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Ukrainian Mathematical Journal

, Volume 58, Issue 12, pp 1847–1875 | Cite as

Problems for partial differential equations with nonlocal conditions. Metric approach to the problem of small denominators

  • V. S. Il’kiv
  • B. I. Ptashnyk
Article

Abstract

A survey of works of the authors and their disciples devoted to the investigation of problems with nonlocal conditions with respect to a selected variable in cylindrical domains is presented. These problems are considered for linear equations and systems of partial differential equations that, in general, are ill posed in the Hadamard sense and whose solvability in certain scales of functional spaces is established for almost all (with respect to Lebesgue measure) vectors composed of the coefficients of the problem and the parameters of the domain.

Keywords

Partial Differential Equation Naukova Dumka Hyperbolic Equation Pseudodifferential Operator Ukrainian National Academy 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    A. M. Nakhushev, “On nonlocal problems with shift and their connection with loaded equations,” Differents. Uravn., 21, No. 1, 92–101 (1985).MathSciNetGoogle Scholar
  2. 2.
    A. V. Bitsadze and A. A. Samarskii, “On some simplest generalizations of linear elliptic boundary-value problems,” Dokl. Akad. Nauk SSSR, 185, No. 4, 739–740 (1969).MathSciNetGoogle Scholar
  3. 3.
    P. N. Vabishchevich, “Nonlocal parabolic problem and inverse heat problem,” Differents. Uravn., 17, No. 7, 1193–1199 (1981).MathSciNetGoogle Scholar
  4. 4.
    I. Ya. Kmit’, “On one nonlocal problem for a quasilinear hyperbolic system of the first order with two independent variables,” Ukr. Mat. Zh., 45, No. 9, 1307–1311 (1993).MathSciNetGoogle Scholar
  5. 5.
    I. Ya. Kmit’ and S. P. Lavrenyuk, “On nonlocal problems for two-dimensional hyperbolic systems,” Usp. Mat. Nauk, 46, No. 6, 149 (1991).Google Scholar
  6. 6.
    M. Ivanchov, Inverse Problem for Equations of Parabolic Type, VNTL, Lviv (2003).Google Scholar
  7. 7.
    A. A. Dezin, General Problems of the Theory of Boundary-Value Problems [in Russian], Nauka, Moscow (1980).zbMATHGoogle Scholar
  8. 8.
    E. A. Grebenikov and Yu. A. Ryabov, Resonances and Small Denominators in Celestial Mechanics [in Russian], Nauka, Moscow (1978).Google Scholar
  9. 9.
    D. G. Bourghin and R. J. Duffin, “The Dirichlet problem for the vibrating string equation,” Bull. Amer. Math. Soc., 45, No. 12, 851–858 (1939).MathSciNetGoogle Scholar
  10. 10.
    C. L. Siegel, “Iterations of analytic functions,” Ann. Math., 43, No. 4, 607–612 (1942).CrossRefGoogle Scholar
  11. 11.
    A. N. Kolmogorov, “On dynamical systems with integral invariant on a torus,” Dokl. Akad. Nauk SSSR, 93, No. 5, 763–766 (1953).zbMATHMathSciNetGoogle Scholar
  12. 12.
    V. I. Arnol’d, “Small denominators and problems of stability of motion in classical and celestial mechanics,” Usp. Mat. Nauk, 18, No. 6(114), 91–192 (1963).Google Scholar
  13. 13.
    J. Moser, “A rapidly convergent iteration method and non-linear differential equations. I,” Ann. Scuola Norm Supper. Pisa Ser. III, 20, No. 2, 265–315 (1966); “A rapidly convergent iteration method and non-linear differential equations. II,” Ann. Scuola Norm Supper. Pisa Ser. III, 20, No. 3, 499–535 (1966).Google Scholar
  14. 14.
    G. A. Baker, Jr., and P. Graves-Morris, Padé Approximants, Addison-Wesley, London (1981).Google Scholar
  15. 15.
    A. V. Groshev, “Theorem on a system of linear forms,” Dokl. Akad. Nauk SSSR, 19, No. 3, 151–152 (1938).Google Scholar
  16. 16.
    V. G. Sprindzhuk, Metric Theory of Diophantine Approximations [in Russian], Nauka, Moscow (1977).zbMATHGoogle Scholar
  17. 17.
    V. Bernik and V. Beresnevich, “On a metrical theorem of W. Schmidt,” Acta Arithm., 75, No. 3, 219–233 (1996).zbMATHMathSciNetGoogle Scholar
  18. 18.
