Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article A Left-Definite Non-Integer-Order Dissipative Operator(Springer Nature, 2026) Ugurlu, EkinIn this paper we consider a non-integer (fractional)-order nonselfadjoint boundary-value problem so that the fractional-order equation is a kind of left-definite equation. We construct a dissipative operator in a Sobolev space H-1(a,b) and we introduce several results on the spectral properties of the related operators. In particular, we construct an inverse operator with the aid of the Dirac-delta function and we apply Krein's theorem to the inverse operator which is compact having a nuclear imaginary component.Article Citation - WoS: 9Citation - Scopus: 8Scattering and Spectral Problems of the Direct Sum Sturm-Liouville Operators(Ministry Communications & High Technologies Republic Azerbaijan, 2017) Allahverdiev, Bilender P.; Uğurlu, Ekin; Ugurlu, Ekin; MatematikIn this paper a space of boundary values is constructed for direct sum minimal symmetric Sturm-Liouville operators and description of all maximal dissipative, maximal accumulative, selfadjoint and other extensions of such a symmetric operator is given in terms of boundary conditions. We construct a selfadjoint dilation of dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function. We prove a theorem on completeness of the system of eigenfunctions and associated functions of the dissipative operators.Article On Singular Fifth-Order Boundary Value Problems With Deficiency Indices (5, 5)(Math Soc Serbia-drustvo Matematicara Srbije, 2022) Uğurlu, Ekin; Ugurlu, Ekin; Tas, Kenan; Taş, Kenan; MatematikThis paper is devoted to introduce a way of construction of the well-defined boundary conditions for the solutions of a singular fifth-order equation with deficiency indices (5, 5). Imposing suitable separated and coupled boundary conditions some properties of the eigenvalues of the problems have been investigated.Article Citation - WoS: 2Citation - Scopus: 1Dependence of Eigenvalues of Some Boundary Value Problems(Tsing Hua Univ, dept Mathematics, 2021) Uğurlu, Ekin; Ugurlu, Ekin; Tas, Kenan; Taş, Kenan; MatematikIn this work we deal with a system of two first-order differential equations containing the same eigenvalue parameter. We consider some suitable separated real and complex coupled boundary conditions, and show that the eigenvalues generated by this system are continuous in an eigenvalue branch. Also we introduce the ordinary and Frechet derivatives of these eigenvalues with respect to some elements of the data.Article Citation - WoS: 1Citation - Scopus: 1Left-Definite Fractional Hamiltonian Systems: Titchmarsh-Weyl Theory(Pergamon-Elsevier Science Ltd, 2025) Ugurlu, EkinHamiltonian systems are useful when formally symmetric boundary value problems generated by ordinary derivatives are considered. However, if the ordinary derivatives are changed by non-integer-order (fractional) derivatives, it is not easy to investigate the corresponding problems. In this paper, we introduce a systematic approach to dealing with fractional boundary value problems by constructing a fractional Hamiltonian system. In particular, we consider a left-definite system, and we construct nested-circles theory (Weyl theory) for this system of equations. Using the Titchmarsh-Weyl function, we prove that at least r-solutions of the 2r-dimensional system of equations should be Dirichlet-integrable on a given interval.Article Variational Approach To a Symmetric Boundary Value Problem Generated by a System of Equations and Separated Boundary Conditions(Wiley, 2024) Ugurlu, EkinThis work provides some information on the eigenvalues and eigenfunctions of a problem which is constructed by a system of equations and symmetric boundary conditions that includes the ordinary second-order Sturm-Liouville boundary value problem. In particular, we show that the problem has an infinite number of discrete eigenvalues with a greatest lower bound and the corresponding eigenfunctions are complete in mean and energy. We introduce the results using the variational approach that enables us to consider only continuous pair functions instead of absolutely continuous pair functions.Article Citation - WoS: 2Citation - Scopus: 2Fractional Hamiltonian Systems: Nested Ellipsoids(Pergamon-elsevier Science Ltd, 2025) Ugurlu, EkinIn this paper, we introduce a singular fractional-order Hamiltonian system with several spectral parameters. Using the inertia indices of the corresponding Hermitian forms we provide a lower bound for the number of linearly independent integrable-square solutions. Moreover, we introduce the Titchmarsh-Weyl function together with an intermediate theorem on the number of the integrable-square solutions. At the end of the paper, we show that 2-sequential and 4-sequential scalar fractional-order differential equations can be embedded into such Hamiltonian systems.Article Citation - WoS: 3Citation - Scopus: 3On the Characteristic Functions and Dirchlet-Integrable Solutions of Singular Left-Definite Hamiltonian Systems(Taylor & Francis Ltd, 2024) Ugurlu, Ekin; Bairamov, Elgiz; Tas, KenanIn this work, a singular left-definite Hamiltonian system is considered and the characteristic-matrix theory for this Hamiltonian system is constructed. Using the results of this theory we introduce a lower bound for the number of Dirichlet-integrable solutions. Moreover we share a relation between the kernel of the solution of the nonhomogeneous boundary value problem and the characteristic-matrix.Article Citation - WoS: 8Citation - Scopus: 8On Some Even-Sequential Fractional Boundary-Value Problems(Springernature, 2024) Ugurlu, EkinIn this paper we provide a way to handle some symmetric fractional boundary-value problems. Indeed, first, we consider some system of fractional equations. We introduce the existence and uniqueness of solutions of the systems of equations and we show that they are entire functions of the spectral parameter. In particular, we show that the solutions are at most of order 1/2. Moreover we share the integration by parts rule for vector-valued functions that enables us to obtain some symmetric equations. These symmetries allow us to handle 2-sequential and 4-sequential fractional boundary-value problems. We provide some expansion formulas for the bilinear forms of the solutions of 2-sequential and 4-sequential fractional equations which admit us to impose some unusual boundary conditions for the solutions of fractional differential equations. We show that the systems of eigenfunctions of 2-sequential and 4-sequential fractional boundary value problems are complete in both energy and mean. Furthermore, we study on the zeros of solutions of 2-sequential fractional differential equations. At the end of the paper we show that 6-sequential fractional differential equation can also be handled as a system of equations and hence almost all the results obtained in the paper can be carried for such boundary-value problems.Article Left-Definite System of First-Order Equations Together With Eigenparameter-Dependent Boundary Conditions(Wiley, 2024) Ugurlu, EkinThis paper provides some information on the eigenvalues and eigenfunctions of some left-definite system of first-order differential equations subject to eigenparameter-dependent boundary conditions. Namely, we show that the pair of solutions of the system of equations satisfying some initial conditions exists and is unique, and this pair is analytic in the spectral parameter of order 1/2. We also introduce Lagrange's formula for the left-definite equation. Using some Prufer angels, we investigate oscillation of zeros of eigenfunctions and asymptotics equations for the eigenvalues of the problem. Moreover, we share some ordinary and Frechet derivatives of eigenvalues and eigenfunctions with respect to some elements of data.