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Existence of positive periodic solutions for Liénard equations with an indefinite singularity of attractive type
Boundary Value Problemsvolume 2018, Article number: 101 (2018)
Abstract
In this paper, we study the periodic problem for the Liénard equation with an indefinite singularity of attractive type
where $f:(0,+\infty )\rightarrow R$ is continuous and may have singularities at zero, r, $\varphi : R\rightarrow R$ are Tperiodic functions, and μ is a positive constant. Using the method of upper and lower functions, we obtain some new results on the existence of positive periodic solutions to the equation.
Introduction
As is well known, differential equations with singularities have a wide range of applications in physics, mechanics, and biology [1–6]. In the past years, many mathematical researchers focused their attention on the equations with singularities [7–20]. As is widely acknowledged, the paper [18] by Lazer and Solimini is a major milestone for the study of periodic problem to secondorder differential equations with singularities. In that paper, the existence of periodic solutions was investigated for the singular equations
(the singularity of attractive type) and
(the singularity of repulsive type), where $h:R\rightarrow R$ is a continuous periodic function. Using topological degree methods, together with the method of lower and upper functions, they obtained that a necessary and sufficient condition for the existence of positive periodic solutions to (1.1) is $\int_{0}^{T}h(s)\,ds>0$. Furthermore, assuming that $\alpha \ge 1$, a necessary and sufficient condition for the existence of positive periodic solutions to (1.2) is $\int_{0}^{T}h(s)\,ds<0$. For $\alpha \in (0,1)$ (weak singularity condition), some equations like (1.2) were given in [18], where $h(t)$ have negative mean values, but the equations have no Tperiodic solution. After that, many papers focused on the periodic problem for some secondorder differential equations with singularities of repulsive type [21–24]. Among them, the singular term was allowed to have a weak singularity (i.e., $\alpha \in (0,1)$). Compared with the singularity of repulsive type, the attractive case did not attract much attention of mathematical researchers. Even so, there are still quite a few papers that focus on the study of periodic solutions for the equations with attractive singularities [7, 16, 17, 20]. For example, Mawhin [7] considered the problem of periodic solutions to the Liénard equation with a attractive singularity suggested by the fundamental example
where $p>1$, $l>0$, and $\mu >0$ are constants, and $h\in L^{\infty }(0,T)$. Using the method of upper and lower functions, they obtained that $\int_{0}^{T}h(s)\,ds>0$ is a necessary and sufficient condition for the existence of positive periodic solutions for equation (1.3). Hakl and Torres [16] obtained sufficient conditions guaranteeing the existence of positive solutions to the periodic problem associated to the equation of Rayleigh–Plesset type
where $g_{1}$, $g_{2}$, σ are nonnegative constants, c, μ, υ, γ are real numbers, and $h_{0}\in L([0,T];R)$. For the case of $\delta =1$, $g_{1}>0$, and $g_{2}=0$, they obtained the following result.
Theorem 1.1
Let $\delta =1$, $g_{1}>0$, and $g_{2}=0$. If $\bar{\varphi }<0$ and
then there exists at least one positive solution to problem (1.4)–(1.5).
It is easy to see that in either equation (1.3) or the equation in problem (1.4)–(1.5), the singular terms are all autonomous. Despite the fact that there are many papers focusing on the equation with nonautonomous singularity term of repulsive type [21–25], equations with nonautonomous singularity term of attractive type seem to receive little attention. We have found only the paper by Hakl and Zamora [17], who studied the following equations with a singularity of attractive type:
where $\lambda >0$, $\delta \in [0,1)$, g, $h\in L([0,T];R)$, and g is a nonnegative function. Using a continuation theorem of coincidence degree theory, they obtained a new result on the existence of positive periodic solutions to (1.6). However, the exponent δ in the power function $x^{\delta }$ is required to satisfy $\delta \in [0,1)$, and there is no friction term of Liénard type $f(x)x'$ in (1.6). For other recent developments and applications in this field, we refer the reader to [16, 23, 26–31]. Inspired by the papers mentioned, the aim of this paper is to study the periodic problem
where f belongs to $C((0,\infty );R)$ and may have singularities at zero, r, $\varphi : R\rightarrow R$ are Tperiodic functions with $r,\varphi \in L([0,T];R)$, and μ is a positive constant. Since $r(t)$ is a timevarying function, (1.7) describes the nature processes more accurately. Observe that in the case where $r(t)$ is equal to zero for some subinterval of $[0,T]$, the singularity can disappear. So the singularity $\frac{r(t)}{x^{\mu }}$ in equation (1.7) is said to be of indefinite attractive type. Using the method of lower and upper functions, we obtain some new results on the existence of positive solutions to boundary value problem (1.7)–(1.8). The significance is that the methods for constructing lower and upper functions in [16] cannot be directly applied to (1.7), since the singularity term $\frac{r(t)}{x^{ \mu }}$ in (1.7) is nonautonomous, and $r(t)$ may be equal to zero at some $t\in [0,T]$.
