Stability in non-autonomous periodic systems with grazing stationary impacts

Author: Marat Akhmet and Aysegul Kivilcim

Abstract: This paper examines impulsive non-autonomous periodic systems whose surfaces of discontinuity and impact functions are not depending on the time variable.
The $W-$map which alters the system with variable moments of impulses to that with fixed moments and facilitates the investigations, is presented. A particular linearizion system with two compartments is utilized to analyze stability of a grazing periodic solution. A significant way to keep down a singularity in linearizion is demonstrated. A concise review on sufficient conditions for the linearizion and stability is presented. An example is given to actualize the theoretical results. PDF |

Iterative methods for generalized split feasibility problems in Banach spaces

Author: Qamrul Hasan Ansari and Aisha Rehan

Abstract: Inspired by the recent work of Takahashi et al. [W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces, Set-Valued Var. Anal., 23 (2015), 205--221], in this paper, we study generalized split feasibility problems (GSFPs) in the setting of Banach spaces. We propose iterative algorithms to compute the approximate solutions of such problems. The weak convergence of the sequence generated by the proposed algorithms is studied. As applications, we derive some algorithms and convergence results for some problems from nonlinear analysis, namely, split feasibility problems, equilibrium problems, etc. Our results generalize several known results in the literature including the results of Takahashi et al. W. Takahashi, H.-K. Xu and J.-C. Yao, Iterative methods for generalized split feasibility problems in Hilbert spaces,
Set-Valued Var. Anal., 23 (2015), 205--221. PDF |

Extending the applicability of Newton's method using Wang's-- Smale's $\alpha$--theory

Author: Ioannis K. Argyros and Santhosh George

Abstract: We improve semilocal convergence results for Newton's method by defining a more precise domain where the Newton iterate lies than in earlier studies using the Smale's $\alpha$-- theory. These improvements are obtained under the same computational cost. Numerical examples are also presented in this study to show that the earlier results cannot apply but the new results can apply to solve equations. PDF |

On the Voronovskaja-type formula for the Bleimann, Butzer and Hahn bivariate operators

Author: Dan BÄƒrbosu and Dan MiclÄƒuÈ™

Abstract: In this paper we present two new alternative ways for the proof of Voronovskaja-type formula of the Bleimann, Butzer and Hahn bivariate operators, using the close connection between the recalled operators and Bernstein bivariate operators, respectively Stancu bivariate operators. PDF |

On an isomorphism lying behind the class number formula

Author: Vlad CriÈ™an

Abstract: Let $p$ be an odd prime such that the Greenberg conjecture holds for the maximal real cyclotomic subfield $\mathbb{K}_1$ of $\mathbb{Q}[ \zeta_p ]$. Let $A_n = (\mathcal{C}(\mathbb{K}_n))_p$ be the $p$-part of the class group of $\mathbb{K}_n$, the $n$-th field in the cyclotomic tower, and let $\underline{E}_n$, $\underline{C}_n$ be the global and cyclotomic units of $\mathbb{K}_n$, respectively. We prove that under this premise, there is some $n_0$ such that for all $m \geq n_0$, the class number formula $\left|\left(\underline{E}_m/\underline{C}_m\right)_p\right|=|A_m|$ hides in fact an isomorphism of $\Lambda[\hbox{Gal}(\mathbb{K}_1/\mathbb{Q})]$-modules. PDF |

Quantitative estimates in uniform and pointwise approximation by Bernstein-Durrmeyer-Choquet operators

Author: Sorin G. Gal and Sorin Trifa

Abstract: For the qualitative results of uniform and pointwise approximation obtained in [Gal-Opris], we present here general quantitative estimates in terms of the modulus of continuity and of a $K$-functional, in approximation by the generalized multivariate Bernstein-Durrmeyer operator $M_{n, \Gamma_{n, x}}$, written in terms of Choquet integrals with respect to a family of monotone and submodular set functions, $\Gamma_{n, x}$, on the standard $d$-dimensional simplex. If $d=1$ and the Choquet integrals are taken with respect to some concrete possibility measures, the estimate in terms of the modulus of continuity is detailed. Examples improving the estimates given by the classical operators also are presented. PDF |

