Journal of Mathematical Sciences

, Volume 233, Issue 6, pp 875–904 | Cite as

Differential Equations with Degenerate Operators at the Derivative Depending on an Unknown Function

  • B. V. LoginovEmail author
  • Yu. B. Rousak
  • L. R. Kim-Tyan


We develop the theory of generalized Jordan chains of multiparameter operator functions A(λ) : E1 → E2, λ ∈ Λ, dimΛ = k, dimE1 = dimE2 = n, where A0 = A(0) is an irreversible operator. For simplicity, in Secs. 1–3, the geometric multiplicity of λ0 is equal to one, i.e., dimN(A0) = 1, N(A0) = span{φ}, dimN*(\( {A}_0^{\ast } \)) = 1, N*(\( {A}_0^{\ast } \)) = span{ψ}, and it is assumed that the operator function A(λ) is linear with respect to λ. In Sec. 4, the polynomial dependence of A(λ) is linearized. However, the results of existence theorems for bifurcations are obtained for the case where there are several Jordan chains. Applications to degenerate differential equations of the form [A0 + R, x)]x′= Bx are provided.


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Authors and Affiliations

  • B. V. Loginov
    • 1
    Email author
  • Yu. B. Rousak
    • 2
  • L. R. Kim-Tyan
    • 3
  1. 1.Ulyanovsk State Technical UniversityUlyanovskRussia
  2. 2.Department of Social ServiceCanberraAustralia
  3. 3.National University of Science and Technology “MISiS”MoscowRussia

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