On the uniqueness of limit cycles for generalized Liénard systems

Hui Zhou; Yueding Yuan

doi:10.1515/math-2022-0558

Article Open Access

On the uniqueness of limit cycles for generalized Liénard systems

Hui Zhou and Yueding Yuan

Published/Copyright: February 14, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, the general Liénard system

d x d t = ϕ ( y ) − F ( x ) , d y d t = − g ( x )

is studied. By using the Filippov transformation, combined with the careful estimation of divergence along the closed orbit, we prove the sufficient conditions for the uniqueness of limit cycles in this system. Our results extend almost all the related existing studies on the Liénard system.

Keywords: generalized Liénard system; limit cycles; uniqueness

MSC 2010: 34A25; 34C07; 34C05; 34C25

1 Introduction

There is extensive literature on the existence and number of limit cycles for planar systems, and most of the research results are related to self-sustaining problems in mathematical models and Hilbert’s 16th problem [1–21]. But it is very difficult to obtain some appropriate conditions to ensure the uniqueness of limit cycles for planar systems [9,15,19–33]. As far as we know, there are some planar systems for which the uniqueness of limit cycles has been considered in recent decades, such as the following Liénard system:

(1.1) d x d t = y − F ( x ) , d y d t = − g ( x ) ,

as well as the more general system

(1.2) d x d t = ϕ ( y ) − F ( x ) , d y d t = − g ( x ) ,

where F ( x ) = ∫ 0 x f ( s ) d s [9,15,22,28–33]. Throughout this article, we assume that f ( x ) and g ( x ) are continuous on ( x 02 , x 01 ) and ϕ ( y ) is continuously differentiable on ( − ∞ , + ∞ ) , where x 02 < 0 < x 01 . Furthermore, we always assume that

(A1) x g ( x ) > 0 for all x ∈ ( − ∞ , 0 ) ∪ ( 0 , + ∞ ) ;

(A2) ϕ ( 0 ) = 0 and ϕ ′ ( y ) > 0 for all y ∈ ( − ∞ , + ∞ ) , y ϕ ( y ) ≠ 0 as y ≠ 0 , and ϕ ′ ( y ) is non-increasing as ∣ y ∣ increases.

Let G ( x ) = ∫ 0 x g ( s ) d s , z 0 i = G ( x 0 i ) , i = 1 , 2 , and z 0 = min { z 01 , z 02 } . We denote by x i ( z ) the inverse function of

z = G ( x ) , ∀ ( − 1 ) i + 1 x ≥ 0 , i = 1 , 2 .

By using the Filippov transformation x = x i ( z ) , i = 1 , 2 , we can reduce the system (1.2) with x ≥ 0 and with x ≤ 0 to the following systems

(1.3) d z d y = F 1 ( z ) − ϕ ( y ) , ∀ z ∈ [ 0 , z 01 ]

and

(1.4) d z d y = F 2 ( z ) − ϕ ( y ) , ∀ z ∈ [ 0 , z 02 ] ,

respectively, where F 1 ( z ) = F ( x 1 ( z ) ) and F 2 ( z ) = F ( x 2 ( z ) ) .

Let L be a closed orbit of the system (1.1) (or the system (1.2)). Then, the arcs L ∩ { ( x , y ) ∣ x ≥ 0 } and L ∩ { ( x , y ) ∣ x ≤ 0 } will be transformed to the integral curves L 1 of (1.3) and the integral curves L 2 of (1.4), respectively. The characteristic exponent γ L of the closed orbit L can be expressed as follows:

(1.5) γ L = − ∫ L f ( x ( t ) ) d t = − ∫ L 1 F 1 ′ ( z ) d y + ∫ L 2 F 2 ′ ( z ) d y ,

where d t > 0 along the integration path and d y > 0 along the curves L 1 and L 2 . It is easy to see from Theorem 4.2.2 of [21] that if γ L < 0 , then L is stable. Therefore, since both adjacent limit cycles cannot be stable, we obtain that if γ L < 0 , then the system (1.1) (or the system (1.2)) has at most one limit cycle. This is one of the most efficient methods for obtaining the uniqueness of the limit cycle of the system (1.1) (or the system (1.2)). By using the aforementioned method, Zeng [28] analyzed the system (1.1) and obtained two sufficient conditions to guarantee the uniqueness of the limit cycles of the system (1.1). A few years later, Zhang [30], Gao and Zhang [33] studied the more general system (1.2). By using the aforementioned method, they obtained some sufficient conditions to guarantee the uniqueness of the limit cycles of the system (1.2) and extend the related results in [28]. But the results need more restrictions on the function F ( x ) , such as F ( x ) needs to satisfy that F 1 ′ ( z ) is nondecreasing as z > a and ( F 1 ( z ) − F 1 ( a ) ) ( z − a ) > 0 as 0 < z − a ≪ 1 .

In this article, we further consider the uniqueness of the limit cycles of the system (1.2). We obtain a new sufficient condition to ensure the uniqueness of limit cycles for the system (1.2), which is different from those sufficient criteria that appeared in [30,33]. For the convenience of the statement, we make the following assumptions on the functions F 1 ( z ) and F 2 ( z ) :

(H1) there exists a real number a ∈ [ 0 , z 0 ] such that F 1 ( z ) ≤ 0 ≤ F 2 ( z ) for all z ∈ ( 0 , a ) , F 1 ( z ) > 0 for all z ∈ ( a , z 01 ) , and F 1 ( z ) ≢ F 2 ( z ) for all z ∈ o + ( 0 ) , where o + ( 0 ) = { z ∣ 0 < z ≪ 1 } ;

(H2) F 2 ′ ( z ) ≤ 0 as F 2 ( z ) < 0 and 0 < z < z 02 ;

(H3) the function F 1 ( z ) F 1 ′ ( z ) is nondecreasing for all z > a (or the function F 2 ( z ) F 2 ′ ( z ) is nondecreasing for all z > 0 and F 1 ( z 01 − 0 ) ≤ F 2 ( z 02 − 0 ) );

(H4) F 1 ′ ( z ) ≥ F 2 ′ ( u ) as F 1 ( z ) = F 2 ( u ) and u ≥ z > a .

Now, we can give a new uniqueness theorem as follows.

Theorem 1.1

Assume that (A1), (A2), and (H1)–(H4) hold. Then, the system (1.2) has at most one limit cycle on ( x 02 , x 01 ) , and it is stable if it exists.

2 Main results

Consider the following equation:

(2.1) d z d y = F i ( z ) − ϕ ( y ) , 0 ≤ z < z 0 .

Here, the function F i ( z ) is continuous on [ 0 , z 0 ) and continuously differentiable on ( 0 , z 0 ) , F i ( 0 ) = 0 , the function ϕ ( y ) is continuously differentiable on ( − ∞ , + ∞ ) , ϕ ( 0 ) = 0 , and ϕ ′ ( y ) > 0 for all y ∈ ( − ∞ , + ∞ ) .

Let L P be the integral curve of equation (2.1) passing through the point P ( z P , y P ) on the isocline y = ϕ − 1 ( F i ( z ) ) , where y P = ϕ − 1 ( F i ( z P ) ) and ϕ − 1 denotes the inverse of ϕ . Let y = ϕ − 1 ( φ P ( z ) ) and y = ϕ − 1 ( φ ˜ P ( z ) ) be the arcs of L P below and above the isocline y = ϕ − 1 ( F i ( z ) ) for all z ∈ [ 0 , z P ] , respectively, that is, y = ϕ − 1 ( φ P ( z ) ) is the equation of the arc

L P ∩ { ( z , y ) ∣ z ≥ 0 , y ≤ ϕ − 1 ( F i ( z ) ) }

and y = ϕ − 1 ( φ ˜ P ( z ) ) is the equation of the arc

L P ∩ { ( z , y ) ∣ z ≥ 0 , y ≥ ϕ − 1 ( F i ( z ) ) } .

