Ulam Stability of Second-Order Periodic Linear Differential Equations with Periodic Damping

Abstract

We addressed the Ulam stability of second-order periodic linear differential equations with periodic damping. The necessary and sufficient conditions for the Ulam stability of second-order periodic linear differential equations were obtained by providing an Ulam stability theorem and an Ulam instability theorem. In particular, when the elastic coefficient remained constant at 0, a very general conclusion was obtained. These results expand on the conclusions in the relevant literature. In addition, for the situation of constant coefficients, the minimum Ulam constant is provided.

1. Introduction

We are concerned with the Ulam stability of the following second-order periodic linear homogeneous equation

$$y^{″}\left(x\right)+f\left(x\right)y^{\prime }\left(x\right)+g\left(x\right)y\left(x\right)=0,$$

(1)

where $f\left(x\right)$ and $g\left(x\right)$ are continuously differentiable $\omega$-periodic functions. In practice, this equation can be used to describe the motion of a spring oscillator with a damping coefficient f and elastic coefficient g.

We say that (1) has Ulam stability (abbreviated as “US”) on ℝ if there is a constant $k>0$ such that, for every $\epsilon >0$ and every $\eta \in C^2(\mathbb{R},\mathbb{R})$ satisfying

$$|{\eta }^{″}\left(x\right)+f\left(x\right){\eta }^{\prime }\left(x\right)+g\left(x\right)\eta \left(x\right)|⩽\epsilon ,$$

there is a solution $y\in C^2(\mathbb{R},\mathbb{R})$ of (1) such that

$$\left|y\right(x)-\eta (x\left)\right|⩽k\epsilon ,\forall x\in \mathbb{R}.$$

The constant k is called the US constant of (1) on ℝ.

The concept of Ulam stability, first applied to functional equations, was posed by Ulam [1,2] and has been well developed subsequently. In 1998, this concept shifted in the field of differential equations [3]. From then onwards, the Ulam stability of differential equations has attracted the attention of many researchers (see [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]). Recently, Fukutaka and Onitsuka [9,10] considered the Ulam stability of the equation

$$y^{\prime }\left(x\right)-\lambda \left(x\right)y\left(x\right)=f\left(x\right)$$

(2)

on ℝ, where $\lambda \left(x\right)$ is a continuous periodic function. Let $I=(a,b)$, $-\infty ⩽a<b⩽\infty$. Fukutaka and Onitsuka [10] established the following theorem.

Theorem 1. 

Let $A\left(x\right)$ and $\Gamma \left(x\right)$ be antiderivatives of $\lambda \left(x\right)$ and $e^{-A\left(x\right)}$ on ℝ, respectively. Suppose that $\lambda \left(x\right)$ is a periodic function with period $\omega >0$ on ℝ. Then, the following holds:

(i)

if $b=\infty$ and ${\int }_0^{\omega }\lambda \left(x\right)dx>0$, then (2) has US with minimum US constant:

$$\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}\Gamma \left(x\right)\right)-\Gamma \left(x\right)\right\}{e}^{A\left(x\right)}$$

on I;

(ii)

if $a=-\infty$ and ${\int }_0^{\omega }\lambda \left(x\right)dx<0$, then (2) has US with minimum US constant:

$$\underset{x\in (0,\omega ]}{max}\left\{\Gamma \left(x\right)-\left(\underset{x\to -\infty }{lim}\Gamma \left(x\right)\right)\right\}{e}^{A\left(x\right)}$$

on I.

In [10], they also obtain a necessary and sufficient condition as follows.

Theorem 2. 

Suppose that $\lambda \left(x\right)$ is a periodic function with period $\omega >0$ on ℝ. Then, (2) has US on ℝ if and only if ${\int }_0^{\omega }\lambda \left(x\right)dx\ne 0$.

