Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems

Hayati Olğar

doi:10.53570/jnt.1799207

Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems

Abstract

In this paper, some spectral properties of the eigenvalues and generalized eigenfunctions of a many-interval boundary-value-jump problem (MIBVJP), which consists of a second-order differential equation defined on a finite number of disjoint intervals under supplementary impulsive conditions and $\lambda$-dependent boundary conditions. Using the theory of operator polynomials in Sobolev space and suitable integral transformations, the basis property of generalized eigenfunctions for the MIBVJP is obtained, and positive definiteness and self-adjointness of the operator polynomial are involved.

Keywords

References

B. P. Belinskiy, J. W. Hiestand, J. V. Matthews, Piecewise uniform optimal design of a bar with an attached mass, Electronic Journal of Differential Equations 2015 (206) (2015) 1--17.
S. Gwak, J. Kim, S. J. Rey, Massless and massive higher spins from anti-de Sitter space waveguide, Journal of High Energy Physics 2016 (2016) 24.
G. Kaoullas, G. C. Georgiou, Start-up and cessation newtonian Poiseuille and Couette flows with dynamic wall slip, Meccanica 50 (2015) 1747--1760.
A. Kawano, A. Morassi, R. Zaera, Detecting a prey in a spider orb-web from in-plane vibration, SIAM Journal on Applied Mathematics 81 (6) (2021) 2297--2322.
Y. Nie, V. Linetsky, Sticky reflecting Ornstein-Uhlenbeck diffusions and the Vasicek interest rate model with the sticky zero lower bound, Stochastic Models 36 (1) (2020) 1--19.
A. Parra-Rodriguez, E. Rico, E. Solano, I. L. Egusquiza, Quantum networks in divergence-free circuit QED, Quantum Science and Technology 3 (2) (2018) 024012.
E. Celik, H. Tunc, M. Sari, An efficient multi-derivative numerical method for chemical boundary value problems, Journal of Mathematical Chemistry 62 (2024) 634--653.
M. K. Kadalbajoo, V. Kumar, B-spline method for a class of singular two-point boundary value problems using optimal grid, Applied Mathematics and Computation 188 (2) (2007) 1856--1869.

P. Roul, U. Warbhe, A novel numerical approach and its convergence for numerical solutions of nonlinear doubly singular boundary value problems, Journal of Computational and Applied Mathematics 226 (2016) 661--676.
P. A. Binding, P. J. Browne, B. A. Watson, Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter I, Proceedings of the Edinburgh Mathematical Society 45 (3) (2002) 631--645.
C. T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 77 (3-4) (1977) 293--308.
B. P. Allahverdiev, H. Tuna, On the resolvent of singular Sturm‐Liouville operators with transmission conditions, Mathematical Methods in the Applied Sciences 43 (7) (2020) 4286--4302.
K. Aydemir, O. Mukhtarov, A new type Sturm-Liouville problem with an abstract linear operator contained in the equation, Quaestiones Mathematicae 45 (12) (2022) 1931--1948.
E. Bairamov, K. Tas, E. Uğurlu, On the characteristic functions and Dirichlet-integrable solutions of singular left-definite Hamiltonian systems, Quaestiones Mathematicae 47 (5) (2024) 983--995.
O. Mukhtarov, M. Yücel, K. Aydemir, A new generalization of the differential transform method for solving boundary value problems, Journal of New Results in Science 10 (2) (2021) 49--58.
O. Mukhtarov, K. Aydemir, Two-linked periodic Sturm–Liouville problems with transmission conditions, Mathematical Methods in the Applied Sciences 44 (18) (2021) 14664--14676.
G. B. Oznur, Y. Aygar, N. D. Aral, An examination of boundary value transmission problem with quadratic spectral parameter, Quaestiones Mathematicae 46 (5) (2023) 871--885.
B. P. Belinskiy, J. P. Dauer, Eigenoscillations of mechanical systems with boundary conditions containing the frequency, Quarterly of Applied Mathematics 56 (3) (1998) 521--541.
M. G. Krein, G. K. Langer, Certain mathematical principles of the theory of damped vibrations of continua, Proceedings of the International Symposium on the Application of the Theory of Functions of a Complex Variable to Continuum Mechanics, Moscow, 1965, pp. 283--322.
A. G. Kostyuchenko, A. A. Shkalikov, Self-adjoint quadratic operator pencils and elliptic problems, Functional Analysis and Its Applications 17 (1983) 109--128.
O. A. Ladyzhenskaia, The boundary value problems of mathematical physics, Springer-Verlag, 1985.
M. V. Keldysh, On eigenvalues and eigenfunctions of some classes of non-self-adjoint equations (in Russian), Doklady Akademii Nauk 77 (1951) 11--14.
A. S. Markus, Introduction to the spectral theory of polynomial pencils, Translation of Mathematical Monographs, American Mathematical Society, 1988.
H. Langer, Factorization of operator pencils, Acta Scientiarum Mathematicarum 38 (1976) 83--96.
H. Olğar, O. Mukhtarov, Weak eigenfunctions of two-interval Sturm-Liouville problems together with interaction conditions, Journal of Mathematical Physics 58 (4) (2017) 042201.
Z. Li, Z. Zheng, J. Qin, The basis property of weak eigenfunctions for Sturm-Liouville problem with boundary conditions dependent rationally on the eigenparameter, Journal of Applied Analysis and Computation 14 (1) (2024) 424--435.
I. C. Gohberg, M. G. Krein, Introduction to the theory of linear non-selfadjoint operators, Translation of mathematical monographs, American Mathematical Society, 1969.

Details

Primary Language

English

Subjects

Operator Algebras and Functional Analysis, Pure Mathematics (Other)

Journal Section

Research Article

Authors

Hayati Olğar ^*
0000-0003-4732-1605
Türkiye

Publication Date

December 31, 2025

Submission Date

October 8, 2025

Acceptance Date

December 4, 2025

Published in Issue

Year 2025 Number: 53

DOI

https://doi.org/10.53570/jnt.1799207

IZ

https://izlik.org/JA64NZ29NZ

Cite

RIS / Bibtex

APA

Olğar, H. (2025). Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems. Journal of New Theory, 53, 77-86. https://doi.org/10.53570/jnt.1799207

AMA

1.Olğar H. Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems. JNT. 2025;(53):77-86. doi:10.53570/jnt.1799207

Chicago

Olğar, Hayati. 2025. “Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems”. Journal of New Theory, nos. 53: 77-86. https://doi.org/10.53570/jnt.1799207.

EndNote

Olğar H (December 1, 2025) Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems. Journal of New Theory 53 77–86.

IEEE

[1]H. Olğar, “Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems”, JNT, no. 53, pp. 77–86, Dec. 2025, doi: 10.53570/jnt.1799207.

ISNAD

Olğar, Hayati. “Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems”. Journal of New Theory. 53 (December 1, 2025): 77-86. https://doi.org/10.53570/jnt.1799207.

JAMA

1.Olğar H. Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems. JNT. 2025;:77–86.

MLA

Olğar, Hayati. “Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems”. Journal of New Theory, no. 53, Dec. 2025, pp. 77-86, doi:10.53570/jnt.1799207.

Vancouver

1.Hayati Olğar. Weak Solutions and Riesz Expansion for Impulsive Boundary Value Problems. JNT. 2025 Dec. 1;(53):77-86. doi:10.53570/jnt.1799207