Multiplicity of periodic solutions for weakly coupled parametrized systems with singularities

1.   Introduction

In 2010, Fonda and Ghirardelli ^[1] established a multiplicity result for the parametrized equation

Here, $g:[0, T]\times\Bbb R\rightarrow \Bbb R$ is a Carathéodory function, and $w:[0, T]\times\Bbb R\rightarrow \Bbb R$ is integrable. Their result generalized the classical result of Lazer and McKenna ^[2], dated 1987, and extended the results of Del Pino et al. ^[3] and Zanini and Zanolin ^[4]. In 2017, Calamai and Sfecci ^[5] extended this multiplicity result to the weakly coupled parametrized system

where $g_i:\Bbb R\times\Bbb R^N\rightarrow \Bbb R$ and $w_i:\Bbb R\rightarrow \Bbb R$ are continuous functions, $x$ is the vector $(x_1, \cdot\cdot\cdot, x_N)$, and $s$ is a real parameter. The functions $g_i$ and $w_i$ are assumed to be $T$-periodic in the time variable. The proof is based on the higher dimensional Poincaré-Birkhoff theorem obtained by Fonda and Ureña ^[6]. Unlike the result in ^[1], Calamai and Sfecci ^[5] do not assume the Lipschitz regularity condition on the functions $g_i$, which is a crucial assumption in ^[1]. Relevant related results can be found in ^{[7,8,9,10,11]}.

On the other hand, Boscaggin et al. ^[12] investigated the parametrized equation (1.1) with a suitable singularity of repulsive type at the origin and linear growth at infinity. They obtained a multiplicity of periodic solutions for Eq (1.1). In order to guarantee uniqueness of the associated Cauchy problems, they also assumed the Lipschitz regularity on the function $g$. For related results on periodic solutions of singularity equations, see also ^{[9,13,14,15,16,17,18,19,20]}. Furthermore, high-order nonlinear systems with nonlinear parameterization and other related stability problems have attracted some authors' attention; see, for instance, ^{[21,22,23,24,25,26]}.

Motivated by the works of ^[12] and ^[5], a natural inquiry arises as to whether weakly coupled parametrized systems with a singularity of repulsive type at the origin and linear growth at infinity possess multiple periodic solutions. In this paper, we will consider the weakly coupled parametrized system

For each index $i = 1, \cdot\cdot\cdot, N$, we assume the following hypotheses hold.

$(H_0)$ The functions $\varphi_i:[0, T]\times(0, +\infty)\rightarrow \Bbb R$ and $w_i:[0, T]\rightarrow \Bbb R$ are continuous, and $s$ is a real parameter.

$(H_1)$ There exists a function $H:[0, T]\times\Bbb R^N\rightarrow \Bbb R$ such that

where $p_i:[0, T]\times\Bbb R^N\rightarrow \Bbb R$ is continuous and bounded.

$(H^i_2)$ There exists a continuous function $f:(0, \delta]\rightarrow \Bbb R$ satisfying

and

$(H^i_3)$ There exists a function $a_i(t)$ such that

$(H^i_4)$ There exist $a^i_\pm$ and an integer $m_i\geq0$ such that

Moreover, the unique solution to

is strictly positive.

Due to the absence of Lipschitz regularity assumptions in our system, uniqueness of the solution cannot be guaranteed for the corresponding Cauchy problem. To address this issue, we employ the higher dimensional Poincaré-Birkhoff theorem obtained by Fonda and Ureña ^[6]. Moreover, the solution trajectories of high-dimensional systems are inherently intricate. To characterize the twisting properties of the solution, we project the high-dimensional system's solution onto the plane and utilize sophisticated phase plane analysis methods.

Throughout the paper, we define $\left[\frac{m_i-1}{2}\right]$ as the greatest integer that is less than $\frac{m_i-1}{2}$. The main results of this paper are presented below.

