does not exist. Hence the function is continuous as well as .
Now we can consider the mapping , which maps Q into itself. Since is continuous and Q is homeomorphic to a convex compact subset of , the Brouwer fixed-point theorem can be applied. Thus, there exists such that .
According to the definitions of the functions , S, and ξ, this yields the result that there exists a non-negative solution to (3) defined on a segment with , positive on the open interval , and such that
Since is non-negative, it is also a solution to the equation
So, the function is a solution to (7) and is defined on the segment with .
Put with λ defined by (6). Then
the procedure of extension can be repeated on , , and so on with . In the same way the solution can be extended to the left. Its restrictions to the neighboring segments satisfy the following equality:
where and hence .
Now we will investigate whether b is greater or less than 1.
Let be the zero of the derivative belonging to the interval . Note that according to the above consideration on changing the sign combinations, we have
In the above notation the solutionsatisfies the following inequalities:
since and on the interval , where only itself changes its sign, while all other with keep the same one. Recall that , which makes to be one of these others. Inequality (9) is proved.
Inequality (10) follows from on the interval , where the derivatives and with keep different signs, while all lower-order derivatives keep the same sign as . □
From the lemma proved it follows that , whence it follows that and .
Now we see that
So, the solution is extended on the half-bounded interval and cannot be extended outside it since
Now consider the function
which is periodic. Indeed, if for some , then
and, according to (8),
The expression in the last parentheses is equal to
So, for all and hence the function is periodic with period .
Now, according to (11), we can express the solution to (7) just as . Multiplying it by we obtain a solution to (3) having the form needed. It still will be a solution to (3) after replacing by arbitrary . □
The substitution produces the following.
For any integer and real there exists a periodic oscillatory function h such that for any satisfying and any the function
is a solution to (1).
For, there exists a constantsuch that any oscillatory solutionto (1) withsatisfies the conditions
for someand, whereandare sequences satisfying, , if, if.
With the help of this theorem, another one can be proved, namely the following.
3 On existence of positive solutions with non-power asymptotic behavior
where and h is a positive periodic non-constant function on R.
A similar result was also proved  about Kneser solutions, i.e. those satisfying as and for . Namely, if , then for any N and there exist an integer and such that and (1) has a solution of the form
where 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 positive solutions.
If, then there existssuch that (1) withhas a solutionsuch that
whereandare periodic positive non-constant functions on R.
Computer calculations give approximate values of α. They are, with the corresponding values of k, as follows:
if , then , ;
if , then , ;
if , then , .
If, then there existssuch that (1) withhas a Kneser solutionsatisfying
with periodic positive non-constant functionson R.
4 Conclusions, concluding remarks, and open problems
So, we give the negative answer to Question 1 and prove the existence of oscillatory solutions with special qualitative properties for Question 2.
It would be interesting to know if positive solutions like (12) exist for and for .
If a positive solution like (12) exists for some , does it follow, for the same n, that such solutions exist for all ?
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The research was supported by RFBR (grant 11-01-00989).
The author declares that she has no competing interests.
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Astashova, I. On quasi-periodic solutions to a higher-order Emden-Fowler type differential equation. Bound Value Probl 2014, 174 (2014). https://doi.org/10.1186/s13661-014-0174-7
- Emden-Fowler type equation
- quasi-periodic solutions
- oscillatory and non-oscillatory solutions