Minimal Usco and Minimal Cusco Maps
Lubica
Hola
Academy of Sciences, Institute of Mathematics Stefˇ anikova 49, 81473 Bratislava,´
Slovakia
author
Dusan
Holy
Department of Mathematics and Computer Science, Faculty of Education,
Trnava University, Priemyselna 4, 918 43 Trnava, Slovakia´
author
text
article
2015
eng
The main aim of this paper is to present a survey of known results concerning minimal usco and minimal cusco maps. We give characterizations of minimal usco and minimal cusco maps in the class of all set-valued maps using quasicontinuous selections. If X is a topological space and Y is a Banach space, there is a bijection between the space of minimal usco maps from X to Y and the space of minimal cusco maps from X to Y. We study this bijection with respect to various topologies on underlying spaces. Some new results are also given.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
125
150
http://www.kjm-math.org/article_13161_9345c4bd651f19c82ac0fcbc44d079f6.pdf
dx.doi.org/10.22034/kjm.2015.13161
Cayley Graphs under Graph Operations II
Nasrin
Malekmohammadi
Department of Pure Mathematics, University of Kashan, Kashan, P.O. Box
87317-51167, Iran
author
Ali Reza
Ashrafi
Department of Pure Mathematics, University of Kashan, Kashan, P.O. Box
87317-51167, Iran
author
text
article
2015
eng
The aim of this paper is to investigate the behavior of Cayley graphs under some graph operations. It is proved that the NEPS, corona, hierarchical, strong, skew and converse skew products of Cayley graphs are again Cayley graphs under some conditions.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
151
163
http://www.kjm-math.org/article_13162_30ac021e4108183dc343b8ef0daeb6bb.pdf
dx.doi.org/10.22034/kjm.2015.13162
Statistical Ergodic Theorems for Markov Semigroups in Spaces with Mixed Norm
Inomjon
Ganiev
Department of Science in Engineering, Faculty of Engineering, International
Islamic University Malaysia, P.O. Box 10, 50728 Kuala-Lumpur, Malaysia
author
Sanobar
Sadaddinova
Department of Mathematics, Tashkent University of Information Technologies , Tashkent, Uzbekistan
author
Umarjon
Ganiev
Department of Physics Fergana Medical College, Fergana, Uzbekistan
author
text
article
2015
eng
This paper describes the semigroups generated by the Markov processes in spaces with mixed norm and proves analogues of statistical ergodic theorems for such semigroups.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
164
173
http://www.kjm-math.org/article_13163_f79e1ed5cb57130e6a0ef97d8461f321.pdf
dx.doi.org/10.22034/kjm.2015.13163
Exponential Stability and Instability in Multiple Delays Difference Equations
S.
Almutairy
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316
USA;
author
M.
Alshammari
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316
USA;
author
Y.
Raffoul
Department of Mathematics, University of Dayton, Dayton, OH 45469-2316
USA;
author
text
article
2015
eng
We use Lyapunov functionals and obtain sufficient conditions that guarantee exponential stability of the zero solution of the difference equation with multiple delays \begin{equation*} x(t+1) = a(t)x(t)+\sum^{k}_{j=1}b_j(t)x(t-h_j). \end{equation*} The novelty of our work is the relaxation of the condition $|a(t)| <1$, in spite of the presence of multiple delays. Using a slightly modified Lyapunov functional, we obtain necessary conditions for the unboundedness of all solutions and for the instability of the zero solution. We provide an example as an application to our obtained results.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
174
184
http://www.kjm-math.org/article_13164_48e6c2523d83b480c06970173928b701.pdf
dx.doi.org/10.22034/kjm.2015.13164
Lipschitz Tensor Product
M.G.
Cabrera-Padilia
Departamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.
author
J.A.
Chavez-Dominguez
Department of Mathematics, University of Oklahoma, Norman, Oklahoma, 7 3019, United States.
author
A.
Jimenez-Vargas
Departamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.
author
M.
Viliegas-Vallecillos
Departamento de Matematicas´ , Universidad de Cadiz´ , 11510 Puerto Real, Spain.
