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

1. Main Results

In this section, we first present a US result of (1).

Theorem 5. 

Suppose that the Riccati equation

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

(1.1)

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

• (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 US constant

  $$\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\mathrm{\Gamma }}_p\left(x\right)\right)-{\mathrm{\Gamma }}_p\left(x\right)\right\}{e}^{{\mathrm{\Lambda }}_p\left(x\right)}\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\mathrm{\Gamma }}_q\left(x\right)\right)-{\mathrm{\Gamma }}_q\left(x\right)\right\}{e}^{{\mathrm{\Lambda }}_q\left(x\right)}$$
  on$\mathbb{R}$;
• (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 US constant

  $$\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\mathrm{\Gamma }}_p\left(x\right)\right)-{\mathrm{\Gamma }}_p\left(x\right)\right\}{e}^{{\mathrm{\Lambda }}_p\left(x\right)}\left[\underset{x\in (0,\omega ]}{max}\left\{{\mathrm{\Gamma }}_q\left(x\right)-\left(\underset{x\to -\infty }{lim}{\mathrm{\Gamma }}_q\left(x\right)\right)\right\}{e}^{{\mathrm{\Lambda }}_q\left(x\right)}\right]$$
  on$\mathbb{R}$;
• (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 US constant

  $$\left[\underset{x\in (0,\omega ]}{max}\left\{{\mathrm{\Gamma }}_p\left(x\right)-\left(\underset{x\to -\infty }{lim}{\mathrm{\Gamma }}_p\left(x\right)\right)\right\}{e}^{{\mathrm{\Lambda }}_p\left(x\right)}\right]\underset{x\in (0,\omega ]}{max}\left\{\left(\underset{x\to \infty }{lim}{\mathrm{\Gamma }}_q\left(x\right)\right)-{\mathrm{\Gamma }}_q\left(x\right)\right\}{e}^{{\mathrm{\Lambda }}_q\left(x\right)}$$
  on$\mathbb{R}$;
• (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 US constant

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

Proof. 

Let $q\left(t\right)=-f\left(x\right)-p\left(x\right)$. Then $p,q$ are continuously differentiable on $\mathbb{R}$ and $f=-p-q$. It follows from (2.1) that

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

Then the equation (1) becomes to the equation

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

which is also equivalent to the equation

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

(1.2)

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 equation

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

Using Theorem 1.1, we get from ${\int }_0^{\omega }p\left(x\right)dx>0$ that

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

Let z be a solution of the equation

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

Using Theorem 1.1 again, one has from ${\int }_0^{\omega }q\left(x\right)dx>0$ that

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

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). Let us first consider the scenario of $g\equiv 0$ and we have the following theorem.

Theorem 6. 

Suppose that ${\int }_0^{\omega }f\left(x\right)dx\ne 0$ and $g\left(x\right)\equiv 0$, then (1) is not Ulam stable.

Proof. 

Set $\overline{f}={\displaystyle \frac{1}{\omega }}{\int }_0^{\omega }f\left(x\right)dx$ and $\tilde{f}=f-\overline{f}$. Then ${\int }_0^{\omega }\tilde{f}\left(x\right)dx=0$ and $\overline{f}\ne 0$. Let $F\left(x\right)$ and $\tilde{F}\left(x\right)$ be antiderivatives of $f\left(x\right)$ and $\tilde{f}\left(x\right)$ on R, respectively. There exist two positive numbers m and M such that

$$m⩽e^{-\tilde{F}\left(x\right)}⩽M$$

for $x\in \mathbb{R}$. For any $\epsilon >0$, let $k>0$ be such that

$$\left|{\displaystyle \frac{f\left(x\right)}{k}}\right|⩽\epsilon .$$

Then the function

$$\eta \left(x\right)=c_1{\int }_0^x{e}^{-F\left(s\right)}ds+c_2+{\displaystyle \frac{x}{k}}$$

solves the equation

$${\eta }^{\prime \prime }+f\left(x\right){\eta }^{\prime }={\displaystyle \frac{f\left(x\right)}{k}}.$$

At the same time, the equation

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

has the general solution

$$y\left(x\right)=c_3{\int }_0^x{e}^{-F\left(s\right)}ds+c_4.$$

Hence, we get

$$|\eta \left(x\right)-y\left(x\right)|=|(c_1-c_3){\int }_0^x{e}^{-F\left(s\right)}ds+(c_2-c_4)+{\displaystyle \frac{x}{k}}|$$

