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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 981401, 9 pages
http://dx.doi.org/10.1155/2011/981401
Research Article

Asymptotic Formula for Oscillatory Solutions of Some Singular Nonlinear Differential Equation

Department of Mathematics, Faculty of Science, Palacký University, 17. Listopadu 12, 771 46 Olomouc, Czech Republic

Received 28 October 2010; Revised 31 March 2011; Accepted 2 May 2011

Academic Editor: Yuri V.Β Rogovchenko

Copyright Β© 2011 Irena Rachůnková and Lukáš Rachůnek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Singular differential equation ( 𝑝 ( 𝑑 ) 𝑒 ξ…ž ) ξ…ž = 𝑝 ( 𝑑 ) 𝑓 ( 𝑒 ) is investigated. Here 𝑓 is Lipschitz continuous on ℝ and has at least two zeros 0 and 𝐿 > 0 . The function 𝑝 is continuous on [0,∞) and has a positive continuous derivative on (0,∞) and 𝑝 ( 0 ) = 0 . An asymptotic formula for oscillatory solutions is derived.

1. Introduction

In this paper, we investigate the equation ξ€· 𝑝 ( 𝑑 ) 𝑒 ξ…ž ξ€Έ ξ…ž = 𝑝 ( 𝑑 ) 𝑓 ( 𝑒 ) , 𝑑 ∈ ( 0 , ∞ ) , ( 1 . 1 ) where 𝑓 satisfies 𝑓 ∈ 𝐿 𝑖 𝑝 l o c βˆƒ ( ℝ ) , 𝑓 ( 0 ) = 𝑓 ( 𝐿 ) = 0 , 𝑓 ( π‘₯ ) < 0 , π‘₯ ∈ ( 0 , 𝐿 ) , ( 1 . 2 )  𝐡 ∈ ( βˆ’ ∞ , 0 ) : 𝑓 ( π‘₯ ) > 0 , π‘₯ ∈  𝐹 ξ‚€ 𝐡 , 0 , ( 1 . 3 ) 𝐡  ξ€œ = 𝐹 ( 𝐿 ) , w h e r e 𝐹 ( π‘₯ ) = βˆ’ π‘₯ 0 𝑓 ( 𝑧 ) d 𝑧 , π‘₯ ∈ ℝ , ( 1 . 4 ) and 𝑝 fulfils [ 𝑝 ∈ 𝐢 0 , ∞ ) ∩ 𝐢 1 𝑝 ( 0 , ∞ ) , 𝑝 ( 0 ) = 0 , ( 1 . 5 ) ξ…ž ( 𝑑 ) > 0 , 𝑑 ∈ ( 0 , ∞ ) , l i m 𝑑 β†’ ∞ 𝑝 ξ…ž ( 𝑑 ) 𝑝 ( 𝑑 ) = 0 . ( 1 . 6 ) Equation (1.1) is a generalization of the equation 𝑒 ξ…ž ξ…ž + π‘˜ βˆ’ 1 𝑑 𝑒 ξ…ž = 𝑓 ( 𝑒 ) , 𝑑 ∈ ( 0 , ∞ ) , ( 1 . 7 ) which arises for π‘˜ > 1 and special forms of 𝑓 in many areas, for example: in the study of phase transitions of Van der Waals fluids [13], in population genetics, where it serves as a model for the spatial distribution of the genetic composition of a population [4, 5], in the homogeneous nucleation theory [6], in the relativistic cosmology for the description of particles which can be treated as domains in the universe [7], in the nonlinear field theory, in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [8]. Numerical simulations of solutions of (1.1), where 𝑓 is a polynomial with three zeros have been presented in [911]. Close problems about the existence of positive solutions can be found in [1214].

Due to 𝑝 ( 0 ) = 0 , (1.1) has a singularity at 𝑑 = 0 .

Definition 1.1. A function 𝑒 ∈ 𝐢 1 [ 0 , ∞ ) ∩ 𝐢 2 ( 0 , ∞ ) which satisfies (1.1) for all 𝑑 ∈ ( 0 , ∞ ) is called a solution of (1.1).

