Positive Periodic Solution for Pipe/Tank Flow Configurations with Friction

Abstract

In this article, we study a periodic boundary value problem related to valveless pumping. The valveless pumping is described by the unidirectional flow of liquid in a system. We establish some conditions for globally asymptotic stability and the existence of a positive periodic solution to the considered equation. Finally, a numerical example shows that the theoretical results in this paper are feasible.

1. Introduction

In 2006, Propst [1] presented a detailed explanation of the pumping effect for flow configurations of one to three rigid tanks connected by rigid pipes. A pump can be considered as a periodic differential system. Propst found the existence of periodic solutions to the corresponding differential equation for a system with two or three tanks. For the simplest structure of one pipe and one tank, Propst proved the existence result of a periodic solution in particular conditions. In general, the structure of one pipe and one tank can be expressed by the following boundary value problem

$$x^{″}+b{x}^{\prime }=\frac{1}{x}(\gamma \left(t\right)-c{\left(u^{\prime }\right)}^2)-d,$$

(1)

$$x\left(0\right)=x\left(T\right),x^{\prime }\left(0\right)=x^{\prime }\left(T\right),$$

(2)

where $b\ge 0,c>1,d>0$ and $\gamma \left(t\right)$ is a continuous periodic function. The specific meaning of the involved coefficients in (1) can be found in [1,2]. Equation (1) is a singular equation. Hakl, Torres and Zamora [3,4] studied the following singular equation:

$$y^{″}+u\left(y\right)y^{\prime }+v(t,y)=w(t,y),$$

where both u and v may have singularity at the origin. Since the study of singular equations is more complex and difficult than the study of regular equations, we attempt to find a more simple and effective method to study Equation (1). In fact, Equation (1) can be transformed into an equivalent regular equation through appropriate variable substitution. Let $x=y^{\mu }$ with $\mu =\frac{1}{c+1}$, and then we transform boundary value problem (1) and (2) to the following boundary value problem

$$y^{″}+a{y}^{\prime }+\delta \left(t\right)y^{\beta }-\rho \left(t\right)y^{\alpha }=0,$$

(3)

$$y\left(0\right)=y\left(T\right),y^{\prime }\left(0\right)=y^{\prime }\left(T\right),$$

(4)

where $\rho \left(t\right)=\frac{\gamma \left(t\right)}{\mu },\delta \left(t\right)=\frac{d}{\mu },\alpha =1-2\mu ,\beta =1-\mu ,\gamma \left(t\right)$ is a continuous periodic function. Since Equation (3) is not a singular equation, many classic research methods, such as the upper and lower solution method and topological degree theory, can be used for studying Equation (3). In [2], Cid, Propst and Tvrdý obtained the following theorems.

Theorem 1. 

Assume that $b>0,c>1,d>0,{\gamma }_{\ast }>0$ and that $\gamma \left(t\right)$ is continuous periodic on $\mathbb{R}$. Then, for BVP (1) and (2), there exists a positive solution if the following condition is satisfied:

$$\frac{(c+1)d^2}{4{\gamma }_{\ast }}<\frac{{\pi }^2}{T^2}+\frac{b^2}{4},$$

where ${\gamma }_{\ast }={min}_{t\in [0,T]}\gamma \left(t\right)$.

Theorem 2. 

Assume that $b>0,c>1,d>0,{\gamma }_{\ast }>0$ and that $\gamma \left(t\right)$ is continuous periodic on $\mathbb{R}$. Then, for BVP (1) and (2), there exists at least one stable positive solution if the following conditions are satisfied:

$$\frac{d^2}{{\gamma }_{\ast }}<\frac{{\pi }^2}{T^2}+\frac{b^2}{4}$$

and

$$(b-1)e^{\ast }<b{e}_{\ast },$$

where ${\gamma }_{\ast }={min}_{t\in [0,T]}\gamma \left(t\right),{\gamma }^{\ast }={max}_{t\in [0,T]}\gamma \left(t\right)$.

After that, Dorociaková et al. [5] considered the existence and stability of a periodic solution for valveless pumping using some fixed point theorems and the inequality technique. However, when using the lower and upper functions method, constructing proper lower and upper functions for BVP (1) and (2) is difficult in [2]. Furthermore, in order to obtain the existence and stability of the periodic solution for BVP (1) and (2), Lemmas 2.2, 2.4 and 2.5 in [2] were used, which greatly increases the difficulty of proof. In [5], sufficient conditions for the existence and exponential stability of a positive periodic solution to BVP (3) and (4) are complicated and are difficult to verify. In Remark 3 of this paper, we find that conditions (39)–(42) are sharp. Therefore, it is important to achieve simple and easily verifiable conditions.

