TY - JOUR

T1 - Solvability of a Boundary Value Problem for Elliptic Differential-Operator Equations of the Second Order with a Quadratic Complex Parameter

AU - Aliev, B. A.

AU - Kerimov, V. Z.

AU - Yakubov, Ya S.

N1 - Publisher Copyright:
© 2020, Pleiades Publishing, Ltd.

PY - 2020/10

Y1 - 2020/10

N2 - Abstract: We study the solvability of the problem for the ellipticsecond-order differential-operator equation λ2u(x)-u"(x) + Au(x) = f(x), xε(0; 1),, in a separable Hilbert space H with the boundaryconditions u'(1)+λBu(0) = f1 and u'(0) = f2, where λ is a complex parameter, A and B are given linear operators in H, the operator A is ᵩ-positive, and f, f1, and f2 are known functions. Sufficient conditions forthe unique solvability of this problem in an appropriate function space are obtained, and an upperbound (coercive if B is a bounded operator and noncoerciveif the operator B is unbounded) is established forthe solution. An application of these abstract results to elliptic boundary value problems is given.

AB - Abstract: We study the solvability of the problem for the ellipticsecond-order differential-operator equation λ2u(x)-u"(x) + Au(x) = f(x), xε(0; 1),, in a separable Hilbert space H with the boundaryconditions u'(1)+λBu(0) = f1 and u'(0) = f2, where λ is a complex parameter, A and B are given linear operators in H, the operator A is ᵩ-positive, and f, f1, and f2 are known functions. Sufficient conditions forthe unique solvability of this problem in an appropriate function space are obtained, and an upperbound (coercive if B is a bounded operator and noncoerciveif the operator B is unbounded) is established forthe solution. An application of these abstract results to elliptic boundary value problems is given.

UR - http://www.scopus.com/inward/record.url?scp=85095948959&partnerID=8YFLogxK

U2 - 10.1134/S00122661200100079

DO - 10.1134/S00122661200100079

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AN - SCOPUS:85095948959

SN - 0012-2661

VL - 56

SP - 1306

EP - 1317

JO - Differential Equations

JF - Differential Equations

IS - 10

ER -