Theorem 1.1. Assume that Hamiltonian system (S) satisfies $(H_0)-(H_1)$ and $(H^i_2)-(H^i_4)$, for every index $i = 1, 2, \cdot\cdot\cdot, N$. Then, there exists $s_0 > 0$ such that, for every $s\geq s_0$, the Hamiltonian system (S) has at least

periodic solutions.

Remark 1.2. Similarly, we can obtain the multiplicity of periodic solutions for system (S) by the Carathéodory type of regularity. Taking $N = 1$, Theorem 1.1 leads to the existence of $1+2(\left[\frac{m_1-1}{2}\right]+1)$ periodic solutions. Namely, if $m_1$ is odd, the Hamiltonian system (S) has at least $m_1+2$ periodic solutions. If $m_1$ is even, there exist $m_1+1$ periodic solutions for the system (S). Without assuming the uniqueness of solutions associated with the Cauchy problem, this result extends Theorem 1.1 in ^[12] to weakly coupled parametrized systems with continuous nonlinearities.

Remark 1.3. The function $\varphi_i(t, x_i)$ in system (S) is singular at the origin and merely continuous without any Lipschitz regularity assumptions. Therefore, Theorem 1.1 extends the results of [5,Theorem 1.3] to the weakly coupled parametrized systems with singularities.

The remaining sections of this paper are organized as follows. Section 2 introduces the basic concept of the $i$-th rotation number and provides some auxiliary lemmas for system (S). In Section 3, we present some auxiliary lemmas for system (P) below and provide a proof of Theorem 1.1.

2.   Preliminaries

If the component $(x_i(t), x'_i(t))$ of $(x(t), x'(t))\in\Bbb R^{2N}$ does not attain the origin, we can transform to the standard polar coordinates as

Thus, if $(x_i(t), x'_i(t))\neq(0, 0)$ for $t\in[\tau_0, \tau_1]$, we can define the $i$-th rotation number of $(x(t), x'(t))$ along that interval as

Here, $\mathrm{Rot}((x_i(t), x'_i(t)); [\tau_0, \tau_1])$ describes clockwise rotations performed by the path of $(x(t), x'(t))$ around the origin in the time interval $[\tau_0, \tau_1]$ and $(x_i, x'_i)$ phase-plane. The modified version of the $i$-th rotation number of $(x(t), x'(t))$ on $[\tau_0, \tau_1]$ is defined as

For more details about the modified rotation numbers, please see ^[11,27]. For $i\in\{1, \cdot\cdot\cdot, N\}$, let

which is a function similar to the "norm" in the phase-plane. For more details on the function, see [12,page 4461].

Remark 2.1. Similar to [11,Theorem 4 and Remark 1], for every integer $j$, we have

Lemma 2.2. There exist $\tilde{s} > 0$ and two positive constant $c_0 < C_0$ such that, for every $s\geq\tilde{s}$, system (S) has a solution $\hat{x} = \hat{x}(s, t)$ whose components satisfy

for every $t\in[0, T]$ and $i\in\{1, \cdot\cdot\cdot, N\}$.

Proof. Consider the truncated function

for $t\in[0, T]$ and $x\in\Bbb R^N$. By $(H_1)$, there exists a constant $M > 0$ such that

for every $(t, x)\in[0, T]\times\Bbb R^N$ and $i = 1, \cdot\cdot\cdot, N$. Consequently, by $(H^i_3)$, we have

uniformly for every $t\in[0, T]$. Consider the system

where $z_i(t) = x_i(t)/s$. By using [5,Lemma 2.3], we have that there are three positive constants $\bar{s}, c_0$ and $C_0$ such that, for every $s\geq\bar{s}$, system (2.2) has a solution $z = z(s, t)$ whose components satisfy $c_0\leq z_i(s, t)\leq C_0$ for every $t\in[0, T]$ and $i\in\{1, \cdot\cdot\cdot, N\}$. Hence, there exists a solution $\hat{x}(s, t)$ of