    V. I. Gorbachuk and M. L. Gorbachuk, Boundary-Value Problems for Operator Differential Equations [in Russian], Naukova Dumka, Kiev (1984).zbMATHGoogle Scholar
  19. 19.
    A. Ya. Khinchin, Continued Fractions [in Russian], Nauka, Moscow (1978).Google Scholar
  20. 20.
    V. Jarnik, “Diophantische Approximationen und Hausdorffsches Mass,” Mat. Sb., 36, No. 3/4, 371–382 (1929).zbMATHGoogle Scholar
  21. 21.
    A. S. Besicovitch, “Sets of fractional dimensions on rational approximations to real numbers,” J. London Math. Soc., 9, 126–131 (1934).zbMATHCrossRefGoogle Scholar
  22. 22.
    N. A. Artem’ev, “Periodic solutions of one class of partial differential equations,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 1, 15–50 (1937).Google Scholar
  23. 23.
    Yu. A. Mitropol’skii, G. P. Khoma, and M. I. Gromyak, Asymptotic Methods for Investigation of Quasiwave Equations of Hyperbolic Type [in Russian], Naukova Dumka, Kiev (1991).Google Scholar
  24. 24.
    B. I. Ptashnik, Ill-Posed Boundary-Value Problems for Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1984).Google Scholar
  25. 25.
    B. I. Ptashnyk, V. S. Il’kiv, and V. M. Polishchuk, Nonlocal Boundary-Value Problems for Partial Differential Equations [in Ukrainian], Naukova Dumka, Kiev (2002).Google Scholar
  26. 26.
    A. M. Samoilenko and B. P. Tkach, Numerical-Analytic Methods in the Theory of Periodic Solutions of Partial Differential Equations [in Russian], Naukova Dumka, Kiev (1992).Google Scholar
  27. 27.
    O. Vejvoda, Partial Differential Equations: Time Periodic Solutions, Noordhoff, Sijthoff (1981).Google Scholar
  28. 28.
    B. I. Ptashnik, “Periodic boundary-value problem for linear hyperbolic equations with constant coefficients,” Mat. Fiz., Issue 12, 117–121 (1972).Google Scholar
  29. 29.
    V. M. Polishchuk and B. I. Ptashnyk, “Periodic boundary-value problem for hyperbolic equations,” in: Proceedings of the Second Conference of Young Scientists of the West Scientific Center of the Ukrainian Academy of Sciences, Section of Mathematical Sciences [in Ukrainian], Uzhhorod (1975), pp. 55–59, Dep. in VINITI, No. 1734-76.Google Scholar
  30. 30.
    V. N. Polishchuk, “Periodic boundary-value problem for linear hyperbolic equations,” Mat. Met. Fiz.-Mekh. Polya, Issue 2, 158–160 (1975).Google Scholar
  31. 31.
    V. N. Polishchuk and B. I. Ptashnik, “On a periodic boundary-value problem for hyperbolic operators that decompose into linear factors of the first order with constant coefficients,” Mat. Met. Fiz.-Mekh. Polya, Issue 3, 6–12 (1976).Google Scholar
  32. 32.
    B. I. Ptashnik, “Periodic boundary-value problem for a hyperbolic operator that decomposes into linear factors of the first order,” Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 11, 985–989 (1973).Google Scholar
  33. 33.
    B. I. Ptashnyk, “Periodic boundary-value problem for polywave operators,” in: Projection-Iterative Methods for the Solution of Differential and Integral Equations [in Ukrainian], Naukova Dumka, Kyiv (1974), pp. 147–154.Google Scholar
  34. 34.
    V. N. Polishchuk, “Periodic boundary-value problem for hyperbolic equations with variable coefficients,” in: Theoretical and Applied Problems of Algebra and Differential Equations [in Russian], Naukova Dumka, Kiev (1976), pp. 60–65.Google Scholar
  35. 35.
    V. M. Polishchuk and B. I. Ptashnyk, “Problem with time-periodic conditions for weakly nonlinear hyperbolic equations,” Mat. Met. Fiz.-Mekh. Polya, 46, No. 3, 7–14 (2003).zbMATHGoogle Scholar
  36. 36.
    V. M. Polishchuk and B. I. Ptashnyk, “Periodic boundary-value problem for weakly nonlinear hyperbolic equations with variable coefficients in the linear part of an operator,” Mat. Met. Fiz.-Mekh. Polya, 48, No. 2, 25–31 (2005).zbMATHGoogle Scholar
  37. 37.
    V. N. Polishchuk and B. I. Ptashnik, “On a periodic boundary-value problem for a system of hyperbolic equations with constant coefficients,” Ukr. Mat. Zh., 30, No. 3, 326–333 (1978).zbMATHMathSciNetGoogle Scholar
  38. 38.
