Abstract
Chapter 4 is the largest one. It is in a sense “barycenter” of the book. Here we consider typical examples of applications of generalized solutions in various branches of pure and applied analysis. In Sect. 4.1, 4.3, and 4.4 we prove theorems on generalized solvability of equations with Hilbert–Schmidt operators and Volterra equations of the first kind and describe their applications in the random processes estimation theory. In Sect. 4.2 the generalized solvability of linear operator equations in classic spaces of sequences is studied. In Sect. 4.5, 4.6, and 4.7 the theorems on generalized solvability of boundary value problems for parabolic and generalized wave equations are proved. Section 4.7 is devoted to discussion of theorems of Lax-Milgram kind in Banach and locally convex spaces.
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Notes
- 1.
We use the following notation of the spaces H 0 k, V 0 k, …: the superscript indicates the number of time derivatives, and the subscript indicates the type of initial conditions.
- 2.
Relation (4.78) can be explained from the viewpoint of distribution. One can show that it is meaningful to consider elements of the space H 0 − 1 as functionals on the space \(\bar{{W}}_{T}^{1}\) (more precisely \({H}_{0}^{-1} \subset {(\bar{{W}}_{T}^{1})}^{{_\ast}}\)), where \(\bar{{W}}_{T}^{1}\) is the completion of L T in the norm \(\|{v\|}_{\bar{{W}}_{T}^{1}}^{2} =\| {v{}_{t}\|}_{{L}_{2}(Q)}^{2}\). Then u (1) ∈ H 0 − 1 should be treated as a functional acting as \(\langle {u}^{(1)},{v\rangle }_{\bar{{W}}_{T}^{1}} = -{(u,{v}_{t})}_{{L}_{2}(Q)}\) for all functions \(v \in \bar{ {W}}_{T}^{1}\), u ∈ H 0 0, including smooth functions u ∈ H 0 0 not satisfying the condition u(0, x) = 0. An important difference of u (1) from the Sobolev generalized derivatives the following: the functional u (1) is defined on functions \(v \in \bar{ {W}}_{T}^{1}\) not necessarily vanishing for t = 0.
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Klyushin, D.A., Lyashko, S.I., Nomirovskii, D.A., Petunin, Y.I., Semenov, V.V. (2012). Applications of the Theory of Generalized Solvability of Linear Equations. In: Generalized Solutions of Operator Equations and Extreme Elements. Springer Optimization and Its Applications(), vol 55. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0619-8_4
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DOI: https://doi.org/10.1007/978-1-4614-0619-8_4
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