Definition 1.2. Let 𝑒 be a solution of (1.1) and let 𝐿 be of (1.2). Denote 𝑒 s u p = s u p { 𝑒 ( 𝑑 ) : 𝑑 ∈ [ 0 , ∞ ) } . If 𝑒 s u p < 𝐿 ( 𝑒 s u p = 𝐿 or 𝑒 s u p > 𝐿 ), then 𝑒 is called a damped solution (a bounding homoclinic solution or an escape solution).

These three types of solutions have been investigated in [1519]. In particular, the existence of damped oscillatory solutions which converge to 0 has been proved in [19].

The main result of this paper is contained in Section 3 in Theorem 3.1, where we provide an asymptotic formula for damped oscillatory solutions of (1.1).

2. Existence of Oscillatory Solutions

Here, we will study solutions of (1.1) satisfying the initial conditions 𝑒 ( 0 ) = 𝐡 , 𝑒 ξ…ž ( 0 ) = 0 , ( 2 . 1 ) with a parameter 𝐡 ≀ 𝐿 . Reason is that we focus our attention on damped solutions of (1.1) and that each solution 𝑒 of (1.1) must fulfil 𝑒 ξ…ž ( 0 ) = 0 (see [19]).

First, we bring two theorems about the existence of damped and oscillatory solutions.

Theorem 2.1 (see [19]). Assume that (1.2)–(1.6) hold. Then for each 𝐡 ∈ [ 𝐡 , 𝐿 ) problem (1.1), (2.1) has a unique solution. This solution is damped.

Theorem 2.2. Assume that (1.2)–(1.6) hold. Further, let there exists π‘˜ 0 ∈ ( 0 , ∞ ) such that 𝑝 ∈ 𝐢 2 ( 0 , ∞ ) , l i m s u p 𝑑 β†’ ∞ | | | | 𝑝 ξ…ž ξ…ž ( 𝑑 ) 𝑝 ξ…ž | | | | ( 𝑑 ) < ∞ , l i m i n f 𝑑 β†’ ∞ 𝑝 ( 𝑑 ) 𝑑 π‘˜ 0 ∈ ] ( 0 , ∞ , ( 2 . 2 ) l i m π‘₯ β†’ 0 + 𝑓 ( π‘₯ ) π‘₯ < 0 , l i m π‘₯ β†’ 0 βˆ’ 𝑓 ( π‘₯ ) π‘₯ < 0 . ( 2 . 3 ) Then for each 𝐡 ∈ [ 𝐡 , 𝐿 ) problem (1.1), (2.1) has a unique solution 𝑒 . If 𝐡 β‰  0 , then the solution 𝑒 is damped and oscillatory with decreasing amplitudes and l i m 𝑑 β†’ ∞ 𝑒 ( 𝑑 ) = 0 . ( 2 . 4 )

Proof. The assertion follows from Theorems 2.3, 2.10 and 3.1 in [19].

Example 2.3. The functions (i) 𝑝 ( 𝑑 ) = 𝑑 π‘˜ , 𝑝 ( 𝑑 ) = 𝑑 π‘˜ l n ( 𝑑 β„“ + 1 ) , π‘˜ , β„“ ∈ ( 0 , ∞ ) , (ii) 𝑝 ( 𝑑 ) = 𝑑 + 𝛼 s i n 𝑑 , 𝛼 ∈ ( βˆ’ 1 , 1 ) , (iii) 𝑝 ( 𝑑 ) = 𝑑 π‘˜ / ( 1 + 𝑑 β„“ ) , π‘˜ , β„“ ∈ ( 0 , ∞ ) , β„“ < π‘˜ satisfy (1.5), (1.6), and (2.2).
The functions (i) 𝑝 ( 𝑑 ) = l n ( 𝑑 + 1 ) , 𝑝 ( 𝑑 ) = a r c t a n 𝑑 , 𝑝 ( 𝑑 ) = 𝑑 π‘˜ / ( 1 + 𝑑 π‘˜ ) , π‘˜ ∈ ( 0 , ∞ ) satisfy (1.5), (1.6), but not (2.2) (the third condition).
The function (i) 𝑝 ( 𝑑 ) = 𝑑 π‘˜ + 𝛼 s i n 𝑑 π‘˜ , 𝛼 ∈ ( βˆ’ 1 , 1 ) , π‘˜ ∈ ( 1 , ∞ ) , satisfy (1.5), (1.6) but not (2.2) (the second and third conditions).