Encouraged by the above work, we study periodic boundary value problem (3) and (4) by using coincidence degree theory. Topological degree theory is often used to study periodic solutions of differential equations; see, for example, quaternion-valued inertial memristor-based neural networks [6]; higher-order delay differential equations [7,8,9]; and fractional multi-point boundary value problems [10]. In this paper, we use the coincidence degree theory in the study of periodic solutions of a strong role combined with the appropriate variable substitution and mathematical analysis to study the periodic solution of pipe/tank flow configurations with friction.

With regard to some other important results about differential equations and systems, see, e.g., [11,12,13,14,15,16,17]. In particular, we transform Equation (3) into a equivalent two-dimensional system through appropriate variable substitution and can then easily study a periodic solution for the above two-dimensional system. Furthermore, we obtain that the periodic solution is stable through using the Lyapunov function method. The main contributions are listed as follows:

(1)

Using appropriate variable substitution, the second-order equation can be transformed into a low-order two-dimensional system so that the dynamic properties of the second-order equation can be conveniently studied.

(2)

Compared with the article [5], the existence and stability conditions of periodic solutions obtained in this paper are easier to verify.

(3)

In [2], strong mathematical analysis skills are required, while the method used in this paper does not require strong mathematical analysis skills.

We organize the remaining research content of this article as follows: Section 2 lists some existing research results and lemmas needed in this paper. In Section 3 and Section 4, we give the main results of this paper. In Section 5, we give a numerical example. Finally, we summarize the research content of the full text and provide a discussion of some issues.

2. Preliminaries

Set U and V be two Banach spaces. Set $M:D\left(M\right)\subset U\to V$ as a Fredholm operator with index zero, which means that $dimKerM=codimImM<+\infty$ and $ImM$ is closed in V. If M is a Fredholm operator with index zero, then there are continuous projectors $G:U\to U,H:V\to V$ such that $M_{D\left(M\right)\cap KerG}:(I-G)X\to ImM$ is invertible, and $ImM=KerH=Im(I-H),ImG=KerM$. Denote, by $F_G$, the inverse of $M_{D\left(M\right)\cap KerG}$.

Let $\Theta$ be a bounded subset of U. A map $R:\overline{\Theta }\to V$ is said to be L-compact in $\overline{\Theta }$ if the operator $F_G(I-H)R\left(\overline{\Theta }\right)$ is relatively compact and $HR\left(\overline{\Theta }\right)$ is bounded.

Lemma 1 

([18]). Assume that U and V are two Banach spaces, and $M:D\left(M\right)\subset U\to V$ is a Fredholm operator with index zero. $R:\overline{\Theta }\to V$ is L-compact on $\overline{\Theta }$, and $\Theta \subset U$ is an open bounded set. If the following conditions are satisfied:

(1) 

$Mx\ne \mu Rx,\forall x\in \partial \Theta \cap D\left(M\right),\forall \mu \in (0,1),$

(2) 

$Rx\notin ImM,\forall x\in \partial \Theta \cap KerM,$

(3) 

$deg\{\Phi HR,\Theta \cap KerM,0\}\ne 0$,

where $\Phi :ImH\to KerM$ is a isomorphic mapping, then the operator equation $Mx=Rx$ has a solution on $\overline{\Theta }\cap D\left(M\right).$

Let $u\left(t\right)=y^{\prime }\left(t\right)+\xi y\left(t\right)$, where $\xi >0$ is a constant. We rewrite Equation (3) as follows:

$$\left\{\begin{array}{c}{y}^{\prime }\left(t\right)=-\xi y\left(t\right)+u\left(t\right),\hfill \\ u^{\prime }\left(t\right)=-au\left(t\right)+(\xi +a)y\left(t\right)-\delta \left(t\right)y^{\beta }\left(t\right)+\rho \left(t\right)y^{\alpha }\left(t\right).\hfill \end{array}\right.$$

(5)

Furthermore, let $y\left(t\right)=e^{v\left(t\right)}$, and we rewrite (5) as follows:

$$\left\{\begin{array}{c}{v}^{\prime }\left(t\right)=-\xi +u\left(t\right)e^{-v\left(t\right)},\hfill \\ u^{\prime }\left(t\right)=-au\left(t\right)+(\xi +a)e^{v\left(t\right)}-\delta \left(t\right)e^{\beta v\left(t\right)}+\rho \left(t\right)e^{\alpha v\left(t\right)}.\hfill \end{array}\right.$$

(6)

It is easy to see that, if $z\left(t\right)={(v\left(t\right),u\left(t\right))}^T$ is a T-periodic solution of system (6), then $y\left(t\right)=e^{v\left(t\right)}$ is a positive T-periodic solution of Equation (3). For studying the periodic solution of Equation (3), we only need to study the periodic solution of system (6).