    V. N. Polishchuk and B. I. Ptashnik, “Periodic solutions of a system of partial differential equations with constant coefficients,” Ukr. Mat. Zh., 32, No. 2, 239–243 (1980).zbMATHMathSciNetGoogle Scholar
  39. 39.
    V. N. Polishchuk, “Problem with nonlocal boundary conditions for hyperbolic equations with variable coefficients,” in: Approximate and Qualitative Methods in the Theory of Differential and Functional Differential Equations [in Russian], Naukova Dumka, Kiev (1979), pp. 54–65.Google Scholar
  40. 40.
    T. P. Hoi, V. M. Polishchuk, and B. I. Ptashnyk, “Nonlocal two-point boundary-value problem for a hyperbolic equation with variable coefficients in a cylindrical domain,” in: Mathematical Methods in Scientific and Engineering Investigations [in Ukrainian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (1996), pp. 62–70.Google Scholar
  41. 41.
    V. A. Il’in and I. A. Shishmarev, “Estimates uniform in a closed domain for eigenfunctions of an elliptic operator and their derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat., 24, No. 6, 883–896 (1960).zbMATHMathSciNetGoogle Scholar
  42. 42.
    V. P. Mikhailov, Partial Differential Equations [in Russian], Nauka, Moscow (1976).Google Scholar
  43. 43.
    T. P. Hoi and B. I. Ptashnyk, “Nonlocal boundary-value problems for systems of linear partial differential equations with variable coefficients,” Ukr. Mat. Zh., 49, No. 11, 1478–1487 (1997).CrossRefGoogle Scholar
  44. 44.
    O. D. Vlasii and B. I. Ptashnyk, “Problem with nonlocal conditions for partial differential equations with variable coefficients,” Ukr. Mat. Zh., 53, No. 10, 1328–1336 (2001).zbMATHGoogle Scholar
  45. 45.
    T. P. Hoi and B. I. Ptashnyk, “Problem with nonlocal conditions for a weakly nonlinear hyperbolic equation with constant coefficients,” in: Nonlinear Boundary-Value Problems of Mathematical Physics and Their Applications, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1996), pp. 74–76.Google Scholar
  46. 46.
    T. P. Hoi and B. I. Ptashnyk, “Problem for weakly nonlinear hyperbolic equations with nonlocal conditions,” Ukr. Mat. Zh., 49, No. 2, 186–195 (1997).CrossRefGoogle Scholar
  47. 47.
    B. I. Ptashnyk, M. M. Symotyuk, and N. M. Zadorozhna, “Problem for quasilinear hyperbolic equations with nonlocal conditions,” Nelin. Gran. Zad., Issue 11, 161–167 (2001).Google Scholar
  48. 48.
    N. M. Zadorozhna, O. M. Mel’nyk, and B. I. Ptashnyk, “Nonlocal boundary-value problem for parabolic equations,” Ukr. Mat. Zh., 46, No. 12, 1621–1627 (1994).zbMATHGoogle Scholar
  49. 49.
    N. M. Zadorozhna and B. I. Ptashnyk, “Nonlocal boundary-value problem for parabolic equations with variable coefficients,” Ukr. Mat. Zh., 47, No. 7, 913–919 (1995).Google Scholar
  50. 50.
    B. I. Ptashnyk and N. M. Zadorozhna, “Nonlocal boundary-value problem for a parabolic equation with variable coefficients,” in: Nonlinear Boundary-Value Problems of Mathematical Physics and Their Applications, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (1994), pp. 164–166.Google Scholar
  51. 51.
    O. M. Medvid’, “Problem with distributed data for factorized partial differential equations,” Mat. Visn. NTSh, 2, 135–146 (2005).Google Scholar
  52. 52.
    O. M. Medvid’ and M. M. Symotyuk, “Problem with integral conditions for linear partial differential equations,” Mat. Met. Fiz.-Mekh. Polya, 46, No. 4, 92–101 (2003).Google Scholar
  53. 53.
    O. M. Medvid’ and M. M. Symotyuk, “Problem with distributed data for partial differential equations,” Mat. Met. Fiz.-Mekh. Polya, 47, No. 4, 155–159 (2004).Google Scholar
  54. 54.
    M. M. Symotyuk, Multipoint Problems for Linear Partial Differential Equations and Pseudodifferential Equations with Partial Derivatives [in Ukrainian], Author’s Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Lviv (2005).Google Scholar
  55. 55.
