## Periodic Solutions for a Singular Liénard Equation with Sign-Changing Weight Functions

Lu Shiping,, Zhou Shile,, Yu Xingchen,

 基金资助: 国家留学基金.  201908320531江苏省研究生科研创新项目.  SJKY19_0957

 Fund supported: the China Scholarship Council Project.  201908320531the Project of Innovation in Scientific Research for Graduate Students of Jiangsu Province.  SJKY19_0957

Abstract

In this paper, we study the existence of positive periodic solutions for a singular Liénard equationwhere $f: (0, +\infty)\rightarrow \mathbb{R}$ is continuous which may have a singularity at $x=0$, $\alpha$ and $\varphi$ are $T$ -periodic functions with $\alpha, \varphi\in L([0, T], \mathbb{R})$, $\mu\in(0, +\infty)$ and $\delta\in(0, 1]$ are constants. The signs of weight functions $\alpha(t)$ and $\varphi(t)$ are allowed to change on $[0, T]$. We prove that the given equation has at least one positive $T$ -periodic solution. The method of proof relies on a continuation theorem of coincidence degree principle.

Keywords： Periodic solution ; Singularity ; Continuation theorem ; Coincidence degree principle

Lu Shiping, Zhou Shile, Yu Xingchen. Periodic Solutions for a Singular Liénard Equation with Sign-Changing Weight Functions. Acta Mathematica Scientia[J], 2021, 41(3): 686-701 doi:

## 1 引言

$$$x''\pm\frac{1}{x^{\mu}} = p(t)$$$

$$$x''(t) = \frac{\alpha(t)}{x^\mu(t)}$$$

$$$\gamma_0<u(t)<M, \max\limits_{t\in[0, T]}|u'(t)|<M_1, \forall u\in \Sigma_1.$$$

$\begin{eqnarray} \Sigma_1& = &\Big\{x\in C_T^1: x''(t)+\lambda f(x(t))x'(t)-\lambda\varphi(t)x(t)+\frac{\lambda\alpha(t)}{x^\mu(t)} = 0, {}\\ &&{\quad} \lambda\in(0, 1], x(t)>0, t\in [0, T]\Big\}. \end{eqnarray}$

$$$x''(t)+f(x(t))x'(t)-\varphi(t)x^\delta(t)+\frac{\alpha(t)}{x^{\mu}(t)} = 0$$$

$$$x''(t)+\lambda f(x(t))x'(t)-\lambda\varphi(t)x^\delta(t)+\frac{\lambda \alpha(t)}{x^{\mu}(t)} = 0, \lambda\in(0, 1],$$$

$$$\rho\Big(RR''+\frac{3}{2}(R')^2\Big) = [P_v-P_{\infty}(t)]+P_{g_0}\Big(\frac{R_0}{R}\Big)^{3k}-\frac{2s}{R}-\frac{4\nu R'}{R}$$$

$$$x''+\frac{4\nu}{x^{\frac{4}{5}}}x'-\frac{5 [P_v-P_{\infty}(t)]}{2\rho}x^{\frac{1}{5}}+\frac{1}{x^{\frac{1}{5}}}\Big(5S-\frac{5 P_{g_0}R_0}{2\rho}\Big) = 0,$$$

## 2 预备定理

(1) $Lx\neq \lambda Nx, \forall x\in \partial\Omega \cap D(L), \lambda\in(0, 1)$;

(2) $Nx\notin{\rm Im }L, \forall x\in \partial\Omega \cap\ker L;$

(3) Brouwer度$\rm {deg} $$(\eta QN, \Omega\cap KerL, 0)\neq 0 . 其中, \eta:{\rm Im }L\rightarrow \ker L 是一个同构映射, 那么方程 Lx = Nx$$ \overline{\Omega}$上至少有一个解.

(H$_1$) $\alpha(t)\le0,$ a.e. $t\in J$; $\alpha(t)>0,$ a.e. $t\in[0, T]\backslash J$, 其中$J\subset [0, T]$是一个闭集.

