Generalized solutions of operator equations and extreme elements
In the first part of this monograph, various notions of generalized solutions of linear equations \(\mathcal{L}u = f \) are introduced. Here, \( \mathcal{L}: D(\mathcal{L}) \to F \) is a linear injective operator between Banach spaces \( E \) and \( F \), with dense domain \( D(\mathcal{L}) \subset E \) and dense range in \( F \). Each of the considered generalized solutions belongs to a completion of \( D(\mathcal{L}) \) with respect to some Hausdorff topology. For example, consider the completion \( \overline{E} \) of \( D(\mathcal{L}) \) with respect to the norm \( \| u \|_{\overline{E}} = \| \mathcal{L} u \|_F \), where \( \| \cdot \|_F \) denotes the norm in the Banach space \( F \). An element \( u \in \overline{E} \) is then called a strong generalized solution if \( \overline{\mathcal{L}} u = f \) is satisfied, where \( \overline{\mathcal{L}}: \overline{E} \to F \) denotes the continuous continuation of \( \mathcal{L} \) with respect to the norms \( \| \cdot \|_{\overline{E}} \) and \( \| \cdot \|_F \). The authors also introduce other kinds of generalized solutions, e.g., weak generalized solutions, strong near-solutions and weak near-solutions. Existence and unique results are presented, respectively, and relations between the various notions are given. Subsequently, applications of the presented theory to special problems are considered, e.g., Volterra integral equations of the first kind, infinite systems of linear equations, partial differential equations of parabolic type, and wave equations. The next chapter is devoted to the approximate computation of classical and generalized solutions of linear operator equations. Special emphasis is put on the Neumann series type iteration method \( \widetilde{x}_k = \mu(I+U+U^2+\dots+ U^k) b, \;k = 0,1,2,\dots, \) where \( U = I -\mu A \) and \( \mu > 0 \), and \( A: H \to H \) is a linear bounded operator in a Hilbert space \( H \). The operator version of this iterative scheme, that is, Schultz's method, is also considered. The chapter that follows considers generalized solutions of linear operator equations in topological vector spaces from a general point of view. The next chapter is devoted to generalized solutions of nonlinear equations \( A(x) = y \). Here, \( A:E \to F \) is a continuous injective mapping between metric spaces \( E \) and \( F \), where \( F \) is complete, and the range of \( A \) is dense in \( F \). Moreover, an element \( \overline{x} \) of the completion of \( E \) with respect to the metric \(\rho^*(x,y) = \rho_F(A(x),A(y)) \) is called a generalized solution of \( A(x) = y \), if it is the limit of a sequence of near-solutions \( x_n \in E \), i.e., \( A(x_n) \) converges to \( y \). Here, \( \rho_F \) denotes the metric of \( F \). The authors consider properties of generalized solutions and their computation, and applications to special nonlinear equations. At the end of this chapter, the theory is extended from metric spaces to uniform spaces. In the final chapter, several notions of generalized solutions of optimization problems \( \varphi(x) \to \max\), \(x \in M \), and \( \varphi(x) \to \min\), \(x \in M \), are considered. Here, \( M \) is a bounded and closed subset of a Banach space \( E \), and \( \varphi: E \to \mathbb{R} \) is a bounded continuous functional. Each considered generalized solution \( x^\ast \) belongs to a completion \( \widetilde{M} \) of the set \( M \) with respect to some Hausdorff topology \( \mathcal{T} \), respectively, and it satisfies \( \overline{\varphi}(x^\ast) = \sup_{x\in M} \varphi(x) \) for maximization problems and \( \overline{\varphi}(x^\ast) = \inf_{x\in M} \varphi(x) \) for minimization problems. Here, the functional \( \overline{\varphi}: \widetilde{M} \to \mathbb{R} \) is obtained by continuous extension of the objective function \( \varphi \) with respect to the considered topology \( \mathcal{T} \). This is a well-written, self-contained and easily readable monograph, with a reasonable number of references to papers and books written in English. It can be recommended to everyone interested in the topic and having some background in functional analysis and topology.
Volterra integral equation of the first kind
partial differential equation of parabolic type
scientific article; zbMATH DE number 5944101