Solvability Conditions for the Nonlocal Boundary-Value Problem for a Differential-Operator Equation with Weak Nonlinearity in the Refined Sobolev Scale of Spaces of Functions of Many Real Variables

We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash–Mozer iterative scheme, we establish the solvability conditions for the posed problem in the Hilbert H¨ormander spaces of functions of several real variables, which f...

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Published in:	Ukrainian mathematical journal Vol. 72; no. 4; pp. 515 - 535
Main Authors:	Il’kiv, V. S., Strap, N. I., Volyanska, I. I.
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2020 Springer Springer Nature B.V
Subjects:	Algebra Analysis Applications of Mathematics Boundary value problems Differential equations Geometry Mathematical analysis Mathematics Mathematics and Statistics Nonlinearity Operators (mathematics) Real variables Statistics
ISSN:	0041-5995, 1573-9376
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Abstract	We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash–Mozer iterative scheme, we establish the solvability conditions for the posed problem in the Hilbert H¨ormander spaces of functions of several real variables, which form a refined Sobolev scale.
AbstractList	We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash–Mozer iterative scheme, we establish the solvability conditions for the posed problem in the Hilbert H¨ormander spaces of functions of several real variables, which form a refined Sobolev scale. We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash-Mozer iterative scheme, we establish the solvability conditions for the posed problem in the Hilbert Hormander spaces of functions of several real variables, which form a refined Sobolev scale.
Audience	Academic
Author	Il’kiv, V. S. Strap, N. I. Volyanska, I. I.
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References	Il’kivVSNytrebychZMPukachPYNonlocal problem with moment conditions for hyperbolic equationsElectron. J. Different. Equat.201720172651937235381386.35268 MikhailetsVAMurachAAHörmander Spaces, Interpolation, and Elliptic Problems2014BerlinWalter de Gruyter10.1515/9783110296891 BertiMBollePCantor families of periodic solutions for completely resonant nonlinear wave equationsDuke Math. J.20061342359419224883410.1215/S0012-7094-06-13424-5 EloeaPWAhmadbBPositive solutions of a nonlinear nth order boundary-value problem with nonlocal conditionsJ. Appl. Math. Lett.200518552152710.1016/j.aml.2004.05.009 VolyanskaIIl’kivVStrapNSolvability conditions of nonlocal boundary-value problem for the differential-operator equation with weak nonlinearityMat. Stud.20185014459390231210.15330/ms.50.1.44-59 B. I. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2002). V. S. Il’kiv and N. I. Strap, “Solvability of the nonlocal boundary-value problem for a system of differential-operator equations in the Sobolev scale of spaces and in a refined scale,” Ukr. Mat. Zh., 67, No. 5, 611–624 (2015); English translation: Ukr. Math. J., 67, No. 5, 690–710 (2015). V. S. Il’kiv and N. I. Strap, “On the solvability of the nonlocal boundary-value problem for the differential-operator equation in a refined Sobolev scale,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 10, No. 2 (2013), pp. 1–23. V. A. Mikhailets and A. A. Murach, “Extended Sobolev scale and elliptic operators,” Ukr. Mat. Zh., 65, No. 3, 392–404 (2013); English translation: Ukr. Math. J., 65, No. 3, 435–447 (2013). BertiMBollePCantor families of periodic solutions of wave equations with Ck nonlinearitiesNonlin. Different. Equat. Appl.20081524727610.1007/s00030-007-7025-5 cr-split#-1798_CR3.1 VA Mikhailets (1798_CR4) 2014 cr-split#-1798_CR6.2 cr-split#-1798_CR6.1 cr-split#-1798_CR3.2 M Berti (1798_CR8) 2006; 134 VS Il’kiv (1798_CR5) 2017; 2017 1798_CR7 PW Eloea (1798_CR2) 2005; 18 M Berti (1798_CR9) 2008; 15 1798_CR1 I Volyanska (1798_CR10) 2018; 50