Set $C_T=\{\vartheta \in C(\mathbb{R},\mathbb{R}):\vartheta (t+T)=\vartheta \left(t\right)\}$ with norm ${\left|\vartheta \right|}_0={max}_{t\in [0,T]}\left|\vartheta \left(t\right)\right|$. Set

$$U=V=\{w={(u(·),v(·))}^T\in C(\mathbb{R},{\mathbb{R}}^2):\omega (t+T)=w\left(t\right)\}$$

with norm $\left|\right|w\left|\right|={max}_{t\in [0,T]}{\left\{\right|u|}_0{,\left|v\right|}_0\}$. Clearly, U and V are Banach spaces.

We define a linear operator by

$$M:D\left(M\right)\subset U\to V,Mz=z^{\prime }\left(t\right)=\left(\begin{array}{c}{v}^{\prime }\left(t\right)\\ u^{\prime }\left(t\right)\end{array}\right).$$

(7)

Furthermore, We define a nonlinear operator by

$$R:U\to V,Rz=\left(\begin{array}{c}-\xi +u\left(t\right)e^{-v\left(t\right)}\\ -au\left(t\right)+(\xi +a)e^{v\left(t\right)}-\delta \left(t\right)e^{\beta v\left(t\right)}+\rho \left(t\right)e^{\alpha v\left(t\right)}\end{array}\right).$$

(8)

We find

$$\mathrm{K}\mathrm{e}\mathrm{r}M={\mathbb{R}}^2,\mathrm{I}\mathrm{m}M=\{\nu |\nu \in V,{\int }_0^T\nu \left(s\right)ds=0\}.$$

Thus, $\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{K}\mathrm{e}\mathrm{r}M=\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{I}\mathrm{m}M=2$, and $ImM$ is closed in V. Therefore, M is a Fredholm operator with index zero. We define a continuous projector G by

$$G:U\to \mathrm{K}\mathrm{e}\mathrm{r}M,Gz=\frac{1}{T}{\int }_0^{T}z\left(s\right)ds$$

and define the other continuous projector H by

$$H:Y\to V/\mathrm{I}\mathrm{m}M,H\psi =\frac{1}{T}{\int }_0^T\psi \left(s\right)ds.$$

Set

$$M_G{=M|}_{D\left(M\right)\cap \mathrm{K}\mathrm{e}\mathrm{r}G}:D\left(M\right)\cap \mathrm{K}\mathrm{e}\mathrm{r}G\to \mathrm{I}\mathrm{m}M,$$

and then

$$M_G^{-1}=F_G:\mathrm{I}\mathrm{m}M\to D\left(M\right)\cap \mathrm{K}\mathrm{e}\mathrm{r}G.$$

We have

$$\left(F_{G}z\right)\left(u\right)={\int }_0^T\Gamma (u,v)z\left(v\right)dv,$$

where

$$\Gamma (u,v)=\left\{\begin{array}{c}\frac{v}{T},0\le v\le u\le T\hfill \\ \frac{v-T}{T},0\le u\le v\le T\hfill \end{array}\right.$$

Therefore, $F_G$ is a completely continuous operator in $\mathrm{I}\mathrm{m}M$. Clearly, $HR\left(\overline{\Theta }\right)$ is bounded on $\overline{\Theta }$. Thus, the operator R is L-compact on $\overline{\Theta }$.

Throughout this paper, we give the following assumptions:

Assumption 1 (H${}_1$). 

$a\ge 0,\delta \left(t\right)$ and $\rho \left(t\right)$ are T-periodic continuous functions with $\delta \left(t\right)\ge 0$ and $\rho \left(t\right)\le 0$.

Assumption 2 (H${}_2$). 

$\beta >1$ and $\alpha >0$.

Assumption 3 (H${}_3$). 

There is a positive constant d such that

$$\sigma \left[-(\xi +a)e^{\sigma \left(t\right)}+\delta \left(t\right)e^{\beta \sigma \left(t\right)}-\rho \left(t\right)e^{\alpha \sigma \left(t\right)}\right]>0for\sigma >d$$

and

$$\sigma \left[(\xi +a)e^{\sigma \left(t\right)}-\delta \left(t\right)e^{\beta \sigma \left(t\right)}+\rho \left(t\right)e^{\alpha \sigma \left(t\right)}\right]>0for\sigma <-d.$$

3. Existence of a Positive Periodic Solution

Theorem 3. 

Suppose assumptions (H${}_1$), (H${}_2$) and (H${}_3$) are satisfied. Then, there exists at least one positive T-periodic solution for Equation (3), provided that

$$\xi {\left(a-a\xi +\xi \right)}^{\frac{1}{\beta -1}}\le N_2,$$

(9)

and

$$\ln \frac{N_2}{\xi }\le N_3,$$

(10)

where $\xi >0$ is a constant, and $N_2$ and $N_3$ are defined by (20) and (22), respectively.