Remark 1.1
In this paper, a function $u:[0,T]\rightarrow (0,+\infty )$ is said to be a positive solution to problem (1.7)–(1.8) if $u:[0,T]\rightarrow R_{+}$ is absolutely continuous together with its first derivative on $[0,T]$ and satisfies (1.8) together with (1.7) almost everywhere on $[0,T]$. From this definition, r, $\varphi : R\rightarrow R$ being Tperiodic, we can easily find that if $u_{0}: [0,T]\rightarrow R$ is a positive solution to boundary value problem (1.7)–(1.8), then $\tilde{u}: R\rightarrow R$, which is a Tperiodic extension of $u_{0}(t)$, is a positive Tperiodic solution to (1.7). Thus, the existence of positive periodic solutions to (1.7) is equivalent to the existence of positive solutions to boundary value problem (1.7)–(1.8).
For convenience, in the end of this introduction, we give some notations used throughout the paper:
$R_{+}=(0,+\infty )$, $R_{+}^{0}=[0,+\infty )$, $[x]_{+}= \max \{x,0 \}$, $[x]_{}=\max \{x,0\}$; $AC^{1}([0,T];R)$ is the set of functions $u:[0,T]\rightarrow R$ such that u and $u'$ are absolutely continuous; $\overline{p}=\frac{1}{T}\int_{0}^{T}\vert p(s) \vert \,ds$ for $p\in L([0,T];R)$.
Preliminary lemmas
The method of lower and upper functions is one of the most widely used methods in nonlinear analysis. Its main idea goes back at least to Picard. Many mathematical researchers have obtained rich results by using this method. For a complete historical review of the method, we refer to the monograph [32]. Now, we give the definitions of upper and lower functions.
Definition 2.1
A function $\alpha \in AC^{1}([0,T];R)$ is called a lower function to problem (1.7)–(1.8) if $\alpha (t)>0$ for every $t\in [0,T]$ and
Definition 2.2
A function $\beta \in AC^{1}([0,T];R)$ is called an upper function to problem (1.7)–(1.8) if $\beta (t)>0$ for every $t\in [0,T]$ and
The following proposition can be found in [32] (or a more general case in [33]).
Proposition 2.1
Let α and β be lower and upper functions to problem (1.7)–(1.8) such that
Then there exists a positive solution u to problem (1.7)–(1.8) such that
We further show some auxiliary results obtained by Hakl, Torres and Zamora [16].
Given $x_{1}\in R_{+}$ and $x_{0}\in R_{+}^{0}$ as fixed constants and the operator $K:C^{1}([0,T];R)\rightarrow C^{1}([0,T];R)$ defined by
we consider the auxiliary problem
where $f\in C(R_{+};R)$ and $q\in L([0,T];R)$. By a solution to problem (2.2)–(2.3) we understand a function $u\in AC^{1}([0,T];R)$ that satisfies (2.2) almost everywhere on $[0,T]$ and (2.3).
Lemma 2.1
([16])
For every solution u to problem
with $\lambda \in (0,1]$, we have the estimate
where $M=\max \{u(t):t\in [0,T]\}$, $m=\min \{u(t):t\in [0,T]\}$.