    O. D. Vlasii and B. I. Ptashnyk, “Problem for partial differential equations unsolved with respect to the leading derivative with nonlocal conditions,” Ukr. Mat. Zh., 55, No. 8, 1022–1034 (2003).CrossRefGoogle Scholar
  56. 56.
    L. I. Komarnyts’ka, “Nonlocal boundary-value problem for an equation with variable coefficients unsolved with respect to the leading derivative,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 40, 17–23 (1994).Google Scholar
  57. 57.
    L. I. Komarnyts’ka and B. I. Ptashnyk, “Problem for a partial differential equation unsolved with respect to the leading time derivative with nonlocal conditions,” in: Boundary-Value Problems with Different Degeneracies and Singularities [in Ukrainian], Chernivtsi (1990), pp. 86–95.Google Scholar
  58. 58.
    M. M. Symotyuk and N. M. Zadorozhna, “Nonlocal boundary-value problem for a nonlinear equation with fractional time derivative with variable coefficients,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 51, 61–70 (1998).Google Scholar
  59. 59.
    M. M. Symotyuk and N. M. Zadorozhna, “Nonlocal boundary-value problem for a differential equation with fractional time derivative with variable coefficients,” Mat. Met. Fiz.-Mekh. Polya, 41, No. 4, 89–94 (1998).zbMATHGoogle Scholar
  60. 60.
    V. S. Il’kiv, “Multipoint nonlocal problem for partial differential equations,” Differents. Uravn., 23, No. 3, 487–492 (1987).MathSciNetGoogle Scholar
  61. 61.
    V. S. Il’kiv, “Extension of a solution of a nonlocal multipoint problem for a partial differential equation with constant coefficients with respect to the time variable,” Mat. Met. Fiz.-Mekh. Polya, 41, No. 4, 78–82 (1998).zbMATHGoogle Scholar
  62. 62.
    V. S. Il’kiv, “Analogs of the Pyartli lemma with absolute constants,” Mat. Met. Fiz.-Mekh. Polya, 42, No. 4, 68–74 (1999).MathSciNetGoogle Scholar
  63. 63.
    V. S. Il’kiv, “Perturbations of a nonlocal problem for differential equations with pseudodifferential coefficients,” Differents. Uravn., 26, No. 11, 1962–1971 (1990).MathSciNetGoogle Scholar
  64. 64.
    V. S. Il’kiv, “Problem with formal initial conditions for differential equations with constant pseudodifferential coefficients,” Ukr. Mat. Zh., 50, No. 7, 877–888 (1998).CrossRefMathSciNetGoogle Scholar
  65. 65.
    V. S. Il’kiv, V. N. Polishchuk, B. I. Ptashnik, and B. O. Salyga, “Nonlocal multipoint problem for pseudodifferential operators with analytic symbols,” Ukr. Mat. Zh., 38, No. 5, 582–587 (1986).MathSciNetGoogle Scholar
  66. 66.
    M. M. Symotyuk, “Problem for pseudodifferential equations with nonlocal multipoint conditions,” Visn. Derzh. Univ. “L’vivs’ka Politekhnika,” Prykl. Mat., No. 411, 280–285 (2000).Google Scholar
  67. 67.
    M. M. Symotyuk, “Problem for an equation with pseudodifferential operators with nonlocal multipoint conditions,” Mat. Met. Fiz.-Mekh. Polya, 43, No. 4, 37–41 (2000).zbMATHGoogle Scholar
  68. 68.
    V. S. Il’kiv, Nonlocal Boundary-Value Problems for Partial Differential Equations and Operator Differential Equations [in Ukrainian], Author’s Abstract of the Doctoral-Degree Thesis (Physics and Mathematics), Kyiv (2006).Google Scholar
  69. 69.
    V. M. Polishchuk, “Problem with nonlocal boundary conditions for hyperbolic systems of differential equations with constant coefficients,” Dopov. Akad. Nauk Ukr. SSR, Ser. A, No. 3, 171–175 (1979).Google Scholar
  70. 70.
    N. M. Zadorozhna, “Problem for systems of parabolic equations of arbitrary order,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 47, 48–55 (1997).Google Scholar
  71. 71.
    V. S. Il’kiv and B. I. Ptashnik, “Problem with nonlocal boundary conditions for a system of partial differential equations with constant coefficients,” Differents. Uravn., 20, No. 6, 1012–1023 (1984).MathSciNetGoogle Scholar
  72. 72.
    V. S. Il’kiv and B. I. Ptashnyk, “Representation and investigation of solutions of a nonlocal problem for systems of partial differential equations,” Ukr. Mat. Zh., 48, No. 2, 184–194 (1996).CrossRefGoogle Scholar
  73. 73.
