Research Article
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Year 2020, Volume: 49 Issue: 1, 338 - 351, 06.02.2020
https://doi.org/10.15672/hujms.552213

Abstract

References

  • [1] N.K. Bari, Trigonometric Series (Russian), Fizmatgiz, Moscow, 1961. English transl.: N.K. Bary, A Treatise on Trigonometric Series. Vols. I, II, MacMillan, New York, 1964.
  • [2] G.D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc. 9 (4), 373-395, 1908.
  • [3] N. Dernek and O. Veliev, On the Riesz basisness of the root functions of the nonselfadjoint Sturm-Liouville operator, Israel J. Math. 145 (1), 113-123, 2005.
  • [4] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomials as potentials, Dokl. Akad. Nauk. 436, 11-13, 2011.
  • [5] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann. 351 (3), 509-540, 2011.
  • [6] P. Djakov and B. Mityagin, Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators, J. Funct. Anal. 263 (8), 2300-2332, 2012.
  • [7] P. Djakov and B. Mityagin, Instability zones of periodic 1-dimensional Schrodinger and Dirac operators, Uspekhi Mat. Nauk 61 (4), 77-182, 2006.
  • [8] N. Dunford, J.T. Schwartz, W.G. Bade and R.G. Bartle, Linear Operators, Wiley- Interscience, New York, 1971.
  • [9] F. Gesztesy and V. Tkachenko, A Schauder and Riesz basis criterion for non-selfadjoint Schrodinger operators with periodic and antiperiodic boundary conditions, J. Differential Equations 253 (2), 400-437, 2012.
  • [10] I. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, 1969.
  • [11] H. Gunes, N.B. Kerimov and U. Kaya, Spectral properties of fourth order differential operators with periodic and antiperiodic boundary conditions, Results Math. 68 (3-4), 501-518, 2015.
  • [12] N. Ionkin and E. Moiceev, Solution of boundary value problem in heat conduction theory with nonlocal boundary conditions, Differ. Equ. 13 (2), 294-304, 1977.
  • [13] B.S. Kashin and A. Saakyan, Orthogonal Series, volume 75, American Mathematical Society, Providence, 2005.
  • [14] N.B. Kerimov and U. Kaya, Some problems of spectral theory of fourth-order differential operators with regular boundary conditions, Arab. J. Math. 3 (1), 49-61, 2014.
  • [15] N.B. Kerimov and U. Kaya, Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions, Math. Methods Appl. Sci. 37 (5), 698-710, 2014.
  • [16] N.B. Kerimov and U. Kaya, Spectral properties of some regular boundary value problems for fourth order differential operators, Cent. Eur. J. Math. 11 (1), 94-111, 2013.
  • [17] N.B. Kerimov and K.R. Mamedov, On the Riesz basis property of the root functions in certain regular boundary value problems, Math. Notes 64 (4), 483-487, 1998.
  • [18] N.B. Kerimov, Unconditional basis property of a system of eigen and associated functions of a fourth-order differential operator, Dokl. Akad. Nauk 286, 803-808, 1986.
  • [19] G. Keselman, On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vyssh. Uchebn. Zaved. Mat. 2, 82-93, 1964.
  • [20] A.A. Kirac, Riesz basis property of the root functions of non-selfadjoint operators with regular boundary conditions, Int. J. Math. Anal. (Ruse) 3 (21-24), 1101-1109, 2009.
  • [21] A.S. Makin, On a class of boundary value problems for the SturmLiouville operator, Differ. Uravn. 35 (8), 1067-1076, 1999.
  • [22] A.S. Makin, Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions, Differ. Equ. 44 (5), 645-658, 2008.
  • [23] A.S. Makin, Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator, Differ. Equ. 44 (3), 341-348, 2008.
  • [24] A.S. Makin, Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math. 73, 71-76, 2006.
  • [25] A.S. Makin, On spectral decompositions corresponding to non-self-adjoint Sturm- Liouville operators, Dokl. Math. 73, 15-18, 2006.
  • [26] A.S. Makin, On the basis property of systems of root functions of regular boundary value problems for the Sturm-Liouville operator, Differ. Equ. 42 (12), 1717-1728, 2006.
  • [27] V. Mikhailov, On Riesz bases in $L_2(0, 1)$, Dokl. Akad. Nauk 144 (5), 981-984, 1962.
  • [28] M.A. Naimark, Linear Differential Operators, 2nd ed. Nauka, Moskow, 1969.
  • [29] A.A. Shkalikov, Bases formed by eigenfunctions of ordinary differential operators with integral boundary conditions, Moscow Univ. Math. Bull. 6, 12-21, 1982.
  • [30] A.A. Shkalikov and O.A. Veliev, On the Riesz basis property of the eigen and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes 85 (5-6), 647-660, 2009.
  • [31] O.A. Veliev, Asymptotic analysis of non-self-adjoint Hill operators, Open Math. J. 11 (12), 2234-2256, 2013.
  • [32] O.A. Veliev and M.T. Duman, The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, J. Math. Anal. Appl. 265 (1), 76-90, 2002.
  • [33] O.A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (1), 195-207, 2010.
  • [34] A. Zygmund, Trigonometric Series, volume 2, Cambridge University Press, Cambridge, 2002.

