2015
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2
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128
Minimal Usco and Minimal Cusco Maps
2
2
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 setvalued 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.
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125
150


Lubica
Hola
Academy of Sciences, Institute of Mathematics Stefˇ anikova 49, 81473 Bratislava,´
Slovakia
Academy of Sciences, Institute of Mathematics
Slovakia (Slovak Rep)


Dusan
Holy
Department of Mathematics and Computer Science, Faculty of Education,
Trnava University, Priemyselna 4, 918 43 Trnava, Slovakia´
Department of Mathematics and Computer Science,
Slovakia (Slovak Rep)
Quasicontinuous function
minimal usco map
minimal cusco map
subcontinuous function
selection
Cayley Graphs under Graph Operations II
2
2
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.
1

151
163


Nasrin
Malekmohammadi
Department of Pure Mathematics, University of Kashan, Kashan, P.O. Box
8731751167, Iran
Department of Pure Mathematics, University
Iran


Ali Reza
Ashrafi
Department of Pure Mathematics, University of Kashan, Kashan, P.O. Box
8731751167, Iran
Department of Pure Mathematics, University
Iran
Cayley graph
corona
Hierarchical product
skew product
converse skew product
NEPS
strong product
Statistical Ergodic Theorems for Markov Semigroups in Spaces with Mixed Norm
2
2
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.
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164
173


Inomjon
Ganiev
Department of Science in Engineering, Faculty of Engineering, International
Islamic University Malaysia, P.O. Box 10, 50728 KualaLumpur, Malaysia
Department of Science in Engineering, Faculty
Malaysia


Sanobar
Sadaddinova
Department of Mathematics, Tashkent University of Information Technologies , Tashkent, Uzbekistan
Department of Mathematics, Tashkent University
Uzbekistan


Umarjon
Ganiev
Department of Physics Fergana Medical College, Fergana, Uzbekistan
Department of Physics Fergana Medical College,
Uzbekistan
Statistical ergodic theorem
Markov semigroup
mixed norm
Exponential Stability and Instability in Multiple Delays Difference Equations
2
2
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(th_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.
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174
184


S.
Almutairy
Department of Mathematics, University of Dayton, Dayton, OH 454692316
USA;
Department of Mathematics, University of
United States


M.
Alshammari
Department of Mathematics, University of Dayton, Dayton, OH 454692316
USA;
Department of Mathematics, University of
United States


Y.
Raffoul
Department of Mathematics, University of Dayton, Dayton, OH 454692316
USA;
Department of Mathematics, University of
United States
Exponential stability
Instability
Lyapunov functional
Lipschitz Tensor Product
2
2
Inspired by ideas of R. Schatten in his celebrated monograph [23] on a theory of crossspaces, we introduce the notion of a Lipschitz tensor product $Xboxtimes E$ 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 $leftlangle text{Lip}0(X,E^*),Xboxtimes Erightrangle$ forms a dual pair.We prove that $Xboxtimes E$ is linearly isomorphic to the linear space of all finiterank 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 $Xboxtimes E$. To ensure the good behavior of a norm on $Xboxtimes E$ with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz crossnorm on $Xboxtimes E$ is defined. We show that the Lipschitz injective norm $varepsilon$, the Lipschitz projective norm $pi$ and the Lipschitz $p$nuclear norm $d_p$ $(1leq pleqinfty)$ are uniform dualizable Lipschitz crossnorms on $Xboxtimes E$. In fact, $varepsilon$ is the least dualizable Lipschitz crossnorm and $pi$ is the greatest Lipschitz crossnorm on $Xboxtimes E$. Moreover, dualizable Lipschitz crossnorms $alpha$ on $Xboxtimes E$ are characterized by satisfying the relation $varepsilonleqalphaleqpi$.In addition, the Lipschitz injective (projective) norm on $Xboxtimes E$ can be identified with the injective (respectively, projective) tensor norm on the Banachspace tensor product of the Lipschitzfree space over $X$ and $E$, but this identification does not hold for the Lipschitz $2$nuclear norm and the corresponding Banachspace tensor norm. In terms of the space $X^#⧆ E^*$, we describe the spaces of Lipschitz compact (finiterank, approximable) operators from $X$ to $E^*$.
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185
218


M.G.
CabreraPadilia
Departamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.
Departamento de Matematicas´ , Universidad
Spain


J.A.
ChavezDominguez
Department of Mathematics, University of Oklahoma, Norman, Oklahoma, 7 3019, United States.
Department of Mathematics, University of
United States


A.
JimenezVargas
Departamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.
Departamento de Matematicas´ , Universidad
Spain


M.
ViliegasVallecillos
Departamento de Matematicas´ , Universidad de Cadiz´ , 11510 Puerto Real, Spain.
Departamento de Matematicas´ , Universidad
Spain
Lipschitz map
tensor product
$p$summing operator
duality
Lipschitz compact operator
$n$Dual Spaces Associated to a Normed Space
2
2
For a real normed space $X$, we study the $n$dual space of $left( X,leftVert cdot rightVert right) $ and show that the space is a Banach space. Meanwhile, for a real normed space $X$ of dimension $dgeq n$ which satisfies property ($G$), we discuss the $n$dual space of $left( X,leftVert cdot,ldots ,cdot rightVert _{G}right) $, where $% leftVert cdot ,ldots ,cdot rightVert _{G}$ is the Gähler $n$norm. We then investigate the relationship between the $n$dual space of $% left( X,leftVert cdot rightVert right) $ and the $n$dual space of $% left( X,leftVert cdot,ldots ,cdot rightVert _{G}right) $. We use this relationship to determine the $n$dual space of $left( X,leftVert cdot ,ldots ,cdot rightVert _{G}right) ~$and show that the space is also a Banach space.
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219
229


Yosafat E.P.
Pangalela
Department of Mathematics and Statistics, University of Otago, PO Box 56,
Dunedin 9054, New Zealand
Department of Mathematics and Statistics,
New Zealand
$n$dual spaces
$n$normed spaces
bounded linear functionals
Toeplitz and Hankel Operators on a Vectorvalued Bergman Space
2
2
In this paper, we derive certain algebraic properties of Toeplitz and Hankel operators defined on the vectorvalued Bergman spaces $L_a^{2, mathbb{C}^n}(mathbb{D})$, where $mathbb{D}$ is the open unit disk in $mathbb{C}$ and $ngeq 1.$ We show that the set of all Toeplitz operators $T_{Phi}, Phiin L_{M_n}^{infty}(mathbb{D})$ 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.
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230
242


Namita
Das
Department of Mathematics, Utkal University, Vanivihar, Bhubaneswar,
751004,, Odisha, India
Department of Mathematics, Utkal University,
India
Bergman space
Toeplitz operators
little Hankel operators
strongoperator topology
finite rank operators
On the Degree of Approximation of Functions Belonging to the Lipschitz Class by (E, q)(C, α, β) Means
2
2
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, $pgeq 1$. In addition, a correct condition in proving of the second mentioned theorem is employed.
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243
252


Xhevat Z.
Krasniqi
University of Prishtina “Hasan Prishtina”, Faculty of Education, Department of Mathematics and Informatics, Avenue “Mother Theresa” 5, Prishtine,
Kosovo.
University of Prishtina “Hasan Prishtina&rdq
Serbia
Lipschitz classes
Fourier series
summability
degree of approximation