Araştırma Makalesi
Some Scattering and Spectral Properties of a Difference Equation with an Interface Condition and Hyperbolic Eigenparameter
Öz
The purpose aim of this study is to investigate the properties of scattering solutions
and the scattering function of an difference equation with the interface conditions and hyperbolic
parameter. We also investigate resolvent operator, Green function, continuous spectrum and the
set of the eigenvalues of this problem. Finally, we present an example to demonstrate the
application of our results. This work is important because the boundary condition is depend on
quadratic hyperbolic parameter. This difference provides a new perspective of the problem.
Anahtar Kelimeler
Scattering function Jost solution asymptotic equation hyperboic parameter resolvent operator continuous spectrum.
Kaynakça
- Adıvar, M., Bairamov, E., Difference equations of second order with spectral singularities, Journal of Mathematical Analysis and Applications, 277(2)(2003), 714–721.
- Adıvar, M., Bairamov, E., Spectral properties of non-selfadjoint difference operators, Journal of Mathematical Analysis and Applications, 261(2)(2001), 461–478.
- Agarwal, R.P., Difference Equations and Inequalities: Theory, Methods, and Applications, New York, Marcel Dekker, 2000.
- Agarwal, R.P., Wong, P.J.Y., Advanced topics in difference equations, Mathematics and Its Applications, vol. 404, Kluwer Academic Dordrecht, 1997(1997).
- Atkinson, F.V., Weiss, G.H., Discrete and Continuous Boundary Problems, American Institute of Physics, 1964.
- Aydemir, K., Olgar, H., Mukhtarov, O.S., Muhtarov, F.S., Differential operator equations with interface conditions in modified direct sum spaces, Filomat, 32(3)(2018), 921–931.
- Aygar, Y., Koprubası, T., A discrete boundary value problem with point interaction, Filomat, 36(18)(2022), 6279–6288.
- Bairamov, E., Cakar, O., Krall, A.M., Non–Selfadjoin difference operators and Jacobi matrices with spectral singularities, Mathematische Nachrichten, 229(1)(2001), 5–14.
- Bairamov, E., Cebesoy, S., Erdal, I., Difference equations with a point interaction, Mathematical Methods in the Applied Sciences, 42(16)(2019), 5498–5508.
- Bairamov, E., Erdal, I., Yardımcı, S., Spectral properties of an impulsive Sturm–Liouville operator, Journal of Inequalities and Applications, (2018), 1–16.
- Bairamov, E., Ugurlu, E., The determinants of dissipative Sturm–Liouville operators with transmission conditions, Mathematical and Computer Modelling, 55(2011), 805–813.
- Birkhoff, G.D., On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Transactions of the American Mathematical Society, 9(2)(1908), 219–231.
- Birkhoff, G.D., Boundary value and expansion problems of ordinary linear differential equations, Transactions of the American Mathematical Society, 9(4)(1908), 373–395.
- Case, K.M., On discrete inverse scattering problems II, Journal of Mathematical Physics, 14(7)(1973), 916–920.
- Glazman, I.A., Direct Methods of Qualitative Spectral Analysis of Singular Diferential Operators, Jerusalem, 1965.
- Guseinov, G.S., The inverse problem of scattering theory for a second-order difference equation on the whole axis, Doklady Akademii Nauk, 231(5)(1976), 1045–1048.
- Guseinov, G.S., On the impulsive boundary value problems for nonlinear Hamiltonian systems, Mathematical Methods in the Applied Sciences, 39(15)(2016), 4496–4503.
- Jiang, G., Lu, Q., Impulsive state feedback control of a predator–prey model, Journal of Computational and Applied Mathematics, 200(1)(2007), 193–207.
- Kelly, W.G., Peterson, Difference Equations, An Introduction with Applications, New York, Academic Press, 1991.
- Koprubası, T., A study of impulsive discrete Dirac system with hyperbolic eigenparameter, Turkish Journal of Mathematics, 45(1)(2021), 540–548.
- Lakshmikantham, V., Bainov, D.D., Simenov, P.S., Theory of Impulsive Differential Equations, Singapore, World Scientific, 1989.
- Leela, S., McRae, F.A., Sivasundaram, S., Controllability of impulsive differential equations, Journal of Mathematical Analysis and Applications, 177(1)(1993), 24–30.
- Lusternik, L.A., Sobolev, V.I., Elements of Functional Analysis, New York, Halsted Press, 1974.