    F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).zbMATHGoogle Scholar
  74. 74.
    P. S. Kazimirs’kyi, Factorization of Matrix Polynomials [in Ukrainian], Naukova Dumka, Kyiv (1981).Google Scholar
  75. 75.
    V. S. Il’kiv, “Nonlocal boundary-value problem for normal anisotropic systems of partial differential equations with constant coefficients,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 54, 84–95 (1999).Google Scholar
  76. 76.
    G. E. Shilov, Mathematical Analysis. Second Special Course [in Russian], Moscow University, Moscow (1965).Google Scholar
  77. 77.
    V. S. Il’kiv, “Nonlocal boundary-value problem for a system of partial differential equations in anisotropic spaces,” Nelin. Gran. Zad., Issue 11, 57–64 (2001).Google Scholar
  78. 78.
    V. S. Il’kiv, “Nonlocal problem for systems of partial differential equations in Sobolev spaces of infinite order,” Mat. Met. Fiz.-Mekh. Polya, 47, No. 4, 115–119 (2004).MathSciNetGoogle Scholar
  79. 79.
    V. S. Il’kiv, “Nonlocal boundary-value problem for systems of partial differential equations of infinite order,” Differents. Uravn., 41, No. 2, 250–257 (2005).MathSciNetGoogle Scholar
  80. 80.
    O. D. Vlasii and B. I. Ptashnyk, “Problem for systems of partial differential equations unsolved with respect to the leading time derivative with nonlocal conditions,” Ukr. Mat. Visn., 1, No. 4, 501–517 (2004).MathSciNetGoogle Scholar
  81. 81.
    V. S. Il’kiv, V. N. Polishchuk, and B. I. Ptashnik, “Nonlocal boundary-value problem for systems of pseudodifferential equations,” in: Methods for Investigation of Differential and Integral Operators [in Russian], Naukova Dumka, Kiev (1989), pp. 75–79.Google Scholar
  82. 82.
    V. S. Il’kiv, “Nonlocal boundary-value problem for an inhomogeneous system of partial differential equations with variable coefficients,” Visn. L’viv. Univ., Ser. Mekh.-Mat., Issue 58, 139–143 (2000).Google Scholar
  83. 83.
    V. S. Il’kiv, Ya. M. Pelekh, and B. O. Salyha, “Nonlocal two-point problem for systems of partial differential equations with variable coefficients,” Visn. Derzh. Univ. “L’vivs’ka Politekhnika,” Prykl. Mat., No. 407, 245–252 (2000).Google Scholar
  84. 84.
    V. S. Il’kiv, “Investigation of a nonlocal boundary-value problem for partial differential equations by the method of optimization in Sobolev spaces,” Mat. Stud., 11, No. 2, 167–176 (1999).zbMATHMathSciNetGoogle Scholar
  85. 85.
    V. Il’kiv, “Incorrect nonlocal boundary-value problem for partial differential equations,” Funct. Anal. Appl., 197, 115–121 (2004).MathSciNetGoogle Scholar
  86. 86.
    V. S. Il’kiv and B. I. Ptashnik, “Ill-posed nonlocal two-point problem for systems of partial differential equations,” Sib. Mat. Zh., 46, No. 1, 119–129 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  87. 87.
    Ya. I. Dasyuk, V. S. Il’kiv, and P. Ya. Pukach, “Nonlocal two-point boundary-value problem for linear partial differential equations,” Visn. Derzh. Univ. “L’vivs’ka Politekhnika,” Prykl. Mat., No. 411, 102–106 (2000).Google Scholar
  88. 88.
    V. S. Il’kiv, “Nonlocal two-point boundary-value problem for a system of inhomogeneous partial differential equations,” Mat. Met. Fiz.-Mekh. Polya, 45, No. 4, 87–94 (2002).zbMATHMathSciNetGoogle Scholar
  89. 89.
    V. S. Il’kiv, “Nonlocal multipoint inhomogeneous problem for systems of partial differential equations with coefficients depending on t,” Mat. Visn. NTSh, 1, 47–58 (2004).zbMATHGoogle Scholar
  90. 90.
    V. S. Il’kiv, “Problem with integral conditions for a system of partial differential equations with variable coefficients,” Visn. Derzh. Univ. “L’vivs’ka Politekhnika,” Prykl. Mat., No. 364, 318–323 (1999).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • V. S. Il’kiv
    • 1
  • B. I. Ptashnyk
    • 2
  1. 1.“L’vivs’ka Politekhnika” National UniversityLviv
  2. 2.Institute for Applied Problems of Mechanics and MathematicsUkrainian National Academy of SciencesLviv

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