Opuscula Mathematica
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ISSN 1232-9274
e-ISSN: 2300-6919
Issue Date
2015
Volume
Vol. 35
Number
No. 2
Description
Journal Volume
Opuscula Mathematica
Vol. 35 (2015)
Projects
Pages
Articles
On small vibrations of a damped Stieltjes string
(2015) Boyko, Olga; Pivovarchik, Vyacheslav
Inverse problem of recovering masses, coefficients of damping and lengths of the intervals between the masses using two spectra of boundary value problems and the total length of the Stieltjes string (an elastic thread bearing point masses) is considered. For the case of point-wise damping at the first counting from the right end mass the problem of recovering the masses, the damping coefficient and the lengths of the subintervals by one spectrum and the total length of the string is solved.
Simple eigenvectors of unbounded operators of the type »normal plus compact«
(2015) Gil', Michael
The paper deals with operators of the form $A=S+B$, where $B$ is a compact operator in a Hilbert space $H$ and $S$ is an unbounded normal one in $H$, having a compact resolvent. We consider approximations of the eigenvectors of $A$, corresponding to simple eigenvalues by the eigenvectors of the operators $A_{n}=S+B_{n}$ ($n=1,2, \ldots$), where $B_n$ is an $n$-dimensional operator. In addition, we obtain the error estimate of the approximation.
On b-vertex and b-edge critical graphs
(2015) Eschouf, Noureddine Ikhlef; Blidia, Mostafa
A $b$-coloring is a coloring of the vertices of a graph such that each color class contains a vertex that has a neighbor in all other color classes, and the $b$-chromatic number $b(G)$ of a graph $G$ is the largest integer $k$ such that $G$ admits a $b$-coloring with $k$ colors. A simple graph $G$ is called $b^{+}$-vertex (edge) critical if the removal of any vertex (edge) of $G$ increases its b-chromatic number. In this note, we explain some properties in $b^{+}$-vertex (edge) critical graphs, and we conclude with two open problems.
Bounded, asymptotically stable, and L1 solutions of Caputo fractional differential equations
(2015) Islam, Muhammad N.
The existence of bounded solutions, asymptotically stable solutions, and $L^1$ solutions of a Caputo fractional differential equation has been studied in this paper. The results are obtained from an equivalent Volterra integral equation which is derived by inverting the fractional differential equation. The kernel function of this integral equation is weakly singular and hence the standard techniques that are normally applied on Volterra integral equations do not apply here. This hurdle is overcomed using a resolvent equation and then applying some known properties of the resolvent. In the analysis Schauder’s fixed point theorem and Liapunov’s method have been employed. The existence of bounded solutions are obtained employing Schauder’s theorem, and then it is shown that these solutions are asymptotically stable by a definition found in [C. Avramescu, C. Vladimirescu, On the existence of asymptotically stable solution of certain integral equations, Nonlinear Anal. 66 (2007), 472–483]. Finally, the $L^1$ properties of solutions are obtained using Liapunov’s method.
A new characterization of convex φ-functions with a parameter
(2015) Micherda, Bartosz
We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by $\Phi$ are isotonic if and only if $\Phi$ is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012.