Lemma 2.2
([16])
For all $x_{1}\in R_{+}$, $x_{0}\in R_{+}^{0}$, and $q\in L([0,T];R)$, there exists a solution $u(t)$ to problem (2.2)–(2.3). Furthermore,
and (2.6) is fulfilled.
Lemma 2.3
If $h,\alpha \in L([0,T],R)$ and $\alpha (t)\ge 0$ for a.e. $t\in [0,T]$, then
Proof
Let
Then we have
Moreover,
By the definition of $h_{n}(t)$ we obtain
where
$E_{}(h)=\{t\in [0,T]: h(t)\le 0\}$, $E_{+}(h)=\{t\in [0,T]: h(t)> 0\}$, and
By the Lebesgue dominated convergence theorem we get
Substituting this into (2.8), we obtain
The proof is complete. □
Main results
Construction of lower function
In this section, we use the notations
where $y\in L([0,T];R)$.
Theorem 3.1
Let $\varphi ,r\in L([0,T],R)$ be such that $\operatorname{ess} \inf \{r(t):t \in [0,T]\}>0$. Then there exists a lower function α to problem (1.7)–(1.8) satisfying $0<\alpha (t)<1$.
Proof
Let $r_{1}>0$ be such that
Consider the periodic problem
By Corollary 2.13 in [16] (with $h_{0}(t)=\varphi (t)$, $\rho_{0}(x)=x$, $g(x)=\frac{r_{1}}{x^{\mu }}, c$ sufficiently large, and $x_{0}\in (0,1)$) there exists a lower function α to periodic problem (3.1) such that $0<\alpha (t)\leq x_{0}$ for $t\in [0,T]$. Obviously, the same α is also a lower function to (1.7)–(1.8). □
Theorem 3.2
Let φ, $r\in L([0,T],R)$, $r(t)\ge 0$ a.e. $t\in [0,T]$ and $\bar{r}>0$. Suppose $\frac{T}{4} \int_{0}^{T}[\varphi (s)]_{}(s)\,ds<1$. Then there exists a lower function α to problem (1.7)–(1.8) such that $0<\alpha (t)<1$ for $t\in [0,T]$.
Proof
Let
Then we have
Obviously, for a constant c satisfying $c>\bar{\varphi}$, there must exist a constant $a_{1}\in (0,1)$ such that, for any $x\in (0,a_{1})$, we have
Since $Y(x)=Y_{+}(x)Y_{}(x)$, we arrive at $Y_{+}(x)< Y_{}(x)$ for $x\in (0,a_{1})$. By a direct calculation, for $x\in (0,a_{1})$, we get
Since $Y(x)=Y_{+}(x)Y_{}(x)$, we obtain the following inequality as $x\rightarrow 0^{+}$:
So, for a sufficiently small constant $a_{1}\in (0,1)$, we have the following equality for $x\in (0,a_{1})$:
Choose a constant $x_{2}\in (0,a_{1})$, and let $x_{1}=(1\frac{T}{4}Y _{+}(x_{2}))x_{2}$. Since $\frac{T}{4}\int_{0}^{T}[\varphi (s)]_{}\,ds<1$ by assumption, we have $1\frac{T}{4}Y_{+}(x_{2})>0$, which ensures that $x_{1}$ make sense. As the first case, we suppose $Y_{+}(x_{2})>0$.
Put
Now we obtain
By Lemma 2.2 there exists a solution u to (2.2)–(2.3) such that (2.6) and (2.7) hold.
where the constants M and m are defined in Lemma 2.1.
Put $\alpha (t)=K(u)(t)=x_{1}+x_{0}(u(t)\min \{u(s):s\in [0,T]\})$. Then equation (2.2) can be written as
for almost every $t\in [0,T]$. Besides, according to (3.3), (3.5), and the definition of $\alpha (t)$, we arrive at
Then, using (3.6), we get
In addition, by the definition of $x_{1}$, $x_{2}$ and by (3.2) we obtain
From the definition of $x_{1}$, $x_{2}$, we arrive at
Using (3.3) and (3.10), we get
Relations of (3.3) and (3.11) imply
By (3.8), (3.12), (3.13) we obtain
Substituting (3.14) into (3.7), we get
By (3.8) we have
Moreover,
Substituting this into (3.15), we arrive at
Consequently, (3.9) and (3.17) ensure that $\alpha (t)$ is a lower function to problem (1.7)–(1.8) such that $0<\alpha (t)<1$ for $t\in [0,T]$.