If $c_1-c_3⩾0$, it is easy to see that ${lim}_{x\to \infty }|\eta \left(x\right)-y\left(x\right)|=\infty$. If $c_1-c_3<0$ and $\overline{f}>0$, for $x<0$,

$$\begin{array}{cc}\hfill \left|\eta \right(x)-y(x\left)\right|& =|(c_1-c_3){\int }_0^x{e}^{-F\left(s\right)}ds+(c_2-c_4)+{\displaystyle \frac{x}{k}}|\hfill \\ \hfill & =|(c_1-c_3){\int }_0^x{e}^{-\overline{f}s-\tilde{F}\left(s\right)}ds+(c_2-c_4)+{\displaystyle \frac{x}{k}}|\hfill \\ \hfill & ⩾(c_1-c_3){\int }_0^x{e}^{-\overline{f}s-\tilde{F}\left(s\right)}ds-|c_2-c_4|-|{\displaystyle \frac{x}{k}}|\hfill \\ \hfill & ⩾m(c_1-c_3){\int }_0^x{e}^{-\overline{f}s}ds-|c_2-c_4|-|{\displaystyle \frac{x}{k}}|\hfill \\ \hfill & ={\displaystyle \frac{m(c_1-c_3)}{\overline{f}}}(1-e^{-\overline{f}x})-|c_2-c_4|-|{\displaystyle \frac{x}{k}}|,\hfill \end{array}$$

which implies ${lim}_{x\to -\infty }|\eta \left(x\right)-y\left(x\right)|=\infty$. If $c_1-c_3<0$ and $\overline{f}<0$, we can obtain ${lim}_{x\to \infty }|\eta \left(x\right)-y\left(x\right)|=\infty$ by using a similar discussion. This completes the proof. □

Immediately after, we will consider the instability of (1) for the general situation.

Theorem 7. 

Suppose that the Riccati equation

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

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 the equation (1) becomes to (7). For any $\epsilon >0$, the function

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

is the general solution of the equation

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

Let

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

then it is the general solution of the equation (7). We obtain that

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

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 have that there exist four numbers $m_p$, $m_q$, $M_p$ and $M_q$ such that

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

and

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

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

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

This implies that, for $x>0$,

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

Then there exists 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 $x>x_1$,

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

So we get ${lim}_{x\to \infty }|\eta \left(x\right)-y\left(x\right)|=\infty$. This indicates that the 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 have that there exist two numbers $m_{\tilde{p}}$ and $M_{\tilde{p}}$ such that

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

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

If $c_1⩾c_3$, for $s>0$,

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

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

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

Then there exists 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 $x>x_1$,

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

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

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

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

This implies that, for $x<0$ and $c_1<c_3$,

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

Then there exists 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 $x<x_2$,

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

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

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

For 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 above discussion, so we omit them. This completes the proof. □

To this extent, we can use Theorem 2.1 and Theorem 2.3 to obtain the following necessary and sufficient conditions.

Theorem 8. 

Suppose that the Riccati equation

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

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

Remark 1. 

Consider the differential equation

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

(1.3)

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 1.4 can not be used to (8). Meanwhile, the theorems of [11] can not be used to (8) yet because (8) has 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 2.1, we know that (8) has US on $\mathbb{R}$. Accordingly, the above theorems are all novel.

Remark 2. 

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

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

and

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

They have the same conclusions under the framework of Theorem 2.1, Theorem 2.3 and Theorem 2.4.

Remark 3. 

For nonhomogeneous equation

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

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

2. Minimal US Constant for Constant Coefficient Differential Equation

In this section, we consider the constant coefficient differential equation

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

(2.1)

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

Theorem 9. 

If $pq\ne 0$, then (9) has US with minimum US constant ${\displaystyle \frac{1}{\left|pq\right|}}$ on$\mathbb{R}$.

Proof. 

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

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

Let k be US constant. If $p>0$ and $q>0$, we use Theorem 2.1 to get that $k={\displaystyle \frac{1}{pq}}$. Similarly, we obtain 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 (9) has US with US constant ${\displaystyle \frac{1}{\left|pq\right|}}$ on $\mathbb{R}$.

Next, we will use the method of contradiction to prove its minimization. Assume that there exists 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 the equation (9). And let $\eta \in C^2(\mathbb{R},\mathbb{R})$ be the solution of the equation

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

We have

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

Since $k_0$ is US constant, so

$$|\eta -y|⩽k_0\epsilon .$$

(2.2)

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

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

Then

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

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 that $z^{\prime }⩾Mx+N$. Thus, we get

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

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 (10) on $\mathbb{R}$. This ends the proof. □

Remark 4. 

Let $p=-q=\lambda$, then the equation (9) becomes to

$$y^{\prime \prime }-{\lambda }^{2}y=0.$$

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