Example 2.4. Let π‘˜ ∈ ( 0 , ∞ ) . (i)The function ξƒ― 𝑓 ( π‘₯ ) = βˆ’ π‘˜ π‘₯ , f o r π‘₯ ≀ 0 , π‘₯ ( π‘₯ βˆ’ 1 ) , f o r π‘₯ > 0 , ( 2 . 5 ) satisfies (1.2) with 𝐿 = 1 , (1.3), (1.4) with 𝐡 = βˆ’ ( 3 π‘˜ ) βˆ’ 1 / 2 and (2.3). (ii)The function ξƒ― 𝑓 ( π‘₯ ) = π‘˜ π‘₯ 2 , f o r π‘₯ ≀ 0 , π‘₯ ( π‘₯ βˆ’ 1 ) , f o r π‘₯ > 0 , ( 2 . 6 ) satisfies (1.2) with 𝐿 = 1 , (1.3), (1.4) with 𝐡 = βˆ’ ( 2 π‘˜ ) βˆ’ 1 / 3 but not (2.3) (the second condition).

In the next section, the generalized Matell's theorem which can be found as Theorem 6.5 in the monograph by Kiguradze will be useful. For our purpose, we provide its following special case.

Consider an interval 𝐽 βŠ‚ ℝ . We write 𝐴 𝐢 ( 𝐽 ) for the set of functions absolutely continuous on 𝐽 and 𝐴 𝐢 l o c ( 𝐽 ) for the set of functions belonging to 𝐴 𝐢 ( 𝐼 ) for each compact interval 𝐼 βŠ‚ 𝐽 . Choose 𝑑 0 > 0 and a function matrix 𝐴 ( 𝑑 ) = ( π‘Ž 𝑖 , 𝑗 ( 𝑑 ) ) 𝑖 , 𝑗 ≀ 2 which is defined on ( 𝑑 0 , ∞ ) . Denote by πœ† ( 𝑑 ) and πœ‡ ( 𝑑 ) eigenvalues of 𝐴 ( 𝑑 ) , 𝑑 ∈ ( 𝑑 0 , ∞ ) . Further, suppose πœ† = l i m 𝑑 β†’ ∞ πœ† ( 𝑑 ) , πœ‡ = l i m 𝑑 β†’ ∞ πœ‡ ( 𝑑 ) ( 2 . 7 ) be different eigenvalues of the matrix 𝐴 = l i m 𝑑 β†’ ∞ 𝐴 ( 𝑑 ) , and let π₯ and 𝐦 be eigenvectors of 𝐴 corresponding to πœ† and πœ‡ , respectively.

Theorem 2.5 (see [20]). Assume that π‘Ž 𝑖 , 𝑗 ∈ 𝐴 𝐢 l o c ξ€· 𝑑 0 ξ€Έ , | | | | ξ€œ , ∞ ∞ 𝑑 0 π‘Ž ξ…ž 𝑖 , 𝑗 | | | | ( 𝑑 ) d 𝑑 < ∞ , 𝑖 , 𝑗 = 1 , 2 , ( 2 . 8 ) and that there exists 𝑐 0 > 0 such that ξ€œ 𝑑 𝑠 R e ( πœ† ( 𝜏 ) βˆ’ πœ‡ ( 𝜏 ) ) d 𝜏 ≀ 𝑐 0 , 𝑑 0 ≀ 𝑠 < 𝑑 , ( 2 . 9 ) or ξ€œ ∞ 𝑑 0 ξ€œ R e ( πœ† ( 𝜏 ) βˆ’ πœ‡ ( 𝜏 ) ) d 𝜏 = ∞ , 𝑑 𝑠 R e ( πœ† ( 𝜏 ) βˆ’ πœ‡ ( 𝜏 ) ) d 𝜏 β‰₯ βˆ’ 𝑐 0 , 𝑑 0 ≀ 𝑠 < 𝑑 . ( 2 . 1 0 ) Then the differential system 𝐱 ξ…ž ( 𝑑 ) = 𝐴 ( 𝑑 ) 𝐱 ( 𝑑 ) ( 2 . 1 1 ) has a fundamental system of solutions π‘₯ ( 𝑑 ) , 𝑦 ( 𝑑 ) such that l i m 𝑑 β†’ ∞ 𝐱 ( 𝑑 ) e βˆ’ ∫ 𝑑 𝑑 0 πœ† ( 𝜏 ) d 𝜏 = π₯ , l i m 𝑑 β†’ ∞ 𝐲 ( 𝑑 ) e βˆ’ ∫ 𝑑 𝑑 0 πœ‡ ( 𝜏 ) d 𝜏 = 𝐦 . ( 2 . 1 2 )