We further consider the case of $Y_{+}(x_{2})=0$. For this case, it is easy to see that
which means
that is,
Obviously, $\alpha (t)=x_{2}$ is a lower function to problem (1.7)–(1.8), and $0<\alpha (t)<1$ for $t\in [0,T]$. □
Construction of upper function
Theorem 3.3
Let r, $\varphi \in L([0,T],R)$ be such that $\operatorname{ess} \sup \{r(t):t \in [0,T]\}<+\infty $. Let, moreover,
Then there exists an upper function $\beta (t)$ to problem (1.7)–(1.8), and $\beta (t)>1$ for $t\in [0,T]$.
Proof
Since $\operatorname{ess} \sup \{r(t):t\in [0,T]\}<+\infty $, there exists a constant $c_{1}\in R$ such that
which results in
Thus there exist $1< a_{1}(\varepsilon )\leq a_{2}(\varepsilon )<+ \infty $ such that
Let $\psi (t)=\varphi (t)\varepsilon $. By Lemma 2.3 we have
namely,
We easily have
Arguing as before, (3.18) is equal to
which implies $\bar{\psi }\geq 0$, that is, $\Psi_{+}\geq \Psi_{} \geq 0$. As a first case, we suppose $\Psi_{}>0$.
Put
From the definition of $q(t)$ we have
By Lemma 2.2 there exists a solution u to problem (2.2)–(2.3) such that (2.6) and (2.7) hold. By (2.6) and (2.7) we get
where the constants M and m are defined in Lemma 2.1.
Let $\beta (t)=K(u)(t)=a_{1}+a_{0}(u(t)\min \{u(s):s\in [0,T]\})$. The function β satisfies
Using (3.24), (3.25), and the definition of $\beta (t)$, we obtain
Putting $a_{2}=a_{1}+\frac{T}{4}a_{1}\Psi_{+} $ and combining with (3.20), we arrive at
The definition of $a_{0}$ and (3.29) imply
From $a_{1}\leq \beta (t)\leq a_{2}$ we get
Substituting (3.30) into (3.26), we arrive at
which, together with (3.19) and (3.27), gives
that is,
Consequently, by Definition 2.2, (3.33), and (3.28), we get that $\beta (t)$ is an upper function to problem (1.7)–(1.8) and $\beta (t)>1$.
Now, we consider the case of $\Psi_{}=0$. For this case, it is easy to see that $\psi (t)\geq 0$ for a.e. $t\in [0,T]$. By Lemma 2.2, when $q(t)=0$, there is a solution u to (2.2)–(2.3) satisfying (3.24) and (3.25). Let $\beta (t)=K(u)(t)=a_{1}+a_{0}(u(t)\min \{u(s):s\in [0,T]\})$. Then $\beta (t)\geq 0$ for all $t\in [0,T]$, and (3.26) can be rewritten as
Since $\beta (t)\psi (t)\geq 0$ for a.e. $t\in [0,T]$, we get
which, together with (3.19), yields
that is,
Consequently, by Definition 2.2, (3.28), and (3.34) we get that $\beta (t)$ is an upper function to problem (1.7)–(1.8) and $\beta (t)>1$. □
Theorem 3.4
Let $\varphi ,r \in L([0,T],R)$, and let $r(t)\ge 0$ for a.e. $t \in [0,T]$. Suppose that
Then there exists an upper function $\beta (t)$ to problem (1.7)–(1.8), and $\beta (t)>1$.
