On Boundary Value Problems for Liénard Type Equation

ANITA KIRICHUKA1, FELIX SADYRBAEV2

1Daugavpils University,

13 Vienibas Street, Daugavpils,

LATVIA

2Institute of Mathematics and Computer Science,

University of Latvia,

Rainis boulevard 29, Riga,

LATVIA

Abstract: The generalized Liénard type differential equation is studied together with the two-point linear

boundary conditions of the Sturm-Liouville type. The existence and multiplicity of solutions are considered.

The existence under suitable conditions is shown to follow from the lower and upper functions theory. For

multiplicity, the polar coordinates approach is used. The multiplicity results are based on the comparison

between behavior of solutions near the trivial one, and solutions near the special one, which is preassumed to

be non-oscillatory. The existence of the latter is required. It is shown also, that these conditions are fulfilled

for a relatively broad class of equations. Some examples are constructed, which are supplied by comments and

illustrations.

Key-Words: - Ordinary differential equations, multiple solutions, existence of solutions, Liénard type

equations, phase portrait, oscillatory behaviors, boundary value problems, variational

equations, heteroclinic trajectories, homoclinic trajectories.

Received: January 27, 2023. Revised: September 29, 2023. Accepted: November 13, 2023. Published: December 6, 2023.

1 Introduction

Boundary value problems (BVPs) for ordinary

differential equations appear often in theoretical

studies, [1], [2], [3], [4], and mathematical

modelling of real-world processes. The existence

of solutions is the main question. The existence of

a solution should be confirmed before performing

some numerical experiments. The linear theory

provides answers to the main theoretical questions

as to the existence and uniqueness of a solution.

Nonlinear problems can be more difficult. The

existence of a solution should be proved in many

cases. Moreover, multiple non-similar solutions

can appear in many practically oriented studies.

The answers typically should be obtained for

particular cases, where the general theory does not

provide recommendations. The equations of

Liènard type can be double-nonlinear

(nonlinearities at f(x) and g(x) as in (1)) and there

is space for rich dynamics of solutions. The

classical Liènard equation is well-studied

qualitatively, focusing on periodic solutions and

bifurcations. The generalized Liènard type

equations are general, and the behavior of

solutions may be quite different. Generalized

equations of Liénard type have been studied in the

works, [5], [6], [7].

In the articles, [2], [8], have studied the

boundary value problems of the form 󰆒󰆒

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.47

Anita Kirichuka, Felix Sadyrbaev

E-ISSN: 2224-2856

437

Volume 18, 2023

󰇛󰇜 󰇛󰇜󰇛󰇜 or 󰆒󰇛󰇜

󰆒󰇛󰇜, where n is a positive integer (most

results concern the cases n = 2, n = 3, n = 5). The

exact estimates of the number of solutions were

obtained for autonomous equations of the form

󰆒󰆒=0 and some results (mostly of a

computational nature) were stated for the case of

󰇛󰇜 being piece-wise constant function, [9].

The phase plane method was used extensively.

Equations of the form:

󰆒󰆒󰇛󰇜󰆒󰇛󰇜 (1)

are a classical object for investigation. The Liènard

and Van-der-Pol equations fall into this class. Both

arose from practice. Equations of the form (1) are

rich in oscillatory behaviors. They are known to

have (under suitable conditions) isolated periodic

solutions. The problem of estimating the number

of limit cycles for the case of polynomial functions

f(x) and g(x) has attracted the attention of

prominent researchers. In contrast, equation (3)

can be reduced to a conservative equation. The

comparison of equations (3) and the shorter

equation:

󰆒󰆒󰇛󰇜 (2)

was made in the work, [10]. We focused on the

case 󰇛󰇜, a > 0, b > 0, and

considered two-point boundary conditions for both

equations. We intended to compare the number of

solutions to the respective BVPs. For this, we

made use of the special change of variables

resulting in eliminating the middle term in (1).

