Abstract
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.
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Aliev, B.A., Kerimov, V.Z. & Yakubov, Y.S. Solvability of a Boundary Value Problem for Elliptic Differential-Operator Equations of the Second Order with a Quadratic Complex Parameter. Diff Equat 56, 1306–1317 (2020). https://doi.org/10.1134/S00122661200100079
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DOI: https://doi.org/10.1134/S00122661200100079