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Multiplicity of positive periodic solutions to thirdorder variable coefficients singular dynamical systems
Boundary Value Problems volume 2023, Article number: 65 (2023)
Abstract
In this paper, by applying a nonlinear alternative principle of Leray–Schauder and Guo–Krasnosel’skii fixed point theorem on compression and expansion of cones, together with truncation technique, we study the existence of multiplicity noncollision periodic solutions to thirdorder singular dynamical systems. By combining the analysis of the sign of Green’s function for a linear equation, we consider the systems where the potential has a repulsive singularity at origin. The socalled strong force condition is not needed, and the nonlinearity may have sign changing behavior. Recent results in the literature, even in the scalar case, are generalized and improved.
1 Introduction
The purpose of this work is to study the existence of noncollision periodic solutions to the thirdorder singular dynamical systems
where \(a\in C({\mathbb{R}}/T{\mathbb{Z}},{\mathbb{R}})\), \(e=(e_{1},\ldots ,e_{n})^{\text{T}}\in C(({\mathbb{R}}/T{\mathbb{Z}}),{ \mathbb{R}}^{n})\), the nonlinearity \(f=(f_{1},\ldots ,f_{n})^{\text{T}}\in C(({\mathbb{R}}/T{\mathbb{Z}}) \times {\mathbb{R}}^{n}\backslash \{0\},{\mathbb{R}}^{n})\) is a continuous vectorvalued function with repulsive singularity at \(x=0\).
Let \(\mathbb{R}_{+}=[0,\infty )\) and \(\mathbb{R}_{+}^{n}=\prod_{i=1}^{n}\mathbb{R}_{+}\). For \(x=(x_{1},\ldots , x_{n})\), \(y=(y_{1},\ldots ,y_{n})\in \mathbb{R}^{n}\), the usual scalar product is denoted by \(\langle x,y \rangle =\sum_{i=1}^{n} x_{i}y_{i}\). We say that (1.1) has a repulsive singularity at the origin if there exists a fixed vector \(v\in {\mathbb{R}}^{n}_{+}\) such that
As usual, by a noncollision nontrivial periodic solution we mean a function \(x=(x_{1},\ldots , x_{n})^{\text{T}}\in C^{3}(({\mathbb{R}}/T{\mathbb{Z}}),{ \mathbb{R}}^{n})\) solving (1.1) such that \(x(t)\neq 0\) for all t and satisfying the periodic boundary conditions
In the pioneering paper [15], Lazer and Solimini investigated the singular equation
where \(\lambda \geq 1\), and h is periodic function with period T; by using the method of upper and lower solutions they proved that a sufficient and necessary condition for the existence of a positive Tperiodic solution is \(\int ^{T}_{0}h(t)\,dt<0\). We say that \(0<\lambda <1\) is the weak force condition for equation (1.3) and \(\lambda \geq 1\) is the strong force condition for ir (the strong force condition was first introduced by Gordon [9]). During the last few decades, the question of existence of noncollision periodic solutions for singular scalar equations and dynamical systems has attracted much attention [1, 5, 15, 21, 22, 25, 27, 28]. For example, in 2019, Jiang [13] investigated a kind of secondorder nonautonomous dynamical systems
By a nonlinear alternative principle of Leray–Schauder and the fixed point theorem in cones the author showed that the singular system (1.4) has at least two positive solutions when the Green’s function is nonnegative.
Singular differential equations and singular dynamical systems have a wide range of applications in biology, physics, and mechanics, such as the nonlinear elasticity [6] and Brillouin focusing system [7]. Usually, the proof is based on either variational approach [23, 29] or topological methods. In particular, degree theory [16, 17], Schauder’s fixed point theorem [14], some fixed point theorems in cones for completely continuous operators [3, 8, 18, 26], and a nonlinear alternative principle of Leray–Schauder type [13, 20] are the most relevant tools.
To avoid collision of the solution with singularity, the strong force condition plays an important role and is standard in the related works. Compared with the strong singularity case, the case of weak singularity was less studied by topological methods [5, 11, 12].
At the same time, some authors began to consider thirdorder singular differential equations and singular dynamical systems [2, 4, 14, 24], for example, the thirdorder differential equation with constant coefficient
with periodic boundary conditions (1.2). Here K is a positive constant, and the nonlinearity \(f(t,x)\) is singular at \(x=0\). In [24], using the Green’s function and fixed point index theory, the existence of multiple positive solutions is obtained. In [14], by using Schauder’s fixed point theorem, together with perturbation technique, the existence of at least one positive solution is established. The main result in [14] is the following.
Theorem 1.1
Let the following three assumptions hold:
 \((A_{1})\):

