The weak eigenfunctions of boundary-value problem with symmetric discontinuities. (English) Zbl 1508.34115
Summary: The main goal of this study is the investigation of discontinuous boundary-value problems for second-order differential operators with symmetric transmission conditions. We introduce the new notion of weak functions for such type of discontinuous boundary-value problems and develop an operator-theoretic method for the investigation of the spectrum and completeness property of the weak eigenfunction systems. In particular, we define some self-adjoint compact operators in suitable Sobolev spaces such that the considered problem can be reduced to an operator-pencil equation. The main result of this paper is that the spectrum is discrete and the set of eigenfunctions forms a Riesz basis of the suitable Hilbert space.
MSC:
| 34L10 | Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators |
| 34L15 | Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
| 34B24 | Sturm-Liouville theory |
| 34A36 | Discontinuous ordinary differential equations |
| 34L05 | General spectral theory of ordinary differential operators |
Keywords:
boundary value problems; transmission conditions; weak eigenfunctions; eigenvalue; completeness; Riesz basisReferences:
| [1] | O. Akcay, The representation of the solution of Sturm-Liouville equation with discontinuity conditions, Acta Math. Sci. Ser. B (Engl. Ed.) 38 (2018), no. 4, 1195-1213. · Zbl 1438.34108 |
| [2] | B. P. Allahverdiev and H. Tuna, Titchmarsh-Weyl theory for Dirac systems with transmission conditions, Mediterr. J. Math. 15 (2018), no. 4, Paper No. 151. · Zbl 1410.34085 |
| [3] | E. Bairamov and Ş. Solmaz, Spectrum and scattering function of the impulsive discrete Dirac systems, Turkish J. Math. 42 (2018), no. 6, 3182-3194. · Zbl 1438.39001 |
| [4] | B. P. Belinskiy and J. P. Dauer, On a regular Sturm-Liouville problem on a finite interval with the eigenvalue parameter appearing linearly in the boundary conditions, Spectral Theory and Computational Methods of Sturm-Liouville Problems (Knoxville 1996), Lecture Notes Pure Appl. Math. 191, Dekker, New York (1997), 183-196. · Zbl 0879.34035 |
| [5] | J. R. Cannon and G. H. Meyer, On a diffusion in a fractured medium, SIAM J. Appl. Math. 3 (1971), 434-448. · Zbl 0266.35002 |
| [6] | R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 2, Springer, Berlin, 1988. · Zbl 0664.47001 |
| [7] | C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 3-4, 293-308. · Zbl 0376.34008 |
| [8] | I. C. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr. 18, American Mathematical Society, Providence, 1969. · Zbl 0181.13504 |
| [9] | P. H. Hung and E. Sánchez-Palencia, Phénomènes de transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl. 47 (1974), 284-309. · Zbl 0286.35007 |
| [10] | M. Kandemir and O. S. Mukhtarov, Nonlocal Sturm-Liouville problems with integral terms in the boundary conditions, Electron. J. Differential Equations 2017 (2017), Paper No. 11. · Zbl 1358.34036 |
| [11] | M. V. Keldyš, On the characteristic values and characteristic functions of certain classes of non-self-adjoint equations, Doklady Akad. Nauk SSSR (N.S.) 77 (1951), 11-14. · Zbl 0045.39402 |
| [12] | N. B. Kerimov and R. G. Poladov, Basis properties of the system of eigenfunctions of the Sturm-Liouville problem with a spectral parameter in the boundary conditions, Dokl. Akad. Nauk 442 (2012), no. 1, 14-19. · Zbl 1246.34026 |
| [13] | E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1978. · Zbl 0368.46014 |
| [14] | O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Appl. Math. Sci. 49, Springer, New York, 1985. · Zbl 0588.35003 |
| [15] | M. R. Lancia and M. A. Vivaldi, On the regularity of the solutions for transmission problems, Adv. Math. Sci. Appl. 12 (2002), no. 1, 455-466. · Zbl 1136.31311 |
| [16] | O. Muhtarov and S. Yakubov, Problems for ordinary differential equations with transmission conditions, Appl. Anal. 81 (2002), no. 5, 1033-1064. · Zbl 1062.34094 |
| [17] | O. S. Mukhtarov and K. Aydemir, Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Math. Sci. Ser. B (Engl. Ed.) 35 (2015), no. 3, 639-649. · Zbl 1340.34333 |
| [18] | O. S. Mukhtarov and K. Aydemir, Basis properties of the eigenfunctions of two-interval Sturm-Liouville problems, Anal. Math. Phys. 9 (2019), no. 3, 1363-1382. · Zbl 1434.34074 |
| [19] | O. S. Mukhtarov and K. Aydemir, Discontinuous Sturm-Liouville problems involving an abstract linear operator, J. Appl. Anal. Comput. 10 (2020), no. 4, 1545-1560. · Zbl 1455.34028 |
| [20] | H. Olǧar and O. S. Mukhtarov, Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions, J. Math. Phys. 58 (2017), no. 4, Article ID 042201. · Zbl 1379.34081 |
| [21] | A. M. Sarsenbi and A. A. Tengaeva, On the basis properties of the root functions of two generalized spectral problems, Differ. Uravn. 48 (2012), no. 2, 294-296. · Zbl 1280.34084 |
| [22] | I. Stakgold, Boundary Value Problems of Mathematical Physics. Vol. II, The Macmillan, New York, 1971. · Zbl 0158.04801 |
| [23] | A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics, The Macmillan, New York, 1963. · Zbl 0111.29008 |
| [24] | H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Math. Libr. 18, North-Holland, Amsterdam, 1978. · Zbl 0387.46033 |
| [25] | E. Uğurlu, Regular third-order boundary value problems, Appl. Math. Comput. 343 (2019), 247-257. · Zbl 1428.34035 |
| [26] | J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary condition, Math. Z. 133 (1973), 301-312. · Zbl 0246.47058 |
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