Proof. 

Consider the following equation:

$$Mz=\mu Rz,\mu \in (0,1).$$

(11)

Take ${\Omega }_1=\{z\in D\left(M\right):Mz=\mu Rz,\mu \in (0,1)\}$, where M and R are expressed by (7) and (8), respectively. For $z={(v\left(t\right),u\left(t\right))}^T\in {\Omega }_1$, it follows, by (11), that

$$v^{\prime }\left(t\right)=-\mu \xi +\mu u\left(t\right)e^{-v\left(t\right)}$$

(12)

and

$$u^{\prime }\left(t\right)=-\mu au\left(t\right)+\mu (\xi +a)e^{v\left(t\right)}-\mu \delta \left(t\right)e^{\beta v\left(t\right)}+\mu \rho \left(t\right)e^{\alpha v\left(t\right)}.$$

(13)

Integrate (12) over $[0,T]$, and then

$${\int }_0^{T}u\left(t\right)e^{-v\left(t\right)}dt=\xi T.$$

(14)

In view of (14), there is $t_0\in [0,T]$ such that

$$u\left(t_0\right)=\xi T{e}^{v\left(t_0\right)}.$$

(15)

On the other hand, by (12), we find

$$e^{v\left(t\right)}{v}^{\prime }\left(t\right)=-\mu \xi e^{v\left(t\right)}+\mu u\left(t\right),$$

i.e.,

$${\left(e^{v\left(t\right)}\right)}^{\prime }=-\mu \xi e^{v\left(t\right)}+\mu u\left(t\right).$$

(16)

For the above $t_0$, by (15) and (16), we find

$$\begin{array}{cc}\hfill e^{v\left(t\right)}& =e^{v\left(t_0\right)}{e}^{-{\int }_{t_0}^t\mu \xi d\tau }+{\int }_{t_0}^t\mu u\left(s\right)e^{{\int }_{t_0}^t\mu \xi d\tau }ds\hfill \\ & =e^{v\left(t_0\right)}{e}^{-\mu \xi (t-t_0)}+\mu e^{\mu \xi (t-t_0)}{\int }_{t_0}^t\mu u\left(s\right)ds\hfill \\ & \le e^{v\left(t_0\right)}+e^{\xi (T-t_0)}(T-t_0){\left|u\right|}_0\hfill \\ & \le \left(\frac{1}{\xi T}+e^{\xi T}T\right){\left|u\right|}_0\hfill \\ & =N_1{\left|u\right|}_0,\hfill \end{array}$$

(17)

where $N_1=\frac{1}{\xi T}+e^{\xi T}T.$ From (13), (17), (H${}_1$) and (H${}_2$), we have

$$\begin{array}{cc}\hfill u\left(t\right)& =u\left(t_1\right)e^{-{\int }_{t_1}^t\mu ad\tau }+{\int }_{t_1}^t\mu [(\xi +a)e^{v\left(s\right)}-\delta \left(s\right)e^{\beta v\left(s\right)}+\rho \left(s\right)e^{\alpha v\left(s\right)}]e^{{\int }_{t_1}^t\mu ad\tau }ds\hfill \\ & =u\left(t_1\right)e^{-\mu a(t-t_1)}+\mu e^{\mu a(t-t_1)}{\int }_{t_1}^t\mu [(\xi +a)e^{v\left(s\right)}-\delta \left(s\right)e^{\beta v\left(s\right)}+\rho \left(s\right)e^{\alpha v\left(s\right)}]ds\hfill \\ & \le {\left|u\right|}_0+e^{a(T-t_1)}(T-t_1)\left[(\xi +a)N_1{\left|u\right|}_0-\check{\delta }{N}_1^{\beta }{\left|u\right|}_0^{\beta }\right],\hfill \end{array}$$

(18)

where $\check{\delta }={min}_{t\in [0,T]}\left|\delta \left(t\right)\right|$. By (18), we have

$$\check{\delta }{N}_1^{\beta }{\left|u\right|}_0^{\beta }\le (\xi +a)N_1{\left|u\right|}_0.$$

(19)

By (19), we have

$${\left|u\right|}_0\le {\left(\frac{(\xi +a)N_1}{\check{\delta }{N}_1^{\beta }}\right)}^{\frac{1}{\beta -1}}=N_2.$$

(20)

From (17) and (20), we have

$$e^{v\left(t\right)}\le N_1{\left|u\right|}_0\le N_1{N}_2.$$

(21)

By (21), we have

$${\left|v\right|}_0\le ln\left(N_1{N}_2\right)=N_3.$$

(22)