\(f(t,u)\) is a nonnegative function on \([0,2\pi ]\times (0,+\infty )\), and \(f(t,u)\) is integrable on \([0,2\pi ]\) for each fixed \(u\in (0,+\infty )\);
 \((A_{2})\):

\(f(t,u)\) is nonincreasing in \(u>0\) for almost all \(t\in [0,2\pi ]\), and
$$ \lim_{u\rightarrow 0^{+}}f(t,u)=+\infty ,\qquad \lim_{u\rightarrow +\infty}f(t,u)=0 $$uniformly for \(t\in [0,2\pi ]\);
 \((A_{3})\):

\(\int ^{2\pi}_{0}f(s,\tau )\,ds<+\infty \) for all \(\tau >0\).
Then equation (1.5) has at least one positive solution if \(K\in (0,\frac{1}{3\sqrt{3}})\).
This paper is mainly motivated by the recent papers [13, 14], but we do not require that all components of the nonlinearity \(f(t,x)\) have a singularity. The new results cover both strong and weak singularities. The structure of the paper is as follows. In Sect. 2, we present a survey on some known results concerning the sign of the Green’s function of the linear equation
associated with periodic boundary conditions (1.2). In Sects. 3 and 4, by employing a nonlinear alternative principle of Leray–Schauder and Guo–Krasnosel’skii’s fixed point theorem, we prove the main existence results for (1.1) under the positiveness of the Green’s function associated with (1.6)–(1.2).
In this paper, we use the following notations. The usual Euclidean norm is denoted by \(x\). More generally, for a fixed vector \(v=(v_{1},\ldots , v_{n})\in \mathbb{R}^{n}_{+}\), we have the welldefined norm
In particular, we get the \(l_{1}\)norm \(x_{v}=x_{1}=\sum_{i=1}^{n}x_{i}\) if \(v=(1,\ldots , 1)\). Let \(\\cdot \\) denote the supremum norm of \(C_{T}=\{x:x\in C(\mathbb{R}/T\mathbb{Z}),\mathbb{R}\}\) and take \(X= C_{T} \times \cdots \times C_{T} (n \text{ copies})\). Then for \(x=(x_{1}, \ldots ,x_{n})\in X\), the natural norm becomes
Obviously, X is a Banach space.
2 Sign of Green’s function and its properties
As we know, it is very complicated to calculate the Green’s function of the thirdorder scalar linear differential equation
with with variable coefficients and periodic boundary conditions (1.2), where \(h\in C(\mathbb{R},\mathbb{R}^{+})\) is a Tperiodic function. In this section, we first discuss the Green’s function of the thirdorder scalar linear differential equation with constant coefficients
where \(A:= \max_{t\in [0,T]}a(t)\). We will use it to investigate the existence of a positive periodic solution for (1.1). In the following, we introduce the Green’s functions of (2.2) and some properties, which can be found in [24]. Let \(A=\rho ^{3}\). Then (2.2) is transformed into
and
Moreover, the solutions of (2.3) can be written as
where
Lemma 2.1
([24])
The boundary value problem (2.4) is equivalent to the integral equation
where
Moreover, for \(G_{2}(t,s)\), if \(\rho \in (0,\frac{2\sqrt{3}\pi}{3T})\), then we have the estimates
The solution of (2.2) can be written as
Thus letting
we can get
Let \(A:= \max_{t\in [0,T]}a(t)\). Since \(G_{1}(t,s)>0\) and \(G_{2}(t,s)\geq 0\), we easily get the following.
Lemma 2.2
Assume that \(0< A<\frac{8\sqrt{3}\pi ^{3}}{9T^{3}}\). Then the Green’s function \(G(t,s)\) associated with the boundary value problem (2.2) is positive for all \((t,s)\in [0,T]\times [0,T]\).
We denote
and thus \(M>m>0\) and \(0<\sigma <1\).
3 Existence result (I)
In this section, we state and prove the first existence result for (1.1). The proof is based on the following nonlinear alternative of Leray–Schauder, which can be found in [19] and has been used in [13, 18].
Lemma 3.1
Let C be a convex subset of a normed linear space E, and let U be an open subset of C with \(0\in U\). Then every compact continuous map \(F:\bar{U}\rightarrow C\) has at least one of the following properties,

