Vol 242, No 3 (2019)

Let X be an unbounded metric space, and let \( \tilde{r} \) be a sequence of positive real numbers tending to infinity. We define the pretangent space \( {\Omega}_{\infty, \tilde{r}}^X \) to X at infinity as a metric space whose points are the equivalence classes of sequences \( \tilde{x}\subset X \) which tend to infinity with the rate \( \tilde{r} \). It is proved that all pretangent spaces are complete and, for every finite metric space Y, there is an unbounded metric space X such that Y and \( {\Omega}_{\infty, \tilde{r}}^X \) are isometric for some \( \tilde{r} \). The finiteness conditions of \( {\Omega}_{\infty, \tilde{r}}^X \) are completely described.

Let ℑ be a monic generalized Jacobi matrix, i.e., a three-diagonal block matrix of a special form. We find conditions for a monic generalized Jacobi matrix ℑ to admit a factorization ℑ = ???????? + αI with ???? and ???? being lower and upper triangular two-diagonal block matrices of special forms. In this case, the shifted parameterless Darboux transformation of ℑ defined by ℑ^(p) = ???????? + αI is shown to be also a monic generalized Jacobi matrix. Analogs of the Christoffel formulas for polynomials of the first and second kinds corresponding to the Darboux transformation ℑ^(p) are found.

The regularized asymptotics of a solution of the Cauchy problem for systems of singularly perturbed ordinary differential equations is constructed. It is shown that a power boundary layer appears in such problems in addition to other boundary layers.

Quasilinear parabolic equations with a degenerate absorption potential are considered. The estimates of all weak solutions of such equations, including large solutions satisfying the blow-up conditions on the parabolic boundary of a domain, are obtained.