Let ${\Omega }_2=\{z\in KerM:Rz\in ImM\}$. If $z\in {\Omega }_2,$ and then $z\in KerM$ and $HRz=0$, which results in

$$u{e}^{-v}=\xi$$

(23)

and

$$-auT+(\xi +a)e^{v}T={\int }_0^T[\delta \left(t\right)e^{\beta v}-\rho \left(t\right)e^{\alpha v}]dt.$$

(24)

From (23) and (24), we have

$$-au+(\xi +a)e^v\ge \check{\delta }{e}^{\beta v}=\check{\delta }\frac{u^{\beta }}{{\xi }^{\beta }}.$$

(25)

Thus, by (9) and (25), we find

$$u\le {\xi }^{\frac{\beta }{\beta -1}}{\left(\frac{a}{\xi }-a+1\right)}^{\frac{1}{\beta -1}}\le N_2.$$

(26)

It follows by (10), (11) and (26) that

$$v=\ln \frac{u}{\xi }\le \ln \frac{N_2}{\xi }\le N_3.$$

(27)

Let $\Omega =\{z={(v,u)}^T:|v{|}_0\le M_1,|u{|}_0\le M_2\}$, where $M_1>N_3,M_2>N_2$. Clearly, ${\Omega }_1,{\Omega }_2\subset \Omega .$ In view of (20), (22), (26) and (27), we verified the conditions (1) and (2) in the Lemma. Now, we claim that the condition (3) of Lemma 1 holds. Let the isomorphic mapping $\Phi :\mathrm{I}\mathrm{m}H\to KerM$ be

$$\Phi (v,u)=\left\{\begin{array}{c}(-u,-v)\mathrm{for}v>d,\hfill \\ (u,-v)\mathrm{for}v<-d.\hfill \end{array}\right.$$

and

$$\mathcal{H}(\omega ,\lambda )=\lambda \omega +\frac{1-\lambda }{T}\Phi HR\omega \mathrm{for}(\omega ,\lambda )\in \Omega \times [0,1].$$

When $v>d$, for $(z,\lambda )\in \partial (\Omega \cap KerM)$, by assumption (H${}_3$), we have

$$\begin{array}{cc}\hfill z^T\mathcal{H}(z,\lambda )& =\lambda (v^2+u^2)+\frac{(1-\lambda )}{T}[auv+u\xi -u^2{e}^{-v}]\hfill \\ & +\frac{(1-\lambda )}{T^2}{\int }_0^{T}v\left[-(\xi +a)e^{v\left(t\right)}+\delta \left(t\right)e^{\beta v\left(t\right)}-\rho \left(t\right)e^{\alpha v\left(t\right)}\right]dt\hfill \\ & >0.\hfill \end{array}$$

Furthermore, when $v<-d$, for $(z,\lambda \in \partial (\Omega \cap KerM)$, by assumption (H${}_3$), we find

$$\begin{array}{cc}\hfill z^T\mathcal{H}(z,\lambda )& =\lambda (v^2+u^2)+\frac{(1-\lambda )}{T}[-auv+u\xi -u^2{e}^{-v}]\hfill \\ & +\frac{(1-\lambda )}{T^2}{\int }_0^{T}v\left[(\xi +a)e^{v\left(t\right)}-\delta \left(t\right)e^{\beta v\left(t\right)}+\rho \left(t\right)e^{\alpha v\left(t\right)}\right]dt\hfill \\ & >0.\hfill \end{array}$$

For $\lambda \in [0,1]$ and $z\in \partial \Omega \cap KerM$, we find $z^T\mathcal{H}(z,\mu )\ne 0.$ Thus,

$$\begin{array}{cc}\hfill deg\left\{\Phi HR,\Omega \cap KerM,0\right\}& =deg\left\{\mathcal{H}(z,0),\Omega \cap KerM,0\right\}\hfill \\ & =deg\left\{\mathcal{H}(z,1),\Omega \cap KerM,0\right\}\ne 0.\hfill \end{array}$$

Hence, the condition (3) of Lemma 1 is satisfied. Applying Lemma 1, there exists a T-periodic solution ${(v,u)}^T$ for system (6). Therefore, there exists a positive T-periodic solution $x\left(t\right)=e^{v\left(t\right)}$ for Equation (3). □

Remark 1. 

The coincidence degree method is one of the main methods to investigate differential equations. When using this method, we should first estimate the prior bound of the solution. In many cases, estimating the prior bound of the solution is the primary difficulty in using this method. For more results about the use of the coincidence degree method, see, e.g., [19,20,21,22,23] and related references.

4. Dynamic Properties of a Positive Periodic Solution

In this section, we consider the dynamic properties of system (5) due to the equivalence of (3) and (5).

Definition 1. 

