Volume 67 , issue 3 ( 2015 ) back
 B-Fredholm spectra and Riesz perturbations 155$-$165 M. Berkani and H. Zariouh

Abstract

Let $T$ be a bounded linear Banach space operator and let $Q$ be a quasinilpotent one commuting with $T$. The main purpose of the paper is to show that we do not have $\sigma_{*}(T+Q)=\sigma_{*}(T)$ where $\sigma_{*}ın\{\sigma_{D},\sigma_{LD}\}$, contrary to what has been announced in the proof of Lemma 3.5 from M. Amouch, {Polaroid operators with SVEP and perturbations of property (gw)}, Mediterr. J. Math. {6} (2009), 461--470, where $\sigma_{D}(T)$ is the Drazin spectrum of $T$ and $\sigma_{LD}(T)$ its left Drazin spectrum. However, under the additional hypothesis $\operatorname{iso}\sigma_{ub}(T)=\emptyset$, the mentioned equality holds. Moreover, we study the preservation of various spectra originating from B-Fredholm theory under perturbations by Riesz operators.

Keywords: B-Fredholm spectrum; Riesz perturbations.

MSC: 47A53, , 47A10, 47A11

 Existence of positive solutions for a class of nonlocal elliptic systems with multiple parameters 166$-$173 Nguyen Thanh Chung and Ghasem Alizadeh Afrouzi

Abstract

In this paper, we study the existence of positive solutions to the following nonlocal elliptic systems $$\cases - M_1\left(ınt_\Omega |\nabla u|^p\,dx\right)\Delta_p u = \alpha_1 a(x)f_1(v) + \beta_1b(x)g_1(u), \quad x ın \Omega,\\ - M_2\left(ınt_\Omega |\nabla v|^q\,dx\right)\Delta_q v = \alpha_2 c(x)f_2(u) + \beta_2d(x)g_2(v), \quad x ın \Omega,\\ u = v = 0, \quad x ın \partial\Omega, \endcases$$ where $\Omega$ is a bounded domain in $\Bbb{R}^N$ with smooth boundary $\partial\Omega$, $1 Keywords: Nonlocal elliptic systems; positive solutions; sub and supersolutions method. MSC: 35D05, 35J60  Topology generated by cluster systems 174$-$184 R. Thangamariappan and V. Renukadevi Abstract In this paper, we prove that$(X,\tau)$and the new topology$(X,\tau_{\Cal E})$have the same semiregularization if${\Cal E}$is a$\pi$-network in$X$with the property${\Cal H}$. Also, we discuss the properties of${\Cal E},\tau_{\Cal E}$and study generalized Volterra spaces and discuss their properties. We show that$\tau_{\Cal E}$coincides with the$\star$-topology for a particular${\Cal E}$. Keywords:$\pi$-network; ideal;$\star$-topology; semiregularization; submaximal and Volterra spaces. MSC: 54A05, 54A10, 54F65, 54E99  Faber polynomial coefficient estimates for a subclass of analytic bi-univalent functions defined by S\u{a}l\u{a}gean differential operator 185$-$193 Serap Bulut Abstract In this work, considering a subclass of analytic bi-univalent functions defined by S\u{a}l\u{a}gean differential operator, we determine estimates for the general Taylor-Maclaurin coefficients of the functions in this class. For this purpose, we use the Faber polynomial expansions. In certain cases, our estimates improve some of existing coefficient bounds. Keywords: Bi-univalent functions; Taylor-Maclaurin series expansion; Coefficient bounds and coefficient estimates; Faber polynomials; S\u{a}l\u{a}gean differential operator. MSC: 30C45, 30C80 $IA$-automorphisms of$p$-groups, finite polycyclic groups and other results 194$-$200 R. G. Ghumde and S. H. Ghate Abstract In this paper, the group$IA(G)$of all$IA$-automorphisms of a group$G$is studied. We prove some results regarding non-triviality, polycyclicity and commutativity of$IA(G)$in addition to proving some basic results. We also prove some results analogous to a result by Schur and a weak form of its converse in the context of$IA$-automorphisms. Keywords: Central automorphism;$IA$-automorphism;$p$-groups. MSC: 20D45, 20D10, 20D15  Fuzzy representable modules and fuzzy attached primes 201$-$211 Naser Zamani, Zeinab Rezaei and Jafar A'zami Abstract Let$M$be a non-zero unitary module over a non-zero commutative ring$R$. A kind of uniqueness theorem for a non-zero fuzzy representable submodule$\mu$of$M$will be proved, and then the set of fuzzy attached primes of$\mu$will be defined. Then among other things, it will be shown that, whenever$R$is Noetherian, a fuzzy prime ideal$\xi$is attached to$\mu$if and only if$\xi$is the annihilator of a fuzzy quotient of$\mu$. The behavior of fuzzy attached primes with fuzzy quotient and fuzzy localization techniques will be studied. Keywords: Fuzzy coprimary submodules; fuzzy coprimary representation; fuzzy attached primes. MSC: 08A72  A note on$I$-convergence and$I^{\star}$-convergence of sequences and nets in topological spaces 212$-$221 Amar Kumar Banerjee and Apurba Banerjee Abstract In this paper, we use the idea of$I$-convergence and$I^{\star}$-convergence of sequences and nets in a topological space to study some important topological properties. Further we derive characterization of compactness in terms of these concepts. We introduce also the idea of$I$-sequentially compactness and derive a few basic properties in a topological space. Keywords:${I}$-convergence;${I}^{\star}$-convergence;${I}$-limit point;${I}$-cluster point;${I}$-sequentially compact. MSC: 54A20, 40A35  On$(n-1,n)$-$\phi$-prime ideals in semirings 222$-$232 Manish Kant Dubey and Poonam Sarohe Abstract Let$S$be a commutative semiring and$T(S)$be the set of all ideals of$S$. Let$\phi\:T(S)\to T(S)\cup \{\emptyset\}$be a function. A proper ideal$I$of a semiring$S$is called an$(n-1,n)$-$\phi$-prime ideal of$S$if$a_{1}a_{2}\cdots a_{n}ın I\setminus \phi(I)$,$a_{1},a_{2},\dots,a_{n}ın S$implies that$a_{1}a_{2}\cdots a_{i-1}a_{i+1}\cdots a_{n}ın I$for some$iın \{1,2,\dots,n\}$. In this paper, we prove several results concerning$(n-1,n)$-$\phi$-prime ideals in a commutative semiring$S$with non-zero identity connected with those in commutative ring theory. Keywords: Semiring;$(n-1,n)$-$\phi$-prime ideal;$\phi$-subtractive ideal;$Q\$-ideal.

MSC: 16Y30, 16Y60