Citation: | Bilender P. Allahverdiev, Hüseyin Tuna. SINGULAR DISCONTINUOUS HAMILTONIAN SYSTEMS[J]. Journal of Applied Analysis & Computation, 2022, 12(4): 1386-1402. doi: 10.11948/20210145 |
We study a discontinuous linear Hamiltonian system in the singular case. For these systems, the Titchmarsh-Weyl theory is established.
[1] | V. Ala and K. R. Mamedov, Basisness of eigenfunctions of a discontinuous Sturm-Liouville operator, J. Adv. Math. Stud., 2020, 13(1), 81-87. |
[2] | B. P. Allahverdiev and H. Tuna, Discontinuous linear Hamiltonian systems, Filomat, 2022, 36(3), 813-827. doi: 10.2298/FIL2203813A |
[3] | B. P. Allahverdiev and H. Tuna, Singular Hahn-Hamiltonian systems, Ufa Mathematical Journal, 2022 (In Press). |
[4] | F. V. Atkinson, Discrete and Continuous Boundary Problems, Acad. Press Inc., New York, 1964. |
[5] | K. Aydemir and O. S. Mukhtarov, Generalized Fourier series as Green's function expansion for multi-interval Sturm-Liouville systems, Mediterr. J. Math., 2017, 14(100), DOI: 10.1007/s00009-017-0901-2. |
[6] | D. Bainov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66, Longman Scientific & Technical, Harlow, 1993. |
[7] | E. Bairamov and E. Uǧurlu, On the characteristic values of the real component of a dissipative boundary value transmission problem, Appl. Math. Comput., 2012, 218, 9657-9663. |
[8] | H. Behncke and D. Hinton, Two singular point linear Hamiltonian systems with an interface condition, Math. Nachr., 2010, 283(3), 365-378. doi: 10.1002/mana.200910032 |
[9] | F. A. Cetinkaya and K. R. Mamedov, A boundary value problem with retarded argument and discontinuous coefficient in the differential equation, Azerb. J. Math., 2017, 7(1), 135-145. |
[10] | R. K. George, A. K. Nandakumaran and A. Arapostathis, A note on controllability of impulsive systems, J. Math. Anal. Appl., 2000, 241, 276-283. doi: 10.1006/jmaa.1999.6632 |
[11] | Z. Guan, G. Chen and T. Ueta, On impulsive control of a periodically forced chaotic pendulum system, IEEE Trans. Automat. Control, 2000, 45, 1724-1727. doi: 10.1109/9.880633 |
[12] | G. S. Guseinov, Boundary value problems for nonlinear impulsive Hamiltonian systems, J. Comput. Appl. Math., 2014, 259, 780-789 doi: 10.1016/j.cam.2013.06.034 |
[13] | G. S. Guseinov, On the impulsive boundary value problems for nonlinear Hamiltonian systems, Math. Meth. Appl. Sci., 2016, 39, 4496-4503. doi: 10.1002/mma.3877 |
[14] | D. B. Hinton and J. K. Shaw, On Titchmarsh-Weyl M(λ)-functions for linear Hamiltonian systems, J. Differ. Equat., 1981, 40(3), 316-342. doi: 10.1016/0022-0396(81)90002-4 |
[15] | D. B. Hinton and J. K. Shaw, Titchmarsh-Weyl theory for Hamiltonian systems, Spectral theory of differential operators, Birmingham, Ala. (1981), North-Holland Math. Stud., North-Holland, Amsterdam-New York, 1981, 55, 219-231. |
[16] | D. B. Hinton and J. K. Shaw, Parameterization of the M(λ) function for a Hamiltonian system of limit circle type, Proc. Roy. Soc. Edinburgh Sect. A, 1983, 93(3-4), 349-360. doi: 10.1017/S0308210500016036 |
[17] | D. B. Hinton and J. K. Shaw, Hamiltonian systems of limit point or limit circle type with both endpoints singular, J. Differ. Equat., 1983, 50, 444-464. doi: 10.1016/0022-0396(83)90071-2 |
[18] | A. M. Krall, Hilbert Space, Boundary Value Problems and Orthogonal Polynomials, Birkhäuser Verlag, Basel, 2002. |
[19] | A. M. Krall, M(λ) theory for singular Hamiltonian systems with one singular point, SIAM J. Math. Anal., 1989, 20(3), 664-700. doi: 10.1137/0520047 |
[20] | A. M. Krall, M(λ) theory for singular Hamiltonian systems with two singular points, SIAM J. Math. Anal., 1989, 20(3), 701-715. doi: 10.1137/0520048 |
[21] | F. R. Lapwood and T. Usami, Free Oscillations of the Earth, Cambridge University Press, Cambridge, 1981. |
[22] | A. Lakmeche and O. Arino, Bifurcation of nontrivial periodic solutions of impulsive differential equations arising from chemotherapeutic treatment, Dyn. Contin. Discrete Impuls. Syst., 2000, 7, 265-287. |
[23] | S. Lenci and G. Rega, Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos Solitons Fractals, 2000, 11, 2453-2472. doi: 10.1016/S0960-0779(00)00030-8 |
[24] | A. V. Likov and Y. A. Mikhailov, The Theory of Heat and Mass Transfer, Translated from Russian by I. Shechtman, Israel Program for Scientific Translations, Jerusalem, 1965. |
[25] | O. N. Litvinenko and V. I. Soshnikov, The Theory of Heterogenous Lines and their Applications in Radio Engineering, Radio, Moscow, 1964 (in Russian). |
[26] | K. R. Mamedov, Spectral expansion formula for a discontinuous Sturm-Liouville problem, Proc. Inst. Math. Mech., Natl. Acad. Sci. Azerb., 2014, 40, 275-282. |
[27] | K. R. Mamedov, On an inverse scattering problem for a discontinuous Sturm-Liouville equation with a spectral parameter in the boundary condition, Bound. Value Probl., 2010, Article ID: 171967, 1-17. |
[28] | K. R. Mamedov and N. Palamut, On a direct problem of scattering theory for a class of Sturm-Liouville operator with discontinuous coefficient, Proc. Jangjeon Math. Soc., 2009, 12(2), 243-251. |
[29] | O. S. Mukhtarov, Discontinuous boundary-value problem with spectral parameter in boundary conditions, Turkish J. Math., 1994, 18, 183-192. |
[30] | O. S. Mukhtarov and K. Aydemir, The eigenvalue problem with interaction conditions at one interior singular point, Filomat, 2017, 31(17), 5411-5420. doi: 10.2298/FIL1717411M |
[31] | O. S. Mukhtarov, H. Olǧar and K. Aydemir, Resolvent operator and spectrum of new type boundary value problems, Filomat, 2015, 29(7), 1671-1680. doi: 10.2298/FIL1507671M |
[32] | S. I. Nenov, Impulsive controllability and optimization problems in population dynamics, Nonlinear Anal., 1999, 36, 881-890. doi: 10.1016/S0362-546X(97)00627-5 |
[33] | H. Olǧar and O. S. Mukhtarov, Weak eigenfunctions of two-Interval Sturm-Liouville problems together with interaction conditions, J. Math. Phys., 2017, 58, 042201, DOI: 10.1063/1.4979615. |
[34] | Y. Yalcin, L. G. Sümer and S. Kurtulan, Discrete-time modeling of Hamiltonian systems, Turkish J. Electric. Eng. Comput. Sci., 2015, 23(1), 149-170. |