(H$_2$) $\bar{\varphi}>h(\delta)\frac{T\overline{\varphi_{+}}(\overline{\varphi_{-}})^{\frac{1}{2}}}{2},$其中, $\delta\in(0, 1]$与方程(1.6)中的$\delta$相同,

$$$x''(t)+\lambda f(x(t))x'(t)-\lambda\varphi(t)x^\delta(t)+\lambda\frac{\alpha(t)}{x^{\mu}(t)} = 0, \lambda\in(0, 1).$$$

$$$D = \Big\{x\in C_T^1:x''+\lambda f(x)x'-\lambda \varphi(t)x^\delta+\frac{\lambda\alpha(t)}{x^\mu} = 0, \lambda \in(0, 1); x(t)>0, \forall t\in[0, T]\Big\},$$$

$$$G(x) = \int_1^x\frac{f(s)}{s^\delta}{\rm d}s, {\qquad} F(x) = \int_1^xf(s){\rm d}s.$$$

## 3 主要结果

$$$\varepsilon\in\Big(0, \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+\delta}}\Big),$$$

$$$\gamma_1\in (\varepsilon, A(\varepsilon))\cap\Big(0, \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+\delta}}\Big),$$$

$$$\sup\limits_{x\in[A(\varepsilon), +\infty)}G(x)<+\infty$$$

$$$\inf\limits_{x\in(0, \gamma_1]}\Big(G(x)-\frac{T\overline{\alpha_-}}{x^{\mu+\delta}}\Big)>\sup\limits_{x\in[A(\varepsilon), +\infty)}G(x)+T\overline{\varphi_+},$$$

令

$$$V = \{x\in C^1_T:x(t)>0, t\in[0, T]; x(t)>\varepsilon, t\in J\},$$$

$$$|u|_\infty<\frac{T\overline{\alpha_{+}}}{2\rho\bar{\varphi}}\Big(\frac{\overline{\varphi_+}}{\gamma_1^\mu}\Big)^\frac{1}{2}+\Big(\frac{\overline{\alpha_{+}}}{\rho\overline{\varphi}\gamma_{1}^{\mu}}\Big)^\frac{1}{2}: = \varrho_0,$$$

(2) $\delta\in(0, 1)$. 由于$\frac{(1+\delta)\delta}{2}<1$以及$\frac{\delta}{2}<1$, 从(3.18) 式我们很容易地知道, 存在常数$\varrho_1>0$使得