whose components satisfy (2.1) for every $t\in[0, T]$ and $i\in\{1, \cdot\cdot\cdot, N\}$. Clearly, (2.1) implies that $\hat{x}_i(s, t)\rightarrow +\infty$ as $s\rightarrow +\infty$ for every $t\in[0, T]$ and $i\in\{1, \cdot\cdot\cdot, N\}$. Then, there exists $\tilde{s} > \bar{s}$ such that, $\hat{x}_i(s, t) > 1$ for every $t\in[0, T]$ and $i\in\{1, \cdot\cdot\cdot, N\}$, and hence $\hat{x}_i(s, t)$ is also a solution of system (S).

Remark 2.3. Note that Remark 2.2 in ^[5] remains valid even when $\nu^{i}_{1}(t) = \nu^{i}_{2}(t) = 0$. Therefore, the conclusion of Lemma 2.3 in ^[5] is also valid when

uniformly for every $t\in[0, T]$. Our proof of Lemma 2.2 relies on this fact.

To simplify the proof below, we define $g_i(t, x) = \varphi_i(t, x_i) + p_i(t, x)$, where $\varphi_i$ and $p_i$ are defined as in $(H_0)-(H_1)$ and $(H^i_2)-(H^i_4)$. Then, the system $(S)$ can be written as

We can verify that $g_i(t, x)$ satisfies the hypotheses $(H_0)'$, $(H^i_2)'$ and $(H^i_3)'$ as follows.

$(H_0)'$ The function $g_i(t, x):[0, T]\times(0, +\infty)^N\rightarrow \Bbb R$ is continuous.

$(H^i_2)'$ There is a continuous function $\tilde{f}:(0, \delta]\rightarrow \Bbb R$ satisfying

and

$(H^i_3)'$ $\lim_{x_i\rightarrow +\infty}\frac{g_i(t, x)}{x_i} = a_i(t)$, uniformly for every $t\in[0, T]$.

Lemma 2.4. For every $s\in\Bbb R$, the solution to the Cauchy problem

is globally defined on $[0, T]$.

Proof. Let $x(t)$ be a solution of the system (2.3). Assume the contrary, that is, there is a component $x_{i_0}(t)$ of $x(t)$ whose maximal interval of definition is $[0, \tau)$ for $\tau < T$. By standard arguments in the theory of initial value problems, we have

By the arguments in [28,Lemma 1], we have

On the other hand, by using the computation in [28,Lemma 2], we find that $\mathrm{Rot}((x_{i_0}(t)-1, x'_{i_0}(t)); [0, \tau])$ is bounded, which is a contradiction with (2.4).

Lemma 2.5. There exists $\hat{R}_s > 0$ such that, if $x:[0, T]\rightarrow \Bbb R^N$ is a solution of (S') with $\mathcal{N}_{i}(x(t), x'(t))\geq\hat{R}_s$ for a certain index $i$ and every $t\in[0, T]$, then

Proof. For simplicity, we assume $m_i = 1$. Take $\hat{R}_s$ such that

Let $x(t)$ be a solution of (S') with $\mathcal{N}_{i}(x(t), x'(t))\geq\hat{R}_s$ for a certain index $i$ and every $t\in[0, T]$. We will show that

That is, in the $(x_i(t), x'_i(t))$-phase plane, $(x(t), x'(t))$ performs more than one turn around $(\hat{x}_i(s, 0), \hat{x}'_i(s, 0))$ in the time interval $[0, T]$.

Writing $(x_i(t)-\hat{x}_i(s, 0), x'_i(t)-\hat{x}'_i(s, 0)$ in polar coordinates,

we can deduce that

for every $t\in[0, T]$. With fixed $\alpha\in((\pi/T)^2, a^{i}_-)$, by $(H^i_3)'$ and $(H^i_4)$, we can choose $d > \hat{x}_i(s, 0)$ such that

for every $t\in[0, T]$ and every $x_i(t)\geq d$.

