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 CMFD, 2016, Volume 59, Pages 119–147 (Mi cmfd290)

Differential equations with degenerate, depending on the unknown function operator at the derivative

B. V. Loginova, Yu. B. Rousakb, L. R. Kim-Tyanc

a Ul'yanovsk State Technical University, Ul'yanovsk, Russia
b Department of Social Service, Canberra, Australia
c National University of Science and Technology "MISIS", Moscow, Russia

Abstract: We develop the theory of generalized Jordan chains of multiparameter operator functions $A(\lambda)\colon E_1\to E_2$, $\lambda\in\Lambda$, $\dim\Lambda=k$, $\dim E_1=\dim E_2=n$, where $A_0=A(0)$ is a noninvertible operator. To simplify the notation, in 1–3 the geometric multiplicity $\lambda_0$ is set to 1, i.e. $\dim N(A_0)=1$, $N(A_0)=\operatorname{span}\{\varphi\}$, $\dim N^\ast(A_0^\ast)=1$, $N^\ast(A_0^\ast)=\operatorname{span}\{\psi\}$, and the operator function $A(\lambda)$ is supposed to be linear with respect to $\lambda$. For the polynomial dependence of $A(\lambda)$, in 4 we consider a linearization. However, the bifurcation existence theorems hold in the case of several Jordan chains as well.
We consider applications to degenerate differential equations of the form $[A_{0}+R(\cdot,x)]x'=Bx$.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 2014/232

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Document Type: Article
UDC: 517.9

Citation: B. V. Loginov, Yu. B. Rousak, L. R. Kim-Tyan, “Differential equations with degenerate, depending on the unknown function operator at the derivative”, Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22–29, 2014). Part 2, CMFD, 59, PFUR, M., 2016, 119–147

Citation in format AMSBIB
\Bibitem{LogRusKim16} \by B.~V.~Loginov, Yu.~B.~Rousak, L.~R.~Kim-Tyan \paper Differential equations with degenerate, depending on the unknown function operator at the derivative \inbook Proceedings of the Seventh International Conference on Differential and Functional-Differential Equations (Moscow, August 22--29, 2014). Part~2 \serial CMFD \yr 2016 \vol 59 \pages 119--147 \publ PFUR \publaddr M. \mathnet{http://mi.mathnet.ru/cmfd290} 

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This publication is cited in the following articles:
1. N. A. Sidorov, D. N. Sidorov, “Skeletnye razlozheniya lineinykh operatorov v teorii neregulyarnykh sistem s chastnymi proizvodnymi”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 20 (2017), 75–95
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