(i)
F has a fixed point in Ū; or

(ii)
There are \(u\in \partial U\) and \(0<\lambda <1\) such that \(x=\lambda Fx\).
Define two functions ω and γ by
and
which is the unique Tperiodic solution of the linear system
Observe that \(u(t)=x(t)+\gamma (t)\) is a Tperiodic solution of (1.1) if the system
has a Tperiodic solution \(x(t)\), since
Theorem 3.2
Assume that \(0< A<\frac{8\sqrt{3}\pi ^{3}}{9T^{3}}\). In addition, suppose that there exists a positive constant \(r>0\) satisfying the following conditions.
 (H_{1}):

For each constant \(L>0\), there exists a continuous function \(\phi _{L}\succ 0\) (which means that \(\phi (t)\geq 0\) for all \(t\in [0,T]\) and it is positive for t in a subset of positive measure) such that
$$ \bigl\langle v,f(t,x)\bigr\rangle \geq \phi _{L}(t)\quad \textit{for } (t,x) \in [0,T] \times {\mathbb{R}}^{n}_{+} \textit{ with } 0< \vert x \vert _{v}\leq L. $$  (H_{2}):

There exist continuous nonnegative functions g and h on \((0,\infty )\) such that
$$ \bigl\langle v,f(t,x)\bigr\rangle \leq g\bigl( \vert x \vert _{v} \bigr)+h\bigl( \vert x \vert _{v}\bigr) $$for all t and \(x\in {\mathbb{R}}^{n}_{+}\) with \(0<x_{v}\leq r+\gamma ^{*}\), where \(g>0\) is nonincreasing, and \(h/g\) is nondecreasing.
 (H_{3}):

We have the following inequality:
$$ \frac{r}{g(\sigma r+\gamma _{*}) \{1+ \frac{h(r+\gamma ^{*})}{g(r+\gamma ^{*})} \}}> \Vert \omega \Vert , $$
where
Then system (1.1) has at least one positive Tperiodic solution.
Proof
Step 1. We first consider a family of systems.
Since (H_{3}) holds, we can choose \(n_{0}\in \{1,2,\ldots \}\) such that \(\frac{1}{n_{0}}<\sigma r +\gamma _{*}\) and
Let \(N_{0}= \{n_{0},n_{0}+1,\ldots \}\) and fix \(n\in N_{0}\). Consider the family of systems
where \(\mu \in [0,1]\), \(\vartheta \in \mathbb{R}_{+}^{n}\) is chosen such that \((v,\vartheta )=1\), with truncation functions
where f̃ is chosen such that \(f_{n}\) are continuous on \([0,T]\times {\mathbb{R}}^{n}\).
So, (3.1) is equivalent to the fixed point problem
where \(\ell = \vartheta /n\), \(\mathcal{T}^{n}\) is defined by
and we have used the fact that
We claim that any fixed point x of (3.3) for all \(\mu \in [0,1]\) must satisfy \(\x\\ne r\). Otherwise, assume that x is a fixed point of (3.3) for some \(\mu \in [0,1]\) such that \(\x\= r\). From Lemma 2.2 we have
Therefore, for all t, we have
So we have
since \(\frac{1}{n} \le \frac{1}{n_{0}} <\sigma r + \gamma _{*}\), which implies that
Thus from (H_{2}) we have
Therefore we have
This is a contradiction to the choice of \(n_{0}\), and thus the claim is proved.
By this claim Lemma 3.1 guarantees that
has a fixed point, denoted by \(x_{n}\), i.e., the system
has a periodic solution \(x_{n}\) with \(\x_{n}\< r\). Since \(\langle v,x_{n}(t)\rangle \geq \langle v,\ell \rangle >0\) for all \(t\in [0,T]\), \(x_{n}\) is in fact a positive Tperiodic solution of (3.4).
Now we show that \(\langle v,x_{n}(t)+\gamma (t)\rangle \) have an uniform positive lower bound, i.e., there exists a constant \(\delta >0\), independent of \(n\in N_{0}\), such that
for all \(n\in N_{0}\). To see this, we know that by (H_{1}) there exists a continuous function \(\phi _{L}(t)\succ 0\) such that
for all t and \(0<\x\\leq L\). Then we have
So we have \(\langle v,x_{n}(t)+\gamma (t)\rangle \geq \delta \) for all n.
Step 2. To pass the solutions \(x_{n}\) of the truncation systems (3.4) to that of the original system (1.1), we need to show that \(\{x_{n}\}_{n\in N_{0}}\) is compact.
Firstm we claim that
for some constant \(H>0\) and all \(n\ge n_{0}\).
Here we denote by \(x_{ni}\) and \(\vartheta _{i}\) the ith components of \(x_{n}\) and ϑ. Since \(x_{ni}\) are Tperiodic solutions of (3.4), we have
for each \(i=1,2,\ldots ,n\).
Multiplying both sides of (3.7) by \(x'_{ni}(t)\) and integrating from 0 to T, we have
Substituting \(\int _{0}^{T} x_{ni}'''(t)x'_{ni}(t)\,dt=\int _{0}^{T}x''_{ni}(t)^{2}\,dt \) into (3.8), we have
where
Using the Writinger inequality, we have
It is easy to see that there is constant \(D>0\) such that
For each \(i=1,\ldots , n\), by the periodic boundary conditions \(x_{ni}(0)=x_{ni}(T)\) we know that there exists a point \(t_{0}\in [0,T]\) such that \(x'_{ni}(t_{0})=0\). Therefore we have
Therefore
Step 3. The facts (3.6) and (3.5) show that \(\{x_{n} \}_{n\in N_{0}}\) is a bounded and equicontinuous family.
Now the Arzelà–Ascoli theorem guarantees that \(\{x_{n} \}_{n\in N_{0}}\) has a subsequence \(\{x_{n_{k}} \}_{k\in {\mathbb{N}}}\) converging uniformly on \([0,T]\) to a function \(x\in X\). Moreover, we have
Furthermore, \(x_{n_{k}}\) satisfies the integral equation
Letting \(k\rightarrow \infty \), we arrive at
Therefore x is a positive Tperiodic solution of (1.1) and satisfies \(0<\x\\leq r\). □
Corollary 3.3
Assume that \(0< A<\frac{8\sqrt{3}\pi ^{3}}{9T^{3}}\), \(b,c\in C[0,T]\) are positive functions, \(e_{1},e_{2}\in C({\mathbb{R}}/T{\mathbb{Z}},{\mathbb{R}})\), \(\alpha ,\beta >0\), and \(\mu \in \mathbb{R}\) is a given positive parameter. Consider the following twodimensional thirdorder nonlinear systems:

