Solvability of a Boundary Value Problem for Elliptic Differential-Operator Equations of the Second Order with a Quadratic Complex Parameter

Autor: Ya. Yakubov, B. A. Aliev, V. Z. Kerimov
Rok vydání: 2020
Předmět:
Zdroj: Differential Equations. 56:1306-1317
ISSN: 1608-3083
0012-2661
DOI: 10.1134/s00122661200100079
Popis: We study the solvability of the problem for the elliptic second-order differential-operator equation $$\lambda ^2 u(x)-u^{\prime {}\prime }(x)+Au(x)=f(x) $$ , $$x\in (0,1)$$ , in a separable Hilbert space $$H$$ with the boundary conditions $$ u^{\prime }(1)+\lambda Bu(0)=f_1$$ and $$u^{\prime }(0)=f_2$$ , where $$\lambda $$ is a complex parameter, $$A$$ and $$B$$ are given linear operators in $$H$$ , the operator $$A$$ is $$\varphi $$ -positive, and $$f $$ , $$f_1$$ , and $$ f_2$$ are known functions. Sufficient conditions for the unique solvability of this problem in an appropriate function space are obtained, and an upper bound (coercive if $$ B$$ is a bounded operator and noncoercive if the operator $$ B$$ is unbounded) is established for the solution. An application of these abstract results to elliptic boundary value problems is given.
Databáze: OpenAIRE
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