If $u^{\ast }\left(t\right)={(u_1^{\ast }\left(t\right),v_1^{\ast }\left(t\right))}^T$ is a periodic solution of system (5) and $u\left(t\right)={(u_1\left(t\right),v_1\left(t\right))}^T$ is any solution of system (5) satisfying

$$\underset{t\to +\infty }{lim}\left(|u_1\left(t\right)-u_1^{\ast }\left(t\right)|+|v_1\left(t\right)-v_1^{\ast }\left(t\right)|\right)=0,$$

then we call $u^{\ast }\left(t\right)$ globally asymptotic stable.

We give the following assumption:

Assumption 4 (H${}_4$). 

Set $f\left(u\right)=u^{\gamma },$ where $u\in \mathbb{R},\gamma >0$. There exists $\mathcal{L}>0$ such that

$$|f\left(u_1\right)-f\left(v_1\right)|\le \mathcal{L}|u_1-v_1|\mathrm{for}\mathrm{each}{u}_1,v_1\in \mathbb{R}.$$

Theorem 4. 

Assume that all conditions of Theorem 3 and (H${}_4$) are satisfied. Then, Equation (3) has a globally asymptotic stable positive periodic solution, provided that

$$1-\frac{1}{{\xi }^2}(\xi +a+\widehat{\delta }\mathcal{L}+\widehat{\rho }\mathcal{L})>0,$$

(28)

where $\widehat{\delta }={max}_{t\in [0,T]}\left|\delta \left(t\right)\right|$ and $\widehat{\rho }={max}_{t\in [0,T]}\left|\rho \left(t\right)\right|$.

Proof. 

Using the conditions of Theorem 3, we obtain that that system (5) has a bounded T-periodic solution $u^{\ast }\left(t\right)={(u_1^{\ast }\left(t\right),v_1^{\ast }\left(t\right))}^T$. Assume $u\left(t\right)={(u_1\left(t\right),v_1\left(t\right))}^T$ to be any solution of system (5). By (5), we find

$$\left\{\begin{array}{cc}{\left[(u_1\left(t\right)-u_1^{\ast }\left(t\right)\right]}^{\prime }\hfill & =-\xi \left[(u_1\left(t\right)-u_1^{\ast }\left(t\right)\right]+\left[(v_1\left(t\right)-v_1^{\ast }\left(t\right)\right],\hfill \\ {\left[(v_1\left(t\right)-v_1^{\ast }\left(t\right)\right]}^{\prime }\hfill & =-a\left[(v_1\left(t\right)-v_1^{\ast }\left(t\right)\right]+(\xi +a)\left[(u_1\left(t\right)-u_1^{\ast }\left(t\right)\right]\hfill \\ & -\delta \left(t\right)\left[(u_1^{\beta }\left(t\right)-{\left(u_1^{\ast }\right)}^{\beta }\left(t\right)\right]+\rho \left(t\right)\left[\right(u_1^{\alpha }\left(t\right)-{\left(u_1^{\ast }\right)}^{\alpha }\left(t\right)].\hfill \end{array}\right.$$

(29)

From the first equation of system (29), we have

$$\begin{array}{cc}\hfill u_1\left(t\right)-u_1^{\ast }\left(t\right)& =\left[u_1\left(t_0\right)-u_1^{\ast }\left(t_0\right)\right]e^{{\int }_{t_0}^t-\xi ds}+{\int }_{t_0}^t{e}^{{\int }_t^s\xi ds}\left[(v_1\left(s\right)-v_1^{\ast }\left(s\right)\right]ds\hfill \\ & =\left[u_1\left(t_0\right)-u_1^{\ast }\left(t_0\right)\right]e^{-\xi (t-t_0)}+{\int }_{t_0}^t{e}^{\xi (s-t)}\left[(v_1\left(s\right)-v_1^{\ast }\left(s\right)\right]ds,\hfill \end{array}$$

(30)

where $t\ge t_0$ and $t_0$ is a given constant. By (30), we have

$$|u_1\left(t\right)-u_1^{\ast }\left(t\right)|\le |u_1\left(t_0\right)-u_1^{\ast }\left(t_0\right)|e^{-\xi (t-t_0)}+\frac{1}{\xi }\left[1-e^{-\xi (t-t_0)}\right]{|v_1-v_1^{\ast }|}_0.$$

(31)