$$$||u||_{\infty}<\max\{\varrho_0, \varrho_1\}: = M_1.$$$

$u $$t_1\in[0, T] 处达到最大值, 那么 u'(t_{1}) = 0. 因此, 从(3.6)式可知 由此得 其中 F 的定义由(2.3)式给出. 因此, 我们有 $$\ \max\limits_{t\in[0, T]}|u'(t)|\leq M_{2}.$$ 定义 X = C_T^1, Y = L([0, T], {{\Bbb R}} ) 以及 此处, D(L) = \{x\in X: x''\in L([0, T], {{\Bbb R}} )\} , \ker L = {{\Bbb R}} , {\rm Im} L = \{y\in Y:\int^T_0y(t){\rm d}t = 0\}. 容易验证 L 是指标为零的Fredholm算子. 此外, 我们定义如下两个连续映射 显然, 我们有 \ker L = {\rm Im } P, \ker Q = {\rm Im } L . L_p = L|_{D(L)\cap \ker P}\rightarrow {\rm Im} L , 则 L_p 拥有逆映射 L_p^{-1}: {\rm Im } L\rightarrow D(L)\cap \ker P . 定义 通过简单的计算, 我们有 $$(K_py)(t) = \int_0^T G(t, s)y(s){\rm d}s, \quad G(t, x) = \left\{\begin{array}{ll} { } \frac{(t-T)s}{T}, & 0\leq s<t\leq T;\\ { } \frac{(s-T)t}{T}, & 0\leq t<s\leq T.\end{array}\right.$$ 定义 $$N:\overline{\Omega}\rightarrow Y, (Nx)(t) = -f(x(t))x'(t)+\varphi(t)x^\delta(t)-\frac{\alpha(t)}{x^{\mu}(t)}.$$ 结合(3.23)和(3.24)式, 我们可以证明算子 K_p(I-Q)N$$ QN$在集合$\overline{\Omega}$上是相对紧的. 因此, $N $$\overline{\Omega} 上是 L -紧的. 此外, 由(3.2)式可知 \gamma_1>\varepsilon , 再根据 V, V_1 的定义, 我们可知 (\partial V\cap \overline{V_1}) = \emptyset . 更进一步地, 我们从 \partial\Omega\subset (\partial V\cap \overline{V_1})\cup(\overline{V}\cap \partial V_1) 可以推出 \partial\Omega\subset\overline{V}\cap \partial V_1 . 根据这个包含关系, 我们可以总结出引理2.2中的条件(1)成立. 事实上, 若其不成立, 则存在 \lambda_0\in (0, 1) 以及 x_0\in \partial\Omega 使得 根据 \partial\Omega\subset\overline{V}\cap \partial V_1 , 我们可得 $$x_0\in \overline{V}\cap \partial V_1,$$ 因此, 下式成立 但是由(3.13), (3.21)和(3.22)式, 可知 再次依据 V_1 的定义, 我们可知 x_0\notin \partial V_1 , 这与(3.25)式矛盾. 如果 x\in(\overline{V}\cap \partial V_1) \cap \ker L , 那么 x(t)\equiv\gamma_{1}$$ x(t)\equiv m_{1}$. 据此, 我们有

$$$QNx\neq0, \forall x\in\partial\Omega\cap \ker L.$$$

$$$\deg\{\eta QN, \Omega\cap \ker L, 0\}\neq 0,$$$

$$$\gamma_2\in (\varepsilon, A(\varepsilon))\cap\Big(0, \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+1}}\Big),$$$

$$$\inf\limits_{x\in(0, \gamma_2]}\Big(F(x)-\frac{T\overline{\alpha_+}}{x^{\mu}}\Big)>\sup\limits_{x\in[A(\varepsilon), +\infty)}F(x),$$$

$$$\gamma_3\in (\varepsilon, A(\varepsilon))\cap\Big(0, \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+1}}\Big),$$$

$$$\sup\limits_{x\in(0, \gamma_3]}\Big(F(x)+\frac{T\overline{\alpha_+}}{x^{\mu}}\Big)<\inf\limits_{x\in[A(\varepsilon), +\infty)}F(x),$$$

(命题1的证明)   设方程(1.6)存在一个$T$ -周期正解$y$, 则

$$$y''(t)+ f(y(t))y'(t)-\varphi(t)y^\delta(t)+\frac{\alpha(t)}{y^{\mu}(t)} = 0.$$$