We first consider the case when $x_i(0) > d$. The proof will be divided into three steps.

Step 1. We claim that there exists $t_1 \in (0, T]$ such that $x_i(t_1) = d$ and $x_i(t) > d$ for every $t \in [0, t_1)$ (see Figure 1).

Suppose, contrary to our claim, that $x_i(t) > d$ for every $t\in[0, T]$. Note that, enlarging $\hat{R}_s$, by $\mathcal{N}_{i}(x(t), x'(t))\geq\hat{R}_s$, we have

for $x_i(t) > d$, where $\eta > 0$ is sufficiently small such that

Combining (2.6), (2.7) with (2.8), we have

that is,

for every $t\in[0, T]$. Notice that

for every $t\in[0, T]$ and $\alpha\in((\pi/T)^2, a^{i}_-)$. Indeed, if $1\leq\alpha < a^{i}_-$, for every $t\in[0, T]$, one has

If $(\pi/T)^2 < \alpha < 1$, we have

for every $t\in[0, T]$. Hence,

By (2.11) and (2.12), we have

which contradicts (2.9).

Step 2. If $x_i(t)\geq d$, by the computation in Step 1, then

On the other hand, if $x_i(t)\in(0, d)$, since $\mathcal{N}_{i}(x(t), x'(t))$ is large for every $t\in[0, T]$, either $x_i(t)$ is near the singularity, or $|x'_i(t)|$ is large. By $(H^i_2)'$, we have

uniformly for every $t\in[0, T]$. Arguing as in [28,Lemma 2], inequality (2.14) holds. Thus, up to enlarging $\hat{R}_s$, we can find $t_2\in(t_1, T]$ such that $x_i(t_2) = d$, $x'_i(t_2) > 0$ and $x_i(t)\in(0, d)$ for every $t\in(t_1, t_2)$ (see Figure 1). Moreover, by a similar argument as that in [28,Lemma 2], we have

Step 3. Note that there are three possible trajectories for the solution when $t > t_2$ (see Figure 1). First, we will prove that there exists $t'\in(t_2, T)$ such that

for the trajectories $(i)$ and $(ii)$. If not, it holds that

Similar to (2.13), we have

and

Combining (2.17), (2.18) with (2.15), we have

which contradicts (2.9).

Second, for the trajectory $(iii)$, we claim that there exists $t_3\in(t_2, T)$ such that $x_i(t_3) = d$, and $x_i(t) > d$ when $t\in(t_2, t_3)$ (see Figure 1). Suppose, contrary to our claim, that $x_i(t) > d$ for every $t\in(t_2, T]$. In this case, the inequalities (2.17)–(2.19) are still valid. That is a contradiction. Moreover, we have

Combining (2.15) with (2.20), we have

Now, we claim that there exists $t_4\in(t_3, T)$ such that

for the trajectory $(iii)$. Suppose, contrary to our claim, that

Therefore, using a similar argument as in [28,Lemma 2], there exists $t_5\in(t_3, t_3+\eta)$ such that $x_i(t_5) = \hat{x}_i(s, 0)$ (see Figure 1). So,

which is a contradiction with (2.22).

By (2.14), (2.16) and (2.21), we have

Using a similar argument as given above, inequality (2.5) is valid for $x_i(0)\leq d$. Moreover, for any positive integer $m_i > 1$, the conclusion can be proved by similar arguments.