- Mamedov, Kh.R., Col, A., On an inverse scattering problem for a class of Dirac operators with spectral parameter in the boundary condition, Journal of Mathematical Analysis and Applications, 393(2)(2012), 470–478.
- Mamedov, Kh.R., Kosar, N.P., Inverse scattering problem for Sturm–Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition, Mathematical methods in the applied sciences, 34(2)(2011), 231–241.
- Marchenko, V.A., Sturm-Liouville Operators And Applications, Birkhauser, Verlag, Basel, (1977.
- Mukhtarov, O.S., Aydemir, K., Two-linked periodic Sturm–Liouville problems with transmission conditions, Mathematical Methods in the Applied Sciences, 44(18)(2021), 14664–14676
- Mukhtarov, O.S., Kadakal, M., Muhtarov, F.S., On discontinuous Sturm-Liouville problems with transmission conditions, Journal of Mathematics of Kyoto University, 44(4)(2004), 779–798.
- Mukhtarov, O.S., Olgar, H., Aydemir, K., Generalized eigenfunctions of one Sturm-Liouville system with symmetric jump conditions, AIP Conference Proceedings, 1726(1)(2016), 020086.
- Mukhtarov, O.S., Tunc, E., Eigenvalue problems for Sturm Liouville equations with transmission conditions, Israel Journal of Mathematics, 144(2004), 367–380.
- Naimark, M.A., Linear Differential Operators 2, Ungar, New York, 1968.
- Nenov, S.I., Impulsive controllability and optimization problems in population dynamics, Nonlinear Analysis: Theory, Methods and Applications, 36(7)(1999), 881–890.
- Oznur, G.B., Aygar, Y., Aral, N.D., An examination of boundary value transmission problem with quadratic spectral parameter, Quaestiones Mathematicae, 46(5)(2023), 871–885.
- Oznur, G.B., Bairamov, E., Scattering theory of the quadratic eigenparameter depending impulsive Sturm-Liouville equations, Turkish Journal of Mathematics, 46(2)(2022), 406–415.
- Oznur, G.B., Spectral singularities of an impulsive Sturm-Liouville operators, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4)(2022), 1080–1094.
Yıl 2024,
Cilt: 16 Sayı: 2, 299 - 308, 31.12.2024
Öz
Kaynakça
- Adıvar, M., Bairamov, E., Difference equations of second order with spectral singularities, Journal of Mathematical Analysis and Applications, 277(2)(2003), 714–721.
- Adıvar, M., Bairamov, E., Spectral properties of non-selfadjoint difference operators, Journal of Mathematical Analysis and Applications, 261(2)(2001), 461–478.
- Agarwal, R.P., Difference Equations and Inequalities: Theory, Methods, and Applications, New York, Marcel Dekker, 2000.
- Agarwal, R.P., Wong, P.J.Y., Advanced topics in difference equations, Mathematics and Its Applications, vol. 404, Kluwer Academic Dordrecht, 1997(1997).
- Atkinson, F.V., Weiss, G.H., Discrete and Continuous Boundary Problems, American Institute of Physics, 1964.
- Aydemir, K., Olgar, H., Mukhtarov, O.S., Muhtarov, F.S., Differential operator equations with interface conditions in modified direct sum spaces, Filomat, 32(3)(2018), 921–931.
- Aygar, Y., Koprubası, T., A discrete boundary value problem with point interaction, Filomat, 36(18)(2022), 6279–6288.
- Bairamov, E., Cakar, O., Krall, A.M., Non–Selfadjoin difference operators and Jacobi matrices with spectral singularities, Mathematische Nachrichten, 229(1)(2001), 5–14.
- Bairamov, E., Cebesoy, S., Erdal, I., Difference equations with a point interaction, Mathematical Methods in the Applied Sciences, 42(16)(2019), 5498–5508.
- Bairamov, E., Erdal, I., Yardımcı, S., Spectral properties of an impulsive Sturm–Liouville operator, Journal of Inequalities and Applications, (2018), 1–16.
- Bairamov, E., Ugurlu, E., The determinants of dissipative Sturm–Liouville operators with transmission conditions, Mathematical and Computer Modelling, 55(2011), 805–813.
- Birkhoff, G.D., On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Transactions of the American Mathematical Society, 9(2)(1908), 219–231.
