Spectral Properties of Discontinuous Boundary Value Problem with Transmission Conditions and Manypoint Singularities

Yıl 2023, , 258 - 269, 31.12.2023

Tevhide Baltürk , Mustafa Kandemir

https://doi.org/10.47000/tjmcs.1138229

Öz

In this study, we investigate a discontinuous Sturm-Liouville boundary value problem on three intervals
with manypoint-transmission conditions in direct sum of Sobolev space. We establish such spectral properties as Fredholmness and coreciveness with respect to the eigenvalue parameter

Anahtar Kelimeler

Boundary-value problem, multipoint-transmission conditions, isomorphism, Fredholm, solvability

Kaynakça

• Agranovich, M.S., Spectral properties of diffraction problems The Generalized Method of Eigenoscillations in the Theory of Diffraction Theory, 1977 (in Russian: translated into English Wiley-VCH, Berlin), 1999.
• Akdoğan, Z., Yakar, A., Demirci, M., Discontinuous fractional Sturm-Liouville problems with transmission conditions, Applied Mathematics and Computation, , 350(2019), 1–10.
• Aliyev, Z.S., Basis properties of a fourth order differential operator with spectral parameter in the boundary condition, Open Mathematics, 8(2)(2010), 378–388.
• Aydemir, K., Boundary value problems with eigenvalue depending boundary and transmission conditions, Boundary Value Problems, 131(2014).
• Aydemir, K., Mukhtarov, O.Sh. Spectrum and Green’s function of a many-interval Sturm-Liouville problem, Z. Naturforsch., 70(5)(2015), 301–308.
• Besow, O.V., Il’in, V.P., Nikolskii, S.M., Integral Pepresentation of Functional and Embedding Theorems, 1, State New York, 1978.
• Borsuk, M., Transmission Problems for Elliptic Second-order Equations in Non-smooth Domains, Springer, Basel AG, 2010.
• Garcia-Huidobra, M., Gupta Chaitan, P., Manasevich, R., Some multipoint boundary value problems of Neumann-Dirichlet type involving a multipoint p-Laplace like operator, J. Math. Anal. Appl., 333(2007), 247–264.
• Imanbaev, N.S., Kanguzhin, B.E., Kalimbetov, B.T., On zeros of the characteristic determinant ofthe spectral problem for a third-order differential operator on a segment with nonlocal boundary conditions, Advances in Difference Equations, 110(2013).
• Kandemir, M., Mukhtarov, O. Sh., Yakubov, Ya. Irregular boundary value problems with discontinuous coefficients and the eigenvalue parameter, Mediterranean Journal of Mathematics, 6(2009), 317–338.
• Kandemir, M., Yakubov, Ya., Regular boundary value problems with a discontinuous coefficient, functional-multipoint conditions, and a linear spectral parameter, Israel Journal of Mathematics, 180(2010), 255–270.
• Kandemir, M., Irregular boundary value problems for elliptic differential-operator equations with discontinuous coefficients and transmission conditions, Kuwait Journal of Science and Engineering, 39(1A)(2012), 71–97.
• Kandemir, M., Mukhtarov, O. Sh., Nonlocal Sturm-Liouville problems with integral terms in the boundary condıtıons, Electronic Journal of Differential Equations, 11(2017), 1–12.
• Kandemir, M., Mukhtarov, O. Sh., Manypoint boundary value problems for elliptic differential-operator equations with interior singularities, Mediterr. J. Math., (2020), 17–35.
• Kato, T., Perturbation Theory for Linear Operators, Sipringer-Verlag, New York Inc., 1966.
• Likov, A.V., Mikhailov, Yu. A., The Theory of Heat and Mass Transfer, Qosenergaizdat, (Russian), 1963.
• Margareth, S.A., Rivera, J.E., Mauricio, S., Villagran, O.V., Transmission problem in thermoelasticity, Hindawi Pub. Cor. B.V.P., ID 190548(2011), 33.
• Mukhtarov, O. Sh., Discontinuous boundary value problem with spectral parameter in boundary conditions, Turkish J. Math., 18(2)(1994), 183–192.
• Mukhtarov, O. Sh., Yakubov, S., Problems for ordinary differential equations with transmission conditions, Applicable Analysis, 81(2002), 1033–1064.
• Mukhtarov, O. Sh., Aydemir, K., Eigenfunction expansion for Sturm-Liouville problems with transmission conditions at one interior point, Acta Mathematica Scienta, 35B(3)(2015), 639–649.
• Rasulov, M.L., Application of Contour Integral Method, (in Russian), Navka, Moskow, 1997.
• Sadybekov, M.A., Turmetov, B. Kh., Solvability of nonlocal boundary-value problems for the Laplace equatıon in the ball, Electronic Journal of Differential Equations, 157(2014), 1–14.
• Shakhmurov, V.B., Linear and nonlinear abstract elliptic equations with VMO coefficients and applications, Fixed point theory and applications, 6(2013), 1–21.
• Titeux, I., Yakubov, Ya., Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients, Math. Models and Methods in Applied Sciences, 7(1997), 1035–1050.
• Triebel, H., Interpolation Theory Function Spaces, Differential Operators, North Holland, Amsterdam, 1978.
• Voitovich, N.N., Katsenelbaum, B.Z., Sivov, A., Generalized Method of Eigen-vibration in the Theory of Diffraction, (Russian), Nauka, Moscow, 1997.
• Yakar, A., Akdoğan, Z., On the fundamental solutions of a discontinuous fractional boundary value problem, Adv. Differ. Equ., 378(2017).
• Yakubov, S., Yakubov, Ya., Differential-operator Equation Ordinary and Partial Differential Equation, Chapman and Hall/CRC, Boca Raton, London State New York Washington, State D. C., 1999.