Izmailov, A. F.; Tret’yakov, A. A. 2-regular solutions of nonlinear problems. Theory and numerical methods. (2-регулярные решения нелинейных задач: теория и численные методы.) (Russian. English summary) Zbl 0965.47050 Moskva: Nauka, Izdatel’skaya Firma Fiziko-Matematicheskaya Literatura (Fizmatlit). 336 p. (1999). Nonlinear operator equations \(F_0(x)=0\) with singular solutions are studied. Usually, a mapping \(F_0:V \to Y\) is considered where \(V\) is a neighbourhood of a given solution \(x_*\) in a Banach space \(X\), \(Y\) is also a Banach space. (Some general considerations are performed in a more general setting.) Singular solutions are discussed, that means solutions \(x_*\) such that \(\operatorname {Im} F_0'(x_*) \neq Y\). The basic idea is to regularize the problem by replacing the original equation \(F_0(x)=0\) with a singular solution \(x_*\) by another equation \(F(x)=0\) having \(x_*\) as a regular solution (i.e., such that \(\operatorname {Im} F'(x_*) = Y\)), and to solve the regularized equation by iterative methods. In general, various approaches are possible. Here a concept of 2-regularity is essentially used. A mapping \(F_0\) is called 2-regular in the point \(x_*\) on the element \(h\) if \(\operatorname {Im} \Psi_2(h) = Y_1 \times Y/Y_1\) where \(Y_1 = \operatorname {Im} F'(x_*)\) and the mapping \(\Psi_2(h)\in\mathcal L(X,Y_1 \times Y/Y_1)\) for \(h \in X\) is defined by \(\Psi_2(h)x = [F'(x_*)x,\pi F''(x_*)(h,x)]\), \(\pi\) is the projector of \(Y\) onto \(Y/Y_1\). In particular, 2-regularity in the solution \(x_*\) on an element \(h\) from \(\operatorname {Ker} F'(x_*)\) plays an essential role. A general regularization construction is given and numerical methods for obtaining 2-regular solutions based on this approach are described. Applications to particular classes of nonlinear problems are given. However, also some cases are studied when the 2-regularity assumption is not fulfilled. Particular attention is paid to the stability aspect. Reviewer: M.Kučera (Praha) Cited in 2 ReviewsCited in 17 Documents MSC: 47J25 Iterative procedures involving nonlinear operators 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 47J05 Equations involving nonlinear operators (general) 49J53 Set-valued and variational analysis 65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization 90C31 Sensitivity, stability, parametric optimization Keywords:nonlinear operator equations; singular solutions; regularization; 2-regularity; iterative methods; stability PDFBibTeX XMLCite \textit{A. F. Izmailov} and \textit{A. A. Tret'yakov}, 2-регулярные решения нелинейных задач: теория и численные методы (Russian). Moskva: Nauka, Izdatel'skaya Firma Fiziko-Matematicheskaya Literatura (Fizmatlit) (1999; Zbl 0965.47050)