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On the Spectral Problem Arising in the Mathematical Model of Bending Vibrations of a Homogeneous Rod

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Abstract

In this paper we consider a spectral problem for ordinary differential equations of fourth order with spectral parameter in the boundary conditions. This problem describes the bending vibrations of a homogeneous rod, in cross-sections of which the longitudinal force acts, on the right end of which a mass is concentrated and on the left end a tracking force acts. We investigate the location of eigenvalues on the real axis, we study the structure of root spaces and oscillation properties of eigenfunctions and we obtain sufficient conditions for the subsystems of root functions of this problem to form a basis in \(L_p,\, 1< p < \infty \).

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References

  1. Aliyev, Z.S.: Basis properties of a fourth order differential operator with spectral parameter in the boundary condition. Cent. Eur. J. Math. 8(2), 378–388 (2010)

    Article  MathSciNet  Google Scholar 

  2. Aliyev, Z.S.: Basis properties in \(L_p\) of systems of root functions of a spectral problem with spectral parameter in a boundary condition. Differ. Equ. 47(6), 766–777 (2011)

    Article  MathSciNet  Google Scholar 

  3. Aliyev, Z.S.: On basis properties of root functions of a boundary value problem containing a spectral parameter in the boundary conditions. Dokl. Math. 87(2), 137–139 (2013)

    Article  MathSciNet  Google Scholar 

  4. Aliyev, Z.S., Dunyamalieva, A.A.: Defect basis property of a system of root functions of a Sturm-Liouville problem with spectral parameter in the boundary conditions. Differ. Equ. 51(10), 1249–1266 (2015)

    Article  MathSciNet  Google Scholar 

  5. Aliyev, Z.S., Dunyamalieva, A.A., Mehraliyev, Y.T.: Basis properties in \(L_p\) of root functions of Sturm–Liouville problem with spectral parameter-dependent boundary conditions. Mediterr. J. Math. 14(3), 1–23 (2017)

    Article  Google Scholar 

  6. Aliyev, Z.S., Guliyeva, S.B.: Properties of natural frequencies and harmonic bending vibrations of a rod at one end of which is concentrated inertial load. J. Differ. Equ. 263(9), 5830–5845 (2017)

    Article  MathSciNet  Google Scholar 

  7. Aliyev, Z.S., Kerimov, N.B.: Spectral properties of the differential operators of the fourth-order with eigenvalue parameter dependent boundary condition. Int. J. Math. Math. Sci. 2012, 456517 (2012)

  8. Azizov, T.Ya., Iokhvidov, I.S.: Linear Operators in Spaces with an Indefinite Metric. John Vielcy, Chichester (1989)

  9. Azizov, T.Ya., Iokhvidov, I.S.: Linear operators in Hilbert spaces with \(G\)-metric. Russ. Math. Surv. 26(4), 45–97 (1971)

    Article  Google Scholar 

  10. Azizov, T.Ya., Iokhvidov, I.S.: A test of whether the root vectors of a completely continuous \(J\)-self-adjoint operator in a Pontrjagin space \(\Pi _{\kappa }\) complete and form a basis. Math. Issled 6(1), 158–161 (1971). (in Russian)

  11. Amara, J., Ben, Vladimirov, A.A.: On a fourth-order problem with spectral and physical parameters in the boundary condition. Izv. Math. 68(4), 645–658 (2004)

    Article  MathSciNet  Google Scholar 

  12. Amara, J.B., Vladimirov, A.A.: On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions. J. Math. Sci. 150(5), 2317–2325 (2008)

    Article  MathSciNet  Google Scholar 

  13. Banks, D.O., Kurowski, G.J.: A Prufer transformation for the equation of a vibrating beam subject to axial forces. J. Differ. Equ. 24, 57–74 (1977)

    Article  MathSciNet  Google Scholar 

  14. Binding, P.A., Browne, P.J., K, Seddici: Sturm–Liouville problems with eigenparameter dependent boundary conditions. Proc. Edinb. Math. Soc. 37, 57–72 (1993)

    Article  MathSciNet  Google Scholar 

  15. Binding, P.A., Browne, P.J.: Application of two parameter eigencurves to Sturm-Liouville problems with eigenparameter dependent boundary conditions. Proc. Roy. Soc. Edinb. Sect. A(125), 1205–1218 (1995)

    Article  MathSciNet  Google Scholar 

  16. Bolotin, B.B.: Vibrations in Technique: Handbook in 6 Volumes, the Vibrations of Linear Systems, I. Engineering Industry, Moscow (1978)