Basis properties of root functions of a regular fourth order boundary value problem

Year 2020, Volume: 49 Issue: 1, 338 - 351, 06.02.2020
https://doi.org/10.15672/hujms.552213

Abstract

In this paper, we consider the following boundary value problem

\[ y^{(4)}+q(x) y=\lambda y,~\ \ \ 0<x<1, \]

\[ y^{\prime\prime\prime}\left(1\right)-\left(-1\right)^{\sigma}y^{\prime\prime\prime}\left(0\right)+\alpha y\left(0\right) =0, \]

\[ y^{(s)}(1) -( -1) ^{\sigma}y^{(s) }( 0) =0,\ \ \ s=\overline{0,2}, \]

where $\lambda $ is a spectral parameter, $q( x)\in L_{1}(0,1)$ is complex-valued function and $\sigma =0,1$. The boundary conditions of this problem are regular but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established. When $\alpha\ne 0$, we proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a Riesz basis in the space $L_{2}( 0,1)$. Furthermore, we show that the system of root functions forms a basis in the space $L_{p}( 0,1)$, $1<p<\infty$ $(p\neq 2)$, under the conditions $\alpha\ne 0$ and $q( x) \in W_{1}^{1}( 0,1)$.

References

  • [1] N.K. Bari, Trigonometric Series (Russian), Fizmatgiz, Moscow, 1961. English transl.: N.K. Bary, A Treatise on Trigonometric Series. Vols. I, II, MacMillan, New York, 1964.
  • [2] G.D. Birkhoff, Boundary value and expansion problems of ordinary linear differential equations, Trans. Amer. Math. Soc. 9 (4), 373-395, 1908.
  • [3] N. Dernek and O. Veliev, On the Riesz basisness of the root functions of the nonselfadjoint Sturm-Liouville operator, Israel J. Math. 145 (1), 113-123, 2005.
  • [4] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomials as potentials, Dokl. Akad. Nauk. 436, 11-13, 2011.
  • [5] P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann. 351 (3), 509-540, 2011.
  • [6] P. Djakov and B. Mityagin, Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators, J. Funct. Anal. 263 (8), 2300-2332, 2012.
  • [7] P. Djakov and B. Mityagin, Instability zones of periodic 1-dimensional Schrodinger and Dirac operators, Uspekhi Mat. Nauk 61 (4), 77-182, 2006.
  • [8] N. Dunford, J.T. Schwartz, W.G. Bade and R.G. Bartle, Linear Operators, Wiley- Interscience, New York, 1971.
  • [9] F. Gesztesy and V. Tkachenko, A Schauder and Riesz basis criterion for non-selfadjoint Schrodinger operators with periodic and antiperiodic boundary conditions, J. Differential Equations 253 (2), 400-437, 2012.
  • [10] I. Gohberg and M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, 1969.
  • [11] H. Gunes, N.B. Kerimov and U. Kaya, Spectral properties of fourth order differential operators with periodic and antiperiodic boundary conditions, Results Math. 68 (3-4), 501-518, 2015.
  • [12] N. Ionkin and E. Moiceev, Solution of boundary value problem in heat conduction theory with nonlocal boundary conditions, Differ. Equ. 13 (2), 294-304, 1977.
  • [13] B.S. Kashin and A. Saakyan, Orthogonal Series, volume 75, American Mathematical Society, Providence, 2005.
  • [14] N.B. Kerimov and U. Kaya, Some problems of spectral theory of fourth-order differential operators with regular boundary conditions, Arab. J. Math. 3 (1), 49-61, 2014.
  • [15] N.B. Kerimov and U. Kaya, Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions, Math. Methods Appl. Sci. 37 (5), 698-710, 2014.
  • [16] N.B. Kerimov and U. Kaya, Spectral properties of some regular boundary value problems for fourth order differential operators, Cent. Eur. J. Math. 11 (1), 94-111, 2013.
  • [17] N.B. Kerimov and K.R. Mamedov, On the Riesz basis property of the root functions in certain regular boundary value problems, Math. Notes 64 (4), 483-487, 1998.
  • [18] N.B. Kerimov, Unconditional basis property of a system of eigen and associated functions of a fourth-order differential operator, Dokl. Akad. Nauk 286, 803-808, 1986.
  • [19] G. Keselman, On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vyssh. Uchebn. Zaved. Mat. 2, 82-93, 1964.
