Di erential Equations with Degenerate, Depending on the Unknown Function Operator at the Derivative

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 a noninvertible operator. To simplify the notation, in Secs. 1-3 the geometric multiplicity λ0 is set to 1, i. e. dimN(A0) = 1, N(A0) = span{ϕ}, dimN∗(A∗0) = 1, N∗(A∗0) = span{ψ}, and the operator function A(λ) is supposed to be linear with respect to λ. For the polynomial dependence of A(λ), in Sec. 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 [A0 + R(·, x)]x*= Bx.