Skip to main content
SpringerLink
Log in

A Problem with Integral Conditions with Respect to Time for Shilov Parabolic Systems of Equations

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

In a domain specified in the form of the Cartesian product of an interval [0,T] and a space ℝp, we study a problem with integral conditions with respect to the time coordinate for Shilov parabolic systems of equations in the class of functions almost periodic with respect to the space variables. We establish a criterion of uniqueness and sufficient conditions for the existence of a solution of the problem. To solve the problem of small denominators encountered in the construction of the solution, we apply the metric approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. M. Gel’fand and G. E. Shilov, Generalized Functions [in Russian], Issue 3: Some Problems of the Theory of Differential Equations, Fizmatgiz, Moscow (1958).

    MATH  Google Scholar 

  2. O. Yu. Danilkina, “On one nonlocal problem for a parabolic equation,” Vestn. Samara Gos. Univ., No. 6(56), 141–154 (2007).

  3. A. M. Kuz’, “A problem with integral conditions with respect to time for a factorized parabolic operator with variable coefficients,” Visn. Nats. Univ. “L’vivs’ka Politekhnika,” Ser. Fiz.-Mat. Nauky, No. 740, 25–33 (2012).

  4. A. M. Kuz’ and B. Yo. Ptashnyk, “A problem with integral conditions with respect to time for a system of equations of the dynamic elasticity theory,” Mat. Met. Fiz.-Mekh. Polya, 56, No. 4, 40–53 (2013); English translation: J. Math. Sci., 208, No. 3, 310–326 (2015).

  5. P. Lankaster, Theory of Matrices, Academic Press, New York (1969).

    Google Scholar 

  6. O. M. Medvid’ and M. M. Symotyuk, “Diophantine approximations of the characteristic determinant of integral problem for a linear partial differential equation,” Nauk. Visn. Chernivtsi Nats. Univ., Ser. Mat., Issue 228, 74–85 (2004).

  7. O. M. Medvid’ and M. M. Symotyuk, “A problem with integral conditions for linear systems of partial differential equations,” Mat. Met. Fiz.-Mekh. Polya, 50, No. 1, 32–39 (2007).

  8. G. Pólya and G. Szegö, Problems and Theorems in Analysis [Russian translation], Part 2, Nauka, Moscow (1978).

    Google Scholar 

  9. L. S. Pontryagin, Ordinary Differential Equations [in Russian], Nauka, Moscow (1982).

    MATH  Google Scholar 

  10. B. Yo. 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).

  11. L. S. Pul’kina, “A nonlocal problem with integral conditions for a hyperbolic equation,” Differents. Uravn., 40, No. 7, 887–892 (2004); English translation: Differ. Equat., 40, No. 7, 947–953 (2004).

  12. Ya. D. Tamarkin, On Some General Problems of the Theory of Ordinary Linear Differential Equations and on the Expansion of Arbitrary Functions in Series [in Russian], Frolov Typolith., Petrograd (1917).

  13. D. K. Faddeev and I. S. Sominskii, Collection of Problems in Higher Algebra [in Ukrainian], Vyshcha Shkola, Kyiv (1971).

    Google Scholar 

  14. L. V. Fardigola, “An integral boundary-value problem in a layer for a system of linear partial differential equations,” Mat. Sb., 186, No. 11, 123–144 (1995); English translation: Sb. Math., 186, No. 11, 1671–1692 (1995).

  15. P. I. Shtabalyuk, “On almost periodic solutions of one problem with nonlocal conditions,” Visn. Derzh. Univ. “L’vivs’ka Politekhnika,” Ser. Dyfer. Rivn. Zastos., No. 286, 153–165 (1995).

  16. M. A. Shubin, “Almost periodic functions and partial differential operators,” Usp. Mat. Nauk, 33, No. 2 (200), 3–47 (1978); English translation: Russ. Math. Surv., 33, No. 2, 1–52 (1978).

  17. G. Avalishvili, M. Avalishvili, and D. Gordeziani, “On integral nonlocal boundary value problems for some partial differential equations,” Bull. Georg. Natl. Acad. Sci., 5, No. 1, 31–37 (2011).

    MathSciNet  MATH  Google Scholar 

  18. A. S. Besicovitch, Almost Periodic Functions, Dover, Cambridge (1954).

    MATH  Google Scholar 

  19. L. Bougoffa, “A coupled system with integral conditions,” Appl. Math. E-Notes, 4, 99–105 (2004).

    MathSciNet  MATH  Google Scholar 

  20. W. Song and W. Gao, “Positive solutions for a second-order system with integral boundary conditions,” Electron. J. Differ. Equat., 2011, No. 13, 1–9 (2011).

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv, Ukraine

    A. M. Kuz’

Authors
  1. A. M. Kuz’
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 57, No. 3, pp. 16 – 28, July – September, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kuz’, A.M. A Problem with Integral Conditions with Respect to Time for Shilov Parabolic Systems of Equations. J Math Sci 217, 149–165 (2016). https://doi.org/10.1007/s10958-016-2963-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-016-2963-2

Search

Navigation