- Birkhoff, G.D., Boundary value and expansion problems of ordinary linear differential equations, Transactions of the American Mathematical Society, 9(4)(1908), 373–395.
- Case, K.M., On discrete inverse scattering problems II, Journal of Mathematical Physics, 14(7)(1973), 916–920.
- Glazman, I.A., Direct Methods of Qualitative Spectral Analysis of Singular Diferential Operators, Jerusalem, 1965.
- Guseinov, G.S., The inverse problem of scattering theory for a second-order difference equation on the whole axis, Doklady Akademii Nauk, 231(5)(1976), 1045–1048.
- Guseinov, G.S., On the impulsive boundary value problems for nonlinear Hamiltonian systems, Mathematical Methods in the Applied Sciences, 39(15)(2016), 4496–4503.
- Jiang, G., Lu, Q., Impulsive state feedback control of a predator–prey model, Journal of Computational and Applied Mathematics, 200(1)(2007), 193–207.
- Kelly, W.G., Peterson, Difference Equations, An Introduction with Applications, New York, Academic Press, 1991.
- Koprubası, T., A study of impulsive discrete Dirac system with hyperbolic eigenparameter, Turkish Journal of Mathematics, 45(1)(2021), 540–548.
- Lakshmikantham, V., Bainov, D.D., Simenov, P.S., Theory of Impulsive Differential Equations, Singapore, World Scientific, 1989.
- Leela, S., McRae, F.A., Sivasundaram, S., Controllability of impulsive differential equations, Journal of Mathematical Analysis and Applications, 177(1)(1993), 24–30.
- Lusternik, L.A., Sobolev, V.I., Elements of Functional Analysis, New York, Halsted Press, 1974.
- Mamedov, Kh.R., Col, A., On an inverse scattering problem for a class of Dirac operators with spectral parameter in the boundary condition, Journal of Mathematical Analysis and Applications, 393(2)(2012), 470–478.
- Mamedov, Kh.R., Kosar, N.P., Inverse scattering problem for Sturm–Liouville operator with nonlinear dependence on the spectral parameter in the boundary condition, Mathematical methods in the applied sciences, 34(2)(2011), 231–241.
- Marchenko, V.A., Sturm-Liouville Operators And Applications, Birkhauser, Verlag, Basel, (1977.
- Mukhtarov, O.S., Aydemir, K., Two-linked periodic Sturm–Liouville problems with transmission conditions, Mathematical Methods in the Applied Sciences, 44(18)(2021), 14664–14676
- Mukhtarov, O.S., Kadakal, M., Muhtarov, F.S., On discontinuous Sturm-Liouville problems with transmission conditions, Journal of Mathematics of Kyoto University, 44(4)(2004), 779–798.
- Mukhtarov, O.S., Olgar, H., Aydemir, K., Generalized eigenfunctions of one Sturm-Liouville system with symmetric jump conditions, AIP Conference Proceedings, 1726(1)(2016), 020086.
- Mukhtarov, O.S., Tunc, E., Eigenvalue problems for Sturm Liouville equations with transmission conditions, Israel Journal of Mathematics, 144(2004), 367–380.
- Naimark, M.A., Linear Differential Operators 2, Ungar, New York, 1968.
- Nenov, S.I., Impulsive controllability and optimization problems in population dynamics, Nonlinear Analysis: Theory, Methods and Applications, 36(7)(1999), 881–890.
- Oznur, G.B., Aygar, Y., Aral, N.D., An examination of boundary value transmission problem with quadratic spectral parameter, Quaestiones Mathematicae, 46(5)(2023), 871–885.
- Oznur, G.B., Bairamov, E., Scattering theory of the quadratic eigenparameter depending impulsive Sturm-Liouville equations, Turkish Journal of Mathematics, 46(2)(2022), 406–415.
- Oznur, G.B., Spectral singularities of an impulsive Sturm-Liouville operators, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 71(4)(2022), 1080–1094.
Toplam 35 adet kaynakça vardır.
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Konular | Matematiksel Yöntemler ve Özel Fonksiyonlar |
| Bölüm | Makaleler |
| Yazarlar | |
| Yayımlanma Tarihi | 31 Aralık 2024 |
| Gönderilme Tarihi | 16 Şubat 2024 |
| Kabul Tarihi | 10 Temmuz 2024 |
| Yayımlandığı Sayı | Yıl 2024 Cilt: 16 Sayı: 2 |