2021-01-28T06:19:02Z
http://www.kjm-math.org/?_action=export&rf=summon&issue=2104
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
Minimal Usco and Minimal Cusco Maps
Lubica
Hola
Dusan
Holy
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.
Quasicontinuous function
minimal usco map
minimal cusco map
subcontinuous function
Selection
2015
08
01
125
150
http://www.kjm-math.org/article_13161_9345c4bd651f19c82ac0fcbc44d079f6.pdf
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
Cayley Graphs under Graph Operations II
Nasrin
Malekmohammadi
Ali Reza
Ashrafi
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.
Cayley graph
corona
Hierarchical product
skew product
converse skew product
NEPS
strong product
2015
08
01
151
163
http://www.kjm-math.org/article_13162_30ac021e4108183dc343b8ef0daeb6bb.pdf
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
Statistical Ergodic Theorems for Markov Semigroups in Spaces with Mixed Norm
Inomjon
Ganiev
Sanobar
Sadaddinova
Umarjon
Ganiev
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.
Statistical ergodic theorem
Markov semigroup
mixed norm
2015
08
01
164
173
http://www.kjm-math.org/article_13163_f79e1ed5cb57130e6a0ef97d8461f321.pdf
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
Exponential Stability and Instability in Multiple Delays Difference Equations
S.
Almutairy
M.
Alshammari
Y.
Raffoul
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.
exponential stability
Instability
Lyapunov functional
2015
08
01
174
184
http://www.kjm-math.org/article_13164_48e6c2523d83b480c06970173928b701.pdf
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
Lipschitz Tensor Product
M.G.
Cabrera-Padilia
J.A.
Chavez-Dominguez
A.
Jimenez-Vargas
M.
Viliegas-Vallecillos
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 $E^*$.
Lipschitz map
tensor product
$p$-summing operator
duality
Lipschitz compact operator
2015
08
01
185
218
http://www.kjm-math.org/article_13165_7d5b50b35f614ef5dfef49db33a649f2.pdf
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
$n$-Dual Spaces Associated to a Normed Space
Yosafat E.P.
Pangalela
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.
$n$-dual spaces
$n$-normed spaces
bounded linear functionals
2015
08
01
219
229
http://www.kjm-math.org/article_13166_e95026b5f6197b67bbfb945b4b49545b.pdf
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
Toeplitz and Hankel Operators on a Vector-valued Bergman Space
Namita
Das
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.
Bergman space
Toeplitz operators
little Hankel operators
strong-operator topology
finite rank operators
2015
08
01
230
242
http://www.kjm-math.org/article_13167_cd3e86baa83715a9bf967ed60c149d34.pdf
Khayyam Journal of Mathematics
Khayyam J. Math.
2015
1
2
On the Degree of Approximation of Functions Belonging to the Lipschitz Class by (E, q)(C, α, β) Means
Xhevat Z.
Krasniqi
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.
Lipschitz classes
Fourier series
summability
degree of approximation
2015
08
01
243
252
http://www.kjm-math.org/article_13168_e62c9334109527d4f323a4bd46193542.pdf