  • [20] A.A. Kirac, Riesz basis property of the root functions of non-selfadjoint operators with regular boundary conditions, Int. J. Math. Anal. (Ruse) 3 (21-24), 1101-1109, 2009.
  • [21] A.S. Makin, On a class of boundary value problems for the SturmLiouville operator, Differ. Uravn. 35 (8), 1067-1076, 1999.
  • [22] A.S. Makin, Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions, Differ. Equ. 44 (5), 645-658, 2008.
  • [23] A.S. Makin, Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator, Differ. Equ. 44 (3), 341-348, 2008.
  • [24] A.S. Makin, Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math. 73, 71-76, 2006.
  • [25] A.S. Makin, On spectral decompositions corresponding to non-self-adjoint Sturm- Liouville operators, Dokl. Math. 73, 15-18, 2006.
  • [26] A.S. Makin, On the basis property of systems of root functions of regular boundary value problems for the Sturm-Liouville operator, Differ. Equ. 42 (12), 1717-1728, 2006.
  • [27] V. Mikhailov, On Riesz bases in $L_2(0, 1)$, Dokl. Akad. Nauk 144 (5), 981-984, 1962.
  • [28] M.A. Naimark, Linear Differential Operators, 2nd ed. Nauka, Moskow, 1969.
  • [29] A.A. Shkalikov, Bases formed by eigenfunctions of ordinary differential operators with integral boundary conditions, Moscow Univ. Math. Bull. 6, 12-21, 1982.
  • [30] A.A. Shkalikov and O.A. Veliev, On the Riesz basis property of the eigen and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes 85 (5-6), 647-660, 2009.
  • [31] O.A. Veliev, Asymptotic analysis of non-self-adjoint Hill operators, Open Math. J. 11 (12), 2234-2256, 2013.
  • [32] O.A. Veliev and M.T. Duman, The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, J. Math. Anal. Appl. 265 (1), 76-90, 2002.
  • [33] O.A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (1), 195-207, 2010.
  • [34] A. Zygmund, Trigonometric Series, volume 2, Cambridge University Press, Cambridge, 2002.
There are 34 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Ufuk Kaya 0000-0003-1278-997X

Esma Kara Kuzu This is me 0000-0003-2692-1070

Publication Date February 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 1

Cite

APA Kaya, U., & Kara Kuzu, E. (2020). Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics, 49(1), 338-351. https://doi.org/10.15672/hujms.552213
AMA Kaya U, Kara Kuzu E. Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics. February 2020;49(1):338-351. doi:10.15672/hujms.552213
Chicago Kaya, Ufuk, and Esma Kara Kuzu. “Basis Properties of Root Functions of a Regular Fourth Order Boundary Value Problem”. Hacettepe Journal of Mathematics and Statistics 49, no. 1 (February 2020): 338-51. https://doi.org/10.15672/hujms.552213.
EndNote Kaya U, Kara Kuzu E (February 1, 2020) Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics 49 1 338–351.
IEEE U. Kaya and E. Kara Kuzu, “Basis properties of root functions of a regular fourth order boundary value problem”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, pp. 338–351, 2020, doi: 10.15672/hujms.552213.
ISNAD Kaya, Ufuk - Kara Kuzu, Esma. “Basis Properties of Root Functions of a Regular Fourth Order Boundary Value Problem”. Hacettepe Journal of Mathematics and Statistics 49/1 (February 2020), 338-351. https://doi.org/10.15672/hujms.552213.
JAMA Kaya U, Kara Kuzu E. Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics. 2020;49:338–351.
MLA Kaya, Ufuk and Esma Kara Kuzu. “Basis Properties of Root Functions of a Regular Fourth Order Boundary Value Problem”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 1, 2020, pp. 338-51, doi:10.15672/hujms.552213.
Vancouver Kaya U, Kara Kuzu E. Basis properties of root functions of a regular fourth order boundary value problem. Hacettepe Journal of Mathematics and Statistics. 2020;49(1):338-51.

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