Clearly, the functions ϕ − 1 ( φ P ( z ) ) and ϕ − 1 ( φ ˜ P ( z ) ) are solutions of the equation

d y d z = 1 F i ( z ) − ϕ ( y )

and satisfy φ P ( z P ) = φ ˜ P ( z P ) = F i ( z P ) , and φ P ( z ) < F i ( z ) < φ ˜ P ( z ) and φ P ′ ( z ) > 0 > φ ˜ P ′ ( z ) for all z ∈ ( 0 , z P ) .

For the sake of convenience, we denote

(2.2) V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) = F i ′ ( z ) F i ( z ) − φ P ( z ) + F i ′ ( z ) φ ˜ P ( z ) − F i ( z ) .

Then, we have

(2.3) ∫ L P F i ′ ( z ) d y = ∫ 0 z P V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z .

It is always assumed in the following six lemmas that φ P ( 0 ) < 0 < φ ˜ P ( 0 ) . It is easy to see that integral (2.3) is convergent. To estimate the sign of the integral (2.3), we first introduce the following lemma.

Lemma 2.1

Let 0 ≤ z 1 < z 2 . If

(2.4) F i ( z 2 ) − F i ( z ) ≥ 0 ( resp. , ≤ 0 )

for all z ∈ ( z 1 , z 2 ) , then

(2.5) ∫ z 1 z 2 V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z ≥ 0 ( resp. , ≤ 0 )

for all z P ≥ z 2 .

Proof

We first consider the case of z P > z 2 . Then, by (2.4), we have

∫ z 1 z 2 F i ′ ( z ) F i ( z ) − φ P ( z ) d z = ∫ z 1 z 2 F i ′ ( z ) − φ P ′ ( z ) F i ( z ) − φ P ( z ) d z + ∫ z 1 z 2 φ P ′ ( z ) F i ( z ) − φ P ( z ) d z = ln F i ( z 2 ) − φ P ( z 2 ) F i ( z 1 ) − φ P ( z 1 ) + ∫ z 1 z 2 φ P ′ ( z ) F i ( z 2 ) − φ P ( z ) d z ≥ ln F i ( z 2 ) − φ P ( z 1 ) F i ( z 1 ) − φ P ( z 1 ) ≥ 0 .

Similarly, we can obtain that

∫ z 1 z 2 F i ′ ( z ) φ ˜ P ( z ) − F i ( z ) d z ≥ 0 .

Thus, (2.5) holds for all z P > z 2 . If z P = z 2 , the proof can be given by taking z 3 → z P − 0 and using L’Hôpital’s Principle (see the proof of (21) and (21’) in [20]). A similar argument gives the case inside the parentheses. The proof is completed.□

Remark 2.1

If F i ( z ) ≢ F i ( z 2 ) on ( z 1 , z 2 ) , then we can obtain the strict inequality in (2.5).

For convenience, we first introduce some notations. Assume that 0 ≤ z 1 < z 2 ≤ z P ≤ z 0 , F i ( z 1 ) < F i ( z 2 ) , and F i ( z ) ≥ F i ( z 1 ) for all z ∈ ( z 1 , z 2 ) . Let

T F i = { z ∈ ( z 1 , z 2 ) ∣ ∃ ξ ∈ ( z , z 2 ) s.t. F i ( z ) > F i ( ξ ) } , N F i = { z ∈ ( 0 , z 0 ) ∣ F i ′ ( z ) = 0 } , R F i = ( z 1 , z 2 ) \ ( T F i ¯ ∪ N F i ) .

Then, the set T F i is the set of right-descending points of the function F i ( z ) on ( z 1 , z 2 ) . Clearly, T F i and R F i are both open subsets of ( z 1 , z 2 ) . For any Lebesgue measurable set E of real numbers, we denote by m ( E ) the measure of E . Now, we can introduce the following two lemmas.

Lemma 2.2

Assume that 0 ≤ z 1 < z 2 and F i ( z 1 ) < F i ( z 2 ) . If

(2.6) F i ( z ) − F i ( z 1 ) ≥ 0

for all z ∈ ( z 1 , z 2 ) , then the function F i ( z ) is strictly increasing on R F i , F i ′ ( z ) > 0 for all z ∈ R F i ,

(2.7) F i ( R F i ) ⊂ ( F i ( z 1 ) , F i ( z 2 ) ) , m ( F i ( R F i ) ) = F i ( z 2 ) − F i ( z 1 ) ,

and

(2.8) ∫ z 1 z 2 V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z ≤ ∫ R F i V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z

for all z P ≥ z 2 .

Proof

By the definition of the set R F i and F i ( z ) ≥ F i ( z 1 ) for all z ∈ ( z 1 , z 2 ) , we have that F i ( R F i ) ⊂ ( F i ( z 1 ) , F i ( z 2 ) ) and F i ′ ( z ) > 0 for all z ∈ R F i . It follows from Riesz’ lemma that T F i is an open subset of ( z 1 , z 2 ) . Let ( c j , d j ) be a component interval of the set T F i . Then, we claim that

(2.9) F i ( c j ) = F i ( d j ) and F i ( z ) ≥ F i ( d j ) for all z ∈ [ c j , d j ] .

Now, we assume, for the sake of contradiction, that there exists a point ξ 0 ∈ [ c j , d j ) such that F i ( ξ 0 ) < F i ( d j ) . We first consider the case of ξ 0 ∈ ( c j , d j ) . It follows from d j ∉ T F i that F i ( d j ) ≤ F i ( z ) for all z ∈ ( d j , z 2 ) . Thus,

F i ( ξ 0 ) < F i ( d j ) ≤ F i ( z )

for all z ∈ ( d j , z 2 ) . On the other hand, ξ 0 ∈ T F i implies that there exists a point ξ 1 ∈ ( ξ 0 , z 2 ) such that F i ( ξ 0 ) > F i ( ξ 1 ) . Therefore, ξ 1 ∈ ( ξ 0 , d j ) . Define

(2.10) ξ 2 = { z ∣ ξ 1 ≤ z ≤ d j and F i ( z ) ≤ F i ( ξ ) for all ξ ∈ [ ξ 1 , d j ] } .

It is obvious that ξ 2 ≠ d j . Thus, ξ 2 ∈ T F i . Furthermore, there exists a point ξ 3 ∈ ( ξ 2 , d j ) such that F i ( ξ 2 ) > F i ( ξ 3 ) , contradicting the definition of ξ 2 . Therefore, we have

(2.11) F i ( ξ 0 ) ≥ F i ( d j ) , ∀ ξ 0 ∈ ( c j , d j ) .

Next, we consider the case of ξ 0 = c j . By (2.11) and the continuity of F i ( z ) , we obtain

lim ξ 0 → c j F i ( ξ 0 ) ≥ lim ξ 0 → c j F i ( d j ) ,

that is, F i ( c j ) ≥ F i ( d j ) . If c j = z 1 , then F i ( c j ) = F i ( z 1 ) ≤ F i ( d j ) by the hypothesis. If c j > z 1 , then c j ∉ T F i . Therefore, F i ( c j ) ≤ F i ( d j ) . Thus, (2.9) holds. Let ( a j , b j ) and ( a k , b k ) be two-component intervals of T F i . We claim that if b j ≤ a k , then F i ( b j ) ≤ F i ( a k ) . In fact, F i ( b j ) > F i ( a k ) implies that b j ∈ T F i , contradicting the definition of ( a j , b j ) . Thus, F i ( z ) is strictly increasing on R F i . By (2.9), we obtain that ∫ c j d j F i ′ ( z ) d z = 0 . Therefore, ∫ T F i F i ′ ( z ) d z = 0 . Thus,

F i ( z 2 ) − F i ( z 1 ) = ∫ z 1 z 2 F i ′ ( z ) d z = ∫ R F i ⋃ T F i F i ′ ( z ) d z = ∫ R F i F i ′ ( z ) d z = m ( F i ( R F i ) ) .