Soon afterwards, Fukutaka and Onitsuka [11] studied Hill’s equation:

$$y^{″}\left(x\right)-({\lambda }^2\left(x\right)-{\lambda }^{\prime }\left(x\right))y\left(x\right)=0$$

(3)

and used Theorem 1 to prove the following theorem:

Theorem 3. 

Let $\Lambda \left(x\right)$, ${\Gamma }^{+}\left(x\right)$ and ${\Gamma }^{-}\left(x\right)$ be antiderivatives of $\lambda \left(x\right)$, $e^{\Lambda \left(x\right)}$ and $e^{-\Lambda \left(x\right)}$ on ℝ, respectively. Suppose that $\lambda \left(x\right)$ is a periodic function with period $\omega >0$ on ℝ. Then, the following holds:

(i)

If ${\int }_0^{\omega }\lambda \left(x\right)dx>0$, then (3) has US with a US constant

$$\left[\underset{x\in (0,\omega ]}{max}\left\{{\Gamma }^{+}\left(x\right)-\left(\underset{x\to -\infty }{lim}{\Gamma }^{+}\left(x\right)\right)\right\}{e}^{-\Lambda \left(x\right)}\right]\left[\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }^{-}\left(x\right)\right)-{\Gamma }^{-}\left(x\right)\right\}{e}^{\Lambda \left(x\right)}\right]$$

on ℝ;

(ii)

If ${\int }_0^{\omega }\lambda \left(x\right)dx<0$, then (3) has US with US constant

$$\left[\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }^{+}\left(x\right)\right)-{\Gamma }^{+}\left(x\right)\right\}{e}^{-\Lambda \left(x\right)}\right]\left[\underset{x\in (0,\omega ]}{max}\left\{{\Gamma }^{-}\left(x\right)-\left(\underset{x\to -\infty }{lim}{\Gamma }^{-}\left(x\right)\right)\right\}{e}^{\Lambda \left(x\right)}\right]$$

on ℝ.

Obviously, the differential Equation (3) is a special form of Equation (1). Cǎdariu, Popa and Raşa [12] discussed (1) and obtained the following result.

Theorem 4. 

Suppose that $K>0$ such that

$$\pm g\left(x\right)⩾{\displaystyle \frac{1}{K}},x\in I$$

(4)

and the Riccati equation $u^{\prime }=-u^2+fu+f^{\prime }\mp g$ has a solution $u\in C^1(I,\mathbb{R})$, with $u\left(a\right)=f\left(a\right)$. Then, Equation (1) is Ulam-stable with the Ulam constant K.

Motivated by Fukutaka and Onitsuka [9,10,11], this paper is devoted to the study of Equation (1) and establishing a US condition without the restriction of (4). Since the equation we are studying has a damping coefficient, our purpose is also to extend Theorem 3.

In the next section, an Ulam stability theorem and Ulam instability theorem are displayed. Furthermore, a necessary and sufficient condition is provided. Section 3 focuses on the minimum Ulam stability constant for a second-order constant coefficient differential equation.

2. Main Results

In this section, we first present the US results of (1).

Theorem 5. 

Suppose that the Riccati equation

$$p^{\prime }=p^2+fp-f^{\prime }+g$$

(5)

has a ω-periodic solution $p\left(x\right)$ satisfying ${\int }_0^{\omega }p\left(x\right)dx\ne 0$. Let ${\Lambda }_p\left(x\right)$, ${\Lambda }_q\left(x\right)$, ${\Gamma }_p\left(x\right)$ and ${\Gamma }_q\left(x\right)$ be antiderivatives of $p\left(x\right)$, $-p\left(x\right)-f\left(x\right)$, $e^{-{\Lambda }_p\left(x\right)}$ and $e^{-{\Lambda }_q\left(x\right)}$ on ℝ, respectively. Then, the following holds:

(i)