This technique was proposed, [11], when studying

isochronous problems. An equation in new

variables has a simpler form and can be (formally)

integrated. This transformation keeps the trivial

solution. This is important because, in various

sources devoted to the study of multiple solutions

of BVP, the following idea was exploited. Imagine

that the oscillatory behavior of solutions can be

measured around the trivial solution. If a

comparison can be made with solutions far away

from the trivial one, some conclusions can be

made about the number of solutions for two-point

boundary value problems. After the reduction of

equation (1) to form (2) using the

above-mentioned variable change, another

comparison can be made, namely, the equation in

question versus the reduced equation. This

approach will be considered in the next sections.

In the article, [10], the equation:

󰆒󰆒󰇛󰇜󰆒󰇛󰇜 (3)

was considered together with the two-point

boundary conditions of the Dirichlet and Neumann

type. The existence of solutions and estimates of

the number of solutions were in focus. The

behavior of solutions, and as a consequence, the

number of solutions heavily depends on the

function f(x). Three types of f(x) were considered,

and for all cases, the comparison was made of the

number of solutions to certain BVP for equation (3)

and Newtonian equation (2). The main conclusion

made in, [10], is that generally, the number of

solutions to the Dirichlet and Neumann problems

for equation (3) is not less than that for equation

(2). In this article, we consider more general the

Sturm-Liouville-type conditions of the form:

󰇛󰇜󰇛󰇜

󰇛󰇜󰇛󰇜 (4)

where all four coefficients are nonnegative but at

least one coefficient in any equation is not zero. Of

course, the Dirichlet and Neumann boundary

conditions 󰇛󰇜󰇛󰇜 󰇛󰇜󰇛󰇜

are included. After the division of the first line in

(4) by 

 one obtains the first condition in

the form 󰇛󰇜󰆒󰇛󰇜, where

󰇡

󰇢󰇣

󰇤 Similarly, the second

condition in (4) can be written in the form

󰇛󰇜󰆒󰇛󰇜,

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.47

Anita Kirichuka, Felix Sadyrbaev

E-ISSN: 2224-2856

438

Volume 18, 2023

where  󰇟󰇠

For instance, the boundary conditions:

󰇛󰇜󰇛󰇜

󰇛󰇜󰆒󰇛󰇜

will be written as:

󰇡

󰇢󰇛󰇜󰇛󰇜󰇛󰇜



󰇛󰇜

󰆒󰇛󰇜

Therefore, our objects of investigation in this

paper are equations (1) and (3) given together with

the boundary conditions of the form:

󰇛󰇜󰇛󰇜

󰇛󰇜󰇛󰇜

(5)

where α[0, π/2], β[π/2, π]. The Dirichlet and

Neumann boundary conditions are included.

In this article, we study equations of the type:

󰆒󰆒󰇛󰇜󰆒󰇛󰇜 and

󰆒󰆒󰇛󰇜󰆒󰇛󰇜

We are interested in the existence of solutions and

the multiplicity. Tools from the general theory, as

well as some specific instruments, are used.

Visualizations in a phase plane are helpful to

understand and explain results.

2 Existence

Suppose all functions in (1) and (3) are continuous.

Sometimes continuous differentiability is needed,

but these cases are commented on consequently.

Since the highest degree of the first order

derivative in (1) and (3) is two, the Bernstein

condition (the quadratic growth with respect to x󰇜,

which ensures boundedness of the first derivative

of a solution, is always fulfilled. The existence of a

solution to the Dirichlet problem:

󰇛󰇜 󰇛󰇜 (6)

follows immediately, if the upper and lower

functions 󰇛󰇜 and 󰇛󰇜 exist such that:

󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜 (7)

and the inequalities:

󰆒󰆒󰇛󰇜󰆒󰇛󰇜 (8)

󰆒󰆒󰇛󰇜󰆒󰇛󰇜 (9)

hold in the interval [a,b] (Theorem 4 and Remark

1 in, [12].

