Abstract

In this article we generalize some definitions and results from ideals in rings to ideals in semirings. Let $R$ be a commutative semiring with identity. