Asymptotic proximity to higher order nonlinear differential equations

Irina Astashova; Miroslav Bartušek; Zuzana Došlá; Mauro Marini

doi:10.1515/anona-2022-0254

Open Access Published by De Gruyter June 14, 2022

Asymptotic proximity to higher order nonlinear differential equations

Irina Astashova , Miroslav Bartušek , Zuzana Došlá and Mauro Marini

From the journal Advances in Nonlinear Analysis

https://doi.org/10.1515/anona-2022-0254

Abstract

The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation.

Keywords: higher order differential equation; unbounded solutions; nonoscillatory solution; asymptotic behavior; topological methods

MSC 2010: Primary 34C10; Secondary 34C15

1 Introduction

Consider the higher order nonlinear equation

(1) u ( n ) + q ( t ) u ( n − 2 ) = r ( t ) ∣ u ∣ λ sgn u , n ≥ 2 , λ > 0 ,

where the functions r and q are continuous and q ( t ) > 0 for t ≥ 1 .

Our aim is to study some asymptotic properties of solutions to (1), by considering (1) as a perturbation of the two-term linear equation

(2) y ( n ) + q ( t ) y ( n − 2 ) = 0 .

Equation (1) is a particular case of the more general equation

(3) u ( n ) + ∑ j = 0 n − 1 a j ( t ) u ( j ) = r ( t ) ∣ u ∣ λ sgn u , n ≥ 2 , λ > 0 ,

where the functions a j , j = 0 , … , n − 1 , are continuous for t ≥ 1 .

By a solution to (1) [(3)] we mean a function u differentiable up to order n , which satisfies (1) [(3)] on [ T x , ∞ ) , T x ≥ 1 and such that

sup { ∣ u ( t ) ∣ : t ≥ T } > 0 for any T ≥ T x .

As usual, a solution u to (1) [(3)] is said to be oscillatory if u changes its sign for arbitrary large t and nonoscillatory if u is different from zero for any sufficiently large t .

These equations have been investigated from different points of view as a generalization of Emden-Fowler-type differential equation

(4) u ( n ) = r ( t ) ∣ u ∣ λ sgn u , λ > 0 , λ ≠ 1 ,

see, for example, [13, Chapter IV], [4, Ch.I], and references therein for more details.

The problem of the proximity of solutions of two differential equations has a long history and has been studied in various directions for a large variety of equations. Here, we recall the monograph [13], previous papers [1,2,3,5,6,7,9,14,16,17,20], and references therein. More precisely, in [1,2] sufficient conditions are obtained for the existence of solutions of (3) which are close in a neighborhood of infinity to any nonzero constant or, more generally to a polynomial of j degree, j = 0 , … , n − 1 . In [3], the problem of asymptotic equivalence of (4) and its perturbations is studied. In [5], an asymptotic classification of solutions of (3) with n = 3 , 4 is given by topological methods. In [6,7], motivated by [12], some asymptotic relationships between (1) and (2) are obtained by using a topological approach. In [9], the existence, uniqueness, and the asymptotic equivalence of a linear differential system and one of its nonlinear perturbation with advanced and retarded argument is considered. In [14], some general relationship between solutions of a linear differential system and its perturbed form with maxima are given. In [16,17], a certain type of asymptotic equivalence between a dynamic equation and its perturbations has been considered. In [20], a type of an asymptotic equivalence between two different differential systems has been examined by using the concept of reflecting functions and these results are applied to the search of periodic solutions.

In particular, in [6,7] an important role is played by the second-order linear equation

(5) h ″ + q ( t ) h = 0 .

If the function q is bounded away from zero, then (5) is oscillatory, that is, any of its solution is oscillatory. Under this additional assumption, in [6, Theorem 1] it is shown that solutions of (2) are, roughly speaking, in asymptotic proximity with solutions of (1) and in [7, Theorem 2, Theorem 3] their boundedness is examined. If the function q tends to some positive constant, then (2) has ( n − 2 ) parametric set of polynomial solutions, and (1) with λ = 1 has ( n − 2 ) parametric set of polynomial-like solutions, see [7, Theorem 5].

Here we consider the case in which the linear equation (5) is nonoscillatory, that is, any of its solution is nonoscillatory. Under this assumption, a direct computation shows that (1) can be written for large t as the two-term equation

h 2 ( t ) x ( n − 2 ) ( t ) h ( t ) ′ ′ = h ( t ) r ( t ) ∣ u ∣ λ sgn u ,

where h is a nonoscillatory solution of (5). Using this fact and an iterative method, which is similar to the one given in [6], we study the existence of a n -parameter set of nonoscillatory solutions to equation (1) with polynomial and noninteger power-law growth, which are, roughly speaking, in asymptotic proximity with solutions of the linear unperturbed equation (2).

As usual, we use the following notation.

Let g i , i = 1 , 2 , 3 be continuous functions and g 3 ≠ 0 near infinity. Then:

The symbol g 1 = O ( g 2 ) means that ultimately ∣ g 1 ( t ) ∣ ≤ M ∣ g 2 ( t ) ∣ for some constant M , that is, there exist T ≥ 0 and M > 0 such that
∣ g 1 ( t ) ∣ ≤ M ∣ g 2 ( t ) ∣ on [ T , ∞ ) ;
The symbol g 1 = o ( g 3 ) means that lim t → ∞ g 1 ( t ) / g 3 ( t ) = 0 ;
The symbol g 1 ∼ g 3 means that lim t → ∞ g 1 ( t ) / g 3 ( t ) = 1 ;
For λ > 0 and u ∈ R the symbol [ u ] ± λ means ∣ u ∣ λ sgn u , for short.