(i)
if \(\beta <1\), then (3.9) has at least one positive periodic solution for each \(\lambda >0\);

(ii)
if \(\beta \ge 1\), then (3.9) has at least one positive periodic solution for each \(0< \lambda < \lambda _{1}\), where \(\lambda _{1}\) is some positive constant;
Proof
We will apply Theorem 3.2. For a fixed vector \(v=(1,1)\), \({(\mathrm{H}_{1})}\) is fulfilled by \(\phi _{L}=b(t)L^{\alpha}\). For \(s\in \mathbb{R}\), \(s>0\), to verify \({(\mathrm{H}_{2})}\), we may take
where
Condition \((\mathrm{H}_{3})\) becomes
for some \(r>0\). So (3.9) has at least one positive periodic solution for
Note that \(\lambda _{1}=\infty \) if \(\beta <1\) and \(\lambda _{1} < \infty \) if \(\beta \ge 1\). So we have (i) and (ii). □
4 Existence result (II)
In this section, by using GuoKrasnosel’skii’s fixed point theorem on compression and expansion of cones, we establish the second existence results for (1.1).
Lemma 4.1
([10])
Let X be a Banach space, and let \(K (\subset X)\) be a cone. Assume that \(\Omega _{1}\), \(\Omega _{2}\) are open subsets of X with \(0\in \Omega _{1}\), \(\bar{\Omega}_{1}\subset \Omega _{2}\), and let
be a completely continuous operator such that either

(i)
\(\ \mathcal{A}u \ \geq \ u \\), \(u\in K\cap \partial \Omega _{1}\), and \(\ \mathcal{A}u \ \leq \ u \\), \(u\in K\cap \partial \Omega _{2}\); or

