Browsing by Author "Ugurlu, Ekin"

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A new method for dissipative dynamic operator with transmission conditions
(Springer Basel Ag, 2018) Uğurlu, Ekin; Ugurlu, Ekin; Tas, Kenan; Taş, Kenan; 238990; 4971
In this paper, we investigate the spectral properties of a boundary value transmission problem generated by a dynamic equation on the union of two time scales. For such an analysis we assign a suitable dynamic operator which is in limit-circle case at infinity. We also show that this operator is a simple maximal dissipative operator. Constructing the inverse operator we obtain some information about the spectrum of the dissipative operator. Moreover, using the Cayley transform of the dissipative operator we pass to the contractive operator which is of the class With the aid of the minimal function we obtain more information on the dissipative operator. Finally, we investigate other properties of the contraction such that multiplicity of the contraction, unitary colligation with basic operator and CMV matrix representation associated with the contraction.
Coordinate-Free Approach for the Characteristic Function of a Fourth-Order Dissipative Operator
(Taylor & Francis inc, 2019) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this article, we investigate some spectral properties of a singular dissipative fourth-order dissipative operator in case at the singular point. For this purpose we construct the characteristic function of both maximal simple dissipative operator and completely non-unitary contraction which is the Cayley transform of the dissipative operator. Using the properties of the characteristic operator-function we obtain the related results of the boundary value problem. Moreover we obtain the selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing eigenfunctions by using coordinate-free approach.
Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator
(Frontiers Media Sa, 2019) Uğurlu, Ekin; Ugurlu, Ekin; Baleanu, Dumitru; Baleanu, Dumitru; 56389; 238990
In this paper we investigate the spectral properties of a third-order differential operator generated by a formally-symmetric differential expression and maximal dissipative boundary conditions. In fact, using the boundary value space of the minimal operator we introduce maximal selfadjoint and maximal non-selfadjoint (dissipative, accumulative) extensions. Using Solomyak's method on characteristic function of the contractive operator associated with a maximal dissipative operator we obtain some results on the root vectors of the dissipative operator. Finally, we introduce the selfadjoint dilation of the maximal dissipative operator and incoming and outgoing eigenfunctions of the dilation.
Dirac systems with regular and singular transmission effects
(Tubitak Scientific & Technological Research Council Turkey, 2017) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this paper, we investigate the spectral properties of singular eigenparameter dependent dissipative problems in Weyl's limit-circle case with finite transmission conditions. In particular, these transmission conditions are assumed to be regular and singular. To analyze these problems we construct suitable Hilbert spaces with special inner products and linear operators associated with these problems. Using the equivalence of the Lax-Phillips scattering function and Sz-Nagy-Foias characteristic functions we prove that all root vectors of these dissipative operators are complete in Hilbert spaces.
Dissipative operator and its Cayley transform
(Tubitak Scientific & Technological Research Council Turkey, 2017) Uğurlu, Ekin; Ugurlu, Ekin; Tas, Kenan; Taş, Kenan; 238990; 4971
In this paper, we investigate the spectral properties of the maximal dissipative extension of the minimal symmetric differential operator generated by a second order differential expression and dissipative and eigenparameter dependent boundary conditions. For this purpose we use the characteristic function of the maximal dissipative operator and inverse operator. This investigation is done by the characteristic function of the Cayley transform of the maximal dissipative operator, which is a completely nonunitary contraction belonging to the class C-0. Using Solomyak's method we also introduce the self-adjoint dilation of the maximal dissipative operator and incoming/outgoing eigenfunctions of the dilation. Moreover, we investigate other properties of the Cayley transform of the maximal dissipative operator.
Extensions of a Minimal Third-Order Formally Symmetric Operator
(Malaysian Mathematical Sciences Soc, 2020) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this paper, we consider some regular boundary value problems generated by a third-order differential equation and some boundary conditions. In particular, we construct maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal operator. Further using Lax-Phillips scattering theory and Sz.-Nagy-Foias characteristic function theory we prove a completeness theorem.
Fourth order differential operators with distributional potentials
(Tubitak Scientific & Technological Research Council Turkey, 2020) Ugurlu, Ekin; Uğurlu, Ekin; Bairamov, Elgiz; 238990
In this paper, regular and singular fourth order differential operators with distributional potentials are investigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators are considered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such that a squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given in the single and direct sum Hilbert spaces.
On a new class of fractional operators
(Springeropen, 2017) Jarad, Fahd; Uğurlu, Ekin; Jarad, Fahd; Ugurlu, Ekin; Abdeljawad, Thabet; Abdeljawad, Thabet; Baleanu, Dumitru; Baleanu, Dumitru; 234808; 238990
This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these operators.