3.   Proof of Theorem 1.1

To prove Theorem 1.1, we introduce the variable $u = (u_1, \cdot\cdot\cdot, u_N)$ as defined by

and we transform system (S') into

where, for every index $i$,

It is clear that $h_i(s, t, u)$ is well defined for every $t\in[0, T]$ and $u_i > -\hat{x}_i(s, t)/s$, and $h_i(s, t, 0)\equiv0$. Consider the Cauchy problem

By (3.1), we find that $x(t)$ is a solution of (2.3) if and only if $u(t)$ solves (3.2), so $u(t)$ is globally defined on $[0, T]$ by Lemma 2.4. If the component $(u_i(t), u'_i(t))$ of $(u(t), u'(t))$ does not attain the origin, we can pass to the standard polar coordinates as

Throughout the rest of the proof, $D(\Gamma^i_s)$ denotes the open bounded region delimited by a Jordan curve $\Gamma^i_s$, $\overline{B}_i(0, \tilde{r})$ denotes the closed ball of radius $\tilde{r}$ centered at the origin, and $\mathcal{D}^i_s$ denotes the set $\{(u, v)\in\Bbb R^N|u_i > -\hat{x}_i(s, 0)/s\}$.

Lemma 3.1. For $i\in\{1, \cdot\cdot\cdot, N\}$, it holds that

uniformly for every $t\in[0, T]$ and $u\in\Bbb R^N$ with $|u_i|\leq\frac{1}{2}c_0$, where $c_0$ is as defined in (2.1).

Proof. From the definition, we have $u_i > -\hat{x}_i(s, t)/s$. So, by (2.1), $h_i(s, t, u)$ is well defined for every $t\in[0, T]$ and $u\in\Bbb R^N$ with $|u_i|\leq\frac{1}{2}c_0$.

By (2.1), we have, for $|u_i|\leq\frac{1}{2}c_0$ and $t\in[0, T]$,

Since $\hat{x}_i(s, t)\rightarrow +\infty$ as $s\rightarrow +\infty$, $su_i+\hat{x}_i(s, t)\rightarrow +\infty$ as $s\rightarrow +\infty$ uniformly for every $t\in [0, T]$ and $|u_i|\leq\frac{1}{2}c_0$. By $(H^{i}_3)'$ and (3.4), the conclusion is thus achieved.

Lemma 3.2. There exist $b$, $\tilde{r}$ with $0 < b < \tilde{r} < c_0/2$ and $s_1\geq \tilde{s}$ such that, for every $s\geq s_1$, if $u:[0, T]\rightarrow \Bbb R^N$ is a solution of (P) with $u_i(0)^2+u'_i(0)^2 = \tilde{r}^2$ for a certain index $i$, then

Proof. We first prove that $r_i(t) < \dfrac{1}{2}c_0$ for every $t\in[0, T]$. On the contrary, suppose that there exists $\bar{t}\in[0, T]$ satisfying

Set

It is clear that $0 < b < \tilde{r} < c_0/2$. By Lemma 3.1, since $r_i(t)\leq c_0/2$ for every $t\in[0, \bar{t}]$, there exists $s_1\geq \tilde{s}$ such that

for every $s\geq s_1$ and $t\in[0, \bar{t}]$. From (P), we get

By a Gronwall argument we have

which contradicts (3.5). By a similar argument as above, we can see that $r_i(t) > b$ for every $t\in[0, T]$.

Lemma 3.3. There exists $s_2\geq s_1$ such that, for every $s\geq s_2$, if $u:[0, T]\rightarrow \Bbb R^N$ is a solution of (P) with $u_i(0)^2+u'_i(0)^2 = \tilde{r}^2$ for a certain index $i$, then

Proof. By $(H^i_4)$, we can fix $\varepsilon > 0$ small such that

By Lemma 3.2, we have $b < \sqrt{u_i(t)^2+u'_i(t)^2} < \frac{1}{2}c_0$ for every $t\in[0, T]$. From (3.3), there exists $s_2\geq s_1$ such that, for every $s\geq s_2$, one has

Therefore,

Lemma 3.4. For every $s\geq s_2$, there exists a strictly star-shaped Jordan curve $\Gamma^i_s$ around the origin such that, if $u:[0, T]\rightarrow \Bbb R^N$ is a solution of (P) with $(u_i(0), u'_i(0))\in\Gamma^i_s$ for a certain index $i$, then

