Two-point nonlocal problem for a weak nonlinear differential operator
equation

I. Volyanska; V. Il’kiv; N. Strap

doi:10.15330/ms.50.1.44-59

In the paper is have studied the solvability of a two-point nonlocal boundary-value problem for the operator-differential equation with weakly nonlinear right-hand side. The proof of the theorems is carried out within the Nash–Moser iterative scheme. In this scheme, the important point is the construction of estimates to the norms of inverse linearized operators arising at each step of this scheme and by the related problem of small denominators. The inverse of linearized operators is obtained by the method developed in the work of M. Berti, P. Bolle (Duke Math. J., 134 (2), 359–419 (2006)). The problem of small denominators is solved by using the metric approach.

1. Arnol’d V.I. Small denominators. I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25, 1961, P. 21–86. (in Russian)

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

3. Berti M., Bolle P., Cantor families of periodic solutions of wave equations with Ck nonlinearities, Nonlinear Differential Equations and Applications, 15 (2008), 247–276.

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

5. Il’kiv V.S. Incorrect nonlocal boundary value problem for partial differential equations, North-Holland Mathematics Studies, 197(C) (2004)б 115–121.

6. Il’kiv, V.S., Nytrebych, Z.M., Pukach, P.Y., Boundary-value problems with integral conditions for a system of Lame equations in the space of almost periodic functions, Electronic Journal of Differential Equations, 2016 (2016)б No.304, 1–12.

7. Il’kiv V.S., Strap N.I. Nonlocal boundary value problem for a differential-operator equation with nonlinear right part in a complex domain, Matematychni Studii, 45 (2016), №2, 170–181. (in Ukrainian)

8. Il’kiv V.S., Strap N.I. Nonlocal boundary value problem for a differential-operator equation with nonlinearity in the spaces of Dirichlet–Taylor series with fixed spectrum, Mathematical Methods and Physicomechanical Fields, 59 (2016), №2, 77–85. (in Ukrainian)

9. Il’kiv V.S., Strap N.I. Solvability of a nonlocal boundary-value problem for the operator-differential equation with weak nonlinearity in a refined scale of Sobolev spaces, Journal of Mathematical Sciences, 218 (2016), №1, 1–15.

10. Il’kiv V.S., Strap N.I. 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, Ukrainian Mathematical Journal, 67 (2015), №5, 690–710.

11. Kalenyuk P.I., Kohut I.V., Nytrebych Z.M. Problem with nonlocal two-point condition in time for a homogeneous partial differential equation of infinite order with respect to space variables, Journal of Mathematical Sciences, 167 (2010), №1, 1–15.

12. Malanchuk O., Nytrebych Z. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables, Open Mathematics, 15 (2017), №1, 101–110.

13. J. Mozer. A rapidly convergent iteration method and non-linear differential equations I. // Annali della Scuola Norm. Super. de Piza ser. III, 20 (1966), №2, 265–315.

14. J. Mozer. A rapidly convergent iteration method and non-linear differential equations II. // Annali della Scuola Norm. Super. de Piza ser. III, 20 (1966), №3, 499–535.

15. Ptashnyk B.Yo., Il’kiv V.S., Kmit’ I.Ya., Polishchuk V.M. Nonlocal boundary value problems for partial differential equations, 415 p. Naukova Dumka, Kyiv (2002). (in Ukrainian)

	Two-point nonlocal problem for a weak nonlinear differential operator equation
Author	I. Volyanska¹, V. Il’kiv,², N. Strap³ i.volyanska@i.ua¹, ilkivv@i.ua², n.strap@i.ua³ 1) Lviv Polytechnic National University, Lviv, Ukraine; 2) Lviv Polytechnic National University, Lviv, Ukraine; 3) College of Oil and Gas, Drohobych, Lviv region, Ukraine
Abstract	In the paper is have studied the solvability of a two-point nonlocal boundary-value problem for the operator-differential equation with weakly nonlinear right-hand side. The proof of the theorems is carried out within the Nash–Moser iterative scheme. In this scheme, the important point is the construction of estimates to the norms of inverse linearized operators arising at each step of this scheme and by the related problem of small denominators. The inverse of linearized operators is obtained by the method developed in the work of M. Berti, P. Bolle (Duke Math. J., 134 (2), 359–419 (2006)). The problem of small denominators is solved by using the metric approach.
Keywords	nonlocal problem; small denominators; metric estimation; Nash–Moser iterative scheme
DOI	doi:10.15330/ms.50.1.44-59
Reference	1. Arnol’d V.I. Small denominators. I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25, 1961, P. 21–86. (in Russian) 2. Berti M., Bolle P., Cantor families of periodic solutions for completely resonant nonlinear wave equations, Duke Math. J., 134 (2006)б №2, 359–419. 3. Berti M., Bolle P., Cantor families of periodic solutions of wave equations with Ck nonlinearities, Nonlinear Differential Equations and Applications, 15 (2008), 247–276. 4. Berti M., Bolle P., Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, Arch. Rational Mech. Anal., 195 (2010), №2, 609–642. 5. Il’kiv V.S. Incorrect nonlocal boundary value problem for partial differential equations, North-Holland Mathematics Studies, 197(C) (2004)б 115–121. 6. Il’kiv, V.S., Nytrebych, Z.M., Pukach, P.Y., Boundary-value problems with integral conditions for a system of Lame equations in the space of almost periodic functions, Electronic Journal of Differential Equations, 2016 (2016)б No.304, 1–12. 7. Il’kiv V.S., Strap N.I. Nonlocal boundary value problem for a differential-operator equation with nonlinear right part in a complex domain, Matematychni Studii, 45 (2016), №2, 170–181. (in Ukrainian) 8. Il’kiv V.S., Strap N.I. Nonlocal boundary value problem for a differential-operator equation with nonlinearity in the spaces of Dirichlet–Taylor series with fixed spectrum, Mathematical Methods and Physicomechanical Fields, 59 (2016), №2, 77–85. (in Ukrainian) 9. Il’kiv V.S., Strap N.I. Solvability of a nonlocal boundary-value problem for the operator-differential equation with weak nonlinearity in a refined scale of Sobolev spaces, Journal of Mathematical Sciences, 218 (2016), №1, 1–15. 10. Il’kiv V.S., Strap N.I. 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, Ukrainian Mathematical Journal, 67 (2015), №5, 690–710. 11. Kalenyuk P.I., Kohut I.V., Nytrebych Z.M. Problem with nonlocal two-point condition in time for a homogeneous partial differential equation of infinite order with respect to space variables, Journal of Mathematical Sciences, 167 (2010), №1, 1–15. 12. Malanchuk O., Nytrebych Z. Homogeneous two-point problem for PDE of the second order in time variable and infinite order in spatial variables, Open Mathematics, 15 (2017), №1, 101–110. 13. J. Mozer. A rapidly convergent iteration method and non-linear differential equations I. // Annali della Scuola Norm. Super. de Piza ser. III, 20 (1966), №2, 265–315. 14. J. Mozer. A rapidly convergent iteration method and non-linear differential equations II. // Annali della Scuola Norm. Super. de Piza ser. III, 20 (1966), №3, 499–535. 15. Ptashnyk B.Yo., Il’kiv V.S., Kmit’ I.Ya., Polishchuk V.M. Nonlocal boundary value problems for partial differential equations, 415 p. Naukova Dumka, Kyiv (2002). (in Ukrainian)
Pages	44-59
Volume	50
Issue	1
Year	2018
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Two-point nonlocal problem for a weak nonlinear differential operator equation