Furthermore, (2.7) holds. It follows from (2.9) and Lemma 2.1 that

∫ c i d i V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z ≤ 0 .

Hence,

∫ T F i V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z ≤ 0 ,

from which (2.8) follows as V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) = 0 on N F i . The proof is completed.□

Lemma 2.3

Let O be an open subset of ( 0 , z 1 ) . Assume that m ( O ) = z 1 and the function v ( z ) ( v ˜ ( z ) ) is continuously differentiable and monotone increasing (monotone decreasing) on O. Suppose that ϕ ( v ( z ) ) < F i ( z ) ( ϕ ( v ˜ ( z ) ) > F i ( z ) ) and

d v ( z ) d z ≥ 1 F i ( z ) − ϕ ( v ( z ) ) resp. , d v ˜ ( z ) d z ≤ 1 F i ( z ) − ϕ ( v ˜ ( z ) )

for all z ∈ O . If there exists z P ≥ z 1 such that

v ( z 1 − 0 ) ≤ ϕ − 1 ( φ P ( z 1 ) ) ( resp. , v ˜ ( z 1 − 0 ) ≥ ϕ − 1 ( φ ˜ P ( z 1 ) ) ) ,

then

v ( z ) ≤ ϕ − 1 ( φ P ( z ) ) ( resp. , v ˜ ( z ) ≥ ϕ − 1 ( φ ˜ P ( z ) ) )

for all z ∈ O .

Proof

Define the function v ∗ : [ 0 , z 1 ] → ( − ∞ , + ∞ ) by

v ∗ ( z ) = v ( z 1 − 0 ) − ∫ z z 1 v ′ ( u ) d u .

Then, this function v ∗ ( z ) is absolutely continuous on [ 0 , z 1 ] and v ∗ ( z ) ≥ v ( z ) for all z ∈ O . Let N 1 = { ( z , y ) ∣ y < ϕ − 1 ( F i ( z ) ) } . Note that 1 / ( F i ( z ) − ϕ ( y ) ) is monotone increasing with respect to y on N 1 , we have

(2.12) v ∗ ( z ) ≤ v ( z 1 − 0 ) − ∫ z z 1 1 F i ( z ) − ϕ ( v ∗ ( z ) ) d z .

In view of the integral inequality theorem, we obtain

v ∗ ( z ) ≤ ϕ − 1 ( φ P ( z ) ) , ∀ z ∈ [ 0 , z 1 ] .

A similar argument gives the case inside the parentheses. The proof is completed.□

Before stating Lemma 2.4, we provide the following definition.

Definition 2.1

We call a point z = Δ a R ( F 1 , F 2 ) point if F 1 ( Δ ) = F 2 ( Δ ) and F 1 ( z ) > F 2 ( z ) for all z ∈ o + ( Δ ) , where o + ( Δ ) = { z ∣ 0 < z − Δ ≪ 1 } .

Now, we introduce Lemma 2.4.

Lemma 2.4

Assume that F 1 ( z ) and F 2 ( z ) satisfy Assumptions (H1)–(H4). If there exists a R ( F 1 , F 2 ) point Δ and Δ > a , then F 1 ( z ) > F 2 ( z ) for all z ∈ ( Δ , z 0 ) .

Proof

We assume for the sake of contradiction that there exists a point Δ 1 ∈ ( Δ , z 0 ) such that

(2.13) F 1 ( Δ 1 ) = F 2 ( Δ 1 ) , F 1 ( z ) > F 2 ( z ) , ∀ z ∈ ( Δ , Δ 1 ) .

Let F 2 ( η ) = min { F 2 ( z ) ∣ Δ ≤ z ≤ Δ 1 } , Δ ≤ η ≤ Δ 1 . By (2.13) and Assumptions (H1)–(H2), we have that F 2 ( η ) ≥ 0 . Let

ξ = max { z ∣ a ≤ z ≤ Δ , F 1 ( z ) = F 2 ( η ) } .

On the other hand, it follows from Assumption (H1) that F 1 ′ ( a + 0 ) > 0 . Therefore,

(2.14) F 1 ( a + 0 ) F 1 ′ ( a + 0 ) > 0 .

By (2.14) and Assumption (H3), we obtain that F 1 ( z ) F 1 ′ ( z ) > 0 for all z > a . Hence, F 1 ( z ) > 0 and F 1 ′ ( z ) > 0 for all z > a . Thus, the inverse function z = F 1 − 1 ( y ) of the function y = F 1 ( z ) on ( ξ , Δ 1 ) is continuously differentiable on ( F 1 ( ξ ) , F 1 ( Δ 1 ) ) . For F 2 ( z ) , we choose the open subset R F 2 of ( η , Δ 1 ) in Lemma 2.2. By Lemma 2.2, we have

(2.15) F 2 ( R F 2 ) ⊂ ( F 1 ( ξ ) , F 1 ( Δ 1 ) ) , m ( F 2 ( R F 2 ) ) = F 1 ( Δ 1 ) − F 1 ( ξ ) .

We denote by z = F 2 − 1 ( y ) the inverse function of y = F 2 ( z ) , z ∈ R F 2 . Then, z = F 2 − 1 ( y ) is continuously differentiable and monotone increasing on F 2 ( R F 2 ) . Therefore, Assumption (H4) implies that

d d y ( F 2 − 1 ( y ) ) ≥ d d y ( F 1 − 1 ( y ) ) , ∀ y ∈ F 2 ( R F 2 ) .

Since m ( F 2 ( R F 2 ) ) = F 1 ( Δ 1 ) − F 1 ( ξ ) , we have that

F 2 − 1 ( F 1 ( Δ 1 ) − 0 ) − F 2 − 1 ( y ) ≥ F 1 − 1 ( F 1 ( Δ 1 ) − 0 ) − F 1 − 1 ( y ) , ∀ y ∈ F 2 ( R F 2 ) ,

that is, F 2 − 1 ( y ) ≤ F 1 − 1 ( y ) for all y ∈ F 2 ( R F 2 ) , contradicting (2.13). The proof is completed.□

Similar to Lemma 2.4, we have the following lemma.

Lemma 2.5

Assume that F 1 ( z ) and F 2 ( z ) satisfy Assumptions (H1)–(H4). For any δ ≥ 0 , let F 1 δ = F 1 ( z − δ ) . If there exists a R ( F 1 δ , F 2 ) point Δ and Δ ≥ a + δ , then F 1 δ ( z ) > F 2 ( z ) for all z ∈ ( Δ , z 0 ) .

For equation (2.1), we obtain the following lemma.

Lemma 2.6

Assume that a ≥ 0 and F i ( a ) = 0 . If F i ( z ) > 0 for all z > a and F i ( z ) F i ′ ( z ) is nondecreasing on ( a , + ∞ ) , then

(2.16) ∫ a z Q V ( F i ( z ) , φ Q ( z ) , φ ˜ Q ( z ) ) d z ≤ ∫ a z P V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z

for all z Q ∈ ( a , z P ) , where the definitions of φ Q ( z ) and φ ˜ Q ( z ) are similar to that of φ P ( z ) and φ ˜ P ( z ) .