If ${\int }_0^{\omega }p\left(x\right)dx>0$ and ${\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx<0$, then (1) has US with a US constant

$$\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }_p\left(x\right)\right)-{\Gamma }_p\left(x\right)\right\}{e}^{{\Lambda }_p\left(x\right)}\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }_q\left(x\right)\right)-{\Gamma }_q\left(x\right)\right\}{e}^{{\Lambda }_q\left(x\right)}$$

on ℝ;

(ii)

If ${\int }_0^{\omega }p\left(x\right)dx>0$ and ${\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx>0$, then (1) has US with a US constant

$$\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }_p\left(x\right)\right)-{\Gamma }_p\left(x\right)\right\}{e}^{{\Lambda }_p\left(x\right)}\left[\underset{x\in (0,\omega ]}{max}\left\{{\Gamma }_q\left(x\right)-\left(\underset{x\to -\infty }{lim}{\Gamma }_q\left(x\right)\right)\right\}{e}^{{\Lambda }_q\left(x\right)}\right]$$

on ℝ;

(iii)

If ${\int }_0^{\omega }p\left(x\right)dx<0$ and ${\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx<0$, then (1) has US with a US constant

$$\left[\underset{x\in (0,\omega ]}{max}\left\{{\Gamma }_p\left(x\right)-\left(\underset{x\to -\infty }{lim}{\Gamma }_p\left(x\right)\right)\right\}{e}^{{\Lambda }_p\left(x\right)}\right]\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }_q\left(x\right)\right)-{\Gamma }_q\left(x\right)\right\}{e}^{{\Lambda }_q\left(x\right)}$$

on ℝ;

(iv)

if ${\int }_0^{\omega }p\left(x\right)dx<0$ and ${\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx>0$, then (1) has US with a US constant

$$\left[\underset{x\in (0,\omega ]}{max}\left\{{\Gamma }_p\left(x\right)-\left(\underset{x\to -\infty }{lim}{\Gamma }_p\left(x\right)\right)\right\}{e}^{{\Lambda }_p\left(x\right)}\right]\left[\underset{x\in (0,\omega ]}{max}\left\{{\Gamma }_q\left(x\right)-\left(\underset{x\to -\infty }{lim}{\Gamma }_q\left(x\right)\right)\right\}{e}^{{\Lambda }_q\left(x\right)}\right]$$

on ℝ.

Proof. 

Let $q\left(x\right)=-f\left(x\right)-p\left(x\right)$. Then, $p,q$ are continuously differentiable on ℝ and $f=-p-q$. It follows from (5) that

$$g=-q^{\prime }+pq.$$

Then, Equation (1) becomes the following

$$y^{″}-(p+q)y^{\prime }+(pq-q^{\prime })y=0,$$

which is also equivalent to the equation

$${(y^{\prime }-qy)}^{\prime }-p(y^{\prime }-qy)=0.$$

(6)

For $\forall \epsilon >0$, we assume that $\eta \in C^2\left(\mathbb{R}\right)$ satisfies

$$|{({\eta }^{\prime }-q\eta )}^{\prime }-p({\eta }^{\prime }-q\eta )|⩽\epsilon ,\forall x\in \mathbb{R}.$$

Let $\phi \left(x\right)={\eta }^{\prime }\left(x\right)-q\left(x\right)\eta \left(x\right)$, $x\in \mathbb{R}$. Then, $\phi \in C^1\left(\mathbb{R}\right)$ and

$$|{\phi }^{\prime }-p\phi |⩽\epsilon ,\forall x\in \mathbb{R}.$$

Now, we prove (i). Let $\varphi$ be a solution of the following equation

$${\varphi }^{\prime }-p\varphi =0.$$

Using Theorem 1, we determine from ${\int }_0^{\omega }p\left(x\right)dx>0$ that

$$|{\eta }^{\prime }-q\eta -\varphi |=|\phi -\varphi |⩽\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }_p\left(x\right)\right)-{\Gamma }_p\left(x\right)\right\}{e}^{{\Lambda }_p\left(x\right)}\epsilon .$$

Let z be a solution of the equation

$$z^{\prime }-qz-\varphi =0.$$

Using Theorem 1 again, one can determine from ${\int }_0^{\omega }q\left(x\right)dx>0$ that

$$|\eta \left(x\right)-z\left(x\right)|⩽\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }_p\left(x\right)\right)-{\Gamma }_p\left(x\right)\right\}{e}^{{\Lambda }_p\left(x\right)}\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\Gamma }_q\left(x\right)\right)-{\Gamma }_q\left(x\right)\right\}{e}^{{\Lambda }_q\left(x\right)}\epsilon .$$

The proof for (ii), (iii), and (iv) is similar to (i), so we omit it.This completes the proof. ☐

Next, we discuss the instability of (1) and necessary and sufficient conditions of stability of (1). Immediately, we consider the instability of (1) for the general situation.

Theorem 6. 

Suppose that the Riccati Equation (5) has a ω-periodic solution $p\left(x\right)$. If ${\int }_0^{\omega }p\left(x\right)dx=0$ or ${\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx=0$, then (1) is not Ulam-stable.

Proof. 

Set $q\left(x\right)=-p\left(x\right)-f\left(x\right)$; then, Equation (1) becomes (6). For any $\epsilon >0$, we consider the following equation

$${({\eta }^{\prime }-q\eta )}^{\prime }-p({\eta }^{\prime }-q\eta )=\epsilon .$$

By multiplying both sides of the above equation by $e^{-{\Lambda }_p\left(x\right)}$ and integrating them, we obtain

$$({\eta }^{\prime }-q\eta )e^{-{\Lambda }_p\left(x\right)}=c_1+\epsilon {\int }_0^x{e}^{-{\Lambda }_p\left(u\right)}du.$$

This deduces the following

$${\eta }^{\prime }-q\eta =c_1{e}^{{\Lambda }_p\left(x\right)}+\epsilon e^{{\Lambda }_p\left(x\right)}{\int }_0^x{e}^{-{\Lambda }_p\left(u\right)}du.$$

By multiplying both sides of the above equation by $e^{-{\Lambda }_q\left(x\right)}$ and integrating them again, we obtain

$$\eta \left(x\right)e^{-{\Lambda }_q\left(x\right)}=c_1{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}ds+c_2+\epsilon {\int }_0^x\left(e^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds$$

So, the function

$$\eta \left(x\right)=\left\{c_1{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}ds+c_2+\epsilon {\int }_0^x\left(e^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds\right\}{e}^{{\Lambda }_q\left(x\right)}$$

is the general solution of the equation.

$${({\eta }^{\prime }-q\eta )}^{\prime }-p({\eta }^{\prime }-q\eta )=\epsilon .$$

Let

$$y\left(x\right)=\left\{c_3{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}ds+c_4\right\}{e}^{{\Lambda }_q\left(x\right)},$$

Then, it is the general solution of Equation (6). We obtain the following:

$$\begin{array}{cc}\hfill \left|\eta \right(x)-y(x\left)\right|& =|(c_1-c_3){\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}ds+(c_2-c_4)\hfill \\ \hfill & +\epsilon {\int }_0^x\left(e^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds|e^{{\Lambda }_q\left(x\right)}\hfill \\ \hfill & =|{\displaystyle \frac{c_2-c_4}{\epsilon }}+{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds|e^{{\Lambda }_q\left(x\right)}\epsilon .\hfill \end{array}$$

If ${\int }_0^{\omega }p\left(x\right)dx=0$ and ${\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx=0$ (i.e., ${\int }_0^{\omega }q\left(x\right)dx=0$), we obtain four numbers $m_p$, $m_q$, $M_p$ and $M_q$ such that

$$m_p⩽{\Lambda }_p\left(x\right)⩽M_p$$

and

$$m_q⩽{\Lambda }_q\left(x\right)⩽M_q$$

for $x\in \mathbb{R}$. For $s>0$,

$$\begin{array}{c}\hfill e^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)⩾e^{m_p-M_q}{e}^{-M_p}s-e^{M_p-m_q}{\displaystyle \frac{|c_1-c_3|}{\epsilon }}.\end{array}$$