This criterion is effective in many cases. For

instance, let the function g(x) be an odd-degree

polynomial with the principal term x2n+1. Then

any sufficiently large positive constant  serves

as the upper function 󰇛󰆒󰆒󰇛󰇜󰆒󰇛󰇜

󰇛󰇜󰇜and, consequently, −  is the lower

function. On the other hand, it is difficult often to

find  and  such that  in the interval

[a,b]. For instance, the problem 󰆒󰆒 󰇛

󰇜 󰇛󰇜󰇛󰇜 has only he trivial

solution 󰇛󰇜 However, the functions

 satisfying (8) and (9) other than the

trivial solution, do not exist. This was shown in,

[12]. As to multiple solutions for BVPs, the

method of upper and lower functions can be used,

if pairs of upper and lower functions can be

constructed. These are rare cases.

The following result follows from known

existence theorems and uses the specific form of

the equations (1) and (3).

Theorem 1. Suppose 󰇛󰇜for x such that

|x| Then the BVPs (1), (4) and (3), (4)

have solutions.

The proof follows from the fact that the functions

 and  are the lower and the upper

functions for these problems. It is essentially that

the straight line

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.47

Anita Kirichuka, Felix Sadyrbaev

E-ISSN: 2224-2856

439

Volume 18, 2023

󰇛󰇜󰇛󰇜

either is vertical or connects the segments󰇝

󰆒󰇞 and 󰇝󰆒󰇞 while the

straight line 󰇛󰇜󰇛󰇜

either is vertical or connects the segments

󰇝󰆒󰇞 and 󰇝󰆒󰇞 It is

also taken into account, that both equations (1) and

(3) satisfy the Bernstein condition (at most the

quadratic growth with respect to 󰆒in both

equations).

3 Multiplicity

Let us pass to polar coordinates using formulas:

󰇛󰇜󰇛󰇜󰇛󰇜

󰇛󰇜󰇛󰇜󰇛󰇜 (10)

the polar system for the equation (3) is:

󰇱󰆒󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜

󰇛󰇜󰇛󰇜󰇛󰇜

󰆒󰇛󰇜

󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜(11)

The boundary conditions (5) in polar coordinates

are:

󰇛󰇜 󰇛󰇜  . (12)

In the work, [8], multiplicity results were

formulated in terms of the variational equations for

the Dirichlet and Neumann boundary conditions.

We wish to do the same for the boundary

conditions (5), or, equivalently, (11).

First, let us introduce the variational equations

for the nonlinear equations (1) and (3), with

respect to the trivial solution 

The variational equations for (1) and (3),

provided that the trivial solution exists (the

necessary condition for this is 󰇛󰇜) are:

󰆒󰆒󰇛󰇜󰆒󰇛󰇜 (13)

and

󰆒󰆒󰇛󰇜 (14)

respectively. Introduce the polar coordinates for

the variational equations using the formulas:

󰇛󰇜󰇛󰇜 󰇛󰇜󰇛󰇜

The following results are true.

Theorem 2. Let the following conditions for the

equation (1) hold:

1) 󰇛󰇜;

2) 󰇛󰇜󰇛󰇜󰇛󰇜

where 󰇛󰇜 corresponds to the

variational equation (13), i=0,1, …;

3) there exists a solution x(t) of the Cauchy

problem (1), ϕ(0)=α such that ϕ(1)< β.

Then there exist at least i nontrivial solutions

of the BVP (1), (5).

Sketch of the proof. The variational equation

around the trivial solution of equation (1) (it exists

due to the condition 1)) is equation (13). Consider

it in polar coordinates together with the initial

conditions 󰇛󰇜 󰇛󰇜 It follows from

condition 2) that the angular function 󰇛󰇜 attains

the values of the form  at least i times.

Look at the main equation (1) written in the polar

coordinates (10). For small r(t) solutions behave

similarly to the solutions of the variational

equation. Therefore, the polar function 󰇛󰇜 takes

i values of the form  while r(0) is small.

Increase r(0) until the value R corresponds to the

solution x(t) from condition 3). The angular

function 󰇛󰇜 changes continuously, and 󰇛󰇜

is in the interval 󰇛󰇜 for r(0)

small enough, while 󰇛󰇜 is less than  for

large enough values r(0). Therefore 󰇛󰇜 takes

all the intermediate values from  to 

Therefore i nontrivial solutions of the BVP (1), (5)

appear.