Proof

It is clear that F i ′ ( z ) > 0 for all z > a . Let k = F i ( z P ) / F i ( z Q ) . Then, k > 1 . First, we prove the following inequalities:

(2.17) v ( u ) = ϕ − 1 ( k − 1 φ P ( F i − 1 ( k F i ( u ) ) ) ) ≥ ϕ − 1 ( φ Q ( u ) ) , v ˜ ( u ) = ϕ − 1 ( k − 1 φ ˜ P ( F i − 1 ( k F i ( u ) ) ) ) ≤ ϕ − 1 ( φ ˜ Q ( u ) ) , a ≤ u ≤ z Q .

In fact, it is easy to see that v ( z Q ) = ϕ − 1 ( φ Q ( z Q ) ) and v ˜ ( z Q ) = ϕ − 1 ( φ ˜ Q ( z Q ) ) . Let z = F i − 1 ( k F i ( u ) ) . Then, F i ( z ) = k F i ( u ) , and the definition of the functions v ( u ) and φ P ( z ) implies that

k ϕ ( v ) = φ P ( F i − 1 ( k F i ( u ) ) ) = φ P ( z ) = ϕ ( y ) .

Therefore, we have

F i ′ ( z ) d z = k F i ′ ( u ) d u , k ϕ ′ ( v ) d v = ϕ ′ ( y ) d y .

Thus, by (2.1), we obtain

d v d u = F i ′ ( u ) ϕ ′ ( v ) ϕ ′ ( y ) F i ′ ( z ) d y d z = F i ′ ( u ) ϕ ′ ( v ) ϕ ′ ( y ) F i ′ ( z ) 1 F i ( z ) − ϕ ( y ) = ϕ ′ ( y ) ϕ ′ ( v ) F i ′ ( u ) F i ′ ( z ) 1 k ( F i ( u ) − ϕ ( v ) ) = ϕ ′ ( y ) ϕ ′ ( v ) F i ′ ( u ) F i ′ ( z ) F i ( u ) F i ( z ) 1 F i ( u ) − ϕ ( v ) ,

that is, the function v ( u ) satisfies the following equation:

(2.18) d v d u = 1 F i ( u ) − ϕ ( v ) ϕ ′ ( y ) F i ( u ) F i ′ ( u ) ϕ ′ ( v ) F i ( z ) F i ′ ( z ) z = F i − 1 ( k F i ( u ) ) , a ≤ u ≤ z Q .

A similar argument proves that the function v ˜ ( u ) also satisfies the above equation.

On the other hand, k > 1 implies that z = F i − 1 ( k F i ( u ) ) ≥ u and y = ϕ − 1 ( k ϕ ( v ) ) ≥ v for all u ∈ ( a , z Q ) . Therefore, by the hypotheses on F i ( z ) and Assumption (A2), we have

ϕ ′ ( y ) ϕ ′ ( v ) ≤ 1 , F i ( u ) F i ′ ( u ) F i ( z ) F i ′ ( z ) ≤ 1

and ϕ ( v ( u ) ) < F i ( u ) for all u ∈ ( a , z Q ) . Then,

(2.19) d v ( u ) d u ≤ 1 F i ( u ) − ϕ ( v ( u ) ) , a ≤ u ≤ z Q .

A similar argument proves that the function v ˜ ( u ) satisfies

(2.20) d v ˜ ( u ) d u ≥ 1 F i ( u ) − ϕ ( v ˜ ( u ) ) , a ≤ u ≤ z Q .

Thus, the differential inequality theorem implies that

(2.21) v ( u ) ≥ ϕ − 1 ( φ Q ( u ) ) , v ˜ ( u ) ≤ ϕ − 1 ( φ ˜ Q ( u ) ) , a ≤ u ≤ z Q .

So, F i ( z ) = k F i ( u ) , k ϕ ( v ) = ϕ ( y ) = φ P ( z ) , k ϕ ( v ˜ ) = ϕ ( y ) = φ ˜ P ( z ) , and (2.21) imply that

(2.22) ∫ a z Q V ( F i ( z ) , φ Q ( z ) , φ ˜ Q ( z ) ) d z ≤ ∫ a z Q V ( F i ( z ) , ϕ ( v ( z ) ) , ϕ ( v ˜ ( z ) ) ) d z = ∫ a z Q V ( F i ( u ) , ϕ ( v ( u ) ) , ϕ ( v ˜ ( u ) ) ) d u = ∫ a z P V 1 k F i ( z ) , 1 k φ P ( z ) , 1 k φ ˜ P ( z ) d z ≤ ∫ a z P V ( F i ( z ) , φ P ( z ) , φ ˜ P ( z ) ) d z

for all z Q ∈ ( a , z P ) . The proof is completed.□

Now, we are in a position to give the detailed proof of Theorem 1.1.

Proof of Theorem 1.1

We give the proof only for the case outside the parentheses. (1.5) implies that we only need to prove

(2.23) ∫ L 1 F 1 ′ ( z ) d y > ∫ L 2 F 2 ′ ( z ) d y ,

where L = L 1 ∪ L 2 is a closed orbit of (1.2). If there are not any R ( F 1 , F 2 ) points on [ a , + ∞ ) , or a = 0 and the origin of coordinates O is a R ( F 1 , F 2 ) point, then system (1.2) has no closed orbits by using Lemma 2.4 and Assumption (H1). Moreover, Lemma 2.4 implies that there exists the unique R ( F 1 , F 2 ) point Δ 0 on ( 0 , z 0 ) and Δ 0 ≥ a , that is,

(2.24) F 1 ( z ) ≤ F 2 ( z ) , ∀ z ∈ ( 0 , Δ 0 )

and

(2.25) F 1 ( z ) > F 2 ( z ) , ∀ z ∈ ( Δ 0 , + ∞ ) .

Let L 1 and L 2 intersect the vertical isocline ϕ ( y ) = F 1 ( z ) and ϕ ( y ) = F 2 ( z ) at A ( z A , y A ) and B ( z B , y B ) , respectively. Then, according to (2.24), (2.25), and the differential inequality theorem, we claim that

(2.26) z A > Δ 0 , z B > Δ 0 .

In fact, we assume for the sake of contradiction that z A ≤ Δ 0 . Let L 1 and L 2 meet the y -axis in point M ( 0 , y M ) and N ( 0 , y N ) where y M < 0 < y N . The paths L 1 and L 2 are described by z 1 ( y ) and z 2 ( y ) for all y ∈ [ y M , y N ] , respectively. Then, z 1 ( y ) ≤ z A ≤ Δ 0 for all y ∈ [ y M , y N ] , and the function z i ( y ) is the solution of the following equation:

d z d y = F i ( z ) − ϕ ( y ) , ∀ y ∈ ( y M , y N ] , z ( y M ) = 0 ,

i = 1 , 2 . Therefore, (2.24) and the differential inequality theorem imply that z 1 ( y ) < z 2 ( y ) for all y ∈ ( y M , y N ] , which is a contradiction. Thus, z A > Δ 0 . A similar argument proves z B > Δ 0 .

Next, we are to prove

(2.27) ϕ ( y A ) = F 1 ( z A ) > ϕ ( y B ) = F 2 ( z B ) .

We assume for the sake of contradiction that F 1 ( z A ) ≤ F 2 ( z B ) . Then, z A < z B . Moreover, let

θ = max { z ∣ z A ≤ z ≤ z B , F 1 ( z A ) = F 2 ( z ) } .

Therefore, F 1 ( z A ) = F 2 ( θ ) > F 2 ( z B ) , contradicting F 1 ( z A ) ≤ F 2 ( z B ) . Thus, (2.27) holds (see Figure 1).

