Skip to main content

Advertisement

On quasi-periodic solutions to a higher-order Emden-Fowler type differential equation

Article metrics

  • 833 Accesses

  • 8 Citations

Abstract

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.

1 Introduction

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

y ( n ) + p 0 | y | k sgny=0,n>2,kR,k>1, p 0 0.
(1)

The fact of the existence of such solutions answers the two questions posed by IT Kiguradze:

Question 1

Can we describe more precisely qualitative properties of oscillatory solutions to (1)?

Question 2

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 [1]. In particular (see Ch. IV, §15), the existence of oscillatory solutions to a generalization of this equation was proved (see also [2] Ch. I, §6.1). In [3] a result was formulated on non-extensibility of oscillatory solutions to (1) with odd n and p 0 >0. In the cases n=3 and n=4 the asymptotic behavior of all oscillatory solutions is described in [4]–[6]. Some results on the existence of blow-up solutions are in [1] (Ch. IV, §16), [2] (Ch. I, §5), [7], [8]. Some results on the existence of some special solutions to this equation are in [2], [4], [5], [7], [9]–[13].

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 [14].

Theorem 1

For any integer n>2 and real k>1 there exists a periodic oscillatory function h such that for any p 0 >0 and x R the function

y(x)= p 0 1 k 1 ( x x ) α h ( log ( x x ) ) ,<x< x ,
(2)

withα= n k 1 is a solution to (1). (See Figure1.)

Figure 1
figure1

A quasi-periodic solution for the equation y + y 3 =0 .

Proof

For any q=( q 0 ,, q n 1 ) R n let y q (x) be the maximally extended solution to the equation

y ( n ) (x)+ | y ( x ) | k =0
(3)

satisfying the initial conditions y ( j ) (0)= q j with j=0,,n1.

For 0j<n put B j = n k n + j ( k 1 ) >1 and β j = 1 B j .

Consider the function N: R n R defined by the formula

N( q 0 ,, q n 1 )= j = 0 n 1 | q j | B j
(4)

and the mapping N ˜ : R n {0} R n {0} defined by the formula

N ˜ ( q ) j =N ( q ) β j q j ,j=0,,n1,

and satisfying the equality N( N ˜ (q))=1 for all q R n {0}.

Next, consider the subset Q R n consisting of all q=( q 0 ,, q n 1 ) R n satisfying the following conditions:

  1. (1)

    q 0 =0,

  2. (2)

    q j 0 for all j{1,,n1},

  3. (3)

    N(q)=1.

The restriction of the projection ( q 0 ,, q n 1 )( q 1 ,, q n 2 ) to the set Q is a homeomorphism of Q onto the convex compact subset

{ ( q 1 , , q n 2 ) : j = 1 n 2 | q j | B j 1  and  q j 0 , j = 1 , , n 2 } R n 2 .

Lemma 1

For anyqQthere exists a q >0such that y q ( a q )=0and y q ( j ) ( a q )<0for allj{1,,n1}.

Proof

Put J=max{jZ:0j<n, q j >0}. This J exists and is positive due to the definition of Q. On some interval (0;ε) all derivatives y q ( j ) (x) with 0jJ are positive. Those with J<jn, due to (3), are negative on the same interval.

While keeping this sign combination, the function y q and its derivatives are bounded, which provides extensibility of y q (x) as the solution to (3) outside the interval (0;ε).

On the other hand, this sign combination cannot take place up to +∞. Indeed, in that case y q (x) would increase providing y q ( n ) (x)< y q ( ε ) k <0 for all x>ε, which is impossible for any positive function on the unbounded interval (0;+).

So, y q (x) must change the sign combination of its derivatives. The only possible combination to be the next one corresponds to the positive derivatives y q ( j ) (x) with 0jJ1 and the negative ones with Jjn.