author
text
article
2015
eng
Inspired by ideas of R. Schatten in his celebrated monograph [23] on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product $X\boxtimes E$ of a pointed metric space $X$ and a Banach space $E$ as a certain linear subspace of the algebraic dual of $\text{Lip}0(X,E^*)$. We prove that $\left\langle \text{Lip}0(X,E^*),X\boxtimes E\right\rangle$ forms a dual pair.We prove that $X\boxtimes E$ is linearly isomorphic to the linear space of all finite-rank continuous linear operators from $(X^\#,\tau_p)$ into $E$, where $X^\#$ denotes the space $\text{Lip}0(X,\mathbb{K})$ and $\tau_p$ is the topology of pointwise convergence of $X^\#$. The concept of Lipschitz tensor product of elements of $X^\#$ and $E^*$ yields the space $X^\#⧆ E^*$ as a certain linear subspace of the algebraic dual of $X\boxtimes E$. To ensure the good behavior of a norm on $X\boxtimes E$ with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on $X\boxtimes E$ is defined. We show that the Lipschitz injective norm $\varepsilon$, the Lipschitz projective norm $\pi$ and the Lipschitz $p$-nuclear norm $d_p$ $(1\leq p\leq\infty)$ are uniform dualizable Lipschitz cross-norms on $X\boxtimes E$. In fact, $\varepsilon$ is the least dualizable Lipschitz cross-norm and $\pi$ is the greatest Lipschitz cross-norm on $X\boxtimes E$. Moreover, dualizable Lipschitz cross-norms $\alpha$ on $X\boxtimes E$ are characterized by satisfying the relation $\varepsilon\leq\alpha\leq\pi$.In addition, the Lipschitz injective (projective) norm on $X\boxtimes E$ can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product of the Lipschitz-free space over $X$ and $E$, but this identification does not hold for the Lipschitz $2$-nuclear norm and the corresponding Banach-space tensor norm. In terms of the space $X^\#⧆ E^*$, we describe the spaces of Lipschitz compact (finite-rank, approximable) operators from $X$ to $E^*$.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
185
218
http://www.kjm-math.org/article_13165_7d5b50b35f614ef5dfef49db33a649f2.pdf
dx.doi.org/10.22034/kjm.2015.13165
$n$-Dual Spaces Associated to a Normed Space
Yosafat E.P.
Pangalela
Department of Mathematics and Statistics, University of Otago, PO Box 56,
Dunedin 9054, New Zealand
author
text
article
2015
eng
For a real normed space $X$, we study the $n$-dual space of $\left( X,\left\Vert \cdot \right\Vert \right) $ and show that the space is a Banach space. Meanwhile, for a real normed space $X$ of dimension $d\geq n$ which satisfies property ($G$), we discuss the $n$-dual space of $\left( X,\left\Vert \cdot,\ldots ,\cdot \right\Vert _{G}\right) $, where $% \left\Vert \cdot ,\ldots ,\cdot \right\Vert _{G}$ is the Gähler $n$-norm. We then investigate the relationship between the $n$-dual space of $% \left( X,\left\Vert \cdot \right\Vert \right) $ and the $n$-dual space of $% \left( X,\left\Vert \cdot,\ldots ,\cdot \right\Vert _{G}\right) $. We use this relationship to determine the $n$-dual space of $\left( X,\left\Vert \cdot ,\ldots ,\cdot \right\Vert _{G}\right) ~$and show that the space is also a Banach space.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
219
229
http://www.kjm-math.org/article_13166_e95026b5f6197b67bbfb945b4b49545b.pdf
dx.doi.org/10.22034/kjm.2015.13166
Toeplitz and Hankel Operators on a Vector-valued Bergman Space
Namita
Das
Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar,
751004,, Odisha, India
author
text
article
2015
eng
In this paper, we derive certain algebraic properties of Toeplitz and Hankel operators defined on the vector-valued Bergman spaces $L_a^{2, \mathbb{C}^n}(\mathbb{D})$, where $\mathbb{D}$ is the open unit disk in $\mathbb{C}$ and $n\geq 1.$ We show that the set of all Toeplitz operators $T_{\Phi}, \Phi\in L_{M_n}^{\infty}(\mathbb{D})$ is strongly dense in the set of all bounded linear operators ${\mathcal L}(L_a^{2, \mathbb{C}^n}(\mathbb{D}))$ and characterize all finite rank little Hankel operators.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
230
242
http://www.kjm-math.org/article_13167_cd3e86baa83715a9bf967ed60c149d34.pdf
dx.doi.org/10.22034/kjm.2015.13167
On the Degree of Approximation of Functions Belonging to the Lipschitz Class by (E, q)(C, α, β) Means
Xhevat Z.
Krasniqi
University of Prishtina “Hasan Prishtina”, Faculty of Education, Department of Mathematics and Informatics, Avenue “Mother Theresa” 5, Prishtine,
Kosovo.
author
text
article
2015
eng
In this paper two generalized theorems on the degree of approximation of conjugate functions belonging to the Lipschitz classes of the type $\text{Lip}\alpha $, $0<\alpha \leq 1$, and $W(L_{p},\xi (t))$ are proved. The first one gives the degree of approximation with respect to the $L_{\infty}$-norm, and the second one with respect to $L_{p}$-norm, $p\geq 1$. In addition, a correct condition in proving of the second mentioned theorem is employed.
Khayyam Journal of Mathematics
Tusi Mathematical Research Group (TMRG) and Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with
The Center of Excellence in Analysis on Algebraic Structures)
2423-4788
1
v.
2
no.
2015
243
252
http://www.kjm-math.org/article_13168_e62c9334109527d4f323a4bd46193542.pdf
dx.doi.org/10.22034/kjm.2015.13168