Differential Equations with Degenerated Variable Operator at the Derivative

Abstract

The theory of Jordan chains for multiparameter operator-functions A(λ):E ₁→E ₂, λ∈Λ, $\operatorname{dim}\varLambda=k$, $\operatorname{dim} E_{1}=\operatorname{dim} E_{2}=n$ is developed. Here A ₀=A(0) is a degenerated operator, $\operatorname{dim}\operatorname{Ker}A_{0}=1$, $\operatorname{Ker}A_{0}=\{\varphi\}$, $\operatorname{Ker}A_{0}^{*}=\{\psi\}$ and the operator-function A(λ) is supposed to be linear in λ. Applications to degenerate differential equations of the form [A ₀+R(⋅,x)]x′=Bx are given.

This work was partially supported by the federal target program “Scientific and scientific-pedagogical personnel of innovative Russia” (Agreement 14.37.21.0373).