Remark. If the additional condition similar to

condition 3) is added with the text replacement

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.47

Anita Kirichuka, Felix Sadyrbaev

E-ISSN: 2224-2856

440

Volume 18, 2023

“ϕ(0)=α+ such that ϕ(1)< β+the

additional i solutions to the BVP are obtained by

considering the initial value 󰇛󰇜 and

repeating the reasoning.

Theorem 3. Let the following conditions for the

equation (3) hold:

1) 󰇛󰇜;

2) 󰇛󰇜󰇛󰇜󰇛󰇜

where 󰇛󰇜 corresponds to the

variational equation (14);

3) there exists a solution x(t) of the Cauchy

problem (3), ϕ(0)=α such that ϕ(1)< β.

Then there exist at least i nontrivial solutions of

the BVP (3), (5).

The proof can be conducted similarly to the

proof of Theorem 2.

In the authors' previous work, the difference

between these two results was discussed

considering the Dirichlet boundary conditions.

4 Corollaries

Let the equation (3) be:

󰆒󰆒󰇛󰇜󰆒󰇛󰇜 (15)

where f(x) is either a positive constant or x.

Theorem 4. Let the following condition for the

equation (15) hold:

󰇛󰇜 󰇛󰇜󰇛󰇜

where 󰇛󰇜 corresponds to the variational

equation 󰆒󰆒

Then there exist at least i nontrivial solutions

of the BVP (15), (5).

Proof. The condition 1) of Theorem 3 is fulfilled.

Condition 3) of Theorem 3 is fulfilled also since

there exists the homoclinic solution with slowly

changing angular function 󰇛󰇜. This function

with 󰇛󰇜 for the time [0,1] does not reach

the value 󰇛󰇜 We assume that α is in

the interval 󰇟󰇜 or β is in the interval 󰇡

󰇤

So the Neumann problem is excluded.

If equation (3) has a phase portrait similar to

Figure 5, where two heteroclinic trajectories form

a bounded region, a result similar to Theorem 4,

can be formulated. Such cases are multiple.

5 Conclusion

Equations of the for (3) may have regions,

surrounded by two heteroclinic or one homoclinic

trajectories. These regions may have complicated

structures, containing the hierarchy of embedded

period annuli, [13]. In simple cases, inside there is a

unique critical point of the type center. Trajectories

near this critical point rotate, and this rotation can

be described in terms of the linearized variational

equation. On the other hand, trajectories passing by

the boundary, slow down and this behavior may be

very different from the behavior near the critical

point. In such cases, boundary value problems may

have multiple solutions. Our examples above are of

this kind. The Dirichlet and Neumann-type

problems have multiple solutions for the

appropriate choice of parameters in the equations.

References:

[1] S. R. Bernfeld. An Introduction to Nonlinear

Boundary Value Problems, Paperback April

19, 2012

[2] W. G. Kelley, A. C. Peterson. The Theory of

Differential Equations: Classical and

Qualitative, (Universitext), 2nd ed. 2010

Edition.

[3] M. A. Krasnosel’skiy, A. I. Perov, A. I.

Povolotskiy. Plane vector fields, Academic

Press, 1966.

[4] R. Reissig, G. Sansone, R. Conti.

Qualitative theory of nonlinear differential

equations, Moscow, “Nauka”, 1974.

(Russian).

[5] M. T. de Bustos, Z. Diab, J. L. G. Guirao, M.

A. López, R. Martínez. Existence of

Periodic Solutions for a Class of the

Generalized Liénard Equations. Symmetry,

2022, 14, 944, https://doi.org/10.3390/

sym14050944.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.47

Anita Kirichuka, Felix Sadyrbaev

E-ISSN: 2224-2856

441

Volume 18, 2023

[6] H. Lucas Koudahoun et al. Exact Classical

and Quantum Mechanics of a Generalized

Singular Equation of Quadratic Liénard

Type. Journal of Mathematics and

Statistics, 2018, vol 14: 183.192, doi:

10.3844/jmssp.2018.183.192.

[7] Jean Akande et al. Theory of Exact

Trigonometric Periodic Solutions to

Quadratic Liénard Type Equations. Journal

of Mathematics and Statistics, 2018, vol. 14:

232.240, doi: 10.3844/jmssp.2018.232.240.