(命题2的证明)  由于条件(H$_1$)成立, 且$\bar{\alpha}>0$, 我们仍用(3.5)式定义集合$V$. 假设$u\in \overline{V} $$u 为(2.1)式的一个 T -周期正解, 则 $$u''(t)+\lambda f(u(t))u'(t)-\lambda\varphi(t)u^\delta(t)+\frac{\lambda\alpha(t)}{u^{\mu}(t)} = 0.$$ 类似于(3.11)式的证明, 可以得到存在常数 \xi\in [0, T] , 使得 $$u(\xi)\geq A(\varepsilon)>\varepsilon.$$ 另一方面, 容易验证存在 t_{3}, t_{4}\in {{\Bbb R}} 满足 $$0<t_{4}-t_{3}\le T, \quad u(t_{3}) = \max\limits_{t\in[0, T]}u(t), \quad u(t_{4}) = \min\limits_{t\in[0, T]}u(t).$$ 根据(3.34)式得到 由此可推出 对(3.33)式在区间 [t_3, t_4] 上积分, 可得 \begin{eqnarray} F(u(t_{4}))& = & F(u(t_{3}))+\int^{t_{4}}_{t_{3}}\varphi(s)u^\delta(s){\rm d}s-\int^{t_{4}}_{t_{3}}\frac{\alpha(s)}{u^{\mu}(s)}{\rm d}s{}\\ &\leq & B_{1}+\int^{T}_{0}\varphi(s)u^\delta(s){\rm d}s-\int^{t_{4}}_{t_{3}}\frac{\alpha(s)}{u^{\mu}(s)}{\rm d}s. \end{eqnarray} 此外, 对(3.33)式在区间 [0, T] 上积分, 可得 将上式代入(3.36)式, 我们有 根据(3.29)式, 我们可以给出下界的估计 $$\min\limits_{t\in[0, T]}u(t) = u(t_{4})>\gamma_{2}.$$ 在(3.33)式两端同时乘以 u^\mu , 且对其在区间 [0, T] 积分, 再根据不等式 我们可以得到 由积分中值定理, 得到存在常数 \eta_0\in[0, T] , 使得 $$\ u(\eta_0)\leq\Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+\delta}}.$$ 结合引理2.1和(3.38)式, 我们得到 $$\ ||u||_{\infty}\leq \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+\delta}}+\frac{\sqrt{T}}{2}\Big(\int_0^T|u'(s)|^2{\rm d}s\Big)^{\frac{1}{2}}.$$ 再次在(3.33)式两端乘以 u , 并对其在区间 [0, T] 上积分, 可得 根据条件 \varphi(t)\geq0 a.e. t\in[0, T] 以及(3.37)式, 我们可以从上式推出 将(3.39)式代入上式, 可得 这意味着存在一个常数 \rho_{3}>0 , 使得 因此, 从(3.39)式可以推得 $$\ ||u||_{\infty}\leq \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+1}}+\frac{\sqrt{T}}{2}\rho_{3}: = M_{3}.$$ 至此, 我们已经证明了(3.37)和(3.40)式成立. 剩下部分的证明与定理3.1中的证明相同. (命题3的证明) 命题2和3的证明十分相似, 唯一的不同点在于方程(2.1)所有可能 T -周期正解的先验下界的估计. 类似地, 我们可以证明(3.34)式成立, 且存在常数 t_3, t_4\in{{\Bbb R}} 满足(3.35)式. 因此, 我们有 另一方面, 对(3.33)式在区间 (t_3, t_4) 进行积分, 可以得到 结合条件(3.31), 我们得到 证毕. 推论3.1 设 \bar{\alpha}>0 , 且下述关系式成立 $$\lim\limits_{x\rightarrow +\infty}F(x)<+\infty,$$ $$\lim\limits_{x\rightarrow 0^+}\Big(F(x)-\frac{T\overline{\alpha_+}}{x^{\mu}}\Big) = +\infty.$$ 那么方程(1.6)至少存在一个 T -周期正解. 显然, 由(3.41)和(3.42)式, 得到 \sup\limits_{x\in[A(\varepsilon), +\infty)}F(x)<+\infty , (3.28)和(3.29)式都成立. 因此, 该推论的结论可由定理3.2中的命题2直接得到. 类似地, 运用定理3.2中的命题3, 我们可以得到下述推论. 推论3.2 设 \bar{\alpha}>0 , 且下述关系式成立 那么方程(1.6)至少存在一个 T -周期正解. 定理3.3 假设 \alpha(t)\ge 0, a.e. t\in[0, T]$$ \bar{\alpha}>0$. 则下述命题成立:

$$$\gamma_4\in \Big(0, \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+\delta}}\Big)$$$

$$$\inf\limits_{x\in(0, \gamma_4]}G(x)>\sup\limits_{x\in[(\frac{\bar{\alpha}}{\bar{\varphi}})^{\frac{1}{\mu+\delta}}, +\infty)}G(x)+T\overline{\varphi_+},$$$

$$$\gamma_5\in \Big(0, \Big(\frac{\bar{\alpha}}{\bar{\varphi}}\Big)^{\frac{1}{\mu+\delta}}\Big),$$$

$$$\sup\limits_{x\in(0, \gamma_5]}G(x)<\inf\limits_{x\in\big[\big(\frac{\bar{\alpha}}{\bar{\varphi}}\big)^{\frac{1}{\mu+\delta}}, +\infty\big)}G(x)-T\overline{\varphi_+},$$$

