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Let $F(u)=h$ be an operator equation in a Banach space $X$, $\|F'(u)-F'(v)\|\leq \omega(\|u-v\|)$, where $\omega\in C([0,\infty))$, $\omega(0)=0$, $\omega(r)>0$ if $r>0$, $\omega(r)$ is strictly growing on $[0,\infty)$. Denote $A(u):=F'(u)$, where $F'(u)$ is the Fr\'{e}chet derivative of $F$, and $A_a:=A+aI.$ Assume that (*) $\|A^{-1}_a(u)\|\leq \frac{c_1}{|a|^b}$, $|a|>0$, $b>0$, $a\in L$. Here $a$ may be a complex number, and $L$ is a smooth path on the complex $a$-plane, joining the origin and some point on the complex $a-$plane, $0 0$ are some suitably chosen constants, $j=2,3,4.$ Existence of a solution $y$ to the equation $F(u)=f$ is assumed. It is also assumed that the equation $F(w_a)+aw_a-f=0$ is uniquely solvable for any $f\in X$, $a\in L$, and $\lim_{|a|\to 0,a\in L}\|w_a-y\|=0.$