Asymptotics of solutions and singular boundary value problems for second-order ordinary differential equations

Sufficient conditions are obtained under which all solutions of the equation $x''=f(t,x,x'), x\in R^n, f\in C^2 ([T,+\infty) \times R^n\times R^n, R^n)$ have asymptotics of the form $\frac{x(t)}{t}=c+o(1), x'(t)=c+o(1)$ for $t\rightarrow+\infty$, and for a fixed $t_0$ for any $x_0,C\in R^n$ there is a solution $\overline{x}(t)$ such that $\overline{x}(t_0)=x_0$ and $\overline{x}(t)$ for $t\rightarrow +\infty$ has an asymptotic of the specified type. The obtained theorems are applied to solving one problem of the theory of gravity.