(命题1的证明)   设方程(1.6) 存在一个$T$ -周期正解$c$, 则

$$$c''(t)+ f(c(t))c'(t)-\varphi(t)c^\delta(t)+\frac{\alpha(t)}{c^{\mu}(t)} = 0.$$$

$$$\int^T_0\frac{\alpha(t)}{c^{\mu+\delta}(t)}{\rm d}t\leq T\bar{\varphi}.$$$

$$$\lim\limits_{x\rightarrow +\infty}G(x)<+\infty,$$$

$$$\lim\limits_{x\rightarrow0^+}G(x) = +\infty.$$$

由(3.53)式知$\sup\limits_{x\in[(\frac{\bar{\alpha}}{\bar{\varphi}})^{\frac{1}{\mu+\delta}}, +\infty)}G(x)<+\infty$. 再根据(3.54)式知, 存在一个常数$\gamma_4\in \big(0, \big(\frac{\bar{\alpha}}{\bar{\varphi}}\big)^{\frac{1}{\mu+\delta}}\big)$, 使得(3.44)式成立. 运用定理3.3中的命题2, 我们可以直接得到结论.

$\bar{\varphi}>0$是方程(1.6)存在$T$ -周期正解的充要条件.

必要性的证明类似于定理3.3中的命题1的证明. 由$\bar{\varphi}>0 $$\delta\in(0, 1) , 易知假设 (H_{2}) 成立. 因此, 由推论3.4 (或3.3)可证明充分性. 接下来证明Rayleigh-Plesset方程(1.9)的 T -周期正解的存在性. 引理3.1[13, 定理3.4] 假设 P_v<\overline{P_\infty} , 且 S<\frac{P_{g_0}R_0}{2\rho} . \nu> 2\rho\overline{[P_\infty(t)-P_v]_+} , 则Rayleigh-Plesset方程(1.9) 至少存在一个 T -周期正解. 引理3.2[12, 定理4.4] 假设 P_v> \overline{P_\infty} , 2\rho S> P_{g_0}R_0 , 如果 $${\rm ess}\inf\Big\{P_\infty(t): t\in[0, T]\Big\}>-\infty,$$ 则Rayleigh-Plesset方程(1.9)至少存在一个 T -周期正解. 易知方程(1.9)是方程(1.6)的特殊形式, 其中 f(x) = \frac{4\nu}{x^{\frac{4}{5}}} , \varphi(t) = \frac{5 [P_v-P_{\infty}(t)]}{2\rho} , \alpha(t)\equiv\big(5S-\frac{5 P_{g_0}R_0}{2\rho}\big) , \mu = \delta = \frac{1}{5} . 显然, G(x) = \int_1^x\frac{f(s)}{s^\delta}{\rm d}s = 4\nu \ln x, x\in(0, +\infty) , 则有 \lim\limits_{x\rightarrow0^+}G(x) = -\infty , 且 \lim\limits_{x\rightarrow +\infty}G(x) = +\infty . 因此, 由推论3.5, 我们可以得到以下结论. 定理3.4 假设 S>\frac{P_{g_0}R_0}{2\rho} . 则Rayleigh-Plesset方程(1.9) 至少存在一个 T -周期正解的充分必要条件为 P_v>\overline{P_\infty} . 注3.1 由于方程(1.6)中的Liénard项 f(x)x' 的系数 f(x) 允许其在 x = 0 具有奇性, 且 \varphi(t)$$ \alpha(t) $$[0, T] 上都可变号, 因此本文完善了文献[15-19]和[23]中的研究工作. 此外, 条件(3.4), (3.29)和(3.31)在估计方程(1.7)周期解的先验正下界过程中, 起着至关重要的作用. 所以 f(x)$$ x = 0$处的奇性有助于方程周期解的存在.

