Ekin Uğurlu

Regular fractional differential equations in the Sobolev space

Abstract 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.

On square integrable solutions of a fractional differential equation

Abstract In this paper we construct the Weyl–Titchmarsh theory for the fractional Sturm–Liouville equation. For this purpose we used the Caputo and Riemann–Liouville fractional operators having the order is between zero and one.

Singular left-definite Hamiltonian systems in the Sobolev space

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

Abstract 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 Weyl’s 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.

Spectral analysis of the direct sum Hamiltonian operators

Abstract 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-Foia¸s 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.