| Summary: | By means of the shooting method together with the maximum principle and the Kneser–Hukahara continuum theorem, the authors present the existence, uniqueness and qualitative properties of solutions to nonlinear second-order boundary value problem on the semi-infinite interval of the following type:
$$
\begin{cases}
y''=f(x,y,y'),& x\in[0,\infty), \\
y'(0)=A,& y(\infty)=B
\end{cases}
$$
and
$$
\begin{cases}
y''=f(x,y,y'),& x\in[0,\infty), \\
y(0)=A,&y(\infty)=B,
\end{cases}
$$
where $A,B\in \mathbb{R}$, $f(x,y,z)$ is continuous on $[0,\infty)\times\mathbb{R}^2$. These results and the matching method are then applied to the search of solutions to the nonlinear second-order non-autonomous boundary value problem on the real line
$$
\begin{cases}
y''=f(x,y,y'), & x\in\mathbb{R} ,\\
y(-\infty)=A,& y(\infty)=B,
\end{cases}
$$
where $A\not=B$, $f(x,y,z)$ is continuous on $\mathbb{R}^3$. Moreover, some examples are given to illustrate the main results, in which a problem arising in the unsteady flow of power-law fluids is included.
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