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Completeness theorem for the dissipative Sturm-Liouville operator on bounded time scales

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Abstract

In this paper we consider a second-order Sturm-Liouville operator of the form

$$l(y): = - [p(t)y^\Delta (t)]^\nabla + q(t)y(t)$$

on bounded time scales. In this study, we construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self-adjoint and other extensions of the dissipative Sturm-Liouville operators in terms of boundary conditions. Using Krein’s theorem, we proved a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales.

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References

  1. R. P. Agarwal, M. Bohner and W. T. Li, Nonoscillation and oscillation theory for functional differential equations, Pure Appl. Math., Dekker, Florida, (2004).

    Google Scholar 

  2. D. R. Anderson, G. Sh. Guseinov and J. Hoffacker, Higher-order self-adjoint boundary-value problems on time scales, J. Comput. Appl. Math., 194(2) (2006), 309–342.

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Merdivenci Atici and G. Sh. Guseinov, On Green’s functions and positive solutions for boundary value problems on time scales, J. Comput. Appl. Math., 141(1-2) (2002), 75–99.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Bairamov and E. Ugurlu, The determinants of dissipative Sturm–Liouville operators with transmission conditions, Math. Comput. Model, 53 (2011), 805–813.

    Article  MathSciNet  MATH  Google Scholar 

  5. E. Bairamov and E. Ugurlu, Krein’s theorems for a dissipative boundary value transmission problem, Complex Anal. Oper. Theory, 7(4) (2013), 831–842.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Bohner and A. Peterson, Dynamic equations on time scales, Birkhäuser, Boston, (2001).

    Book  MATH  Google Scholar 

  7. M. Bohner and A. Peterson, (Eds.), Advances in dynamic equations on time scales, Birkhäuser, Boston, (2003).

    Book  MATH  Google Scholar 

  8. J. Eckhardt and G. Teschl, Sturm-Liouville operators with measure-valued coefficients, J. Anal. Math., 120 (2013), 151–224.

    Article  MathSciNet  MATH  Google Scholar 

  9. I. C. Gohberg and M.G. Krein, Introduction to the theory of linear nonselfadjoint operators, Amer. Math. Soc., Providence, (1969).

    Google Scholar 

  10. M. L. Gorbachuk and V. I. Gorbachuk, Boundary value problems for operator differential equations, Naukova Dumka, Kiev, (1984); English transl. Birkhauser Verlag, (1991).

    MATH  Google Scholar 

  11. G. Guseinov, Completeness theorem for the dissipative Sturm-Liouville operator, Doga-Tr. J. Math., 17 (1993), 48–54.

    MathSciNet  MATH  Google Scholar 

  12. G. Guseinov and H. Tuncay, The determinants of perturbation connected with a dissipative Sturm-Liouville operator, J. Math. Anal. Appl., 194 (1995), 39–49.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Guseinov, Completeness of the eigenvectors of a dissipative second order difference operator: dedicated to Lynn Erbe on the occasion of his 65th birthday, In honor of Professor Lynn Erbe, J. Difference Equ. Appl., 8(4) (2002), 321–331.

    Article  MathSciNet  Google Scholar 

  14. G. Sh. Guseinov, Self-adjoint boundary value problems on time scales and symmetric Green’s functions, Turkish J. Math., 29(4) (2005), 365–380.

    MathSciNet  MATH  Google Scholar 

  15. S. Hilger, Analysis on measure chains - a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Huseynov, Limit point and limit circle cases for dynamic equations on time scales, Hacet. J. Math. Stat., 39 (2010), 379–392.

    MathSciNet  MATH  Google Scholar 

  17. A. N. Kochubei, Extensions of symmetric operators and symmetric binary relations, Mat. Zametki, 17 (1975), 41–48; English transl. in Math. Notes, 17 (1975), 25-28.

    MathSciNet  Google Scholar 

  18. M. G. Krein, On the indeterminate case of the Sturm-Liouville boundary problem in the interval (0,8); 1Ş, Izv. Akad. Nauk SSSR Ser. Mat., 16(4) (1952), 293–324 (in Russian).

    MathSciNet  Google Scholar 

  19. V. Lakshmikantham, S. Sivasundaram and B. Kaymakcalan, Dynamic systems on measure chains, Kluwer Academic Publishers, Dordrecht, (1996).

    Book  MATH  Google Scholar 

  20. M. A. Naimark, Linear differential operators, 2nd edn., (1968), Nauka, Moscow, English transl. of 1st. edn., New York, 1,2, (1969).

    MATH  Google Scholar 

  21. B. P. Rynne, L 2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl., 328 (2007), 1217–1236.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Mehmet Akif Ersoy University, Burdur, Turkey

    Hüseyin Tuna

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  1. Hüseyin Tuna
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Tuna, H. Completeness theorem for the dissipative Sturm-Liouville operator on bounded time scales. Indian J Pure Appl Math 47, 535–544 (2016). https://doi.org/10.1007/s13226-016-0196-1

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  • DOI: https://doi.org/10.1007/s13226-016-0196-1

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