The same arguments show that the new sign combination must also change and finally, after J changes, we arrive at the case with y q (x)>0 and y q ( j ) (x)<0 with 1jn. Now, contrary to the previous cases, the function y q (x) does not increase, but its first derivative is negative and decreases (recall that n>2). Hence this sign combination also cannot take place on an unbounded interval and therefore it must change to the case with all negative y q ( j ) (x), 0jn. By the way, the function y q (x) must vanish at some point a q >0, which completes the proof of Lemma 1. □

Note that a q is not only the first positive zero of y q (x), but the only positive one. Indeed, all y q ( j ) (x) with 0<j<n are negative at a q , whence, according to (3), all y q ( j ) (x) with 0j<n decrease and are negative for all x> a q in the domain of y q (x).

To continue the proof of Theorem 1, consider the function ξ:q a q taking each qQ to the first positive zero of the function y q . To prove its continuity, we apply the implicit function theorem. The function ξ(q) can be considered as a local solution X(q) to the equation S 0 (q,X)=0, where

S:(q,x) ( S 0 ( q , x ) , S 1 ( q , x ) , , S n 1 ( q , x ) ) = ( y q ( x ) , y q ( x ) , , y q ( n 1 ) ( x ) )

is the C 1 ‘solution’ mapping defined on a domain including R n ×{0}. The necessary for the implicit function theorem condition S 0 X ( q 0 ,, q n 1 , a q )0 is satisfied since the left-hand side of the last inequality is equal to y q ( a q )<0. Besides, any function X(q) implicitly defined near a point ( q 0 , a q 0 ) must be positive in some its neighborhood. Hence locally X(q) must be equal to ξ(q), but neither to a non-positive zero of y q (x) nor to a non-first positive one, which does not exist. Hence the function ξ(q) is continuous as well as X(q).

Now we can consider the mapping S ˜ :q N ˜ (S(q,ξ(q))), which maps Q into itself. Since S ˜ is continuous and Q is homeomorphic to a convex compact subset of R n 2 , the Brouwer fixed-point theorem can be applied. Thus, there exists q ˆ Q such that S ˜ ( q ˆ )= q ˆ .

According to the definitions of the functions N ˜ , S, and ξ, this yields the result that there exists a non-negative solution y ˆ (x)= y q ˆ (x) to (3) defined on a segment [0; a 1 ] with a 1 = a q ˆ , positive on the open interval (0; a 1 ), and such that

λ β j y ˆ ( j ) ( a 1 )= y ˆ ( j ) (0),j=0,,n1,
(5)

with

λ=N ( S ( q ˆ , ξ ( q ˆ ) ) ) = j = 0 n 1 | y ˆ ( j ) ( a 1 ) | B j >0.
(6)

Since y ˆ (x) is non-negative, it is also a solution to the equation

y ( n ) (x)+ | y ( x ) | k sgny(x)=0.
(7)

Note that for any solution y 1 (x) to (7) the function y 2 (x)= b α y 1 (bx+c) with arbitrary constants b>0 and c is also a solution to (7). Indeed, we have α+n=kα and y 2 ( j ) (x)= b α + j y 1 ( j ) (bx+c) for all j=0,,n, whence

y 2 ( n ) ( x ) + | y 2 ( x ) | k sgn y 2 ( x ) = b α + n y 1 ( n ) ( b x + c ) b k α | y 1 ( b x + c ) | k sgn y 1 ( b x + c ) = b k α ( y 1 ( n ) ( b x + c ) + | y 1 ( b x + c ) | k sgn y 1 ( b x + c ) ) = 0 .

So, the function z(x)= b α y ˆ (bx a 1 b) is a solution to (7) and is defined on the segment [ a 1 ; a 2 ] with a 2 = a 1 + a 1 b .