It follows from z A > Δ 0 ≥ a that F 1 ( z ) ≤ F 1 ( z A ) for all z ∈ [ 0 , z A ] . If F 2 ( z B ) ≤ 0 , Assumptions (H1) and (H2) imply that F 2 ( z ) ≥ F 2 ( z B ) for all z ∈ [ 0 , z B ] . Therefore, by Lemma 2.1, (2.23) holds. Hence, we can assume that

(2.28) ϕ ( y ) = F 2 ( z B ) > 0 .

Let y = ϕ − 1 ( φ i ( z ) ) and y = ϕ − 1 ( φ ˜ i ( z ) ) be the arcs of L i below and above the isocline y = ϕ − 1 ( F i ( z ) ) for all z ∈ [ 0 , z 0 i ] , i = 1 , 2 , respectively. It is clear that

(2.29) φ 2 ( z ) ≤ φ 1 ( z ) < 0 < φ ˜ 2 ( z ) ≤ φ ˜ 1 ( z ) , ∀ z ∈ [ 0 , a ] .

Choose the point C ( z C , y C ) on the isocline y = ϕ − 1 ( F 1 ( z ) ) , where z C = F 1 − 1 ( ϕ ( y B ) ) and y C = y B . We denote by L 3 the integral curve of (1.3) passing through the point C . Let y = ϕ − 1 ( φ 3 ( z ) ) and y = ϕ − 1 ( φ ˜ 3 ( z ) ) be the equations of the arcs of L 3 below and above the isocline y = ϕ − 1 ( F 1 ( z ) ) for all z ∈ [ 0 , z 01 ] , respectively. Next, we shall prove (2.23) and the proof is divided in three steps as follows.

Step 1. we prove that z C < min { z A , z B } .

By (2.27) and Assumption (H1), we have that z C < z A . Now, we prove z C < z B . We assume for the sake of contradiction that z C ≥ z B . Note that F 2 ( z B ) = ϕ ( y B ) = F 1 ( z C ) . Hence, it follows from Lemma 2.4 that F 1 ( z ) ≤ F 2 ( z ) for all z ∈ ( a , z B ) . Therefore, (2.24) and (2.25) imply that z B ≤ Δ 0 , contradicting (2.26).

Step 2. See Figure 2. For F 1 ( z ) and F 2 ( z ) , we choose the open subsets R F 1 C of ( a , z C ) and R F 2 B of ( 0 , z B ) in Lemma 2.2, respectively. It is clear that

(2.30) m ( F 1 ( R F 1 C ) ) = m ( F 2 ( R F 2 B ) ) = ϕ ( y B ) .

We denote by z = F 1 − 1 ( y ) and z = F 2 − 1 ( y ) the inverse functions of y = F 1 ( z ) , z ∈ R F 1 C and y = F 2 ( z ) , z ∈ R F 2 B , respectively. Then, z = F 1 − 1 ( y ) and z = F 2 − 1 ( y ) are continuously differentiable and monotone increasing on the open sets R F 1 C and R F 2 B , respectively. Again, let

(2.31) S C = F 1 − 1 ( F 1 ( R F 1 C ) ∩ F 2 ( R F 2 B ) ) .

It is easy to see that S C is an open set, S C ⊂ R F 1 C , m ( S C ) = m ( R F 1 C ) , and that the function F 2 − 1 ( F 1 ( u ) ) is continuously differentiable and monotone increasing on S C . Define

(2.32) v 2 ( u ) = ϕ − 1 ( φ 2 ( F 2 − 1 ( F 1 ( u ) ) ) ) , v ˜ 2 ( u ) = ϕ − 1 ( φ ˜ 2 ( F 2 − 1 ( F 1 ( u ) ) ) ) , u ∈ S C .

Then, it is easy to see that v 2 ( u ) is monotone increasing and v ˜ 2 ( u ) monotone decreasing on S C , and that they both satisfy the following equation:

(2.33) d v d u = 1 F 1 ( u ) − ϕ ( v ) F 1 ′ ( u ) F 2 ′ ( z ) z = F 2 − 1 ( F 1 ( u ) ) , ∀ u ∈ S C .

Next, we shall prove

(2.34) v 2 ( u ) ≤ ϕ − 1 ( φ 3 ( u ) ) , v ˜ 2 ( u ) ≥ ϕ − 1 ( φ ˜ 3 ( u ) ) , ∀ u ∈ S C .

First, we let

F 2 ( η ) = min { F 2 ( z ) ∣ z ∈ [ Δ 0 , z B ] }

and

ξ = max { z ∣ z ∈ [ a , Δ 0 ] , F 1 ( z ) = F 2 ( η ) } .

Then, we claim that

(2.35) ( u − F 2 − 1 ( F 1 ( u ) ) ) ( ξ − u ) ≥ 0 , ∀ u ∈ S C .

In fact, if u ∈ S C ∪ ( a , ξ ) , then u ≤ Δ 0 . (2.24) and the definition of the function F 2 − 1 imply that F 2 − 1 ( F 1 ( u ) ) ≤ u . Therefore, (2.35) holds for all u ∈ S C ∪ ( a , ξ ) . If u ∈ S C ∪ ( Δ 0 , z C ) , then a similar argument proves that (2.35) holds in this case. If u ∈ S C ∪ [ ξ , Δ 0 ] , then F 1 ( u ) ≥ F 1 ( ξ ) = F 2 ( η ) . The definition of the function F 2 − 1 implies that

F 2 − 1 ( F 1 ( u ) ) ≥ η ≥ Δ 0 ≥ u .

Thus, (2.35) holds for all u ∈ S C .

So, we have

(2.36) ( ϕ − 1 ( φ 2 ( u ) ) − v 2 ( u ) ) ( ξ − u ) ≥ 0 , ( ϕ − 1 ( φ ˜ 2 ( u ) ) − v ˜ 2 ( u ) ) ( ξ − u ) ≤ 0 , ∀ u ∈ S C .

Note that ( ξ , z C ) ⊂ R F 1 C . Hence,

m ( S C ∩ ( ξ , z C ) ) = m ( R F 1 C ∩ ( ξ , z C ) ) = z C − ξ .

Thus, it follows from Assumption (H4) that F 1 ′ ( u ) ≥ F 2 ′ ( z ) > 0 , where z = F 2 − 1 ( F 1 ( u ) ) and u ∈ S C ∩ ( ξ , z C ) . Again note that

ϕ − 1 ( φ 3 ( z C ) ) = ϕ − 1 ( φ 2 ( z B ) ) ≥ v 2 ( z C − 0 ) , ϕ − 1 ( φ ˜ 3 ( z C ) ) = ϕ − 1 ( φ ˜ 2 ( z B ) ) ≤ v ˜ 2 ( z C − 0 ) .

Hence, by (1.3), (2.33), and Lemma 2.3, we obtain

(2.37) v 2 ( u ) ≤ ϕ − 1 ( φ 3 ( u ) ) , v ˜ 2 ( u ) ≥ ϕ − 1 ( φ ˜ 3 ( u ) ) , ∀ u ∈ S C ∩ ( ξ , z C ) .

Furthermore, it follows from (2.36) and (2.37) that

ϕ − 1 ( φ 3 ( ξ + 0 ) ) ≥ v 2 ( ξ + 0 ) ≥ ϕ − 1 ( φ 2 ( ξ + 0 ) ) , ϕ − 1 ( φ ˜ 3 ( ξ + 0 ) ) ≤ v ˜ 2 ( ξ + 0 ) ≤ ϕ − 1 ( φ ˜ 2 ( ξ + 0 ) ) .

Therefore,

(2.38) ϕ − 1 ( φ 3 ( ξ ) ) ≥ ϕ − 1 ( φ 2 ( ξ ) ) , ϕ − 1 ( φ ˜ 3 ( ξ ) ) ≤ ϕ − 1 ( φ ˜ 2 ( ξ ) ) .

