Nonlocal boundary value problem for a differential-operator equation
with nonlinear right part in a complex domain

V. S. Ilkiv; N. I. Strap

doi:10.15330/ms.45.2.170-181

The paper is devoted to investigation of nonlocal boundary value problem for a differential-operator equation with the nonlinear right part and with the operator

$B=(B_1,\dotsc,B_p)$ , where

$B_j\equiv z_j\frac{\partial}{\partial z_j}$ ,

$j=1,\dotsc,p$ , --- operators of the generalized differentiation on complex variable

$z_j$ . This problem is incorrect in the Hadamard sense and its sobvability related to the small denominators. By using of the Nash-Mozer iteration scheme the conditions of the sobvability of the problem in the scale of spaces of functions of several complex variables are established.

1. Ptashnyk B.Yo., Il’kiv V.S., Kmit’ I.Ya., Polishchuk V.M. Nonlocal boundary value problems for partial differential equations, Kiev: Naukova Dumka, 2002. – 415 р. (in Ukrainian)

2. Eloea Paul W., Ahmadb Bashir, Positive solutions of a nonlinear n-th order boundary value problem with nonlocal conditions, J. Applied Mathematics Letters, 18 (2005), №5, 521–527.

3. Borok V.M., Fardigola L.V. Nonlocal well-posed boundary-value problems in a layer. Mathematical notes, 48 (1990), №1, 20–25. (in Russian)

4. Kalenyuk P.I., Nytrebych Z.M., Kohut I.V. Problem with nonlocal two-point condition in time variable for homogeneous partial differential equation of infinite order in spatial variables. Math.Methods Phys. Mech. Fields, 51 (2008), №4, 17–26. (in Ukrainian)

5. Berti M., Bolle P., Cantor families of periodic solutions of wave equations with

$C^k$ nonlinearities, Nonlinear Differential Equations and Applications, 15 (2008), 247–276.

6. Berti M., Bolle P., Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, Arch. Rational Mech. Anal. 195, (2010), №2, 609–642.

7. Berti M., Bolle P., Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J. 134, (2006), №2, 359–419.

	Nonlocal boundary value problem for a differential-operator equation with nonlinear right part in a complex domain (in Ukrainian)
Author	V. S. Ilkiv, N. I. Strap ilkivv@i.ua, n.strap@i.ua National University Lviv Polytechnic
Abstract	The paper is devoted to investigation of nonlocal boundary value problem for a differential-operator equation with the nonlinear right part and with the operator $B=(B_1,\dotsc,B_p)$ , where $B_j\equiv z_j\frac{\partial}{\partial z_j}$ , $j=1,\dotsc,p$ , --- operators of the generalized differentiation on complex variable $z_j$ . This problem is incorrect in the Hadamard sense and its sobvability related to the small denominators. By using of the Nash-Mozer iteration scheme the conditions of the sobvability of the problem in the scale of spaces of functions of several complex variables are established.
Keywords	differential equation; generalized differentiation; small denominators; the Nash-Moser scheme
DOI	doi:10.15330/ms.45.2.170-181
Reference	1. Ptashnyk B.Yo., Il’kiv V.S., Kmit’ I.Ya., Polishchuk V.M. Nonlocal boundary value problems for partial differential equations, Kiev: Naukova Dumka, 2002. – 415 р. (in Ukrainian) 2. Eloea Paul W., Ahmadb Bashir, Positive solutions of a nonlinear n-th order boundary value problem with nonlocal conditions, J. Applied Mathematics Letters, 18 (2005), №5, 521–527. 3. Borok V.M., Fardigola L.V. Nonlocal well-posed boundary-value problems in a layer. Mathematical notes, 48 (1990), №1, 20–25. (in Russian) 4. Kalenyuk P.I., Nytrebych Z.M., Kohut I.V. Problem with nonlocal two-point condition in time variable for homogeneous partial differential equation of infinite order in spatial variables. Math.Methods Phys. Mech. Fields, 51 (2008), №4, 17–26. (in Ukrainian) 5. Berti M., Bolle P., Cantor families of periodic solutions of wave equations with $C^k$ nonlinearities, Nonlinear Differential Equations and Applications, 15 (2008), 247–276. 6. Berti M., Bolle P., Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, Arch. Rational Mech. Anal. 195, (2010), №2, 609–642. 7. Berti M., Bolle P., Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J. 134, (2006), №2, 359–419.
Pages	170-181
Volume	45
Issue	2
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Nonlocal boundary value problem for a differential-operator equation with nonlinear right part in a complex domain (in Ukrainian)