[8] A. Kirichuka, The number of solutions to

the boundary value problem with

linear-quintic and linear-cubic-quintic

nonlinearity, WSEAS Transactions on

Mathematics, vol. 19(64), pp.589-597,

2020,

http://dx.doi.org/10.37394/23206.2020.19.6

[9] E.Ellero and F.Zanolin. Homoclinic and

heteroclinic solutions for a class of

second-order non-autonomous ordinary

differential equations: multiplicity results

for stepwise potentials. Boundary Value

Problems, 2013, vol.167,

https://doi.org/10.1186/1687-2770-2013-16

[10] A. Kirichuka, F. Sadyrbaev. Boundary

Value Problems for Liènard-Type Equations

with Quadratic Dependence on the

‘Velocity’, Abstract and Applied Analysis,

Vol. 2022, Article ID 9228511,

https://doi.org/10.1155/2022/9228511.

[11] M. Sabatini. On the period function of ′′ 

󰇛󰇜′󰇛󰇜. Journal of Differential

Equations, vol.196 (2004), pp.151-168,

https://doi.org/10.1016/S0022-0396(03)000

67-6.

[12] Y. Klokov, N. Vasil’ev. Foundations of the

theory of Boundary Value Problems for

Ordinary Differential Equations, Riga,

Zinatne, 1978 (in Russian).

[13] A. Gritsans, I. Yermachenko. On the

Maximum Number of Period Annuli for

Second Order Conservative Equations,

Mathematical Modelling and Analysis, Vol.

26, Issue 4, 612–630, 2021,

https://doi.org/10.3846/mma.2021.13979.

APPENDIX

We provide here some visual descriptions of

equations of the form (3), which can be studied by

the above-used approach. It is significant that

following the half-ray 󰇛󰇜 (the first of the

boundary conditions) one meat a trajectory

entering a saddle point. Then slowly changing

solutions of equation (3) exist in the vicinity of

such trajectory. For the estimates of the number of

solutions to BVPs the properties of the equations

of variations are also essential.

Consider equation (15) with different

functions 󰇛󰇜, let us represent the phase

portraits.

The phase portrait of equation (15), a = 50, b = 25,

f (x) =µ = 1 is depicted in Figure 1.

Fig. 1:

The phase portrait of x´´+f(x)x´

+(ax− bx

)=0, a

50, b

25, f (x) =µ = 1.

The phase portrait of equation (15), a = 50, b =

25,

f (x) =(x2 – 1) is depicted in Figure 2.

Fig. 2:

The phase portrait of x´´+f(x)x´

+(ax− bx

)=0, a

50, b

25, f (x)= x

– 1.

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.47

Anita Kirichuka, Felix Sadyrbaev

E-ISSN: 2224-2856

442

Volume 18, 2023

The phase portrait of equation (15), a = 50, b =

25, f (x) =1-x3 is depicted in Figure 3.

Fig. 3:

The phase portrait of x´´+f(x)x´

+(ax− bx

)=0, a

50, b

25, f (x)= 1-x

The phase portrait of equation (15), a = 50, b =

25, f (x) =ex is depicted in Figure 4.

Fig. 4:

The phase portrait of x´´+f(x)x´

+(ax− bx

)=0, a =

50, b = 25, f (x)= e

The phase portrait of equation (15), a = 50, b =

25, f (x) =x is depicted in Figure 5.

Fig. 5:

The phase portrait of x´´+f(x)x´

+(ax− bx

)=0, a

50, b

25, f (x)=

The phase portrait of equation (15), a = 50, b =

25,

f (x) =x2 is depicted in Figure 6.

Fig. 6:

The phase portrait of x´´+f(x)x´

+(ax− bx

)=0, a

50, b

25, f (x)=

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

All authors have contributed equally to the

creation of this article.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflict of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.

en_US

WSEAS TRANSACTIONS on SYSTEMS and CONTROL

DOI: 10.37394/23203.2023.18.47

Anita Kirichuka, Felix Sadyrbaev

E-ISSN: 2224-2856

443

Volume 18, 2023