Abstract: The aim of this article is to improve the results of Piri et al. [Fixed
Point Theory and Applications 2014, 2014:210] by introducing new types of
contractions say Suzuki-Berinde type $F$-contractions and Suzuki type
rational $F$-contractions. We also establish a common fixed point theorem
for a sequence of multivalued mappings. An example is also given to support our main results. PDF |

Bezier variant of genuine-Durrmeyer type operators based on Polya distribution

Author: Trapti Neer, Ana Maria Acu and P. N. Agrawal

Abstract: In this paper we introduce the Bezier variant of genuine-Durrmeyer type operators having P\'{o}lya basis functions. We give a global approximation theorem in terms of second order modulus of continuity, a direct approximation theorem by means of the Ditzian-Totik modulus of smoothness and a Voronovskaja type theorem by using the Ditzian-Totik modulus of smoothness. The rate of convergence for functions whose derivatives are of bounded variation is obtained. Further, we show the rate of convergence of these operators to certain functions by illustrative graphics using the Maple algorithms. PDF |

Hardy-Littlewood-Polya theorem of majorization in the framework of generalized convexity

Author: Constantin P. Niculescu and Ionel RovenÈ›a

Abstract: Based on a new concept of generalized relative convexity, a large extension of Hardy-Littlewood-Polya theorem of majorization is obtained. Several applications escaping the classical framework of convexity are included. PDF |

Blow up of solutions for 3D quasi-linear wave equations with positive initial energy

Author: Amir Peyravi

Abstract: In this paper we investigate blow up property of solutions for a
system of nonlinear wave equations with nonlinear dissipations and
positive initial energy in a bounded domain in $\mathbb{R}^{3}$.
Our result improves and extends earlier results in the literature
such as the ones in [Zhou, J. and Mu, C., The lifespan for 3D
quasilinear wave equations with nonlinear damping terms, Nonlinear
Anal., 74 (2011), 5455--5466] and [Piskin, E., Uniform decay and
blow-up of solutions for coupled nonlinear Klein-Gordon equations
with nonlinear damping terms, Math. Meth. Appl. Scie., 37 (2014),
No. 18, 3036--3047] in which the nonexistence results obtained only
for negative initial energy or the one in [Ye, Y., Global existence
and nonexistence of solutions for coupled nonlinear wave equations
with damping and source terms, Bull. Korean Math. Soc., 51 (2014),
No. 6, 1697--1710] where blow up results have been not addressed.
Estimate for the lower bound of the blow up time is also given. PDF |

Existence of non-trivial complex unit neighborhoods

Author: Pablo Piniella

Abstract: First, we briefly mention the basic definitions and results on unit neighborhoods of zero. Next, we show the existence of certain non-trivial complex unit neighborhoods of zero. We expose a generalization of the construction method used on the mentioned particular case. Since this construction may not lead to a unit neighborhood of zero, we develop some necessary conditions. Finally, we describe our heuristic use of Wolfram Mathematica to prove the existence of non-trivial complex unit neighborhoods. PDF |

Abstract: In this paper we introduce and study the concept of normality
degree of a finite group $G$. This quantity measures the
probability of a random subgroup of $G$ to be normal. Explicit
formulas are obtained for some particular classes of finite
groups. Several limits of normality degrees are also computed. PDF |

Author: Liping Yuan, Tudor Zamfirescu and Yue Zhang

Abstract: A set $S$ in $\mathbb{R}^{d}$ is called $it$-convex if, for any two
distinct points in $S$, there exists a third point in $S$, such that
one of the three points is equidistant from the others.
In this paper we first investigate nondiscrete $it$-convex sets,
then discuss about the $it$-convexity of the eleven Archimedean
tilings, and treat subsequently finite subsets of the square
lattice. Finally, we obtain a lower bound on the number of isosceles
triples contained in an $n$-point $it$-convex set. PDF |