\begin{equation}\label{4.5} \begin{aligned} \left| {r_ + ^{(0)}(\bar \xi ) - r_ - ^{(0)}(\bar \xi )} \right| &= \left| {r_ + ^{(0)}(\bar \xi ;\xi ,\eta ) - r_ - ^{(0)}(\bar \xi ;\xi ,\eta )} \right|\\ &\le \int_0^\xi {\left\{ {\left( {\frac{{\kappa \sqrt {1 - {t^2}} t}}{F} + \frac{{G\left( {1 - {t^2}} \right)}}{{2c{F^2}(S + {a_0} + {a_1}t)}}} \right)t} \right.} \\ \left. { + \left( {\frac{{\kappa \sqrt {1 - {t^2}} t}}{F} + \frac{{G\left( {1 - {t^2}} \right)}}{{2c{F^2}({R^{(0)}} - {a_0} + {a_1}t)}}} \right)t} \right\}dt\\ &\le \int_0^\xi {\left\{ {\frac{{2\kappa \sqrt {1 - {t^2}} {t^2}}}{F} + \frac{{R + S + 2{a_1}t}}{{({R^{(0)}} - {a_0} + {a_1}t)({S^{(0)}} + {a_0} + {a_1}t)}}} \right\}} dt\\ &\le \int_0^\xi {\left\{ {2M{t^2} + Mt} \right\}} dt \le \int_0^\xi {2Mt} dt \le M{\xi ^2}. \end{aligned} \end{equation} 如何换行对齐

可以使用align环境，并在需要换行的地方添加 \\ 符号。具体代码如下：

\begin{align}
\left| {r_ + ^{(0)}(\bar \xi ) - r_ - ^{(0)}(\bar \xi )} \right| &= \left| {r_ + ^{(0)}(\bar \xi ;\xi ,\eta ) - r_ - ^{(0)}(\bar \xi ;\xi ,\eta )} \right|\\
&\le \int_0^\xi {\left\{ {\left( {\frac{{\kappa \sqrt {1 - {t^2}} t}}{F} + \frac{{G\left( {1 - {t^2}} \right)}}{{2c{F^2}(S + {a_0} + {a_1}t)}}} \right)t} \right.} \nonumber \\
&\quad \left. { + \left( {\frac{{\kappa \sqrt {1 - {t^2}} t}}{F} + \frac{{G\left( {1 - {t^2}} \right)}}{{2c{F^2}({R^{(0)}} - {a_0} + {a_1}t)}}} \right)t} \right\}dt \nonumber\\
&\le \int_0^\xi {\left\{ {\frac{{2\kappa \sqrt {1 - {t^2}} {t^2}}}{F} + \frac{{R + S + 2{a_1}t}}{{({R^{(0)}} - {a_0} + {a_1}t)({S^{(0)}} + {a_0} + {a_1}t)}}} \right\}} dt \nonumber\\
&\le \int_0^\xi {\left\{ {2M{t^2} + Mt} \right\}} dt \le \int_0^\xi {2Mt} dt \le M{\xi ^2}.
\end{align}

输出效果如下：

$$
\begin{aligned}
\left| {r_ + ^{(0)}(\bar \xi ) - r_ - ^{(0)}(\bar \xi )} \right| &= \left| {r_ + ^{(0)}(\bar \xi ;\xi ,\eta ) - r_ - ^{(0)}(\bar \xi ;\xi ,\eta )} \right|\\
&\le \int_0^\xi {\left\{ {\left( {\frac{{\kappa \sqrt {1 - {t^2}} t}}{F} + \frac{{G\left( {1 - {t^2}} \right)}}{{2c{F^2}(S + {a_0} + {a_1}t)}}} \right)t} \right.} \\
&\quad \left. { + \left( {\frac{{\kappa \sqrt {1 - {t^2}} t}}{F} + \frac{{G\left( {1 - {t^2}} \right)}}{{2c{F^2}({R^{(0)}} - {a_0} + {a_1}t)}}} \right)t} \right\}dt\\
&\le \int_0^\xi {\left\{ {\frac{{2\kappa \sqrt {1 - {t^2}} {t^2}}}{F} + \frac{{R + S + 2{a_1}t}}{{({R^{(0)}} - {a_0} + {a_1}t)({S^{(0)}} + {a_0} + {a_1}t)}}} \right\}} dt\\
&\le \int_0^\xi {\left\{ {2M{t^2} + Mt} \right\}} dt \le \int_0^\xi {2Mt} dt \le M{\xi ^2}.
\end{aligned}
$$