On the solutions of a fractional boundary value problem
(Tubitak Scientific & Technological Research Council Turkey, 2018) Uğurlu, Ekin; Ugurlu, Ekin; Baleanu, Dumitru; Baleanu, Dumitru; Tas, Kenan; Taş, Kenan; 238990; 56389; 4971
This paper is devoted to showing the existence and uniqueness of solution of a regular second-order nonlinear fractional differential equation subject to the ordinary boundary conditions. The Banach fixed point theorem is used to prove the results.
Regular fractional differential equations in the sobolev space
(Walter de Gruyter Gmbh, 2017) Uğurlu, Ekin; Ugurlu, Ekin; Baleanu, Dumitru; Baleanu, Dumitru; Tas, Kenan; Taş, Kenan; 238990; 56389; 4971
In this paper regular fractional Sturm-Liouville boundary value problems are considered. In particular, new inner products are described in the Sobolev space and a symmetric operator is established in this space.
Regular fractional dissipative boundary value problems
(Springer, 2016) Baleanu, Dumitru; Baleanu, Dumitru; Ugurlu, Ekin; Uğurlu, Ekin; 56389; 238990
In this manuscript we present a regular dissipative fractional operator associated with a fractional boundary value problem. In particular, we present two main dissipative boundary value problems and one of them contains the spectral parameter in the boundary conditions. To construct the associated dissipative operator we present a direct sum Hilbert space.
Regular third-order boundary value problems
(Elsevier Science inc, 2019) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In 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.
Singular conformable sequential differential equations with distributional potentials
(Natl inquiry Services Centre Pty Ltd, 2019) Baleanu, Dumitru; Baleanu, Dumitru; Jarad, Fahd; Jarad, Fahd; Ugurlu, Ekin; Uğurlu, Ekin; 56389; 234808; 238990
In this paper we consider the singular conformable sequential equation with distributional potentials. We present Weyl's theory in the frame of conformable derivatives. Moreover we give two theorems on limit-point case.
Singular Dirac systems in the Sobolev space
(Tubitak Scientific & Technological Research Council Turkey, 2017) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this paper we construct Weyl's theory for the singular left-definite Dirac systems. In particular, we prove that there exists at least one solution of the system of equations that lies in the Sobolev space. Moreover, we describe the behavior of the solution belonging to the Sobolev space around the singular point.
Singular Hamiltonian system with several spectral parameters II: Odd-order case
(Academic Press inc Elsevier Science, 2019) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this paper we deal with a singular Hamiltonian system of odd-order with several spectral parameters and we investigate the behavior of the solution of this system at singular point with the aid of the characteristic function theory. Moreover, some results have been introduced for the Weyl-Titchmarsh function for some special Hamiltonian systems of odd-order with several spectral parameters. (C) 2019 Elsevier Inc. All rights reserved.
Singular left-definite Hamiltonian systems in the Sobolev space
(int Scientific Research Publications, 2017) Uğurlu, Ekin; Ugurlu, Ekin; Taş, Kenan; Tas, Kenan; Baleanu, Dumitru; Baleanu, Dumitru; 238990; 4971; 56389
This paper is devoted to construct Weyl's theory for the singular left-definite even-order Hamiltonian systems in the corresponding Sobolev space. In particular, it is proved that there exist at least n-linearly independent solutions in the Sobolev space for the 2n-dimensional Hamiltonian system. (C) 2017 All rights reserved.
Singular multiparameter dynamic equations with distributional potentials on time scales
(Natl inquiry Services Centre Pty Ltd, 2017) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this paper, we consider both singular single and several multiparameter second order dynamic equations with distributional potentials on semi-infinite time scales. At first we construct Weyls theory for the single singular multiparameter dynamic equation with distributional potentials and we prove that the forward jump of at least one solution of this equation must be squarely integrable with respect to some multiple function which is of one sign and nonzero on the given time scale. Then using the obtained results for the single dynamic equation with several parameters, we investigate the number of the products of the squarely integrable solutions of the singular several equations with distributional potentials and several parameters.
Some singular third-order boundary value problems
(Wiley, 2020) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In 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.
Spectral analysis of the direct sum hamiltonian operators
(Natl inquiry Services Centre Pty Ltd, 2016) Allahverdiev, Bilender P.; Uğurlu, Ekin; Ugurlu, Ekin; 238990
In this paper we investigate the deficiency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foias characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces.
The spectral analysis of a nuclear resolvent operator associated with a second order dissipative differential operator
(Springer Heidelberg, 2017) Ugurlu, Ekin; Uğurlu, Ekin; Bairamov, Elgiz; 238990
In this paper, we introduce a new approach for the spectral analysis of a linear second order dissipative differential operator with distributional potentials. This approach is related with the inverse operator. We show that the inverse operator is a non-selfadjoint trace class operator. Using Lidskii's theorem, we introduce a complete spectral analysis of the second order dissipative differential operator. Moreover, we give a trace formula for the trace class integral operator.