Put b= λ k 1 n k with λ defined by (6). Then

b α + j = λ k 1 n k ( n k 1 + j ) = λ n + ( k 1 ) j n k = λ β j ,

whence, taking into account (5), we obtain z ( j ) ( a 1 )= λ β j y ˆ ( j ) (0)= y ˆ ( j ) ( a 1 ). Thus, z(x) can be used to extend the solution y ˆ (x) on [0; a 2 ]. Since z(x) satisfies the conditions similar to (5), namely,

λ β j z ( j ) ( a 2 )= λ β j b α + j y ˆ ( j ) ( a 1 )= z ( j ) ( a 1 ),

the procedure of extension can be repeated on [0; a 3 ], [0; a 4 ], and so on with a s + 1 = a s + a s a s 1 b . In the same way the solution y ˆ (x) can be extended to the left. Its restrictions to the neighboring segments satisfy the following equality:

y ˆ (x)= b α y ˆ ( b ( x a s ) + a s 1 ) ,
(8)

where x[ a s ; a s + 1 ] and hence b(x a s )+ a s 1 [ a s 1 ; a s ].

Now we will investigate whether b is greater or less than 1.

Let a j , s be the zero of the derivative y ˆ ( j ) (x) belonging to the interval ( a s 1 ; a s ). Note that according to the above consideration on changing the sign combinations, we have

a j + 1 , s < a j , s << a 0 , s = a s < a n 1 , s + 1 < a n 2 , s + 1 <.

Lemma 2

In the above notation the solutiony(x)= y ˆ (x)satisfies the following inequalities:

|y( a 1 , s )|<|y( a n 1 , s + 1 )|,
(9)
|y( a j + 1 , s )|<|y( a j , s )|,0<j<n1.
(10)

Proof

Indeed,

1 k + 1 ( | y ( a n 1 , s + 1 ) | k + 1 | y ( a 1 , s ) | k + 1 ) = a 1 , s a n 1 , s + 1 y ( x ) | y ( x ) | k sgn y ( x ) d x = a 1 , s a n 1 , s + 1 y ( x ) y ( n ) ( x ) d x = y ( x ) y ( n 1 ) ( x ) | a 1 , s a n 1 , s + 1 + a 1 , s a n 1 , s + 1 y ( x ) y ( n 1 ) ( x ) d x > 0

since y ( a 1 , s )= y ( n 1 ) ( a n 1 , s + 1 )=0 and y (x) y ( n 1 ) (x)>0 on the interval ( a 1 , s ; a n 1 , s + 1 ), where only y(x) itself changes its sign, while all other y ( j ) (x) with 0<j<n keep the same one. Recall that n>2, which makes y (x) to be one of these others. Inequality (9) is proved.

Inequality (10) follows from y(x) y (x)>0 on the interval ( a j + 1 , s , a j , s ), where the derivatives y ( j ) (x) and y ( j + 1 ) (x) with 0<j<n1 keep different signs, while all lower-order derivatives keep the same sign as y ( j ) (x). □

From the lemma proved it follows that | y ˆ ( a 1 , s )|<| y ˆ ( a 1 , s + 1 )|= b α | y ˆ ( a 1 , s )|, whence it follows that b>1 and a s a s 1 =b( a s + 1 a s )> a s + 1 a s .

Now we see that

s = 0 ( a s + 1 a s )= a 1 s = 0 b s =and s = 0 ( a s + 1 a s )= a 1 s = 0 b s = a <.

So, the solution y ˆ (x) is extended on the half-bounded interval (; a ) and cannot be extended outside it since

lim sup x a | y ˆ (x)|= lim s + | y ˆ ( a 1 , s )|=| y ˆ ( a 1 , 0 )| lim s + b s α =+.

Now consider the function

h(t)= e t α y ˆ ( a e t ) ,
(11)

which is periodic. Indeed, if a e t [ a s ; a s + 1 ] for some sZ, then

h(t+logb)= e t α b α y ˆ ( a b e t )

and, according to (8),

h(t)= e α t y ˆ ( a e t ) = e α t b α y ˆ ( b a b e t b a s + a s 1 ) .

