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

We develop the theory of generalized Jordan chains of multiparameter operator functions $A(\lambda)\colon E_1\to E_2$, $\lambda\in\Lambda$, $\dim\Lambda=k$, $\dim E_1=\dim E_2=n$, 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 $[A_{0}+R(\cdot,x)]x'=Bx$.