Second, it follows from the definition of ξ that F 1 ( z ) ≤ F 2 ( z ) for all z ∈ ( a , ξ ) . Again, since φ 2 ( a ) ≤ φ 3 ( a ) < 0 , (2.38) and applying the differential inequality theorem to (1.3) and (1.4) imply that

(2.39) ϕ − 1 ( φ 3 ( u ) ) ≥ ϕ − 1 ( φ 2 ( u ) ) , ϕ − 1 ( φ ˜ 3 ( u ) ) ≤ ϕ − 1 ( φ ˜ 2 ( u ) ) , ∀ u ∈ ( a , ξ ) .

Thus, (2.34) follows from (2.36), (2.37), and (2.39).

Step 3. Let

S B = F 2 − 1 ( F 1 ( R F 1 C ) ∩ F 2 ( R F 2 B ) ) .

Then, S B is an open set, S B ⊂ R F 2 B and m ( S B ) = m ( R F 2 B ) . By (2.31), S B = F 2 − 1 ( F 1 ( S C ) ) . Hence, since m ( S C ) = m ( R F 1 C ) and R F 1 C = ( a , z C ) , applying Lemma 2.2 and (2.34) implies that

(2.40) ∫ L 2 F 2 ′ ( z ) d y ≤ ∫ R F 2 B V ( F 2 ( z ) , φ 2 ( z ) , φ ˜ 2 ( z ) ) d z = ∫ S B V ( F 2 ( z ) , φ 2 ( z ) , φ ˜ 2 ( z ) ) d z = ∫ S C V ( F 1 ( u ) , ϕ ( v 2 ( u ) ) , ϕ ( v ˜ 2 ( u ) ) ) d u < ∫ S C V ( F 1 ( u ) , φ 3 ( u ) , φ ˜ 3 ( u ) ) d u = ∫ a z C V ( F 1 ( u ) , φ 3 ( u ) , φ ˜ 3 ( u ) ) d u .

Moreover, by Assumption (H3) and Lemma 2.6, we have

(2.41) ∫ a z C V ( F 1 ( u ) , φ 3 ( u ) , φ ˜ 3 ( u ) ) d u ≤ ∫ a z A V ( F 1 ( u ) , φ 1 ( u ) , φ ˜ 1 ( u ) ) d u .

In addition, it follows from Lemma 2.1 that

(2.42) ∫ 0 a V ( F 1 ( u ) , φ 1 ( u ) , φ ˜ 1 ( u ) ) d u ≥ 0 .

Thus, (2.23) follows from (2.40), (2.41), and (2.42). The proof is completed.□

$Figure 1 The arcs of the integral curve L i {L}_{i} below and above the isoline ϕ ( y ) = F i ( z ) \phi (y)={F}_{i}\left(z) , i = 1 , 2 i=1,2 .$

Figure 1

The arcs of the integral curve L i below and above the isoline ϕ ( y ) = F i ( z ) , i = 1 , 2 .

$Figure 2 The arcs of the integral curve L 1 , L 2 {L}_{1},{L}_{2} , and L 3 {L}_{3} below and above the isoline ϕ ( y ) = F i ( z ) \phi (y)={F}_{i}\left(z) .$

Figure 2

The arcs of the integral curve L 1 , L 2 , and L 3 below and above the isoline ϕ ( y ) = F i ( z ) .

Remark 2.2

F 1 ′ ( z ) = d F d x 1 d x 1 d z = f ( x 1 ) g ( x 1 ) ,

we have

F 1 ( z ) F 1 ′ ( z ) = F ( x 1 ) f ( x 1 ) g ( x 1 ) ,

where x 1 = G − 1 ( z ) ≥ 0 . Define a ∗ by a = G ( a ∗ ) and a ∗ ≥ 0 . Then, Assumption ( H3 ∗ ) implies that Assumptions (H3) holds:

( H3 ∗ ) the function F ( x ) f ( x ) g ( x ) is nondecreasing for all x ∈ ( a ∗ , x 01 ) (or the function F ( x ) f ( x ) g ( x ) is nonincreasing for all x ∈ ( x 02 , 0 ) and F ( x 01 − 0 ) ≤ F ( x 02 + 0 ) ).

3 Example

In this section, we give an example to show the application of Theorem 1.1 in Section 1. In [19,34,35], Yuan et al. studied the following system

(3.1) d x d t = x ( 1 + A 1 x − A 3 x 2 − ( A 2 + x ) y ) , d y d t = A 0 y ( x 2 − 1 ) ,

where A 0 > 0 , A 1 > 0 , A 3 > 0 , A 2 ∈ ( − ∞ , + ∞ ) , and ( x , y ) ∈ Ω ∗ , where Ω ∗ = { ( x , y ) ∣ x ≥ 0 , y ≥ 0 } . There are at most three equilibrium points: O ( 0 , 0 ) , R 0 ( x 0 , 0 ) , and R ( 1 , y ∗ ) , where y ∗ = − k h > 0 , k = A 3 − A 1 − 1 , h = A 2 + 1 , and x 0 > 0 satisfies the equation 1 + A 1 x − A 3 x 2 = 0 . The variable change of

x → e x , y → y ∗ e − y , d t → ( A 2 + x ) d t

transfers system (3.1) to the following Liénard system

(3.2) d x d t = ϕ ( y ) − F ( x ) , d y d t = − g ( x ) ,

where ϕ ( y ) = y ∗ ( 1 − e − y ) , F ( x ) = y ∗ + A 3 e 2 x − A 1 e x − 1 A 2 + e x , g ( x ) = A 0 ( e 2 x − 1 ) A 2 + e x , − ∞ < x < ln x 0 , and − ∞ < y < ∞ .

Let G ( x ) = ∫ 0 x g ( s ) d s , z 0 i = G ( x 0 i ) , i = 1 , 2 , and z 0 = min { z 01 , z 02 } . We denote by x i ( z ) the inverse function of

z = G ( x ) , ∀ ( − 1 ) i + 1 x ≥ 0 , i = 1 , 2 .

In addition, F i ( z ) = F ( x i ( z ) ) , i = 1 , 2 .

Next, we prove that the functions F 1 ( z ) and F 2 ( z ) satisfy Assumptions (H1)–(H4) if k < 0 , h > 1 and δ > 0 , where δ = A 1 A 2 − 1 − A 3 ( 2 A 2 + 1 ) .

In fact, it is easy to see that

F ( x ) = A 3 ( e x − 1 ) ( e x − 1 − η ) ( e x − 1 + h ) − 1 ,

where η = δ A 3 h > 0 . Therefore, the equation F ( x ) = 0 has a unique positive solution ln ( 1 + η ) . G ( ln ( 1 + η ) ) is the real number a in Assumption (H1). It is easy to see that F ( x ) < 0 for all x ∈ ( 0 , ln ( 1 + η ) ) , F ( x ) > 0 for all x ∈ ( − ∞ , 0 ) ∪ ( ln ( 1 + η ) , + ∞ ) , g ( x ) < 0 for all x ∈ ( − ∞ , 0 ) , and g ( x ) > 0 for all x ∈ ( 0 , + ∞ ) . Thus, G ′ ( x ) = g ( x ) < 0 for all x ∈ ( − ∞ , 0 ) implies that the function x 2 ( z ) is strictly decreasing on ( 0 , + ∞ ) and x 2 ( z ) < 0 for all z ∈ ( 0 , + ∞ ) . Hence, F 2 ( z ) > 0 for all z ∈ ( 0 , + ∞ ) and system (3.2) satisfies Assumption (H2).