The expression in the last parentheses is equal to

b ( a a s ) b e t + a s 1 =b a s + 1 a s 1 b 1 b e t + a s 1 = a s a s 1 1 b 1 + a s 1 b e t = a b e t .

So, h(t+logb)=h(t) for all tR and hence the function h(t) is periodic with period 2logb.

Now, according to (11), we can express the solution y ˆ (x) to (7) just as y ˆ (x)= ( a x ) α h(log( a x)). Multiplying it by p 0 1 k 1 we obtain a solution to (3) having the form needed. It still will be a solution to (3) after replacing a by arbitrary x R.  □

The substitution xx produces the following.

Corollary 1

For any integer n>2 and real k>1 there exists a periodic oscillatory function h such that for any p 0 R satisfying ( 1 ) n p 0 >0 and any x R the function

y(x)=| p 0 | 1 k 1 ( x x ) α h ( log ( x x ) ) , x <x<,

is a solution to (1).

Note that the following theorem was earlier proved in [4], [5].

Theorem 2

Forn=3, there exists a constantB(0,1)such that any oscillatory solutiony(x)to (1) with p 0 <0satisfies the conditions

( 1 ) x i + 1 x i x i x i 1 = B 1 , i = 2 , 3 , , ( 2 ) y ( x i + 1 ) y ( x i ) = B α , i = 1 , 2 , 3 , , ( 3 ) y ( x i + 1 ) y ( x i ) = B α + 1 , i = 1 , 2 , 3 , , ( 4 ) | y ( x i ) | = M ( x i x ) α , i = 1 , 2 , 3 , ,

for someM>0and x , where x 1 < x 2 << x i <and x 1 < x 2 << x i <are sequences satisfyingy( x j )=0, y ( x j )=0, y(x)0ifx( x i , x i + 1 ), y (x)0ifx( x i , x i + 1 ).

With the help of this theorem, another one can be proved, namely the following.

Theorem 3

Forn=3and any realk>1there exists a periodic oscillatory function h such that the functionsy(x)= p 0 1 k 1 | x x | α h(log|x x |)withα= n k 1 and arbitrary x are solutions, respectively, to (1) with p 0 <0if defined on(; x )and to (1) with p 0 >0if defined on( x ;+).

3 On existence of positive solutions with non-power asymptotic behavior

For (1) with p 0 =1 it was proved [11] that for any N and K>1 there exist an integer n>N and kR such that 1<k<K and (1) has a solution of the form

y= ( x x ) α h ( log ( x x ) ) ,
(12)

where α= n k 1 and h is a positive periodic non-constant function on R.

A similar result was also proved [11] about Kneser solutions, i.e. those satisfying y(x)0 as x and ( 1 ) j y ( j ) (x)>0 for 0j<n. Namely, if p 0 = ( 1 ) n 1 , then for any N and K>1 there exist an integer n>N and kR such that 1<k<K and (1) has a solution of the form

y(x)= ( x x ) α h ( log ( x x ) ) ,

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.

Theorem 4

[13]

If12n14, then there existsk>1such that (1) with p 0 =1has a solutiony(x)such that

y ( j ) (x)= ( x x ) α j h j ( log ( x x ) ) ,j=0,1,,n1,

whereα= n k 1 and h j are periodic positive non-constant functions on R.

Remark

Computer calculations give approximate values of α. They are, with the corresponding values of k, as follows:

if n=12, then α0.56, k22.4;

if n=13, then α1.44, k10.0;

if n=14, then α2.37, k6.9.

Corollary 2

If12n14, then there existsk>1such that (1) with p 0 = ( 1 ) n 1 has a Kneser solutiony(x)satisfying

y ( j ) (x)= ( x x 0 ) α j h j ( log ( x x 0 ) ) ,j=0,1,,n1,

with periodic positive non-constant functions h j on R.

4 Conclusions, concluding remarks, and open problems

  1. 1.

    So, we give the negative answer to Question 1 and prove the existence of oscillatory solutions with special qualitative properties for Question 2.