3. Asymptotic Formula

In order to derive an asymptotic formula for a damped oscillatory solution 𝑒 of problem (1.1), (2.1), we need a little stronger assumption than (2.3). In particular, the function 𝑓 ( π‘₯ ) / π‘₯ should have a negative derivative at π‘₯ = 0 .

Theorem 3.1. Assume that (1.2)–(1.6), and (2.2) hold. Assume, moreover, that there exist πœ‚ > 0 and 𝑐 > 0 such that 𝑓 ( π‘₯ ) π‘₯ [ ] ∈ 𝐴 𝐢 βˆ’ πœ‚ , πœ‚ , l i m π‘₯ β†’ 0 𝑓 ( π‘₯ ) π‘₯ = βˆ’ 𝑐 . ( 3 . 1 ) Then for each 𝐡 ∈ [ 𝐡 , 𝐿 ) problem (1.1), (2.1) has a unique solution 𝑒 . If 𝐡 β‰  0 , then the solution 𝑒 is damped and oscillatory with decreasing amplitudes such that l i m s u p 𝑑 β†’ ∞ √ | | | | 𝑝 ( 𝑑 ) 𝑒 ( 𝑑 ) < ∞ . ( 3 . 2 )

Proof. We have the following steps:Step 1 (construction of an auxiliary linear differential system). Choose 𝐡 ∈ [ 𝐡 , 𝐿 ) , 𝐡 β‰  0 . By Theorem 2.2, problem (1.1), (2.1) has a unique oscillatory solution 𝑒 with decreasing amplitudes and satisfying (2.4). Having this solution 𝑒 , define a linear differential equation 𝑣 ξ…ž ξ…ž + 𝑝 ξ…ž ( 𝑑 ) 𝑣 𝑝 ( 𝑑 ) ξ…ž = 𝑓 ( 𝑒 ( 𝑑 ) ) 𝑒 ( 𝑑 ) 𝑣 , ( 3 . 3 ) and the corresponding linear differential system π‘₯ ξ…ž 1 = π‘₯ 2 , π‘₯ ξ…ž 2 = 𝑓 ( 𝑒 ( 𝑑 ) ) π‘₯ 𝑒 ( 𝑑 ) 1 βˆ’ 𝑝 ξ…ž ( 𝑑 ) π‘₯ 𝑝 ( 𝑑 ) 2 . ( 3 . 4 ) Denote ξ€· π‘Ž 𝐴 ( 𝑑 ) = 𝑖 , 𝑗 ξ€Έ ( 𝑑 ) 𝑖 , 𝑗 ≀ 2 = βŽ› ⎜ ⎜ ⎝ 0 1 𝑓 ( 𝑒 ( 𝑑 ) ) 𝑒 βˆ’ 𝑝 ( 𝑑 ) ξ…ž ( 𝑑 ) 𝑝 ⎞ ⎟ ⎟ ⎠  ξƒͺ ( 𝑑 ) , 𝐴 = 0 1 βˆ’ 𝑐 0 . ( 3 . 5 ) By (1.6), (2.4), and (3.1), 𝐴 = l i m 𝑑 β†’ ∞ 𝐴 ( 𝑑 ) . ( 3 . 6 ) Eigenvalues of 𝐴 are numbers √ πœ† = 𝑖 𝑐 and √ πœ‡ = βˆ’ 𝑖 𝑐 , and eigenvectors of 𝐴 are √ π₯ = ( 1 , 𝑖 𝑐 ) and √ 𝐦 = ( 1 , βˆ’ 𝑖 𝑐 ) , respectively. Denote ξ‚΅ 𝑝 𝐷 ( 𝑑 ) = ξ…ž ( 𝑑 ) ξ‚Ά 2 𝑝 ( 𝑑 ) 2 + 𝑓 ( 𝑒 ( 𝑑 ) ) 𝑒 ( 𝑑 ) , 𝑑 ∈ ( 0 , ∞ ) . ( 3 . 