A problem with polynomial asymptotic initial data at the point at infinity (Q2704020)

Problems with initial data at singular points of ordinary differential equations are little studied in the general case. One of possible approaches to the problem of singular points is to define some functions to which the desired solutions must asymptotically tend in some neighborhood of singular points.NEWLINENEWLINENEWLINEHere, the author considers the case in which there are solutions behaving like algebraic polynomials in a neighborhood at infinity. A function \(f\) is said to asymptotically approximate the polynomial \(p_m(t)\) as \(t\to+\infty\) if NEWLINE\[NEWLINE\lim_{t\to+\infty} (x(t)- p_m(t))^{(k)}= 0,\quad k= 0,1,\dots, m.NEWLINE\]NEWLINE The author studies the following problem with initial data at infinity: given a polynomial \(p_m(t)\) of degree \(\leq m\), find a solution to the equation NEWLINE\[NEWLINEx^{(m+ 1)}= f(t, x,x',\dots, x^{(m)})\tag{1}NEWLINE\]NEWLINE that asymptotically approximates the polynomial \(p_m(t)\) as \(t\to+\infty\).NEWLINENEWLINENEWLINEThe author proves an existence and uniqueness theorem on problem (1). In the first part of the paper, the author considers the following problem: what conditions should be imposed on the right-hand side in equation (1) to guarantee that all solutions to this equation asymptotically approximate polynomials of degree not greater than a predefined value as \(t\to+\infty\).