  2. 2.

    It would be interesting to know if positive solutions like (12) exist for n15 and for 5n11.

  3. 3.

    If a positive solution like (12) exists for some k 0 >1, does it follow, for the same n, that such solutions exist for all k> k 0 ?

References

  1. 1.

    Kiguradze IT, Chanturia TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluver Academic, Dordreht; 1993.

  2. 2.

    Astashova, IV: Qualitative properties of solutions to quasilinear ordinary differential equations. In: Astashova, IV (ed.) Qualitative Properties of Solutions to Differential Equations and Related Topics of Spectral Analysis: scientific edition, M.: UNITY-DANA, pp. 22-290 (2012) (in Russian)

  3. 3.

    Kondratiev VA, Samovol VS: On some asymptotic properties to solutions for the Emden-Fowler type equations. Differ. Uravn. 1981, 17(4):749-750. (in Russian)

  4. 4.

    Astashova, IV: On the asymptotic behavior of the oscillating solutions for some nonlinear differential equations of the third and fourth order. In: Reports of the Extended Sessions of the I. N. Vekua Institute of Applied Mathematics, 3(3), 9-12, Tbilisi (1988) (in Russian)

  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. 10.1007/s10958-005-0066-6

  6. 6.

    Astashova, I: On Asymptotic Behavior of Solutions to a Forth Order Nonlinear Differential Equation. In: Mathematical Methods in Finance and Business Administration. Proceedings of the 1st WSEAS International Conference on Pure Mathematics (PUMA ’14), Tenerife, Spain, January 10-12, pp. 32-41 (2014). ISBN:978-960-474-360-5

  7. 7.

    Astashova, IV: Asymptotic behavior of solutions of certain nonlinear differential equations. In: Reports of Extended Session of a Seminar of the I. N. Vekua Institute of Applied Mathematics, 1(3), 9-11, Tbilisi (1985) (in Russian)

  8. 8.

    Kiguradze IT: Blow-up Kneser solutions of nonlinear higher-order differential equations. Differ. Equ. 2001, 31(6):768-777. 10.1023/A:1019282318238

  9. 9.

    Kiguradze IT: An oscillation criterion for a class of ordinary differential equations. Differ. Equ. 1992, 28(2):180-190.

  10. 10.

    Kiguradze IT, Kusano T: On periodic solutions of even-order ordinary differential equations. Ann. Mat. Pura Appl. 2001, 180(3):285-301. 10.1007/s102310100012

  11. 11.

    Kozlov VA: On Kneser solutions of higher order nonlinear ordinary differential equations. Ark. Mat. 1999, 37(2):305-322. 10.1007/BF02412217

  12. 12.

    Kusano T, Manojlovic J: Asymptotic behavior of positive solutions of odd order Emden-Fowler type differential equations in the framework of regular variation. Electron. J. Qual. Theory Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-45

  13. 13.

    Astashova IV: On power and non-power asymptotic behavior of positive solutions to Emden-Fowler type higher-order equations. Adv. Differ. Equ. 2013.

  14. 14.

    Astashova I: On existence of quasi-periodic solutions to a nonlinear higher-order differential equation. Abstracts of International Workshop on the Qualitative Theory of Differential Equations 2013, 16-18. [http://www.rmi.ge/eng/QUALITDE-2013/Astashova_workshop_2013.pdf] http://www.rmi.ge/eng/QUALITDE-2013/Astashova_workshop_2013.pdf

Download references

Acknowledgements

The research was supported by RFBR (grant 11-01-00989).

Author information

Correspondence to Irina Astashova.

Additional information

Competing interests

The author declares that she has no competing interests.

Authors’ original submitted files for images

Below are the links to the authors’ original submitted files for images.

Authors’ original file for figure 1

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Keywords

  • Emden-Fowler type equation
  • quasi-periodic solutions
  • oscillatory and non-oscillatory solutions