On Spectral Properties of Discontinuous Differential Operator with Second Order

Abstract

In this work, we consider the spectral problem for a second-order discontinuous differential operator with a spectral parameter in the boundary condition in $L_p,  1<p<\infty$. We study a method for establishing the basicity of eigenfunctions for such a problem.  Such spectral problems arise while one solves the problem of a loaded string fixed at both ends with a load placed in the between ends of the string by the Fourier method.

Keywords

• spectral problem,eigenfucntions,completeness,minimality

Kaynakça

1. [1] D .L. Russel, On exponential bases for the Sobolev spaces over an interval ,J. Math. Anal. Appl., 87(2) (1982), 528-550.
2. [2] Z. G. Huseynov, A.M. Shykhammedov, On bases of sines and cosines in Sobolev spaces, Appl. Math. Lett., 25(3) (2012), 275-278.
3. [3] B. T. Bilalov, T. B. Gasymov, On basicity of a part of a system with infinite defect, Trans. NAS Azerb.,28(7) (2007), 53-59.
4. [4] T. B. Gasymov, On necessary and sufficient conditions of basicity of some defective systems in Banach space, Trans. NAS Azerb., 26(1)(2006), 65-70.
5. [5] B. T. Bilalov, Bases of exponentials, sines, and cosines, Differ. Uravn., 39(5)(2003), 619-622.
6. [6] B. T. Bilalov, T.R. Muradov, Defect bases of Banach spaces, Proc. IMM NASA., 22(30) (2005), 23-26.
7. [7] X. He, H. Volkmer, Riesz bases of solutions of Sturm–Liouville equations, J. Fourier Anal. Appl., 7(3) (2001), 297-307.
8. [8] A. A. Huseynli, On the stability of basisness in Lp(1 < p < 1of cosines and sines, Turk. J. Math., 35(1) (2011), 47-54.

9. [9] A. N. Tichonoff, A. A. Samarskii , Equations of mathematical physics, M. 1977. 736 p. (Russian).
10. [10] F.V. Atkinson, Discrete and Continuous Boundary Problems, Moscow, Mir (1968).
11. [11] A.N. Tikhonov, Samarskii, A.A., Equations of Mathematical Physics, Mosk. Gos. Univ., Moscow (1999); Dover, New York (2011).
12. [12] T. Iwaniec, C. Sbordone, On the integrability of the Jacobian under minimal hypothesis, Arch. Ration. Mech. Anal., 119 (1992), 129–143.
13. [13] L. Boccardo, Quelques problemes de Dirichlet avec donnees dans de grands espaces de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 325(1997), 1269–1272.
14. [14] C. Capone, A. Fiorenza, G. E. Karadzhov, Grand Orlicz spaces and global integrability of the Jacobian, Math. Scand., 102 (2008), 131–148.
15. [15] N. Fusco, P. L. Lions, C. Sbordone, Sobolev imbedding theorems in borderline cases, Proc. Amer. Math. Soc., 124(2) (1996), 561 – 565.
16. [16] L. Greco, T. Iwaniec, C. Sbordone, Inverting the p-harmonic operator, Manuscripta Math., 92(2)(1997), 249 – 258.
17. [17] T. Iwaniec, P. Koskela, J. Onninen, Mappings of finite distortion: Monotonicity and Continuity, Invent. Math., 144 (2001), 507 – 531.
18. [18] C. Sbordone, Grand Sobolev spaces and their applications to variational problems, Le Matematiche, 2(51) (1996), 335 – 347.
19. [19] B. Stroffolini, A stability result for p-harmonic systems with discontinuous coefficients, Electronic Journal of Diff. Equations, 2 (2001), 1 – 7.
20. [20] A. Fiorenza, B. Gupta, P. Jain, Themaximal theorem for weighted grand Lebesgue spaces, Studia Math., 188(2) (2008), 123–133.
21. [21] V. Kokilashvili, A. Meskhi, Trace inequalities for integral operators with fractional order in grand Lebesgue spaces, Studia Math., 210 (2012), 159–176.
22. [22] A. Fiorenza, G. E. Karadzhov, Grand and small Lebesgue spaces and their analogues, Z. Anal. Anwen., , 23(4) (2004), 657–681.
23. [23] A. Fiorenza, A. Mercaldo, J.M. Rakotoson, Regularity and comparison result in grand Sobolev spaces for parabolic equations with measure data, Appl. Math. Lett., 14(8) (2001), 979–981.
24. [24] A. Fiorenza, A. Mercaldo, J. M. Rakotoson, Regularity and uniqueness results in grand Sabolev spaces for parabolic equations with measure data, Discrete Contin. Dyn. Syst., 8(4) (2003), 893–906.
25. [25] A. Fiorenza, Duality and reflexivity in grand Lebesgue spaces, Collect. Math., 51(2) (2000), 131–148.
26. [26] A. Fiorenza, B. Gupta, P. Jain, The maximal theorem for weighted grand Lebesgue spaces, Studia Math., 188(2) (2008), 123–133.
27. [27] A. Fiorenza, j. M. Rakotoson, On small Lebesgue spaces and their applications, Comptes Rendus Math., 334(1) (2002), 23–26.
28. [28] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Nonstandard Function Spaces: Variable Exponent Lebesgue and Amalgam Spaces, vol. 1. Springer, Heidelberg, 2016.
29. [29] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Nonstandard Function Spaces: Variable Exponent H´’older, Morrey–Campanato and Grand Spaces, vol. 2. Springer, Heidelberg, 2016.
30. [30] R. E. Castillo, H. Rafeiro, An introductory course in Lebesgue spaces, Springer, Switzerland, 2016.
31. [31] B. T. Bilalov, T. B. Gasymov, On basicity of eigenfunctions of second-order discontinuous differential operator, Ufa Mathematical J., (2017), 9.1.
32. [32] B. Muckenhoupt, B., Hardy’s Inequality with Weights, Studia Math., 44 (1972), 31–38.
33. [33] Gasymov, B. Telman, J. M. Shakhrizad, On the convergence of spectral expansions for one discontinuous problem with spectral parameter in the boundary condition, Trans. NAS Azerb., 26(4) (2006), 103-116.
34. [34] V. Kokilashvili, M. Alexander, A note on the boundedness of the Hilbert transform in weighted grand Lebesgue spaces, Georgian Mathematical J., 16(3) (2009), 547-551. 50