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Torres P J . Mathematical Models with Singularities: A Zoo of Singular Creatures. Paris: Atlantis Press, 2015

Lazer A C , Solimini S .

On periodic solutions of nonlinear differential equations with singularities

Proc Amer Math Soc, 1987, 99 (1): 109- 114

Habets P , Sanchez L .

Periodic solutions of some Liénard equations with singularities

Proc Amer Math Soc, 1990, 109, 1035- 1044

Zhang M .

Periodic solutions of Liénard equations with singular forces of repulsive type

J Math Anal Appl, 1996, 203 (1): 254- 269

Jebelean P , Mawhin J .

Periodic solutions of singular nonlinear perturbations of the ordinary p -Laplacian

J Adv Nonlinear Stud, 2002, 2 (3): 299- 312

Torres P J .

Weak singularities may help periodic solutions to exist

J Differ Equa, 2007, 232, 277- 284

Chu J , Torres P J , Zhang M .

Periodic solutions of second order non-autonomous singular dynamical systems

J Differential Equations, 2007, 239, 196- 212

Li X , Zhang Z .

Periodic solutions for second order differential equations with a singular nonlinearity

Nonlinear Anal, 2008, 69, 3866- 3876

Hakl R , Torres P J , Zamora M .

Periodic solutions of singular second-order differential equations: the repulsive case

Topol Method Nonl Anal, 2012, 39, 199- 220

Chu J , Liang Z , Liao F , Lu S .

Existence and stability of periodic solutions for relativistic singular equations

Commun Pure Appl Anal, 2017, 16 (2): 591- 609

Wang Z .

Periodic solutions of Liénard equation with a singularity and a deviating argument

Nonlinear Anal: Real World Appl, 2014, 16, 227- 234

Cheng Z , Ren J .

Positive solutions for fourth-order singular nonlinear differential equation with variable-coefficient

Math Methods Appl Sci, 2016, 39, 2251- 2274

Hakl R , Torres P J , Zamora M .

Periodic solutions of singular second-order differential equations: Upper and lower functions

Nonlinear Anal, 2011, 74, 7078- 7093

Hakl R , Torres P J , Zamora M .

Periodic solutions of singular second-order differential equations: the repulsive case

Topol Method Nonl Anal, 2012, 39, 199- 220

Hakl R , Torres P J .

On periodic solutions of second-order differential equations with attractive-repulsive singularities

J Differential Equations, 2010, 248 (1): 111- 126

Chu J , Torres P J , Wang F .

Twist periodic solutions for differential equations with a combined attractive-repulsive singularity

J Math Anal Appl, 2016, 437, 1070- 1083

Bravo J L , Torres P J .

Periodic solutions of a singular equation with indefinite weight

Adv Nonlinear Stud, 2010, 10 (4): 927- 938

Hakl R , Zamora M .

Periodic solutions to second-order indefinite singular equations

J Differential Equations, 2017, 263, 451- 469

Urena A J .

Periodic solutions of singular equations

Topol Methods Nonlinear Anal, 2016, 47, 55- 72

Pishkenari H N , Behzad M , Meghdari A .

Nonlinear dynamic analysis of atomic force microscopy under deterministic and random excitation

Chaos Solitons Fractals, 2008, 37 (3): 748- 762

Torres P J , Zhang M .

A monotone iterative scheme for a nonlinear second order equation based on a generalized antimaximum principle

Math Nachr, 2003, 251, 101- 107

Towers I , Malomed B A .

Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity

J Opt Soc Amer B, 2002, 19 (3): 537- 543

Lu S , Guo Y , Chen L .

Periodic solutions for Liénard equation with an indefinite singularity

Nonlinear Analysis: Real World Appl, 2019, 45, 542- 556

Gaines R E , Mawhin J L . Coincidence Degree, and Nonlinear Differential Equations. Berlin: Springer, 1997

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