Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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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: 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.Article Citation - Scopus: 1On the Zeros of Solutions of Ordinary and Fractional Differential Equations(Wiley, 2023) Ugurlu, EkinThis paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second-order Sturm-Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.Article Citation - WoS: 1Citation - Scopus: 1The Spectral Analysis of a System of First-Order Equations With Dissipative Boundary Conditions(Wiley, 2021) Ugurlu, EkinThis paper aims to share some completeness theorems related with a boundary value problem generated by a system of equations and non-self-adjoint (dissipative) boundary conditions. Indeed, we consider a system of equations that contains a continuous analogous of the orthogonal polynomials on the unit circle. Constructing the characteristic function of the related dissipative operator, we share some completeness theorems. Moreover, we give an explicit form of the self-adjoint dilation of the dissipative operator.Article The Characteristic Matrix Function of a Dissipative Hamiltonian Operator(Wiley, 2021) Ugurlu, EkinIn this paper, we consider a singular dissipative even-order Hamiltonian operator with a finite number of transmission conditions. Using coordinate-free approach, we construct the characteristic matrix-function of the Cayley transform of the dissipative operator. Using the equivalence of completeness property of root functions of Cayley transform and dissipative operator, we prove some completeness theorems. Moreover, we construct an explicit form of the resolvent operator of dissipative operator.Article Citation - WoS: 11Citation - Scopus: 11Some Singular Third-Order Boundary Value Problems(Wiley, 2020) Ugurlu, EkinIn this paper, we consider some singular formally symmetric (self-adjoint) boundary value problems generated by a singular third-order differential expression and separated and coupled boundary conditions. In particular, we consider that the minimal symmetric operator generated by the third-order differential expression has the deficiency indices (3,3). We investigate same spectral properties related with these problems, and we introduce a method to find the resolvent operator.Article Citation - WoS: 39Citation - Scopus: 40Regular Third-Order Boundary Value Problems(Elsevier Science inc, 2019) Ugurlu, EkinIn this paper, we consider some boundary value problems generated by third-order formally symmetric (self-adjoint) regular differential expression and separated, real-coupled and complex-coupled boundary conditions. It is shown that these problems generate self-adjoint operators. Moreover, the dependence of eigenvalues of these problems on the data are studied and some derivatives of the eigenvalues with respect to some elements of data are introduced. (C) 2018 Elsevier Inc. All rights reserved.