Methods for solving nonlinear operator equations with singular Fredholm derivatives (Q1974734)

The author describes an application of the 2-regularity theory to nonlinear mappings whose derivatives are Fredholm operators. The equation \(F(x)=0\) is considered where \(x_0\) be its solution. The finding of the singular solution \(x_0\) by conventional methods is problematic. The author introduces the mapping \[ \Phi_2(x)= F(x)+ PF'(x)h, \] where \(P\) orthoprojector onto \((\operatorname {Im}F'(x_0))^\perp\), \(h\in \operatorname {Ker}F'(x_0)\), and considers the equation \(\Phi_2(x) = 0\). Then \(x_0\) is also a solution to the last equation. Moreover, under assumption of 2-regularity of the mapping \(F(\cdot)\) the point \(x_0\) is a nonsingular solution to equation \(\Phi_2(x)= 0\).