From G ′ ( x ) = g ( x ) > 0 for all x ∈ ( 0 , + ∞ ) , we obtain that the function x 1 ( z ) is strictly increasing on ( 0 , + ∞ ) . In addition, 0 < x 1 ( z ) < ln ( 1 + η ) for all z ∈ ( 0 , a ) . Therefore, F 1 ( z ) < 0 for all z ∈ ( 0 , a ) . On the other hand, x 1 ( z ) > ln ( 1 + η ) for all z ∈ ( a , + ∞ ) implies that F 1 ( z ) > 0 for all z ∈ ( a , + ∞ ) . Thus, system (3.2) satisfies Assumption (H1).

Since a = G ( ln ( 1 + η ) ) in Assumption (H1), we have that ln ( 1 + η ) is the real number a ∗ in Assumption ( H3 ∗ ). Let e x − 1 = e u . Then, x ∈ ( a ∗ , + ∞ ) implies e u > η . Let

F 3 ( u ) = A 3 e u ( e u − η ) ( e u + h ) − 1 ,

g 3 ( u ) = A 0 e u ( e u + 2 ) ( e u + h ) − 1 ,

and

(3.3) f 3 ( u ) = F 3 ′ ( u ) = A 3 e u ( e u + h ) − 2 [ ( e u + h ) 2 − h ( h + η ) ] .

Then,

(3.4) F ( x ) = F 3 ( u ) , g ( x ) = g 3 ( u ) , f ( x ) = f 3 ( u ) e x − u

and

(3.5) F 3 ( u ) f 3 ( u ) g 3 ( u ) = A 0 − 1 A 3 2 H 1 ( u ) ,

where

H 1 ( u ) = e u ( e u − η ) [ ( e u + h ) 2 − h ( h + η ) ] ( e u + 2 ) ( e u + h ) 2 .

It is easy to see that

(3.6) H 1 ′ ( u ) = e u [ H 2 ( u ) + H 3 ( u ) ] ,

where

H 2 ( u ) = 2 ( e u − η ) [ ( e u + h ) 3 − h 2 ( η + h ) ] ( e u + 2 ) 2 ( e u + h ) 3

and

H 3 ( u ) = e u [ ( e u + h ) 2 − h ( h + η ) ] ( e u + 2 ) ( e u + h ) 2 + 2 h ( h + η ) ( e u − η ) ( e 2 u + e u ) ( e u + 2 ) 2 ( e u + h ) 3 .

Therefore, by (3.5) and (3.6), we have

(3.7) d d x F ( x ) f ( x ) g ( x ) = d d x F 3 ( u ) f 3 ( u ) g 3 ( u ) e x − u = e 2 ( x − u ) d d u F 3 ( u ) f 3 ( u ) g 3 ( u ) + F 3 ( u ) f 3 ( u ) g 3 ( u ) d d x [ e x − u ]

= A 3 2 ( e u + 1 ) A 0 e 2 u [ ( e u + 1 ) H 1 ′ ( u ) − H 1 ( u ) ] = A 3 2 ( e u + 1 ) A 0 e 2 u [ H 4 ( u ) + H 5 ( u ) ] ,

where

H 4 ( u ) = e u ( e u + 1 ) [ ( e u + h ) 2 − h ( h + η ) ] ( e u + 2 ) ( e u + h ) 2 + 2 h e u ( h + η ) ( e u + 1 ) 2 ( e u − η ) ( e u + 2 ) 2 ( e u + h ) 3 , H 5 ( u ) = ( 2 e u + 1 ) ( e u − η ) [ ( e u + h ) 3 − h 2 ( h + η ) ] ( e u + 2 ) 2 ( e u + h ) 3 + h e u ( h + η ) ( e u − η ) ( e u + 2 ) 2 ( e u + h ) 3 .

Thus, it is easy to see that H 4 ( u ) > 0 and H 5 ( u ) > 0 for all u ∈ ( ln η , + ∞ ) . Hence, by (3.7), we obtain

d d x F ( x ) f ( x ) g ( x ) > 0

for all x ∈ ( a ∗ , + ∞ ) . So, the system (3.2) satisfies Assumption ( H3 ∗ ), and by Remark 2.2, the system (3.2) satisfies Assumption (H3).

By the definition of the functions F i ( z ) and x i ( z ) , we have

(3.8) d F i ( z ) d z = d F ( x i ( z ) ) d z = d F d x i d x i ( z ) d z = f ( x i ) 1 d z d x i = f ( x i ) 1 d G ( x i ) d x i = f ( x i ) g ( x i ) ,

where i = 1 , 2 , x 1 ( z ) ≥ 0 and x 2 ( z ) ≤ 0 for all z ∈ ( 0 , + ∞ ) . Note that a = G ( ln ( 1 + η ) ) . If z ∈ ( a , + ∞ ) , then z = G ( x ) implies that x 1 ( z ) ∈ ( ln ( 1 + η ) , + ∞ ) . Let u = ln ( e x 1 − 1 ) . Then, e u > η for all z ∈ ( a , + ∞ ) . Therefore, by (3.3), we obtain that f 3 ( u ) > 0 for all z ∈ ( a , + ∞ ) , and (3.4) implies that f ( x 1 ) > 0 for all z ∈ ( a , + ∞ ) . It is easy to see that g ( x ) > 0 for all x ∈ ( 0 , + ∞ ) . Thus, by (3.8), we have that F 1 ′ ( z ) > 0 for all z ∈ ( a , + ∞ ) . A similar argument gives F 2 ′ ( z ) ≤ 0 for all z ∈ ( 0 , + ∞ ) . Hence, system (3.2) satisfies Assumption (H4).

It is easy to see that the system (3.2) satisfies Assumptions (A1)–(A2). So, Theorem 1.1 is applicable in the system (3.2). We have

Theorem 3.1

If k < 0 , h > 1 , and δ > 0 , then system (3.1) has at most one limit cycle around the positive equilibrium point R ( 1 , y ∗ ) , and it is stable if it exists.

Remark 3.1

It is easy to prove that the conditions in Theorem 3.1 cannot guarantee

d d x f ( x ) g ( x ) ≥ 0

for all x ∈ ( a ∗ , ln x 0 ) and F ( ln x 0 ) > F ( − ∞ ) . Thus, Theorem 3.1 cannot be proved by employed Zhang’s theorem and the results in [30].

Funding information: This work was supported by the Natural Science Foundation of China (No. 11561068), China Postdoctoral Science Foundation (No. 2016M592442), Scientific Research Fund of Hunan Provincial Education Department (No. 21A0596), and Hunan Provincial Natural Science Foundation (No. 2022JJ30107).
Author contributions: All authors contributed equally to the writing of this article and read and approved the final manuscript.
Conflict of interest: The authors state that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

References

[1] A. Bakhshalizadeh, R. Asheghi, H. R. Z. Zangeneh, and M. E. Gashti, Limit cycles near an eye-figure loop in some polynomial Liénard systems, J. Math. Anal. Appl. 455 (2017), 500–515. 10.1016/j.jmaa.2017.05.064Search in Google Scholar

[2] H. Giacomini and M. Grau, Transversal conics and the existence of limit cycles, J. Math. Anal. Appl. 428 (2015), 563–586. 10.1016/j.jmaa.2015.03.015Search in Google Scholar

[3] L. G. Guo, P. Yu, and Y. F. Chen, Bifurcation analysis on a class of Z(2)-equivariant cubic switching systems showing eighteen limit cycles, J. Differential Equations 266 (2019), 1221–1244. 10.1016/j.jde.2018.07.071Search in Google Scholar

[4] L. L. Li and J. M. Yang, On the number of limit cycles for a quintic Liénard system under polynomial perturbations, J. Appl. Anal. Comput. 9 (2019), 2464–2481. 10.11948/20190221Search in Google Scholar