References_xml	– reference: V. S. Il’kiv and N. I. Strap, “Solvability of the nonlocal boundary-value problem for a system of differential-operator equations in the Sobolev scale of spaces and in a refined scale,” Ukr. Mat. Zh., 67, No. 5, 611–624 (2015); English translation: Ukr. Math. J., 67, No. 5, 690–710 (2015). – reference: B. I. Ptashnyk, V. S. Il’kiv, I. Ya. Kmit’, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kyiv (2002). – reference: Il’kivVSNytrebychZMPukachPYNonlocal problem with moment conditions for hyperbolic equationsElectron. J. Different. Equat.201720172651937235381386.35268 – reference: V. S. Il’kiv and N. I. Strap, “On the solvability of the nonlocal boundary-value problem for the differential-operator equation in a refined Sobolev scale,” in: Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences [in Ukrainian], 10, No. 2 (2013), pp. 1–23. – reference: MikhailetsVAMurachAAHörmander Spaces, Interpolation, and Elliptic Problems2014BerlinWalter de Gruyter10.1515/9783110296891 – reference: BertiMBollePCantor families of periodic solutions for completely resonant nonlinear wave equationsDuke Math. J.20061342359419224883410.1215/S0012-7094-06-13424-5 – reference: EloeaPWAhmadbBPositive solutions of a nonlinear nth order boundary-value problem with nonlocal conditionsJ. Appl. Math. Lett.200518552152710.1016/j.aml.2004.05.009 – reference: VolyanskaIIl’kivVStrapNSolvability conditions of nonlocal boundary-value problem for the differential-operator equation with weak nonlinearityMat. Stud.20185014459390231210.15330/ms.50.1.44-59 – reference: V. A. Mikhailets and A. A. Murach, “Extended Sobolev scale and elliptic operators,” Ukr. Mat. Zh., 65, No. 3, 392–404 (2013); English translation: Ukr. Math. J., 65, No. 3, 435–447 (2013). – reference: BertiMBollePCantor families of periodic solutions of wave equations with Ck nonlinearitiesNonlin. Different. Equat. Appl.20081524727610.1007/s00030-007-7025-5 – ident: 1798_CR1 – volume: 15 start-page: 247 year: 2008 ident: 1798_CR9 publication-title: Nonlin. Different. Equat. Appl. doi: 10.1007/s00030-007-7025-5 – ident: #cr-split#-1798_CR6.1 doi: 10.1007/s11253-015-1108-y – volume: 50 start-page: 44 issue: 1 year: 2018 ident: 1798_CR10 publication-title: Mat. Stud. doi: 10.15330/ms.50.1.44-59 – ident: #cr-split#-1798_CR6.2 doi: 10.1007/s11253-015-1108-y – ident: 1798_CR7 – volume-title: Hörmander Spaces, Interpolation, and Elliptic Problems year: 2014 ident: 1798_CR4 doi: 10.1515/9783110296891 – volume: 18 start-page: 521 issue: 5 year: 2005 ident: 1798_CR2 publication-title: J. Appl. Math. Lett. doi: 10.1016/j.aml.2004.05.009 – ident: #cr-split#-1798_CR3.2 doi: 10.1007/s11253-013-0787-5 – volume: 2017 start-page: 1 issue: 265 year: 2017 ident: 1798_CR5 publication-title: Electron. J. Different. Equat. – volume: 134 start-page: 359 issue: 2 year: 2006 ident: 1798_CR8 publication-title: Duke Math. J. doi: 10.1215/S0012-7094-06-13424-5 – ident: #cr-split#-1798_CR3.1 doi: 10.1007/s11253-013-0787-5
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Snippet	We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash–Mozer iterative scheme,... We study the solvability of the nonlocal boundary-value problem for a differential equation with weak nonlinearity. By using the Nash-Mozer iterative scheme,...
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SubjectTerms	Algebra Analysis Applications of Mathematics Boundary value problems Differential equations Geometry Mathematical analysis Mathematics Mathematics and Statistics Nonlinearity Operators (mathematics) Real variables Statistics
Title	Solvability Conditions for the Nonlocal Boundary-Value Problem for a Differential-Operator Equation with Weak Nonlinearity in the Refined Sobolev Scale of Spaces of Functions of Many Real Variables
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