Comments to the problems of small perturbations of linear equations and linear term of the spectral characteristics of a Fredholm operator

In the monograph [1] and the article [1,2] the problem on perturbation of linear equation by small linear summand of the form $(B-\varepsilon A)x=h$ were investigated with closely defined on $D_{B}$ Fredholmian operator $B:E_{1}\supset D_{B}\rightarrow E_{2}$, $D_{A}\supset D_{B}$, or $A\in L\{E_{1},E_{2}\}$, $\varepsilon\in\mathbb{C}^{1}$ - small parameter, $E_{1}$ and $E_{2}$ - are Banach spaces. The application of the results [3,4] formulated in the form of the lemma on the biorthogonality of generalized Jordan chains allows to give some retainings of the results [1,2]. This problem is considered here in the general case of sufficiently smooth (analytic) by $\varepsilon$ operator-function $A(\varepsilon)$. It is given also the application of the biorthogonality lemma and branching equation in the root subspaces to the problem on perturbation of Fredholm points in $C$-spectrum of the operator $A(0)$.