By the use of the second equation in system (29), we obtain

$$\begin{array}{cc}\hfill v_1\left(t\right)-v_1^{\ast }\left(t\right)& =\left[v_1\left(t_0\right)-v_1^{\ast }\left(t_0\right)\right]e^{{\int }_{t_0}^t-ads}+{\int }_{t_0}^t{e}^{{\int }_t^{s}ads}[(\xi +a)\left[(u_1\left(s\right)-u_1^{\ast }\left(s\right)\right]\hfill \\ & -\delta \left(s\right)\left[(u_1^{\beta }\left(s\right)-{\left(u_1^{\ast }\right)}^{\beta }\left(s\right)\right]+\rho \left(t\right)\left[\right(u_1^{\alpha }\left(s\right)-{\left(u_1^{\ast }\right)}^{\alpha }\left(s\right)]]ds\hfill \\ & =\left[v_1\left(t_0\right)-v_1^{\ast }\left(t_0\right)\right]e^{-a(t-t_0)}+{\int }_{t_0}^t{e}^{a(s-t)}[(\xi +a)\left[(u_1\left(s\right)-u_1^{\ast }\left(s\right)\right]\hfill \\ & -s\left(s\right)\left[(u_1^{\beta }\left(s\right)-{\left(u_1^{\ast }\right)}^{\beta }\left(s\right)\right]+\rho \left(s\right)\left[\right(u_1^{\alpha }\left(s\right)-{\left(u_1^{\ast }\right)}^{\alpha }\left(s\right)]]ds\hfill \end{array}$$

which, together with assumption (H${}_4$), results in

$$|v_1\left(t\right)-v_1^{\ast }\left(t\right)|\le |v_1\left(t_0\right)-v_1^{\ast }\left(t_0\right)|e^{-a(t-t_0)}+\frac{1}{\xi }\left[1-e^{-a(t-t_0)}\right](\xi +a+\widehat{\delta }\mathcal{L}+\widehat{\rho }\mathcal{L}){|u_1-u_1^{\ast }|}_0.$$

(32)

From (31) and (32), we have

$$\begin{array}{cc}& \left[1-\frac{1}{{\xi }^2}(1-e^{-\xi (t-t_0)})(1-e^{-a(t-t_0)})(\xi +a+\widehat{\delta }\mathcal{L}+\widehat{\rho }\mathcal{L})\right]{|u_1-u_1^{\ast }|}_0\hfill \\ & \le |u_1\left(t_0\right)-u_1^{\ast }\left(t_0\right)|e^{-\xi (t-t_0)}+\frac{1}{\xi }\left[1-e^{-\xi (t-t_0)}\right]|v_1\left(t_0\right)-v_1^{\ast }\left(t_0\right)|e^{-a(t-t_0)}.\hfill \end{array}$$

(33)

Let $t\to +\infty$ for (33), and then

$$\underset{t\to +\infty }{lim}\underset{t\in \mathbb{R}}{max}|u_1\left(t\right)-u_1^{\ast }\left(t\right)|=0.$$

(34)

By (34), we find

$$\underset{t\to +\infty }{lim}|u_1\left(t\right)-u_1^{\ast }\left(t\right)|=0.$$

(35)

In view of (35), let $t\to +\infty$ for (32), and then

$$\underset{t\to +\infty }{lim}|v_1\left(t\right)-v_1^{\ast }\left(t\right)|=0.$$

(36)

By (35) and (36), we have

$$\underset{t\to +\infty }{lim}[|u_1\left(t\right)-u_1^{\ast }\left(t\right)|+|v_1\left(t\right)-v_1^{\ast }\left(t\right)|]=0.$$

(37)

In view of (37), there exists a globally asymptotic stable solution $u^{\ast }\left(t\right)={(u_1^{\ast }\left(t\right),v_1^{\ast }\left(t\right))}^T$ for system (5). Thus, there exists a globally asymptotic stable solution $u_1^{\ast }\left(t\right)$ for Equation (3). □

Remark 2. 

In [2], the authors indicated that a pump belongs to a periodically forced differential system. Hence, studying the asymptotic properties of periodic solutions has important significance to pipe/tank flow configurations. In [2], to obtain an asymptotically stable positive solution of Equation (3), constructing upper and lower and functions is very difficult. In [5], exponential stability of a periodic solution for (3) was obtained. However, the obtained conditions in [5] are strong and are not easy to verify. In this article, we obtain conditions for the existence of periodic solutions that are easy to verify.

5. Example

Equation (3) has wide applications in pipe/tank configurations. Consider the following nonlinear differential system:

$$\left\{\begin{array}{c}{y}^{\prime }\left(t\right)=-\xi y\left(t\right)+u\left(t\right),\hfill \\ u^{\prime }\left(t\right)=-au\left(t\right)+(\xi +a)y\left(t\right)-\delta \left(t\right)y^{\beta }\left(t\right)+\rho \left(t\right)y^{\alpha }\left(t\right),\hfill \end{array}\right.$$