7 ) Then eigenvalues of 𝐴 ( 𝑑 ) have the form 𝑝 πœ† ( 𝑑 ) = βˆ’ ξ…ž ( 𝑑 ) + √ 2 𝑝 ( 𝑑 ) 𝑝 𝐷 ( 𝑑 ) , πœ‡ ( 𝑑 ) = βˆ’ ξ…ž ( 𝑑 ) βˆ’ √ 2 𝑝 ( 𝑑 ) 𝐷 ( 𝑑 ) , 𝑑 ∈ ( 0 , ∞ ) . ( 3 . 8 ) We see that l i m 𝑑 β†’ ∞ πœ† ( 𝑑 ) = πœ† , l i m 𝑑 β†’ ∞ πœ‡ ( 𝑑 ) = πœ‡ . ( 3 . 9 ) Step 2 (verification of the assumptions of Theorem 2.5). Due to (1.6), (2.4), and (3.1), we can find 𝑑 0 > 0 such that 𝑒 ξ€· 𝑑 0 ξ€Έ | | 𝑒 | | ξ€· 𝑑 β‰  0 , ( 𝑑 ) ≀ πœ‚ , 𝐷 ( 𝑑 ) < 0 , 𝑑 ∈ 0 ξ€Έ , ∞ . ( 3 . 1 0 ) Therefore, by (3.1), π‘Ž 2 1 ( 𝑑 ) = 𝑓 ( 𝑒 ( 𝑑 ) ) 𝑒 ( 𝑑 ) ∈ 𝐴 𝐢 l o c ξ€· 𝑑 0 ξ€Έ , , ∞ ( 3 . 1 1 ) and so | | | | ξ€œ ∞ 𝑑 0 ξ‚΅ 𝑓 ( 𝑒 ( 𝑑 ) ) ξ‚Ά 𝑒 ( 𝑑 ) ξ…ž | | | | = | | | | d 𝑑 l i m 𝑑 β†’ ∞ 𝑓 ( 𝑒 ( 𝑑 ) ) βˆ’ 𝑓 ξ€· 𝑒 ξ€· 𝑑 𝑒 ( 𝑑 ) 0 ξ€Έ ξ€Έ 𝑒 ξ€· 𝑑 0 ξ€Έ | | | | = | | | | 𝑓 ξ€· 𝑒 ξ€· 𝑑 βˆ’ 𝑐 βˆ’ 0 ξ€Έ ξ€Έ 𝑒 ξ€· 𝑑 0 ξ€Έ | | | | < ∞ . ( 3 . 1 2 ) Further, by (2.2), π‘Ž 2 2 ( 𝑑 ) = βˆ’ 𝑝 ξ…ž ( 𝑑 ) / 𝑝 ( 𝑑 ) ∈ 𝐢 1 ( 𝑑 0 , ∞ ) . Hence, due to (1.6), | | | | ξ€œ ∞ 𝑑 0 ξ‚΅ 𝑝 ξ…ž ( 𝑑 ) ξ‚Ά | | | | = | | | | 𝑝 ( 𝑑 ) d 𝑑 l i m 𝑑 β†’ ∞ 𝑝 ξ…ž ( 𝑑 ) βˆ’ 𝑝 𝑝 ( 𝑑 ) ξ…ž ξ€· 𝑑 0 ξ€Έ 𝑝 ξ€· 𝑑 0 ξ€Έ | | | | = 𝑝 ξ…ž ξ€· 𝑑 0 ξ€Έ 𝑝 ξ€· 𝑑 0 ξ€Έ < ∞ . ( 3 . 1 3 ) Since π‘Ž 1 1 ( 𝑑 ) ≑ 0 and π‘Ž 1 2 ( 𝑑 ) ≑ 1 , we see that (2.8) is satisfied. Using (3.8) we get R e ( πœ† ( 𝑑 ) βˆ’ πœ‡ ( 𝑑 ) ) ≑ 0 . This yields ξ€œ 𝑑 𝑠 R e ( πœ† ( 𝜏 ) βˆ’ πœ‡ ( 𝜏 ) ) d 𝜏 = 0 < 𝑐 0 , 𝑑 0 ≀ 𝑠 < 𝑑 , ( 3 . 