Equation (3) has been studied in the literature as a perturbation of the linear differential equation

(6) y ( n ) + ∑ j = 0 n − 1 a j ( t ) y ( j ) = 0 .

We start by referring Sobol’s result [18] obtained for equation (6).

Theorem A

[18, Theorem 1] Under the condition

(7) ∫ 1 ∞ t n − j − 1 ∣ a j ( t ) ∣ d t < ∞ for j = 0 , 1 , … , n − 1 ,

equation (6) has a fundamental family of solutions y j satisfying for large t the conditions

(8) y j ( t ) = t j ( 1 + o ( 1 ) ) for j = 0 , 1 , … , n − 1 .

In this paper, it is also proved that the derivatives of such solutions also have a power-law growth.

This result has been extended by I. Kiguradze to equation (4) and to a more general higher order equation.

Theorem B

[11, Theorem 1] Let λ ≥ 1 and 1 ≤ k ≤ n . In order equation (4) has solutions u i ( t ) ( i = 1 , 2 , … , k ) such that for t → ∞

u i ( t ) ∼ c i t i − 1 , u i ′ ( t ) ∼ ( i − 1 ) c i t i − 2 , … u i ( i − 1 ) ( t ) ∼ ( i − 1 ) ! c i ≠ 0 ,

it is sufficient, and if r does not change its sign, it is also necessary that

∫ 1 ∞ ∣ r ( t ) ∣ t n − 1 + ( λ − 1 ) ( k − 1 ) d t < ∞ .

The next result follows from [13, Corollary 8.2] applying to equation

(9) u ( n ) = q ( t , u , u ′ , … , u ( n − 1 ) ) ,

where q ( t , x 1 , … , x n ) is a continuous function on [ 1 , ∞ ) × R n .

Theorem C

Let there exist l ∈ { 1 , … , n } , μ ∈ ( 0 , ∞ ) , μ ≠ 0 , and a continuous function q ∗ : [ T , ∞ ) → R + , T ≥ 1 , such that

∣ q ( t , x 1 , … , x n ) ∣ ≤ q ∗ ( t ) for t ≥ T , x k − α ( l − 1 ) ! ( l − k ) ! t l − k ≤ μ t l − k ( k = 1 , … , l ) , ∣ x k ∣ ≤ μ t l − k ( k = l + 1 , … , n ) ,

and

∫ T ∞ t n − l q ∗ ( t ) d t < ∞ .

Then equation (9) possesses a solution u having the asymptotic representation

u ( k − 1 ) ( t ) = α ( l − 1 ) ! ( l − k ) ! t l − k + o ( t l − k ) ( k = 1 , … , l ) , u ( k − 1 ) ( t ) = o ( t l − k ) ( k = l + 1 , … , n ) .

Using this result it is easy to give conditions if (3) has solutions with a polynomial power-law growth at infinity. These results are presented in Section 3, along with a discussion on the proximity between equations (1) and (2). Finally, in Section 4 some comments, examples, and suggestions for future research studies are presented.

2 Asymptotic representation of solutions to equation (1)

Our main result in this section concerns the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior and reads as follows.

Theorem 1

Let the second-order differential equation (5) be nonoscillatory. Assume that for some real number m ∈ [ 0 , n − 1 ]

(10) ∫ 1 ∞ t n − 1 + m λ + i q ∣ r ( t ) ∣ d t < ∞ ,

where i q = 0 in the case

(11) ∫ 1 ∞ t q ( t ) d t < ∞

and i q = 1 in the case

(12) ∫ 1 ∞ t q ( t ) d t = ∞ .

Then for any solution y to (2) such that y ( t ) = O ( t m ) there exists a solution u to (1) such that for large t

(13) u ( i ) ( t ) = y ( i ) ( t ) + ε i ( t ) , i = 0 , 1 , … , n − 1 ,

where all ε i are functions of bounded variation and lim t → ∞ ε i ( t ) = 0 .

To prove Theorem 1, we use a similar approach to that given in [6], which is based on the induction method, an iterative process, and suitable estimates for solutions to (2).

Equations (2) and (5) are strictly related. When q ( t ) ≡ 0 , the space of solutions to (2) has a fundamental system of solutions consisting of

t j , j = 0 , 1 , … , n − 1 .

In the general case, it is easy to see that the space of solutions to (2) has a fundamental system of solutions consisting of

(14) Γ 1 , Γ 2 , t j , j = 0 , 1 , … , n − 3 ,

where t j are missing in case n = 2 , and

(15) Γ 1 ( t ) = ∫ 1 t ( t − s ) n − 3 h 1 ( s ) d s , Γ 2 ( t ) = ∫ 1 t ( t − s ) n − 3 h 2 ( s ) d s ,

in case n ≥ 3 ,

Γ 1 ( t ) = h 1 ( t ) , Γ 2 ( t ) = h 2 ( t ) in case n = 2 ;

h 1 , h 2 are two independent solutions to (5), see [6].

Let h 1 and h 2 be two linearly independent solutions to (5) with Wronskian d > 0 . Put

(16) w ( s , t ) = h 1 ( s ) h 2 ( t ) − h 1 ( t ) h 2 ( s ) , z ( s , t ) = ∂ ∂ t w ( s , t ) .

Lemma 1

Let (5) be nonoscillatory.

If (11) holds, then there exists a constant K > 0 such that
∣ w ( s , t ) ∣ ≤ K s , ∣ z ( s , t ) ∣ ≤ K s for s ≥ t ≥ 1 ,
and
Γ 1 ( t ) = O ( t n − 2 ) , Γ 2 ( t ) = O ( t n − 1 ) .
If (12) holds, then there exists a constant K > 0 such that
∣ w ( s , t ) ∣ ≤ K s 2 , ∣ z ( s , t ) ∣ ≤ K s for s ≥ t ≥ 1 ,
and
Γ 1 ( t ) = O ( t n − 1 ) , Γ 2 ( t ) = O ( t n − 1 ) .

Proof

Let h 1 , h 2 be two linearly independent solutions to (5) with Wronskian d > 0 . According to [8], Theorem 1(i 2 ), (i 3 ), and Theorem 2(i 2 ), (i 3 ), we can choose

lim t → ∞ ∣ h 1 ( t ) ∣ = c 1 > 0 , lim t → ∞ h 1 ′ ( t ) = 0 , lim t → ∞ ∣ h 2 ( t ) ∣ = ∞ , lim t → ∞ ∣ h 2 ′ ( t ) ∣ = c 2 > 0

in case (11). Hence, ∣ h 2 ( t ) ∣ ≤ c 2 t and from (16) we obtain ∣ w ( s , t ) ∣ ≤ K s for t ≥ 1 and K = c 1 c 2 . Similarly,

lim t → ∞ ∣ h i ( t ) ∣ = ∞ , lim t → ∞ h i ′ ( t ) = 0 , i = 1 , 2 ,

in case (12). From these, (15), and (16) the conclusions follow.□

The following lemma is a modification of [6, Lemma 1], where it has been stated for (1) in the case that (5) is oscillatory.

Lemma 2

Let equation (5) be nonoscillatory and let { u k } k ∈ N be a sequence of continuous functions on [ t 0 , ∞ ) , t 0 ≥ 1 . Consider for k ≥ 2 and t ≥ t 0 the sequence { α k } is given by

α k ( t ) = ( − 1 ) n + 1 d ∫ t T k ( σ − t ) n − 3 ( n − 3 ) ! ∫ σ T k r ( s ) [ u k − 1 ( s ) ] ± λ w ( s , σ ) d s d σ ,

in case n ≥ 3 and

α k ( t ) = − 1 d ∫ t T k r ( s ) [ u k − 1 ( s ) ] ± λ w ( s , t ) d s

in case n = 2 , where T k = t 0 + k .

If (11) holds, then, for t ≥ t 0 ,
(17) ∣ α k ( i ) ( t ) ∣ ≤ M ∫ t T k s n − 1 − i ∣ r ( s ) ∣ ∣ u k − 1 ( s ) ∣ λ d s , i = 0 , 1 , … , n − 3 ,

(18) ∣ α k ( i ) ( t ) ∣ ≤ M ∫ t T k s ∣ r ( s ) ∣ ∣ u k − 1 ( s ) ∣ λ d s , i = n − 2 , n − 1 ;
If (12) holds, then, for t ≥ t 0 ,
(19) ∣ α k ( i ) ( t ) ∣ ≤ M ∫ t T k s n − i ∣ r ( s ) ∣ ∣ u k − 1 ( s ) ∣ λ d s , i = 0 , 1 , … , n − 3 , ∣ α k ( n − 2 ) ( t ) ∣ ≤ M ∫ t T k s 2 ∣ r ( s ) ∣ ∣ u k − 1 ( s ) ∣ λ d s , ∣ α k ( n − 1 ) ( t ) ∣ ≤ M ∫ t T k s ∣ r ( s ) ∣ ∣ u k − 1 ( s ) ∣ λ d s ,
where M = K / d and K is given in Lemma 1. Note that (17) and (19) are missing if n = 2 .

Proof

Suppose (11) holds, case (12) can be treated similarly. For n ≥ 3 , t ≥ t 0 , and i = 1 , 2 , … , n − 3 we have

(20) α k ( i ) ( t ) = ( − 1 ) n + i + 1 1 d ∫ t T k ( σ − t ) n − 3 − i ( n − 3 − i ) ! ∫ σ T k r ( s ) [ u k − 1 ( s ) ] ± λ w ( s , σ ) d s d σ .

Hence, for n ≥ 2

(21) α k ( n − 2 ) ( t ) = − 1 d ∫ t T k r ( s ) [ u k − 1 ( s ) ] ± λ w ( s , t ) d s ,

(22) α k ( n − 1 ) ( t ) = − 1 d ∫ t T k r ( s ) [ u k − 1 ( s ) ] ± λ z ( s , t ) d s ,

(23) α k ( n ) ( t ) = r ( t ) [ u k − 1 ( t ) ] ± λ − q ( t ) α k ( n − 2 ) ( t ) .

If n = 2 the statement follows from (21), (22), and Lemma 1.

Let n ≥ 3 . From (20) and Lemma 1(i) we have

∣ α k ( n − 3 ) ( t ) ∣ ≤ K d ∫ t T k s 2 ∣ r ( s ) ∣ ∣ u k − 1 ∣ λ d s

and the conclusion holds for i = n − 3 . Furthermore, for i = 0 , 1 , … , n − 4 we obtain

(24) ∣ α k ( i ) ( t ) ∣ ≤ ∫ t T k ∣ α k ( i + 1 ) ( s ) ∣ d s ≤ K d ∫ t T k ( s − t ) n − 1 − i ∣ r ( s ) ∣ ∣ u k − 1 ∣ λ d s .

Thus, the conclusion holds for these i . Finally, the estimates for i = n − 2 , n − 1 , follow from (21), (22), and Lemma 1(i).□

Proof of Theorem 1

The idea of the proof is similar to that of Theorem 1 in [6].

Suppose (11) holds, i.e., i q = 0 . Since y ( t ) = O ( t m ) , there exists L > 0 such that for t ≥ 1

(25) ∣ y ( t ) ∣ < L t m .

Put L ¯ = L + 1 / 2 and choose t 0 > 1 such that

(26) L ¯ K d ∫ t 0 ∞ s n − 1 + λ m ∣ r ( s ) ∣ d s ≤ 1 2 ,

where K is given by Lemma 1. Put T k = t 0 + k . On [ t 0 , ∞ ) , consider the sequence { u k } defined by

u 1 ( t ) = y ( t ) , u k ( t ) = y ( t ) + ( − 1 ) n + 1 d ∫ t T k ( σ − t ) n − 3 ( n − 3 ) ! ∫ σ T k r ( s ) [ u k − 1 ( s ) ] ± λ ( s , σ ) d s d σ

in case n ≥ 3 , and

u k ( t ) = y ( t ) − 1 d ∫ t T k r ( s ) [ u k − 1 ( s ) ] ± λ w ( s , t ) d s

in case n = 2 .

For the functions α k defined on [ t 0 , ∞ ) in Lemma 2, we have

α k ( i ) ( t ) = u k ( i ) ( t ) − y ( i ) ( t ) , i = 0 , … , n − 1 ,

and, in view of (23),

(27) α k ( n ) ( t ) = r ( t ) [ u k − 1 ( t ) ] ± λ − q ( t ) u k ( n − 2 ) ( t ) + q ( t ) y k ( n − 2 ) ( t ) = u k ( n ) ( t ) − y ( n ) ( t ) .

Thus, for t ≥ t 0 we have

(28) u k ( n ) ( t ) + q ( t ) u k ( n − 2 ) ( t ) = r ( t ) [ u k − 1 ( t ) ] ± λ .

We show that for all i = 0 , … , n the sequences { u k ( i ) } are uniformly bounded and equicontinuous on each finite subinterval of [ t 0 , ∞ ) .

First, let us show that

∣ u k ( t ) − y ( t ) ∣ ≤ 1 2 , t ∈ [ t 0 , ∞ ) .

Clearly, this holds for k = 1 . We proceed by induction and assume that

∣ u k − 1 ( t ) − y ( t ) ∣ ≤ 1 2 , t ∈ [ t 0 , ∞ ) .

From this and (25), we have ∣ u k − 1 ( t ) ∣ < L ¯ t m . In view of Lemma 2(i) and (25), we obtain

∣ α k ( t ) ∣ ≤ K L ¯ λ d ∫ t T k s n − 1 + λ m ∣ r ( s ) ∣ d s .

Thus, in view of (26), we obtain

(29) ∣ u k ( t ) − y ( t ) ∣ ≤ K L ¯ λ d ∫ t ∞ s n − 1 + λ m ∣ r ( s ) ∣ d s ≤ 1 2 .

Similarly, using again Lemma 2(i),

(30) ∣ u k ( i ) ( t ) − y ( i ) ( t ) ∣ = ∣ α k ( i ) ( t ) ∣ ≤ M 1 ∫ t ∞ s n − 1 − i + λ m ∣ r ( s ) ∣ d s

for i = 1 , … , n − 2 , and n ≥ 3 , and

(31) ∣ u k ( n − 1 ) ( t ) − y ( n − 1 ) ( t ) ∣ = ∣ α k ( n − 1 ) ( t ) ∣ ≤ M 1 ∫ t ∞ s 1 + λ m ∣ r ( s ) ∣ d s

for n ≥ 2 , where M 1 = K L ¯ / d and K is given by Lemma 1.

Hence, { u k ( i ) } , i = 0 , 1 , … , n − 1 , are uniformly bounded on each finite subinterval of [ t 0 , ∞ ) . Moreover, in view of (28), the same holds for { u k ( n ) } . Then { u k ( i ) } , i = 0 , … , n − 1 , are equicontinuous on each finite subinterval in [ t 0 , ∞ ) and, from (28), the same holds for { u k ( n ) } . Hence, { u k } admits a converging subsequence { u k j } such that { u k j ( i ) } , i = 0 , … , n , uniformly converge to a function u ( i ) on each finite subinterval of [ t 0 , ∞ ) .

Again from (28) we obtain that u is a solution to (1). From (10), (30), and (31), using the Lebesgue dominated convergence theorem, we obtain

∫ t 0 ∞ ∣ u ( i ) ( s ) − y ( i ) ( s ) ∣ d s < ∞ , i = 1 , 2 , … , n − 1 ,

whence all u ( i ) − y ( i ) , i = 0 , … , n − 2 , are of bounded variation in a neighborhood of infinity. Taking into account (27), we obtain

∣ u ( n ) ( t ) − y ( n ) ( t ) ∣ ≤ ∣ r ( t ) ∣ ∣ u λ ( t ) ∣ + q ( t ) ∣ u ( n − 2 ) ( t ) − y ( n − 2 ) ( t ) ∣ .

Since ∣ u ( t ) ∣ < L ¯ t m , we obtain

∣ u ( n ) ( t ) − y ( n ) ( t ) ∣ ≤ L ¯ λ ∣ r ( t ) ∣ t λ m + q ( t ) ∣ u ( n − 2 ) ( t ) − y ( n − 2 ) ( t ) ∣ .

Thus, u ( n − 1 ) − y ( n − 1 ) is also of bounded variation in a neighborhood of infinity. Finally, from (29), (30), and (31) we obtain (13).

If (12) holds, then the proof is similar, we use Lemma 2(ii) instead of Lemma 2(i).□

From Theorem 1 we obtain the following.

Corollary 1

Let (5) be nonoscillatory and n ≥ 3 . Assume that (10) holds with i q as in Theorem 1 and some m ∈ { 0 , 1 , … , n − 3 } . Then for any polynomial Q with deg Q ≤ m there exists a solution u of (1) such that for large t

u ( i ) ( t ) = Q ( i ) ( t ) + ε i ( t ) , i = 0 , 1 , … , n − 1 ,

where all ε i are functions of bounded variation and lim t → ∞ ε i ( t ) = 0 .

Proof

Since Q is a solution of (2), the conclusion follows from Theorem 1.□

Corollary 2

Let (5) be nonoscillatory. Assume that

(32) ∫ 1 ∞ t ( n − 1 ) ( λ + 1 ) + i q ∣ r ( t ) ∣ d t < ∞ ,

where i q is as in Theorem 1. Then for any polynomial Q with deg Q ≤ n − 3 in case n ≥ 3 , Q ≡ 0 in case n = 2 , there exist solutions u to (1) such that for large t

u ( i ) ( t ) = ( c 1 Γ 1 ( t ) + c 2 Γ 2 ( t ) + Q ( t ) ) ( i ) + ε i ( t ) , i = 0 , … , n − 1 ,

where Γ 1 and Γ 2 are given by (15), c 1 and c 2 are arbitrary constants, and ε i are functions of bounded variation and lim t → ∞ ε i ( t ) = 0 .

Proof

Take the fundamental system of solutions to (2) given in (14). Applying Theorem 1 for m = n − 1 , we obtain the conclusion.□

We complete this section by considering the special case of (1)

(33) u ( n ) ( t ) + ε t 2 u ( n − 2 ) ( t ) = r ( t ) ∣ u ∣ ± λ ,

and the related linear equation

(34) y ( n ) ( t ) + ε t 2 y ( n − 2 ) ( t ) = 0 ,

where ε ∈ ( 0 , 1 / 4 ) . Then the estimations given in Lemma 1(ii) can be improved and we obtain the following results.

Theorem 2

Let ε ∈ ( 0 , 1 / 4 ) , and, for some real number m ∈ [ 0 , n − 1 ] ,

(35) ∫ 1 ∞ t n − 1 + m λ ∣ r ( t ) ∣ d t < ∞ .

Then for any solution y to (34) such that y ( t ) = O ( t m ) there exists a solution u to (33) such that for large t

u ( i ) ( t ) = y ( i ) ( t ) + ε i ( t ) , i = 0 , 1 , … , n − 1 ,

where all ε i are functions of bounded variation and lim t → ∞ ε i ( t ) = 0 .

Proof

The corresponding second-order linear equation

h ″ + ε t 2 h = 0

is nonoscillatory and has a fundamental system of solutions h 1 ( t ) = t μ 1 , h 2 ( t ) = t μ 2 , where

μ 1 = ( 1 − 1 − 4 ε ) / 2 , μ 2 = ( 1 + 1 − 4 ε ) / 2 ,

see, e.g., [19, Chapter 2.1]. Take the fundamental system of solutions to (34) given in (14). If n ≥ 3 , then

(36) Γ 1 ( t ) = ∫ 1 t ( t − s ) n − 3 s μ 1 d s = O ( t β ) ,

(37) Γ 2 ( t ) = ∫ 1 t ( t − s ) n − 3 s μ 2 d s = O ( t γ ) ,

where

β = n − 3 2 − 1 − 4 ε 2 , γ = n − 3 2 + 1 − 4 ε 2 .

If n = 2 ,

(38) Γ 1 ( t ) = h 1 ( t ) = O ( t β ) , Γ 2 ( t ) = h 2 ( t ) = O ( t γ ) .

Furthermore,

∣ w ( s , t ) ∣ = h 1 ( s ) h 2 ( t ) − h 1 ( t ) h 2 ( s ) ≤ 2 s for s ≥ t ≥ 1 .

Now we proceed by the similar way as in the proof of Theorem 1 replacing the estimation of w from Lemma 1(ii) by ∣ w ( s , t ) ∣ ≤ 2 s .□

From Theorem 2 we obtain immediately the following.

Corollary 3

Assume that

∫ 1 ∞ t n − 1 + γ λ ∣ r ( t ) ∣ d t < ∞ ,

where γ = n − 3 2 + 1 − 4 ε 2 .

Then for any polynomial Q with deg Q ≤ n − 3 in case n ≥ 3 , Q ≡ 0 in case n = 2 , there exist solutions u of (33), such that for large t

u ( i ) ( t ) = ( c 1 Γ 1 ( t ) + c 2 Γ 2 ( t ) + Q ( t ) ) ( i ) + ε i ( t ) , i = 0 , … , n − 1 ,

where Γ 1 and Γ 2 are given by (36), (37), and (38), c 1 and c 2 are arbitrary constants, and ε i are functions of bounded variation and lim t → ∞ ε i ( t ) = 0 .

Proof

We apply Theorem 2 with m = γ .□

3 Asymptotic representation of solutions to equation (3)

In this section, we study some asymptotic properties of equation (3) and discuss the proximity of this equation to the unperturbed linear equation (6). We start by presenting conditions under which equation (3) has solutions with asymptotic growth of polynomial type. This result can be obtained as a corollary of Theorem C.

Corollary 4

Suppose that the continuous functions a 0 , … , a n − 1 satisfy (7) and for some integer number m ∈ { 0 , 1 , … , n − 1 }

(39) ∫ 1 ∞ t n − 1 + ( λ − 1 ) m ∣ r ( t ) ∣ d t < ∞ .

Then for any C ≠ 0 there exists a solution u to equation (3) satisfying for large t

(40) u ( j ) ( t ) = C m ! ( m − j ) ! t m − j ( 1 + o ( 1 ) ) for j = 0 , 1 , … , m , u ( j ) ( t ) = o ( t m − j ) for j = m + 1 , … , n − 1 .

In particular, if

(41) ∫ 1 ∞ t w ∣ r ( t ) ∣ d t < ∞ ,

where w = λ ( n − 1 ) if λ ≥ 1 and w = n − 1 if λ < 1 , then for any solution y of (6) there exists a solution u of (3)such that for large t

(42) u ( t ) = y ( t ) ( 1 + o ( 1 ) ) ,

(43) u ( i ) ( t ) = y ( i ) ( t ) ( 1 + o ( 1 ) ) + o ( t − 1 ) , i = 1 , … , n − 1 .

Proof

In Theorem C, choose l = m + 1 , k = j + 1 , α = C , μ = 1 . Put

q ( t , x 1 , … , x n ) = r ( t ) ∣ x 1 ∣ ± λ − ∑ k = 1 n a k − 1 ( t ) x k ,

q ∗ ( t ) = c 1 t λ ( l − 1 ) ∣ r ( t ) ∣ + c 2 ∑ k = 1 n ∣ a k − 1 ( t ) ∣ t l − k ,

and c 1 = ( 1 + ∣ α ∣ ) λ , c 2 = 1 + ∣ α ∣ ( l − 1 ) ! .

We claim that

∣ q ( t , x 1 , … , x n ) ∣ ≤ q ∗ ( t )

for t ≥ 1 and

∣ x k − α ( l − 1 ) ! ( l − k ) ! t l − k ∣ ≤ t l − k , k = 1 , … , l , ∣ x k ∣ ≤ t l − k , k = l + 1 , … , n .

Indeed, we have

∣ q ( t , x 1 , … , x n ) ∣ ≤ ∣ r ( t ) ∣ ( t l − 1 + ∣ α ∣ t l − 1 ) λ + ∑ k = 1 l ∣ a k − 1 ( t ) ∣ t l − k + ∣ α ∣ ( l − 1 ) ! ( l − k ) ! t l − k + ∑ k = l + 1 n ∣ a k − 1 ( t ) ∣ t l − k ≤ c 1 t λ ( l − 1 ) ∣ r ( t ) ∣ + c 2 ∑ k = 1 n ∣ a k − 1 ( t ) ∣ t l − k = q ∗ ( t ) .

Thus, using (7) and (39)

∫ 1 ∞ t n − l q ∗ ( t ) d t = c 1 ∫ 1 ∞ t n − l + λ ( l − 1 ) ∣ r ( t ) ∣ d t + c 2 ∫ 1 ∞ ∑ k = 1 n ∣ a k − 1 ( t ) ∣ t n − k d t < ∞ .

Hence, both relations in (40) hold.

In order to complete the proof, let us show that (42) and (43) are satisfied. Assume that (41) holds. Then (39) is valid for m = 0 , 1 , … , n − 1 . Indeed, denoting w m = n − 1 + ( λ − 1 ) m , we have

max m = 0 , 1 , … , n − 1 w m = w n − 1 = λ ( n − 1 ) if λ ≥ 1 , max m = 0 , 1 , … , n − 1 w m = w 0 = n − 1 if λ < 1 .

Let y be an arbitrary solution of (6). Then according to the proved part of Corollary 4 with r ( t ) ≡ 0 , there exist a fundamental system of solutions to (6), say y 0 ( t ) , … , y n − 1 ( t ) such that

(44) y l ( j ) ( t ) = m ! ( m − j ) ! t m − j ( 1 + o ( 1 ) ) for j = 0 , 1 , … , l y l ( j ) ( t ) = o ( t m − j ) for j = l + 1 , … , n − 1 ,

where l ∈ 0 , 1 , … , n − 1 , and

y ( t ) = ∑ l = 0 n − 1 c l y l ( t ) , t ≥ 1 ,

where c l , l = 0 , 1 , … , n − 1 are suitable constants. Let m ∈ { 0 , 1 , … , n − 1 } be such that c m ≠ 0 , c l = 0 for l > m . Using (44) we have

y ( j ) ( t ) = ∑ l = 0 m c l y l ( j ) ( t ) = c m m ! ( m − j ) ! t m − j ( 1 + o ( 1 ) ) for j = 1 , … , m , y ( j ) ( t ) = ∑ l = 0 m c l y l ( j ) ( t ) = o ( t m − j ) for j = m + 1 , … , n − 1 .

From here and (40), equation (3) has a solution u with the same asymptotics, letting C = c m . Therefore, for large t

u ( k − 1 ) ( t ) = y ( k − 1 ) ( t ) ( 1 + o ( 1 ) ) for k = 1 , … , m , u ( k − 1 ) ( t ) = y ( k − 1 ) ( t ) + o ( t m − k ) = y ( k − 1 ) ( t ) + o ( t − 1 ) for k = m + 1 , … , n .

If k = 1 , then (42) holds, in other cases we obtain (43).□

Observe that if y ( i ) for some i ∈ { 1 , … , n − 1 } is bounded away from zero, then (43) is equivalent to

u ( i ) ( t ) = y ( i ) ( t ) ( 1 + o ( 1 ) ) .

