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

Abstract

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.