Proof
Put $z(t,x)=\varphi (t) \frac{r(t)}{x^{\mu +1}}$, $(t,x)\in [0,T]\times (0,+\infty )$. Then
Using the definition of $Z(x)$, we easily obtain that $Z(x)\rightarrow T\bar{\varphi }$ as $x\rightarrow +\infty $. Furthermore, by Lemma 2.3 we get
The condition $r(t)>0$, a.e. $t\in [0,T]$, gives that
Also, (3.36) implies that there exist a constant $\varepsilon _{0}\in (0,1)$ (small enough) and a constant $\rho \in (1,+\infty )$ (large enough) such that
and
We consider the case $Z_{}(a_{1})>0$ as the first case. Put $a_{1}\in (\rho ,+\infty )$ and $a_{2}= a_{1}+\frac{T}{4}a_{1}Z_{+}(a _{1})$. Then $a_{2}\in (\rho ,+\infty )$. Let
Clearly,
By Lemma 2.2 there exists a solution u to (2.2)–(2.3) satisfying (2.6) and (2.7). In view of (2.6) and (2.7), we get
where the constants M and m are defined in Lemma 2.1. Put
Then (2.2) can be rewritten as
a.e. $t\in [0,T]$. So
From (3.37), (3.38), and the definition of $a_{2}$ it follows that
Using (3.48) and the definition of $a_{0}$, we get
Because of $a_{1}\leq \beta (t)\leq a_{2}$, we have
Substituting this into (3.45), we obtain
a.e. $t\in [0,T]$, that is,
By the inequality $\beta (t)\ge a_{1}$, $t\in [0,T]$ (see (3.46)), we get that
From (3.51) and (3.47) we see that $\beta (t)$ is an upper function to problem (1.7)–(1.8), and $\beta (t)>1$ for all $t\in [0,T]$.
Now, we consider the remaining case $Z_{}(a_{1})=0$. Under this situation, we have $z(t,a_{1})\ge 0$ for $t\in [0,T]$, which implies
that is,
Obviously, $\beta (t)=a_{1}$ is an upper function to problem (1.7)–(1.8). □
Existence theorems
Theorem 3.5
Let $\varphi ,r\in L([0,T],R)$ be such that
Let, moreover,
Then there exists at least one positive Tperiodic solution to problem (1.7).
Proof
By Theorems 3.1 and 3.3 there exist a lower function $0<\alpha (t)<1$ and an upper function $\beta (t)>1$. Therefore, the result can be obtained directly from Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.1
Consider the following secondorder differential equation:
where $\mu \in (0,+\infty )$ is a constant.
From the equation we see that $\varphi (t)=2\sin {t}$ and $r(t)=2+\sin {t}$; obviously, $r, \varphi \in L([0,T],R)$, and $1\leq r(t)\leq 3$. By direct calculation we get
and we arrive at
Consequently, all the conditions of Theorem 3.5 are satisfied. So by this theorem we get that there exists at least one positive Tperiodic solution to equation (3.52).
Theorem 3.6
Let $\varphi , r\in L([0,T],R)$ be such that
and
Then there exists at least one positive Tperiodic solution to problem (1.7).
Proof
By Theorems 3.1 and 3.4 there exist a lower function $0<\alpha (t)<1$ and an upper function $\beta (t)>1$. Therefore, the result is a direct consequence of Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.2
Consider the following secondorder differential equation:
where $\mu \in (0,+\infty )$ is a constant.
From the equation we see that $\varphi (t)=1\frac{1}{2} \sin {t}$ and $r(t)=1+\frac{1}{2}\cos {t}\geq 0$; obviously, $r, \varphi \in L([0,T],R)$. By direct calculation we have
and we arrive at
Consequently, all the conditions of Theorem 3.6 are satisfied, and so by this theorem there exists at least one positive Tperiodic solution to equation (3.53).
Theorem 3.7
Let $\varphi , r\in L([0,T],R)$. Suppose that the following assumptions hold:

(1)
$r(t)\ge 0$ for a.e. $t\in [0,T]$, $\bar{r}>0$, and $\operatorname{ess} \sup \{r(t)\}<+\infty $;

(2)
$\frac{T}{4}\int_{0}^{T}[\varphi (s)]_{}\,ds<1$;

(3)
$\frac{T}{4}\int_{0}^{T}[\varphi (t)]_{+}\,ds\int_{0}^{T}[\varphi (t)]_{}\,ds \leq \int_{0}^{T}[\varphi (t)]_{}\,ds\int_{0}^{T}[\varphi (t)]_{+}\,ds$.