This implies that, for $x>0$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{c_2-c_4}{\epsilon }}& +{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds\hfill \\ \hfill & ⩾{\displaystyle \frac{1}{2}}{e}^{m_p-M_q}{e}^{-M_p}{x}^2-e^{M_p-m_q}{\displaystyle \frac{|c_1-c_3|}{\epsilon }}x+{\displaystyle \frac{c_2-c_4}{\epsilon }}.\hfill \end{array}$$

Then, there is a $x_1>0$ such that, for $x⩾x_1$,

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{e}^{m_p-M_q}{e}^{-M_p}{x}^2-e^{M_p-m_q}{\displaystyle \frac{|c_1-c_3|}{\epsilon }}x+{\displaystyle \frac{c_2-c_4}{\epsilon }}>x.\end{array}$$

Hence, for $x>x_1$,

$$\begin{array}{c}\hfill |{\displaystyle \frac{c_2-c_4}{\epsilon }}+{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds|e^{{\Lambda }_q\left(x\right)}\epsilon >e^{m_q}x\epsilon .\end{array}$$

So, we obtain ${lim}_{x\to \infty }|\eta \left(x\right)-y\left(x\right)|=\infty$. This indicates that Equation (1) is not Ulam-stable.

If ${\int }_0^{\omega }p\left(x\right)dx\ne 0$ and ${\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx=0$ (i.e., ${\int }_0^{\omega }q\left(x\right)dx=0$). Let $p\left(x\right)=\overline{p}+\tilde{p}\left(x\right)$, where $\overline{p}={\displaystyle \frac{1}{\omega }}{\int }_0^{\omega }p\left(x\right)dx$. Let us first consider ${\int }_0^{\omega }p\left(x\right)dx>0$. Then, ${\int }_0^{\omega }\tilde{p}\left(x\right)dx=0$ and $\overline{p}>0$. We obtain two numbers $m_{\tilde{p}}$ and $M_{\tilde{p}}$ such that

$$m_{\tilde{p}}⩽{\Lambda }_{\tilde{p}}\left(x\right)⩽M_{\tilde{p}}$$

for $x\in \mathbb{R}$.

If $c_1⩾c_3$, for $s>0$,

$$\begin{array}{cc}\hfill e^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)& ⩾e^{\overline{p}s+m_{\tilde{p}}-M_q}{\int }_0^s{e}^{-\overline{p}u-M_{\tilde{p}}}du\hfill \\ \hfill & =e^{\overline{p}s+m_{\tilde{p}}-M_q-M_{\tilde{p}}}{\displaystyle \frac{1}{\overline{p}}}(1-e^{-\overline{p}s})\hfill \\ \hfill & ={\displaystyle \frac{1}{\overline{p}}}{e}^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}(e^{\overline{p}s}-1).\hfill \end{array}$$

This implies that, for $x>0$ and $c_1⩾c_3$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{c_2-c_4}{\epsilon }}& +{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds\hfill \\ \hfill & ⩾e^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}\left({\displaystyle \frac{1}{{\overline{p}}^2}}(e^{\overline{p}x}-1)-{\displaystyle \frac{1}{\overline{p}}}x\right)+{\displaystyle \frac{c_2-c_4}{\epsilon }}.\hfill \end{array}$$

Then, there is a $x_1>0$ such that, for $x⩾x_1$,

$$\begin{array}{c}\hfill e^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}\left({\displaystyle \frac{1}{{\overline{p}}^2}}(e^{\overline{p}x}-1)-{\displaystyle \frac{1}{\overline{p}}}x\right)+{\displaystyle \frac{c_2-c_4}{\epsilon }}>x.\end{array}$$

Hence, for $x>x_1$,

$$\begin{array}{c}\hfill |{\displaystyle \frac{c_2-c_4}{\epsilon }}+{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds|e^{{\Lambda }_q\left(x\right)}\epsilon >e^{m_q}x\epsilon .\end{array}$$

So, we obtain ${lim}_{x\to \infty }|\eta \left(x\right)-y\left(x\right)|=\infty$.