(38)

where $T=2\pi ,\xi =2,a=1.9,\delta \left(t\right)=0.75\times {10}^{-3}>0,\rho \left(t\right)=sint-2<0,\alpha =0.1$ and $\beta =1.5.$ Clearly, assumptions (H${}_1$) and (H${}_2$) hold. From $\delta \left(t\right)>0$ and $\rho \left(t\right)<0$, assumption (H${}_3$) holds. After a simple calculation, we have

$$N_1=\frac{1}{\xi T}+e^{\xi T}T\approx 1.79\times {10}^6,N_2={\left(\frac{(\xi +a)N_1}{\check{\delta }{N}_1^{\beta }}\right)}^{\frac{1}{\beta -1}}\approx 16,N_3=ln\left(N_1{N}_2\right)\approx 17.17$$

Thus,

$${\xi }^{\frac{\beta }{\beta -1}}{\left(\frac{a}{\xi }-a+1\right)}^{\frac{1}{\beta -1}}\approx 0.02\le N_2=16,$$

and

$$\ln \frac{N_2}{\xi }\approx 5.05\le 17.17=N_3.$$

Thus, all conditions of Theorem 3 are satisfied. Hence, there is a positive periodic solution for (3). Furthermore, choosing $\mathcal{L}=1\times {10}^{-3}$, we have

$$1-\frac{1}{{\xi }^2}(\xi +a+\widehat{s}\mathcal{L}+\widehat{r}\mathcal{L})=0.03>0$$

and condition (28) holds. Hence, all conditions of Theorem 4 hold. From Theorem 4, there is a globally asymptotic stable positive periodic solution ${(y,u)}^T$ for (38). Hence, there is globally asymptotic stable positive periodic solution x.

System (38) has a globally asymptotic stable positive periodic solution ${(y,u)}^T$, i.e., Equation (3) has a globally asymptotic stable positive periodic solution. Figure 1 supports this conclusion.

Remark 3. 

In [5], the authors obtained the following results for (3).

Theorem 5. 

Assume that $\rho \in C([t_0,\infty ),(0,\infty )),a>0,0<\alpha <\beta <1,c>0$, and there is function $\vartheta \in C([t_0,\infty ),\mathbb{R})$ and constants $L,l$ such that the following conditions are satisfied:

$$0<l\le exp\left({\int }_{t_0}^t[-a+\vartheta \left(s\right)]ds\right)\le L\mathrm{for}t\ge t_0,$$

(39)

$${\int }_t^{t+T}[-a+\vartheta \left(s\right)]ds=0\mathrm{for}t\ge t_0,$$

(40)

$$\begin{array}{cc}\hfill \vartheta \left(t\right)exp\left({\int }_{t_0}^t[-a+\vartheta \left(s\right)]ds\right)& ={\int }_{t_0}^t[\delta \left(s\right)exp\left(\alpha {\int }_{t_0}^s[-a+\vartheta \left(v\right)]dv\right)\hfill \\ & -\rho \left(s\right){\int }_{t_0}^{t}exp\left(\beta {\int }_{t_0}^s[-a+\vartheta \left(v\right)]dv\right)]ds\mathrm{for}t\ge t_0.\hfill \end{array}$$

(41)

In addition, there are constants $L_{\ast },l_{\ast }>0$ such that $l_{\ast }\le l,L_{\ast }\ge L$ and

$$\alpha l_{\ast }^{\alpha -1}\delta \left(t\right)-\beta L_{\ast }^{\beta -1}\rho \left(t\right)\le 0\mathrm{for}t\ge t_0,$$

(42)

where $\delta \left(t\right)=\frac{c}{\mu },\rho \left(t\right)=\frac{e\left(t\right)}{\mu }$. Then, there is an exponentially stable positive periodic solution for (3).

Clearly, conditions (39)–(41) are strong and difficult to verify. In [5], the authors obtained the existence of positive periodic solution of Equation (3) by using Schauder’s fixed point theorem. Furthermore, they obtained exponential stability of a positive periodic solution of Equation (3) by using the Lyapunov function method. The numerical simulation in [5] supports these conclusions. In the present paper, under more relaxed conditions, the numerical simulation supports our conclusions. Therefore, the results of this paper improve and enhance the results of [5].

6. Conclusions and Discussion

This paper considered a positive periodic solution of a nonlinear equation related to valveless pumping. The second-order equation reduced to a two-dimensional system via a linear variable transformation. We obtained some results for a periodic solution to (3). A simulation example demonstrated the validity of our results.

Since Equation (3) is a singular second-order equation, in future research work, we will focus on studying how the singularity of the equation affects the existence of the solution. In recent years, stochastic differential systems have received great attention. Hu, Zhu and Karimi [24] studied the Razumikhin stability theorem for a class of impulsive stochastic delay differential systems by using the stochastic analysis technique and Razumikhin approach. In [14], the authors dealt with the exponential stability of an impulsive stochastic food chain system with time-varying delays. The stability problems of stochastic nonlinear delay systems with exogenous disturbances and event-triggered feedback control were investigated in [17]. In the future, we will study stochastic pipe/tank flow configurations with friction.