1 4 ) for any positive constant 𝑐 0 . Consequently (2.9) is valid.Step 3 (application of Theorem 2.5). By Theorem 2.5 there exists a fundamental system 𝐱 ( 𝑑 ) = ( π‘₯ 1 ( 𝑑 ) , π‘₯ 2 ( 𝑑 ) ) , 𝐲 ( 𝑑 ) = ( 𝑦 1 ( 𝑑 ) , 𝑦 2 ( 𝑑 ) ) of solutions of (3.4) such that (2.12) is valid. Hence l i m 𝑑 β†’ ∞ π‘₯ 1 ( 𝑑 ) e βˆ’ ∫ 𝑑 𝑑 0 πœ† ( 𝜏 ) d 𝜏 = 1 , l i m 𝑑 β†’ ∞ 𝑦 1 ( 𝑑 ) e βˆ’ ∫ 𝑑 𝑑 0 πœ‡ ( 𝜏 ) d 𝜏 = 1 . ( 3 . 1 5 ) Using (3.8) and (3.10), we get ξ‚΅ βˆ’ ξ€œ e x p 𝑑 𝑑 0 ξ‚Ά ξ‚΅ ξ€œ πœ† ( 𝜏 ) d 𝜏 = e x p 𝑑 𝑑 0 ξ‚΅ 𝑝 ξ…ž ( 𝜏 ) βˆ’ √ 2 𝑝 ( 𝜏 ) ξ‚Ά ξ‚Ά  1 𝐷 ( 𝜏 ) d 𝜏 = e x p 2 l n 𝑝 ( 𝑑 ) 𝑝 ξ€· 𝑑 0 ξ€Έ ξƒͺ ξ‚΅ ξ€œ e x p βˆ’ 𝑖 𝑑 𝑑 0  | | | | ξ‚Ά , 𝐷 ( 𝜏 ) d 𝜏 ( 3 . 1 6 ) and, hence, | | | e βˆ’ ∫ 𝑑 𝑑 0 πœ† ( 𝜏 ) d 𝜏 | | | = ξƒŽ 𝑝 ( 𝑑 ) 𝑝 ξ€· 𝑑 0 ξ€Έ ξ€· 𝑑 , 𝑑 ∈ 0 ξ€Έ . , ∞ ( 3 . 1 7 ) Similarly | | | e βˆ’ ∫ 𝑑 𝑑 0 πœ‡ ( 𝜏 ) d 𝜏 | | | = ξƒŽ 𝑝 ( 𝑑 ) 𝑝 ξ€· 𝑑 0 ξ€Έ ξ€· 𝑑 , 𝑑 ∈ 0 ξ€Έ . , ∞ ( 3 . 1 8 ) Therefore, (3.15) implies 1 = l i m 𝑑 β†’ ∞ | | | π‘₯ 1 ( 𝑑 ) e βˆ’ ∫ 𝑑 𝑑 0 πœ† ( 𝜏 ) d 𝜏 | | | = l i m 𝑑 β†’ ∞ | | π‘₯ 1 | | ξƒŽ ( 𝑑 ) 𝑝 ( 𝑑 ) 𝑝 ξ€· 𝑑 0 ξ€Έ , 1 = l i m 𝑑 β†’ ∞ | | | 𝑦 1 ( 𝑑 ) e βˆ’ ∫ 𝑑 𝑑 0 πœ‡ ( 𝜏 ) d 𝜏 | | | = l i m 𝑑 β†’ ∞ | | 𝑦 1 | | ξƒŽ ( 𝑑 ) 𝑝 ( 𝑑 ) 𝑝 ξ€· 𝑑 0 ξ€Έ . ( 3 . 1 9 ) Step 4 (asymptotic formula). In Step 1, we have assumed that 𝑒 is a solution of (1.1), which means that 𝑒 ξ…ž ξ…ž 𝑝 ( 𝑑 ) + ξ…ž ( 𝑑 ) 𝑒 𝑝 ( 𝑑 ) ξ…ž ( 𝑑 ) = 𝑓 ( 𝑒 ( 𝑑 ) ) , f o r 𝑑 ∈ ( 0 , ∞ ) . ( 3 . 2 0 ) Consequently 𝑒 ξ…ž ξ…ž 𝑝 ( 𝑑 ) + ξ…ž ( 𝑑 ) 𝑒 𝑝 ( 𝑑 ) ξ…ž 𝑓 ( 𝑑 ) = ( 𝑒 ( 𝑑 ) ) 𝑒 ( 𝑑 ) 𝑒 ( 𝑑 ) , f o r 𝑑 ∈ ( 0 , ∞ ) , ( 3 . 2 1 ) and, hence, 𝑒 is also a solution of (3.3). This yields that there are 𝑐 1 , 𝑐 2 ∈ ℝ such that 𝑒 ( 𝑑 ) = 𝑐 1 π‘₯ 1 ( 𝑑 ) + 𝑐 2 𝑦 1 ( 𝑑 ) , 𝑑 ∈ ( 0 , ∞ ) . Therefore, l i m s u p 𝑑 β†’ ∞ √ | | | | ≀ ξ€· | | 𝑐 𝑝 ( 𝑑 ) 𝑒 ( 𝑑 ) 1 | | + | | 𝑐 2 | | ξ€Έ  𝑝 ξ€· 𝑑 0 ξ€Έ < ∞ . ( 3 . 2 2 )

