Asymptotics of solutions to differential equations near singular points (Q2783076)

Conditions are obtained under which all solutions to the normal system of equations \(x^\prime(t)=f(t,x(t))\), \(t\in[t_0,\infty)\), asymptotically or strongly asymptotically converge to vector polynomials as the argument tends to infinity. It is proved, that if \(f\) is \(m\)-times continuously differentiable, a necessary and sufficient condition, that the solution \(x(t)\) asymptotically approaches to some vector polynomial of degree at most \(m\) as \(t\to+\infty\), is the convergence of the iterated integral \( \left( I_{m+1}\left({\partial\over\partial t}+ f(t,x(t))\nabla\right)^m f(t,x(t))\right)(t_0), \) with \((I_k x)(t)= \int_t^{+\infty} dt_1 \int_{t_1}^{+\infty} dt_2 \dots \int_{t_{k-1}}^{+\infty} x(t_k) dt_k . \) In a certain sense, this work concerns the inverse problem in the theory of the asymptotics of the solutions. These questions are dealt with substantially a less number of works as compared with those devoted to direct methods for establishing the asymptotics of the solutions to preassigned equations. Similar to the classical Cauchy problem with the initial data at a regular point, the study of the inverse problem deals with sufficient conditions on the right-hand side, that guarantee the existence of solutions with given asymptotics at a singular point. Further, for the system of the form \(Lx=f\), where \(L\) is a first-order linear differential operator, conditions are found, under which all its solutions \(L\)-asymptotically converge to the solutions to the homogeneous system \(Lx=0\) as the argument tends to the singular point of the former system.NEWLINENEWLINEFor the entire collection see [Zbl 0981.00017].