Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801Minimal Usco and Minimal Cusco Maps1251501316110.22034/kjm.2015.13161ENLubicaHolaAcademy of Sciences, Institute of Mathematics Stefˇ anikova 49, 81473 Bratislava,´
SlovakiaDusanHolyDepartment of Mathematics and Computer Science, Faculty of Education,
Trnava University, Priemyselna 4, 918 43 Trnava, Slovakia´Journal Article20141204The 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.http://www.kjm-math.org/article_13161_9345c4bd651f19c82ac0fcbc44d079f6.pdfDepartment of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801Cayley Graphs under Graph Operations II1511631316210.22034/kjm.2015.13162ENNasrinMalekmohammadiDepartment of Pure Mathematics, University of Kashan, Kashan, P.O. Box
87317-51167, IranAli RezaAshrafiDepartment of Pure Mathematics, University of Kashan, Kashan, P.O. Box
87317-51167, IranJournal Article20141208The 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.http://www.kjm-math.org/article_13162_30ac021e4108183dc343b8ef0daeb6bb.pdfDepartment of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801Statistical Ergodic Theorems for Markov Semigroups in Spaces with Mixed Norm1641731316310.22034/kjm.2015.13163ENInomjonGanievDepartment of Science in Engineering, Faculty of Engineering, International
Islamic University Malaysia, P.O. Box 10, 50728 Kuala-Lumpur, MalaysiaSanobarSadaddinovaDepartment of Mathematics, Tashkent University of Information Technologies , Tashkent, UzbekistanUmarjonGanievDepartment of Physics Fergana Medical College, Fergana, UzbekistanJournal Article20140526This paper describes the semigroups generated by the Markov processes in spaces with mixed norm and proves analogues of statistical ergodic theorems for such semigroups.http://www.kjm-math.org/article_13163_f79e1ed5cb57130e6a0ef97d8461f321.pdfDepartment of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801Exponential Stability and Instability in Multiple Delays Difference Equations1741841316410.22034/kjm.2015.13164ENS.AlmutairyDepartment of Mathematics, University of Dayton, Dayton, OH 45469-2316
USA;M.AlshammariDepartment of Mathematics, University of Dayton, Dayton, OH 45469-2316
USA;Y.RaffoulDepartment of Mathematics, University of Dayton, Dayton, OH 45469-2316
USA;Journal Article20140709We 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.http://www.kjm-math.org/article_13164_48e6c2523d83b480c06970173928b701.pdfDepartment of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801Lipschitz Tensor Product1852181316510.22034/kjm.2015.13165ENM.G.Cabrera-PadiliaDepartamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.J.A.Chavez-DominguezDepartment of Mathematics, University of Oklahoma, Norman, Oklahoma, 7 3019, United States.A.Jimenez-VargasDepartamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.M.Viliegas-VallecillosDepartamento de Matematicas´ , Universidad de Cadiz´ , 11510 Puerto Real, Spain.Journal Article20151104Inspired 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 $Xboxtimes E$ 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 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 $Xboxtimes E$. 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 cross-norm 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 cross-norms on $Xboxtimes E$. In fact, $varepsilon$ is the least dualizable Lipschitz cross-norm and $pi$ is the greatest Lipschitz cross-norm on $Xboxtimes E$. Moreover, dualizable Lipschitz cross-norms $alpha$ on $Xboxtimes E$ are characterized by satisfying the relation $varepsilonleqalphaleqpi$.<br />In addition, the Lipschitz injective (projective) norm on $Xboxtimes E$ 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 $E^*$.http://www.kjm-math.org/article_13165_7d5b50b35f614ef5dfef49db33a649f2.pdfDepartment of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801$n$-Dual Spaces Associated to a Normed Space2192291316610.22034/kjm.2015.13166ENYosafat E.P.PangalelaDepartment of Mathematics and Statistics, University of Otago, PO Box 56,
Dunedin 9054, New ZealandJournal Article20150816For 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.http://www.kjm-math.org/article_13166_e95026b5f6197b67bbfb945b4b49545b.pdfDepartment of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801Toeplitz and Hankel Operators on a Vector-valued Bergman Space2302421316710.22034/kjm.2015.13167ENNamitaDasDepartment of Mathematics, Utkal University, Vanivihar, Bhubaneswar,
751004,, Odisha, IndiaJournal Article20141107In 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 $ngeq 1.$ 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.http://www.kjm-math.org/article_13167_cd3e86baa83715a9bf967ed60c149d34.pdfDepartment of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures and Tusi Mathematical Research Group)Khayyam Journal of Mathematics2423-47881220150801On the Degree of Approximation of Functions Belonging to the Lipschitz Class by (E, q)(C, α, β) Means2432521316810.22034/kjm.2015.13168ENXhevat Z.KrasniqiUniversity of Prishtina “Hasan Prishtina”, Faculty of Education, Department of Mathematics and Informatics, Avenue “Mother Theresa” 5, Prishtine,
Kosovo.Journal Article20150604In 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.http://www.kjm-math.org/article_13168_e62c9334109527d4f323a4bd46193542.pdf