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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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).
P. W. Eloea and B. Ahmadb, “Positive solutions of a nonlinear nth order boundary-value problem with nonlocal conditions,” J. Appl. Math. Lett., 18, No. 5, 521–527 (2005).
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).
V. A. Mikhailets and A. A. Murach, Hörmander Spaces, Interpolation, and Elliptic Problems, Walter de Gruyter, Berlin (2014).
V. S. Il’kiv, Z. M. Nytrebych, and P. Y. Pukach, “Nonlocal problem with moment conditions for hyperbolic equations,” Electron. J. Different. Equat., 2017, No. 265, 1–9 (2017).
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.
M. Berti and P. Bolle, “Cantor families of periodic solutions for completely resonant nonlinear wave equations,” Duke Math. J., 134, No. 2, 359–419 (2006).
M. Berti and P. Bolle, “Cantor families of periodic solutions of wave equations with Ck nonlinearities,” Nonlin. Different. Equat. Appl., 15, 247–276 (2008).
I. Volyanska, V. Il’kiv, and N. Strap, “Solvability conditions of nonlocal boundary-value problem for the differential-operator equation with weak nonlinearity,” Mat. Stud., 50, No. 1, 44–59 (2018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 4, pp. 452–466, April, 2020.
Rights and permissions
About this article
Cite this article
Il’kiv, V.S., Strap, N.I. & Volyanska, I.I. 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. Ukr Math J 72, 515–535 (2020). https://doi.org/10.1007/s11253-020-01798-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-020-01798-7