If y ( i ) = o ( t − 1 ) , then (43) is equivalent to

u ( i ) ( t ) = y ( i ) ( t ) + o ( t − 1 ) .

Remark 1

If (7) and (41) hold, then for any C ≠ 0 and every m = 0 , 1 , … , n − 1 there exist solutions y 1 , … , y n of equation (6) with different asymptotic representation (40), and by Corollary 4 there exists n -parametric set of solutions of equation (3).

Remark 2

In [12, Section 8.3, p. 163], there is an example of Emden-Fowler-type equation (4) having solution u with a noninteger power-law asymptotic behavior. Thus, the corresponding linear equation y ( n ) = 0 has no solution y satisfying lim t → ∞ ∣ u ( t ) − y ( t ) ∣ = 0 .

Remark 3

Applying Corollary 4 to equation (1), we obtain the following comparison between Corollary 4 and Theorem 1: Condition (7) of Corollary 4 reads as condition (11) of Theorem 1. If (39) holds but not (10), Corollary 4 is applicable while Theorem 1 is not applicable. The same holds if q changes its sign. If m is not integer or (12) holds, Theorem 1 is applicable, while the application of Corollary 4 is not possible.

Remark 4

In [2, Theorem 2.4] it is proved that if λ > 1 , (7) and (32) with i q = 0 hold, then there exist solutions u to equation (3) such that for large t

(45) u ( t ) = ∑ j = 0 n − 1 C j y j ( t ) + o ( 1 ) ,

where C j are arbitrary constants and the functions y j are a fundamental system of solutions to equation (6) such that for large t

y j ( t ) = t j j ! ( 1 + o ( 1 ) ) .

Corollary 2 extends this result for equation (1).

4 Examples and suggestions

The following examples illustrate Theorems 1, 2, and Corollary 4.

Example 1

Let λ > 0 and consider the nonlinear equation for t ≥ 1

(46) u ( 4 ) + 1 t 2 log e t u ( 2 ) = e − t ( t 2 log e t + 1 ) ( 1 + e − t ) λ t 2 log e t ∣ u ∣ ± λ .

A standard calculation shows that

(47) u ( t ) = t + e − t

is a solution of (46). Setting

q ( t ) = 1 t 2 log e t , r ( t ) = e − t ( t 2 log e t + 1 ) ( 1 + e − t ) λ t 2 log e t ,

we obtain that (5) is nonoscillatory and (12) is valid. Moreover, we have for any σ > 0

∫ 1 ∞ t σ r ( t ) d t < ∞ .

Thus, all the assumptions of Theorem 1 are verified with m = 1 and so equation (46) has a solution u such that for any large t

u ( i ) ( t ) = y ( i ) ( t ) + ε i ( t ) , i = 0 , … , 3 ,

where ε i are functions of bounded variation such that lim t → ∞ ε i ( t ) = 0 and y ( t ) = t , as solution (47) illustrates.

Example 2

Consider the nonlinear equation for t ≥ 1

(48) u ( 3 ) + 3 16 1 t 2 u ′ = cos t t 5 ∣ u ∣ ± 1 / 2 .

A standard calculation shows that a fundamental system of solutions of the linear equation

y ( 3 ) + 3 16 1 t 2 y ′ = 0

is given by

y 1 ( t ) = 1 , y 2 ( t ) = t 5 / 4 , y 3 = t 7 / 4 .

Since

∫ 1 ∞ t − 2 ∣ cos t ∣ d t < ∞ ,

condition (35) is satisfied for n = 3 , λ = 2 − 1 , r ( t ) = t − 5 cos t , and any m ∈ { 0 , 1 , 2 } . Hence, from Theorem 2 equation (48) has a solution u k such that for any large t

u k ( i ) ( t ) = y k ( i ) ( t ) + ε k , i ( t ) , k , i = 0 , 1 , 2 ,

where ε k , i are functions of bounded variation such that lim t → ∞ ε k , i ( t ) = 0 .

Example 3

Consider the nonlinear equation for t ≥ 1

(49) u ( 3 ) + 1 t ( 2 t − 1 ) u ( 2 ) − 1 t 2 ( 2 t − 1 ) u ′ = r ( t ) ∣ u ∣ ± λ ,

where 0 < λ < 1 and r is a continuous function for t ≥ 1 such that

(50) ∫ 1 ∞ t 2 ∣ r ( t ) ∣ d t < ∞ .

A standard calculation shows that the corresponding linear equation to (49), that is, the equation

y ( 3 ) + 1 t ( 2 t − 1 ) y ( 2 ) − 1 t 2 ( 2 t − 1 ) y ′ = 0 , t ≥ 1 ,

has the fundamental system of solutions consisting of

y 1 ( t ) = 1 , y 2 ( t ) = t 2 , y 3 ( t ) = ∫ 1 t s log 2 s 2 s − 1 d s .

Denoting by a i , i = 0 , 1 , 2 , the functions

a 0 ( t ) = 0 , a 1 ( t ) = − 1 t 2 ( 2 t − 1 ) , a 2 ( t ) = 1 t ( 2 t − 1 ) ,

it is immediate to verify that (7) holds for all j = 0 , 1 , 2 . From (50), also conditions (39) and (41) are valid, and so, in virtue of Corollary 4, there exist three solutions u i , i = 1 , 2 , 3 , of (49) such that

u 1 ( t ) = y 1 ( t ) ( 1 + o ( 1 ) ) = 1 + o ( 1 ) , u 2 ( t ) = y 2 ( t ) ( 1 + o ( 1 ) ) = t 2 ( 1 + o ( 1 ) ) , u 3 ( t ) = y 3 ( t ) ( 1 + o ( 1 ) ) = ∫ 1 t s log 2 s 2 s − 1 d s ( 1 + o ( 1 ) ) + o ( 1 ) .