    Google Scholar 

  17. Fulton, C.T.: Two-point boundary value problems with eigenvalue parameter in the boundary conditions. Proc. R. Soc. Edinb. Sect. A 77(3–4), 293–308 (1977)

    Article  MathSciNet  Google Scholar 

  18. Il’in, V.A.: Unconditional basis property on a closed interval of systems of eigen-and associated functions of a second-order differential operator. Dokl. Akad. Nauk SSSR 273(5), 1048–1053 (1983)

    MathSciNet  Google Scholar 

  19. Kapustin, N.Yu., Moiseev, E.I.: On the basis property in the space \(L_p\) of systems of eigenfunctions corresponding to two problems with spectral parameter in the boundary condition. Differ. Equ. 36(10), 1357–1360 (2000)

  20. Kapustin, N.Yu.: On a spectral problem arising in a mathematical model of torsional vibrations of a rod with pulleys at the ends. Differ. Equ. 41(10), 1490–1492 (2005)

    Article  MathSciNet  Google Scholar 

  21. Kerimov, N.B., Aliev, Z.S.: On oscillation properties of the eigenfunctions of a fourth order differential operator. Trans. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. 25(4), 63–76 (2005)

    MathSciNet  Google Scholar 

  22. Kerimov, N.B., Aliev, Z.S.: The oscillation properties of the boundary value problem with spectral parameter in the boundary condition. Trans. Acad. Sci. Azerb. Ser. Phys. Tech. 25(7), 61–68 (2005)

    MathSciNet  Google Scholar 

  23. Kerimov, N.B., Aliev, Z.S.: Basis properties of a spectral problem with spectral parameter in the boundary condition. Sb. Math. 197(10), 1467–1487 (2006)

    Article  MathSciNet  Google Scholar 

  24. Kerimov, N.B., Aliev, Z.S.: On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in a boundary condition. Differ. Equ. 43(7), 905–915 (2007)

    Article  MathSciNet  Google Scholar 

  25. Kerimov, N.B., Mirzoev, V.S.: On the basis properties of a spectral problem with spectral parameter in a boundary condition. Sib. Math. J. 44(5), 813–816 (2003)

    Article  MathSciNet  Google Scholar 

  26. Kerimov, N.B., Poladov, R.G.: Basis properties of the system of eigenfunctions in the Sturm–Liouville problem with a spectral parameter in the boundary conditions. Dokl. Math. 85(1), 8–13 (2012)

    Article  MathSciNet  Google Scholar 

  27. Meleshko, S.V., Pokornyi, Y.V.: On a vibrational boundary value problem. Differ. Uravn. 23(8), 1466–1467 (1987). (in Russian)

    MATH  Google Scholar 

  28. Moiseev, E.I., Kapustin, N.Yu.: On singularities of the root space of a spectral problem with spectral parameter in a boundary condition. Dokl. Math. 385(1), 20–24 (2002)

  29. Möller, M., Zlnsou, B.: Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions. Quaestion. Math. 34(3), 393–406 (2011)

    Article  MathSciNet  Google Scholar 

  30. Naimark, M.A.: Linear Differential Operators. Ungar, New York (1967)

    MATH  Google Scholar 

  31. Russakovskii, E.M.: Operator treatment of boundary problems with spectral parameters entering via polynomials in the boundary conditions. Funct. Anal. Appl. 9(4), 358–359 (1975)

    Article  Google Scholar 

  32. Shkalikov, A.A.: Boundary value problems for ordinary differential equations with a parameter in the boundary conditions. J. Sov. Math. 33, 1311–1342 (1986)

    Article  Google Scholar 

  33. Walter, J.: Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Math. Z. 133(4), 301–312 (1973)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are deeply grateful to the referee, whose comments and requests contributed to a significant improvement in the text and in the transparency of the obtained results.

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Authors and Affiliations

  1. Baku State University, Z. Khalilov str., 23, 1148, Baku, Azerbaijan

    Ziyatkhan S. Aliyev & Faiq M. Namazov

  2. Institute of Mathematics and Mechanics NAS of Azerbaijan, B. Vahabzadeh str., 9, 1141, Baku, Azerbaijan

    Ziyatkhan S. Aliyev

Authors
  1. Ziyatkhan S. Aliyev
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  2. Faiq M. Namazov
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Correspondence to Ziyatkhan S. Aliyev.

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Communicated by M. Langer.

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Aliyev, Z.S., Namazov, F.M. On the Spectral Problem Arising in the Mathematical Model of Bending Vibrations of a Homogeneous Rod. Complex Anal. Oper. Theory 13, 3675–3693 (2019). https://doi.org/10.1007/s11785-019-00924-z

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