(ii)
\(\ \mathcal{A}u \ \leq \ u \\), \(u\in K\cap \partial \Omega _{1}\), and \(\ \mathcal{A}u \ \geq \ u \\), \(u\in K\cap \partial \Omega _{2}\).
Then \(\mathcal {A}\) has a fixed point in \(K\cap (\bar{\Omega}_{2}\setminus \Omega _{1})\).
Let \(X= C_{T} \times \cdots \times C_{T} (n \text{ copies})\) and define
where σ is as in (2.5).
We can readily verify that K is a cone in the Banach space X. Define the operator
for \(x\in X\) and \(t\in [0,T]\). Then Φ is well defined and maps X into K.
Indeed, for \(t\in \mathbb{R}\) and \(x\in X\), we have
On the other hand,
Thus
This implies that \(\Phi (X)\subset K\). It is easy to prove \(\Phi :X\rightarrow K\) is completely continuous.
Theorem 4.2
Assume that \(0< A<\frac{8\sqrt{3}\pi ^{3}}{9T^{3}}\) and \((\mathrm{H}_{1})\)–\((\mathrm{H}_{3})\) hold. In addition, we assume that the following two conditions are satisfied:
 (H_{4}):

There exist continuous nonnegative functions g and \(h_{1}\) such that
$$\begin{aligned} \bigl\langle v,f(t,x)\bigr\rangle \geq g_{1}\bigl( \vert x \vert _{v}\bigr)+h_{1}\bigl( \vert x \vert _{v} \bigr) \quad \textit{for all } (t,x)\in [0,T]\times \mathbb{R}_{+}^{n} \backslash \{0 \}, \end{aligned}$$where \(g_{1}>0\) is nonincreasing, and \(h_{1}/g_{1}\) is nondecreasing.
 (H_{5}):

There exists \(R >r\) such that
$$\begin{aligned} \Vert \omega \Vert g_{1}\bigl(R+\gamma ^{*}\bigr) \biggl\{ 1+ \frac{h_{1}(\sigma R+\gamma _{*})}{g_{1}(\sigma R+\gamma _{*})} \biggr\} \geq R. \end{aligned}$$
Then, besides the solution x constructed in Theorem 3.2, problem (1.1)–(1.2) has another positive Tperiodic solution x̃ with \(r< \\tilde{x}\gamma \\leq R\).
Proof
Let K be a cone in X defined by (4.1). Define
First, we claim that \(\\Phi x\\leq \x\\) for \(x\in K\cap \partial \Omega _{1}\). Indeed, if \(x\in K\cap \partial \Omega _{1}\), then \(\x\=r\), and we have
Thus
Next, we prove that \(\\Phi x\\geq \x\ \) for \(x\in K\cap \partial \Omega _{2} \). Indeed, if \(x\in K\cap \partial \Omega _{2}\), then \(\x\= R\), and we have
Thus
Now Lemma 4.1 guarantees that Φ has at least one fixed point \(\tilde{x}\in K\cap (\bar{\Omega}_{2} \backslash \Omega _{1})\) with \(r\leq \\tilde{x}\\leq R\). □
Let us consider again example (3.9) in Corollary 3.3.
Corollary 4.3
Assume in (3.9) that \(0< A<\frac{8\sqrt{3}\pi ^{3}}{9T^{3}}\), \(b(t)>0\) and \(c(t)>0\) for all \(t\in [0,T]\), and \(\beta >1\). Then, for each μ with \(0<\lambda <\lambda _{1}\), where \(\lambda _{1}\) is given as in Corollary 3.3, problem (3.9) has at least two different positive solutions.
To verify \({(\mathrm{H}_{4})}\), for \(s\in \mathbb{R}\), \(s>0\), we may take
where
If \(\beta >1\), then condition \({(\mathrm{H}_{5})}\) becomes
Since \(\beta >1\), the righthand side goes to 0 as \(R\rightarrow +\infty \). Thus, for any given \(0<\lambda <\lambda _{1}\), it is always possible to find \(R\gg r\) such that (4.2) is satisfied. Thus (3.9) has an additional positive periodic solution x̃.
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Acknowledgements
We would like to express our great thanks to the referees for their valuable suggestions. Shengjun Li was supported by Hainan Provincial Natural Science Foundation of China (Grant No. 120RC450), the National Natural Science Foundation of China (Grant No. 11861028), Key Laboratory of Engineering Modeling and Statistical Computation of Hainan Province. Fang Zhang was supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20201447), and Science and Technology Innovation Talent Support Project of Jiangsu Advanced Catalysis and Green Manufacturing Collaborative Innovation Center (Grant No. ACGM20221002).
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Li, S., Zhang, F. Multiplicity of positive periodic solutions to thirdorder variable coefficients singular dynamical systems. Bound Value Probl 2023, 65 (2023). https://doi.org/10.1186/s13661023017501
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DOI: https://doi.org/10.1186/s13661023017501