Proof. Let $x(t)$ be a solution of (S'). In the $i$-th half plane $\{x_i > 0\}$, choose a suitable closed rectangle $\mathcal{K}_s$ such that $(\hat{x}_i(s, t), \hat{x}'_i(s, t))\in\mathcal{K}_s$ for every $t\in [0, T]$. Without loss of generality, we assume that if $\mathcal{N}_{i}(x(t), x'(t))\geq\hat{R}_s$ for large enough $\hat{R}_s$, then $(x(t), x'(t))\not\in\mathcal{K}_s$. Since $x(t)$ is globally defined on $[0, T]$, by the elastic property in [11,Lemma 6], there exists $\tilde{R}_s\geq\hat{R}_s$ such that, for any solution $x(t)$ of (S'),

Let $\Gamma^i_s = \{(u_i, u'_i)\in D^{i}_s:\mathcal{N}_{i}(su+\hat{x}(s, 0), su'+\hat{x}'(s, 0)) = \tilde{R}_s\}$. Suppose that $u:[0, T]\rightarrow \Bbb R^N$ is a solution of (P) with $(u_i(0), u'_i(0))\in\Gamma^i_s$ for some index $i$. Based on (3.1), we have $\mathcal{N}_{i}(x(0), x'(0)) = \tilde{R}_s$, which implies that $\mathcal{N}_{i}(x(t), x'(t))\geq\hat{R}_s$ for all $t\in[0, T]$.

Using Proposition 2.2 in ^[12], we can obtain the independence of

with respect to $\lambda\in[0, 1]$ in the $(x_i, x'_i)$ phase-plane. By applying (3.1), Lemma 2.5 and the definition of $i$-th rotation number, we have

Proof of Theorem 1.1. By (2.1), there exists $\hat{r}$ such that

For any $(u_i, u'_i)\in\Gamma^i_s$, if $u_i\geq-\hat{r}$, by the definition of $\mathcal{N}_{i}(su+\hat{x}_i(s, 0), su'+\hat{x}'_i(s, 0))$, we have

So,

for large enough $\tilde{R}_s$ as in (3.6). On the other hand, if $u_i < -\hat{r}$, inequality (3.7) clearly holds for any $(u_i, u'_i)\in\Gamma^i_s$. Recall $\tilde{r} < c_0/2$, so $\overline{B}_i(0, \tilde{r})\subset D(\Gamma^i_s)$.

We will prove the multiplicity of periodic solutions by the Poincaré-Birkhoff theorem stated in [5,Theorem 3.1], which is a simplified version of [6,Theorem 1.2]. Note that there exist $[\frac{m_i-1}{2}]+1$ integers in the interval $[\frac{m_i-1}{2}, m_i]$. Choose $s_0 = s_2$ and fix $s\geq s_0$. Taking $\Omega = (\overline{D}(\Gamma^i_s)\backslash B(0, \tilde{r}))^N$, the number of possible choices of the values $(l_1, l_2, \cdot\cdot\cdot, l_N)$ in [5,Theorem 3.1] is $\prod\limits^N_{i = 1}\left(\left[\frac{m_i-1}{2}\right]+1\right)$ by Remark 2.1 and Lemmas 3.3 and 3.4. Applying [5,Theorem 3.1], there exist $N+1$ periodic solutions for system (P) for every $(l_1, l_2, \cdot\cdot\cdot, l_N)$. Coming back to the system (S'), by (3.1), there exist $(N+1)\prod\limits^N_{i = 1}\left(\left[\frac{m_i-1}{2}\right]+1\right)$ periodic solutions for system ($S'$). The pivot solution to ($S'$) is provided by Lemma 2.2. The proof is thus concluded.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11901507, 12101337, 12071410) and Qing Lan Project of the Jiangsu Higher Education Institutions of China.

Conflict of interest

The authors declare there is no conflict of interest.