If $c_1<c_3$, for $s<0$,

$$\begin{array}{cc}\hfill -e^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)& =e^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_3-c_1}{\epsilon }}+{\int }_s^0{e}^{-{\Lambda }_p\left(u\right)}du\right)\hfill \\ \hfill & ⩾e^{\overline{p}s+m_{\tilde{p}}-M_q}{\int }_s^0{e}^{-\overline{p}u-M_{\tilde{p}}}du\hfill \\ \hfill & =e^{\overline{p}s+m_{\tilde{p}}-M_q-M_{\tilde{p}}}{\displaystyle \frac{1}{\overline{p}}}(e^{-\overline{p}s}-1)\hfill \\ \hfill & =e^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}{\displaystyle \frac{1}{\overline{p}}}(1-e^{\overline{p}s}).\hfill \end{array}$$

Thisimplies that, for $x<0$ and $c_1<c_3$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{c_2-c_4}{\epsilon }}& +{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds\hfill \\ \hfill & ={\displaystyle \frac{c_2-c_4}{\epsilon }}-{\int }_x^0{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds\hfill \\ \hfill & ⩾{\int }_x^0{e}^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}{\displaystyle \frac{1}{\overline{p}}}(1-e^{\overline{p}s})ds+{\displaystyle \frac{c_2-c_4}{\epsilon }}\hfill \\ \hfill & =e^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}\left(-{\displaystyle \frac{1}{{\overline{p}}^2}}(1-e^{\overline{p}x})-{\displaystyle \frac{1}{\overline{p}}}x\right)+{\displaystyle \frac{c_2-c_4}{\epsilon }}.\hfill \end{array}$$

Noting that $\overline{p}>0$, we obtain a $x_2<0$ such that, for $x<x_2$,

$$\begin{array}{c}\hfill e^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}\left(-{\displaystyle \frac{1}{{\overline{p}}^2}}(1-e^{\overline{p}x})-{\displaystyle \frac{1}{\overline{p}}}x\right)+{\displaystyle \frac{c_2-c_4}{\epsilon }}>e^{m_{\tilde{p}}-M_q-M_{\tilde{p}}}\left(-{\displaystyle \frac{1}{2\overline{p}}}x\right).\end{array}$$

Hence, for $x<x_2$,

$$\begin{array}{c}\hfill |{\displaystyle \frac{c_2-c_4}{\epsilon }}+{\int }_0^x{e}^{{\Lambda }_p\left(s\right)-{\Lambda }_q\left(s\right)}\left({\displaystyle \frac{c_1-c_3}{\epsilon }}+{\int }_0^s{e}^{-{\Lambda }_p\left(u\right)}du\right)ds|e^{{\Lambda }_q\left(x\right)}\epsilon >e^{m_q+m_{\tilde{p}}-M_q-M_{\tilde{p}}}\left(-{\displaystyle \frac{1}{2\overline{p}}}x\right)\epsilon .\end{array}$$

So, we obtain ${lim}_{x\to -\infty }|\eta \left(x\right)-y\left(x\right)|=\infty$.

Hence, for ${\int }_0^{\omega }p\left(x\right)dx>0$ and ${\int }_0^{\omega }q\left(x\right)dx=0$, Equation (1) is not Ulam-stable.