Remark 3.2. Due to (2.2) and (3.2), we have for a solution 𝑒 of Theorem 3.1 ξ€· 𝑑 𝑒 ( 𝑑 ) = 𝑂 βˆ’ π‘˜ 0 / 2 ξ€Έ , f o r 𝑑 ⟢ ∞ . ( 3 . 2 3 )

Example 3.3. Let π‘˜ ∈ ( 1 , ∞ ) . (i)The functions 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ βˆ’ 1 ) and 𝑓 ( π‘₯ ) = π‘₯ ( π‘₯ βˆ’ 1 ) ( π‘₯ + 2 ) satisfy all assumptions of Theorem 3.1. (ii) The functions 𝑓 ( π‘₯ ) = π‘₯ 2 π‘˜ βˆ’ 1 ( π‘₯ βˆ’ 1 ) and 𝑓 ( π‘₯ ) = π‘₯ 2 π‘˜ βˆ’ 1 ( π‘₯ βˆ’ 1 ) ( π‘₯ + 2 ) satisfy (1.2)–(1.4) but not (3.1) (the second condition).

Example 3.4. Consider the initial problem ξ€· 𝑑 2 𝑒 ξ…ž ξ€Έ ξ…ž = 𝑑 2 𝑒 ( 𝑒 βˆ’ 5 ) ( 𝑒 + 1 0 ) , 𝑒 ( 0 ) = βˆ’ 3 , 𝑒 ξ…ž ( 0 ) = 0 . ( 3 . 2 4 ) Here 𝐿 0 = βˆ’ 1 0 , 𝐿 = 5 and we can check that 𝐡 < βˆ’ 3 . Further, all assumptions of Theorems 2.2 and 3.1 are fulfilled. Therefore, by Theorem 2.2, there exists a unique solution 𝑒 of problem (3.24) which is damped and oscillatory and converges to 0. By Theorem 3.1, we have l i m s u p 𝑑 β†’ ∞ 𝑑 | | | | ξ‚€ 1 𝑒 ( 𝑑 ) < ∞ , t h a t i s , 𝑒 ( 𝑑 ) = 𝑂 𝑑  , f o r 𝑑 ⟢ ∞ . ( 3 . 2 5 ) The behaviour of the solution 𝑒 ( 𝑑 ) and of the function 𝑑 𝑒 ( 𝑑 ) is presented on Figure 1.

981401.fig.001
Figure 1

Remark 3.5. Our further research of this topic will be focused on a deeper investigation of all types of solutions defined in Definition 1.2. For example, we have proved in [15, 19] that damped solutions of (1.1) can be either oscillatory or they have a finite number of zeros or no zero and converge to 0. A more precise characterization of behaviour of nonoscillatory solutions are including their asymptotic formulas in as open problem. The same can be said about homoclinic solutions. In [17] we have found some conditions which guarantee their existence, and we have shown that if 𝑒 is a homoclinic solution of (1.1), then l i m 𝑑 β†’ ∞ 𝑒 ( 𝑑 ) = 𝐿 . In order to discover other existence conditions for homoclinic solutions, we would like to estimate their convergence by proper asymptotic formulas.

Acknowledgments

The authors thank the referees for comments and suggestions. This paper was supported by the Council of Czech Government MSM 6198959214.

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