A similar asymptotic representation holds for the derivatives of u i ′ , u i ″ , i = 1 , 2 , 3 . For instance, we have

u 2 ′ ( t ) = 2 t ( 1 + o ( 1 ) ) + o ( t − 1 ) = 2 t ( 1 + o ( 1 ) ) , u 2 ″ ( t ) = 2 ( 1 + o ( 1 ) ) + o ( t − 1 ) = 2 + o ( 1 ) ,

and similarly for the first and second derivative of u 1 and u 3 .

Open problems.

(1) Condition (35) in Theorem 2 is weaker than (10) in Theorem 1. This is due to the fact that it was possible to explicitly calculate the fundamental system of solutions of (5), and from these solutions, to obtain a fundamental system of (2). Since an explicit expression of solutions of (5) is possible also for particular choices of q , like, for example, the Euler equation

h ″ + 1 4 t 2 h = 0 ,

see, e.g., [19, Chapter 2.1], it should be interesting to study whenever Theorem 2 remains to hold for equations of the type

u ( n ) + φ ( t ) u ( n − 2 ) = r ( t ) ∣ u ∣ ± λ ,

where φ is a continuous function for t ≥ 1 and asymptotic representations of solutions of the second-order linear equation

h ″ + φ ( t ) h = 0

are known.

(2) The study on the proximity of solutions could be extended to the one between delay and neutral equations and the corresponding linear equations without delay or neutral argument, by applying the proximity results obtained in Section 2. More precisely, this study could be accomplished in two steps. First, by considering the possible proximity between the nonlinear equation (1) (or (3)) and the corresponding nonlinear equation with functional argument and later to apply Theorem 1 or Theorem 2. Concerning the first step, observe that the functional argument may produce a loss of proximity. To illustrate this fact, consider the Emden-Fowler equation without deviating argument

(51) x ″ ( t ) = b ( t ) ∣ x ( t ) ∣ ± λ ,

where b is a continuous function for t ≥ 1 and the corresponding equation with deviating argument

(52) x ″ ( t ) = b ( t ) ∣ x ( τ ( t ) ) ∣ ± λ , τ ( t ) < t .

When λ > 1 and b is positive, equation (51) always has Kneser solutions, that is, solutions x such that x ( t ) > 0 , x ′ ( t ) < 0 for large t . Moreover, if, in addition,

∫ 1 ∞ s b ( s ) d s < ∞ ,

then (51) does not have Kneser solutions which tend to zero as t → ∞ , see, e.g., [13, Section 13.2]. If τ ( t ) < t , then this result may fail for (52), see, e.g., [15].

In the sublinear case, that is, 0 < λ < 1 , it is known that there might exist equations of type (51) without Kneser solutions. For instance, if

(53) liminf t → ∞ t 2 b ( t ) > 0 ,

then (51) does not have Kneser solution, see [13, Corollary 17.3]. On the other hand, the equation

x ″ ( t ) = 1 t 2 log t ∣ x ( τ ( t ) ) ∣ ± λ , t ≥ 2 ,

has Kneser solutions, as [10, Corollary 2] illustrates.

In virtue of these facts, it should be interesting to show if an analogous type of proximity between Kneser solutions arises for equations (1) and

w ( n ) + q ( t ) w ( n − 2 ) = r ( t ) ∣ w ( τ ( t ) ) ∣ λ sgn w ( τ ( t ) ) .

Similarly, the same question concerns the proximity between Kneser solutions for equations (3) and

w ( n ) + ∑ j = 0 n − 1 a j ( t ) w ( j ) = r ( t ) ∣ w ( τ ( t ) ) ∣ λ sgn w ( τ ( t ) ) ,

where n is even. Here the meaning of Kneser solution is that the solution x satisfies for large t

x ( t ) > 0 , x ( i ) ( t ) x ( i + 1 ) ( t ) < 0 , i = 0 , 1 , … , n − 1 .

Acknowledgments

The authors thank the referees for their valuable comments and suggestions, which improve the value of the article.

Funding information: The research of I. Astashova has been supported by RSF (Project 20-11-20272), the research of M. Bartušek and Z. Došlá has been supported by the grant GA20-11846S of the Czech Science Foundation. The research of M. Marini has been partially supported by Gnampa, National Institute for Advanced Mathematics (INdAM) of Italian National Research Council.
Conflict of interest: The authors state no conflict of interest.

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Received: 2021-11-16

Revised: 2022-04-14

Accepted: 2022-05-02

Published Online: 2022-06-14

This work is licensed under the Creative Commons Attribution 4.0 International License.

Asymptotic proximity to higher order nonlinear differential equations

Abstract

1 Introduction

Theorem A

Theorem B

Theorem C

2 Asymptotic representation of solutions to equation (1)

Theorem 1

Lemma 1

Proof

Lemma 2

Proof

Proof of Theorem 1

Corollary 1

Proof

Corollary 2

Proof

Theorem 2

Proof

Corollary 3

Proof

3 Asymptotic representation of solutions to equation (3)

Corollary 4

Proof

Remark 1

Remark 2

Remark 3

Remark 4

4 Examples and suggestions

Example 1

Example 2

Example 3

Acknowledgments

References

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