Asymptotics of solutions of a class of linear differential equations

Abstract

РАЗДЕЛ XXII. МАТЕМАТИКА Конечная Н.Н. Асимптотика решений одного класса линейных дифференциальных уравнений Северный (Арктический) федеральный университет имени М.В. Ломоносова (Россия, Архангельск

Linear differential operators with distribution coefficients of various singularity orders

“…That research was continued in Konechnaja et al, 25,26 where the solution asymptotics as $$$ x\to \infty $$$ were investigated for the two‐term differential equation $$$$ {\left(p(x){y}^{(n)}\right)}^{(n)}+q(x)y=\lambda y,\kern0.30em x>0. $$$$ The analogous solution asymptotics for some other classes of higher order differential equations with distribution coefficients were obtained in Konechnaja and Mirzoev 18 and Konechnaya 27 …”

“…The matrices F i 0 ,i 2 coincide with the ones provided in Konechnaya. 27 In particular, the matrix F 0,0 corresponds to the well-known regular case (see Everitt and Marcus 30, Appendix A ), and F 2,1 was obtained in Vladimirov. 28 For clarity, denote  i 0 ,i 2 ∶=  I .…”

“…The analogous solution asymptotics for some other classes of higher order differential equations with distribution coefficients were obtained in Konechnaja and Mirzoev 18 and Konechnaya. 27 Relying on the ideas of Neiman-Zade and Shkalikov 21 and Vladimirov, 28 Vladimirov 29 has developed an alternative approach to construction of regularization matrices for differential operators represented by bilinear forms. The construction of Vladimirov 29 can be applied to a wider class of operators than Mirzoev and Shkalikov.…”

“…In Konechnaya, 27 the regularization matrices have been provided for the differential expression $$$$ {\left(p{y}^{\prime \prime}\right)}^{\prime \prime }-{\left({q}^{\left(\alpha \right)}{y}^{\prime}\right)}^{\prime }+{r}^{\left(\beta \right)}y, $$$$ where $$$ p,q,r $$$ are regular functions, $$$ \alpha \in \left\{0,1\right\},\beta \in \left\{0,1,2\right\} $$$. In the case $$$ p\equiv 1 $$$, () is equivalent to the differential expression () for $$$ n=4,I=\left({i}_0,{i}_1,{i}_2\right),\Sigma =\left({\sigma}_0,{\sigma}_1,{\sigma}_2\right) $$$ with $$$ {\sigma}_1=0 $$$: …”