The situation of ${\int }_0^{\omega }p\left(x\right)dx<0$ and ${\int }_0^{\omega }q\left(x\right)dx=0$ and the situation of ${\int }_0^{\omega }p\left(x\right)dx=0$ and ${\int }_0^{\omega }q\left(x\right)dx\ne 0$ are similar to the above discussion, so we omit them. This completes the proof. ☐

To this extent, we can use Theorems 5 and 6 to obtain the following necessary and sufficient conditions.

Theorem 7. 

Suppose that the Riccati Equation (5) has a ω-periodic solution $p\left(x\right)$. Then, (1) has US on ℝ if and only if ${\int }_0^{\omega }p\left(x\right)dx{\int }_0^{\omega }(p\left(x\right)+f\left(x\right))dx\ne 0$.

When elastic coefficient $g\equiv 0$, the Riccati Equation (5) has a periodic solution $p=-f$. Then, $q=-f-p=0$ and the following theorem is true when using Theorem 6.

Theorem 8. 

Suppose that $g\left(x\right)\equiv 0$; then, (1) is not Ulam-stable.

Remark 1. 

Consider the differential equation

$$y^{″}-(1+cosx)y^{\prime }+({\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}cosx)y=0.$$

(7)

We have $f\left(x\right)=-(1+cosx)$ and $g\left(x\right)={\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}cosx$. Since $-{\displaystyle \frac{1}{4}}⩽g\left(x\right)⩽{\displaystyle \frac{3}{4}}$, Theorem 4 cannot be used for (7). Meanwhile, the theorems of [11] cannot be used for (7) yet because (7) has a damping coefficient. The Riccati equation

$$p^{\prime }=p^2+fp-f^{\prime }+g=p^2-(1+cosx)p+{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{2}}cosx-sinx$$

has a $2\pi$-periodic solution $p\left(x\right)={\displaystyle \frac{1}{2}}+cosx$. Let $q\left(x\right)={\displaystyle \frac{1}{2}}$. By using Theorem 5, we know that (7) has US on ℝ. Accordingly, the above theorems are all novel.

Remark 2. 

The results of the above theorems have symmetry. In fact, we can consider two symmetric equations

$$y^{″}-(p+q)y^{\prime }+(pq-q^{\prime })y=0$$

and

$$y^{″}-(p+q)y^{\prime }+(pq-p^{\prime })y=0.$$

They have the same conclusions under the framework of Theorems 5–7.

Remark 3. 

For a nonhomogeneous equation

$$y^{″}\left(x\right)+f\left(x\right)y^{\prime }\left(x\right)+g\left(x\right)y=h\left(x\right),$$

we can obtain the analogues by using a similar approach as in [11].

3. Minimal US Constant for Constant Coefficient Differential Equation

In this section, we consider the constant coefficient differential equation

$$y^{″}-(p+q)y^{\prime }+pqy=0,$$

(8)

where $p,q\in \mathbb{R}$. We have the following theorem.

Theorem 9. 

If $pq\ne 0$, then (8) has US with a minimum US constant ${\displaystyle \frac{1}{\left|pq\right|}}$ on ℝ.

Proof. 

According to the definitions of ${\Lambda }_p$, ${\Lambda }_q$, ${\Gamma }_p$ and ${\Gamma }_q$, we set

$${\Lambda }_p\left(x\right)=px,{\Lambda }_q\left(x\right)=qx,{\Gamma }_p\left(x\right)=-{\displaystyle \frac{1}{p}}{e}^{-px},{\Gamma }_q\left(x\right)=-{\displaystyle \frac{1}{q}}{e}^{-qx}.$$

Let k be US constant. If $p>0$ and $q>0$, we use Theorem 5 to determine that $k={\displaystyle \frac{1}{pq}}$. Similarly, we determine that $k={\displaystyle \frac{1}{-pq}}$ if $p>0$ and $q<0$ or $p<0$ and $q>0$ and $k={\displaystyle \frac{1}{pq}}$ if $p<0$ and $q<0$. So, (8) has US with a US constant ${\displaystyle \frac{1}{\left|pq\right|}}$ on ℝ.