[5] J. Llibre, R. Ramirez, V. Ramirez, and N. Sadovskaia, The 16th Hilbert problem restricted to circular algebraic limit cycles, J. Differential Equations 260 (2016), 5726–5760. 10.1016/j.jde.2015.12.019Search in Google Scholar

[6] N. Li, M. A. Han and V. G. Romanovski, Cyclicity of some Liénard systems, Commun. Pure Appl. Anal. 14 (2015), 2127–2150. 10.3934/cpaa.2015.14.2127Search in Google Scholar

[7] J. Mawhin and G. Villari, Periodic solutions of some autonomous Liénard equations with relativistic acceleration, Nonlinear Anal. 160 (2017), 16–24. 10.1016/j.na.2017.05.001Search in Google Scholar

[8] K. Murakami, A concrete example with multiple limit cycles for three dimensional Lotka-Volterra systems, J. Math. Anal. Appl. 457 (2018), 1–9. 10.1016/j.jmaa.2017.07.076Search in Google Scholar

[9] S. Pérez-González, J. Torregrosa, and P. J. Torres, Existence and uniqueness of limit cycles for generalized j-Laplacian Liénard equations, J. Math. Anal. Appl. 439 (2016), 745–765. 10.1016/j.jmaa.2016.03.004Search in Google Scholar

[10] S. G. Ruan and D. M. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 61 (2001), 1445–1472. 10.1137/S0036139999361896Search in Google Scholar

[11] Y. Tian, M. A. Han, and F. F. Xu, Bifurcations of small limit cycles in Liénard systems with cubic restoring terms, J. Differential Equations 267 (2019), 1561–1580. 10.1016/j.jde.2019.02.018Search in Google Scholar

[12] G. Tigan, Using Melnikov functions of any order for studying limit cycles, J. Math. Anal. Appl. 448 (2017), 409–420. 10.1016/j.jmaa.2016.11.021Search in Google Scholar

[13] J. F. Wang, X. Y. Chen, and L. H. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl. 469 (2019), 405–427. 10.1016/j.jmaa.2018.09.024Search in Google Scholar

[14] J. F. Wang, and L. H. Huang, Limit cycles bifurcated from a focus-fold singularity in general piecewise smooth planar systems, J. Differential Equations 304 (2021), 491–519. 10.1016/j.jde.2021.10.006Search in Google Scholar

[15] D. M. Xiao and Z. F. Zhang, On the existence and uniqueness of limit cycles for generalized Liénard systems, J. Math. Anal. Appl. 343 (2008), 299–309. 10.1016/j.jmaa.2008.01.059Search in Google Scholar

[16] Y. Q. Xiong, M. A. Han, and D. M. Xiao, The maximal number of limit cycles bifurcating from a Hamiltonian triangle in quadratic systems, J. Differential Equations 280 (2021), 139–178. 10.1016/j.jde.2021.01.016Search in Google Scholar

[17] P. Yu and F. Li, Bifurcation of limit cycles in a cubic-order planar system around a nilpotent critical point, J. Math. Anal. Appl. 453 (2017), 645–667. 10.1016/j.jmaa.2017.04.019Search in Google Scholar

[18] P. Yu, M. A. Han, and X. Zhang, Eighteen limit cycles around two symmetric foci in a cubic planar switching polynomial system, J. Differential Equations 275 (2021), 939–959. 10.1016/j.jde.2020.11.001Search in Google Scholar

[19] Y. D. Yuan, H. B. Chen, C. X. Du, and Y. J. Yuan, The limit cycles of a general Kolmogorov system, J. Math. Anal. Appl. 392 (2012), 225–237. 10.1016/j.jmaa.2012.02.065Search in Google Scholar

[20] X. W. Zeng, An existence and uniqueness theorem of limit cycles of Liénard equation, Acta Math. Sinica 21 (1978), 263–269 (in Chinese). Search in Google Scholar

[21] Z. F. Zhang, T. R. Ding, W. Z. Huang, and Z. X. Dong, Qualitative theory of differential equation, in: Modern Mathematics Basic Series, Science Press, Beijing, 1997 (in Chinese). Search in Google Scholar

[22] T. Carletti and G. Villari, A note on existence and uniqueness of limit cycles for Liénard systems, J. Math. Anal. Appl. 307 (2005), 763–773. 10.1016/j.jmaa.2005.01.054Search in Google Scholar

[23] J. F. Huang and H. H. Liang, A uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities, J. Math. Anal. Appl. 457 (2018), 498–521. 10.1016/j.jmaa.2017.08.008Search in Google Scholar

[24] J. F. Huang, H. H. Liang, and J. Llibre, Non-existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities, J. Differential Equations 265 (2018), 3888–3913. 10.1016/j.jde.2018.05.019Search in Google Scholar

[25] H. H. Liang and J. F. Huang, On the uniqueness and expression of limit cycles in planar polynomial differential system via monotone iterative technique, Applicable Analisis 101 (2022), 3365–3388. 10.1080/00036811.2020.1849629Search in Google Scholar

[26] C. Z. Li and J. Llibre, Uniqueness of limit cycles for Liénard differential equations of degree four, J. Differential Equations 252 (2012), 3142–3162. 10.1016/j.jde.2011.11.002Search in Google Scholar

[27] L. J. Yang and X. W. Zeng, An upper bound for the amplitude of limit cycles in Liénard systems with symmetry, J. Differential Equations 258 (2015), 2701–2710. 10.1016/j.jde.2014.12.021Search in Google Scholar

[28] X. W. Zeng, On the uniqueness of limit cycle of Liénard equation, Sci. China A 25 (1982), no. 6, 583–592 (in Chinese). Search in Google Scholar

[29] X. W. Zeng, Z. F. Zhang, and S. Z. Gao, On the uniqueness of the limit cycles of the generalized Liénard equation, Bull. London Math. Soc. 26 (1994), 213–247. 10.1112/blms/26.3.213Search in Google Scholar

[30] P. G. Zhang, The problem of the uniqueness of limit cycle for the equations dxdt=φ(y)−F(x),dydt=−g(x), J. Zhejiang Univ. (Natural Sci.) 24 (1990), no. 3, 433–448 (in Chinese). Search in Google Scholar

[31] P. G. Zhang and S. Q. Zhao, On the uniqueness of limit cycle of the general Liénard equation, Acta Math. Sinica 47 (2004), no. 6, 1193–1200 (in Chinese). Search in Google Scholar

[32] Z. F. Zhang, Proof of the uniqueness theorem of limit cycles of generalized Liénard equations, Appl. Anal. 23 (1986), 63–76. 10.1080/00036818608839631Search in Google Scholar

[33] Z. F. Zhang and S. Z. Gao, On the uniqueness of limit cycle of one type of nonlinear differential equations, J. Peking Univ. (Natural Sci.) 22 (1986), no. 1, 1–13 (in Chinese). Search in Google Scholar

[34] H. B. Chen, Existence and uniqueness of limit cycle of a cubic Kolmogorov’s differential system, J. Hunan Agricult. Univ. 22 (1996), no. 1, 78–81 (in Chinese). Search in Google Scholar

[35] X. C. Huang and L. Zhu, Limit cycles in a general Kolmogorov model, J. Nonlinear Anal. 60 (2005), 1393–1414. 10.1016/j.na.2004.11.003Search in Google Scholar

Received: 2022-07-31

Revised: 2022-12-30

Accepted: 2023-01-20

Published Online: 2023-02-14

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

Articles in the same Issue

https://doi.org/10.1515/math-2022-0558

Keywords for this article

generalized Liénard system; limit cycles; uniqueness

Creative Commons

BY 4.0