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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)

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
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