Next, we will use the method of contradiction to prove its minimization. Assume that there is a US constant $k_0$ satisfying $0<k_0<{\displaystyle \frac{1}{\left|pq\right|}}$. Let $y\in C^2(\mathbb{R},\mathbb{R})$ be the solution of Equation (8), and let $\eta \in C^2(\mathbb{R},\mathbb{R})$ be the solution of the equation

$${\eta }^{″}-(p+q){\eta }^{\prime }+pq\eta =\epsilon .$$

We have

$$({\eta }^{″}-y^{″})-(p+q)({\eta }^{\prime }-y^{\prime })+pq(\eta -y)=\epsilon .$$

since $k_0$ is US constant, so

$$|\eta -y|⩽k_0\epsilon .$$

(9)

Denote $\eta -y$ by z. We obtain

$$z^{″}-(p+q)z^{\prime }+pqz=\epsilon .$$

Then,

$$\begin{array}{cc}\hfill z^{″}-(p+q)z^{\prime }& =-pqz+\epsilon \hfill \\ \hfill & ⩾-\left|pq\right|k_0\epsilon +\epsilon \hfill \\ \hfill & =|pq|\epsilon ({\displaystyle \frac{1}{\left|pq\right|}}-k_0)\hfill \\ \hfill & \triangleq M>0.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill z^{\prime }-(p+q)z⩾{\int }_{x_0}^{x}Mds-z^{\prime }\left(x_0\right)+(p+q)z\left(x_0\right).\end{array}$$

This deduces that

$$\begin{array}{cc}\hfill z^{\prime }& =(p+q)z+Mx-M{x}_0-z^{\prime }\left(x_0\right)+(p+q)z\left(x_0\right)\hfill \\ \hfill & ⩾-|p+q|k_0\epsilon +Mx-M{x}_0-z^{\prime }\left(x_0\right)+(p+q)z\left(x_0\right).\hfill \end{array}$$

Put $N=-|p+q|k_0\epsilon -M{x}_0-z^{\prime }\left(x_0\right)+(p+q)z\left(x_0\right)$, so we obtain $z^{\prime }⩾Mx+N$. Thus, we obtain

$$z\left(x\right)⩾{\displaystyle \frac{1}{2}}M{x}^2-{\displaystyle \frac{1}{2}}M{x}_0^2+Nx-N{x}_0+z\left(x_0\right),$$

which implies $li{m}_{x\to \infty }z\left(x\right)=\infty$, i.e., $li{m}_{x\to \infty }(\eta \left(x\right)-y\left(x\right))=\infty$. This contradicts (9) on ℝ. This ends the proof. ☐

Remark 4. 

For a nonhomogeneous equation

$$y^{″}-(p+q)y^{\prime }+pqy=h\left(x\right),$$

(10)

where $p,q\in \mathbb{R}$ and $h\in C\left(\mathbb{R}\right)$, we can use a treatment similar to the proof of Theorem 9 to determine that, if $pq\ne 0$, then (10) has US with a minimum US constant ${\displaystyle \frac{1}{\left|pq\right|}}$ on ℝ. More generally, if ${\alpha }^2-4\beta >0$, the equation

$$y^{″}+\alpha y^{\prime }+\beta y=h\left(x\right)$$

has US with a minimum US constant ${\displaystyle \frac{1}{\left|\beta \right|}}$ on ℝ. Therefore, the US of the equation is closely related to the elastic coefficient. For the case of constant coefficients, the best US constant depends only on the elastic coefficient.

Remark 5. 

Let $p=-q=\lambda$, then Equation (8) becomes

$$y^{″}-{\lambda }^{2}y=0.$$

In this situation, the minimum US constant is ${\displaystyle \frac{1}{\left|pq\right|}}={\displaystyle \frac{1}{{\lambda }^2}}$ on ℝ. This states that the above theorem extends Theorem 7 of [11].