Then there exists at least one positive Tperiodic solution to problem (1.7).
Proof
By Theorems 3.2 and 3.3 there exist a lower function $0<\alpha (t)<1$ and an upper function $\beta (t)>1$. Therefore, the result is a direct consequence of Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.3
Consider the following secondorder differential equation:
where $\mu \in (0,+\infty )$ is a constant, and $r: R\rightarrow R$ is Tperiodic with
From equation (3.54) we see that $\varphi (t)=5+2\sin {t}$. Obviously, φ, $r\in L([0,T],R)$, and condition (1) is satisfied. By direct calculation we have
This inequality implies that condition (2) is satisfied. Also, we can get the equalities
and
which results in
Consequently, all the conditions of Theorem 3.7 are satisfied, so by this theorem there exists at least one positive Tperiodic solution to equation (3.54).
Theorem 3.8
Let φ, $r\in L([0,T],R)$ satisfy $\bar{r}>0$. Assume that the following conditions hold:

(1)
$r(t)\ge 0$ for a.e. $t\in [0,T]$;

(2)
$\frac{T}{4}\int_{0}^{T}[\varphi (s)]_{}(s)\,ds<1$;

(3)
$\frac{T}{4}\int_{0}^{T}[\varphi (s)]_{+}\,ds\int_{0}^{T}[\varphi (s)]_{}\,ds< \int_{0}^{T}[\varphi (s)]_{}\,ds\int_{0}^{T}[\varphi (s)]_{+}\,ds$.
Then there exists at least one positive Tperiodic solution to problem (1.7).
Proof
By Theorems 3.2 and 3.4 there exist a lower function $0<\alpha (t)<1$ and an upper function $\beta (t)>1$. Therefore, the result follows immediately from Proposition 2.1 and Remark 1.1 in Sect. 1. □
Example 3.4
Consider the following secondorder differential equation:
where $f\in C((0,+\infty ),R)$, $\mu \in (0,+\infty )$ is a constant, and $r: R\rightarrow R$ is Tperiodic defined as
From (3.55) we see that $\varphi (t)=1\frac{1}{2} \sin {t}$. Obviously, φ, $r\in L([0,T],R)$, and condition (1) is satisfied. By direct calculation we have
This implies that condition (2) is satisfied. Furthermore, we can also get the equalities
and
which results in
Consequently, all the conditions of Theorem 3.8 are satisfied, so that by this theorem there exists at least one positive Tperiodic solution to equation (3.55).
Remark 3.1
Obviously, the conclusions associated with Examples 3.1–3.4 can be obtained neither by using the results of [7] nor by using the results of [16] (see Theorem 1.1), since the singularity term $\frac{r(t)}{x^{\mu }}$ is nonautonomous. Furthermore, even if $f(x)\equiv 0$ for $x\in (0,+\infty )$, the above conclusion associated with Example 3.4 cannot be deduced from the main theorem of [17] (Theorem 1 of [17]). This is due to the fact that the condition of $\delta \in [0,1)$ is required in [17].
Conclusions
In this paper, we study the periodic problem for Liénard equations with a singularity of attractive type in the case of $r(t)\ge 0$ for a.e. $t\in [0,T]$. The proofs of main results are based on the method of upper and lower functions. It is interesting that the singularity term $\frac{r(t)}{x^{\mu }}$ in (1.7) is nonautonomous, which generalizes the corresponding results in the known literature where $r(t)$ is a constant function. In the next research, we will continue to study the periodic problem to the singular equation like (1.7) where $r(t)$ is a changingsign function.
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Acknowledgements
The authors are grateful to anonymous referees for their constructive comments and suggestions, which have greatly improved this paper.
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The authors are gratefully acknowledge support from NSF of China (No. 11271197).
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Correspondence to Shiping Lu.
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Keywords
 Boundary value problem
 Periodic solution
 Singularity
 Upper and lower functions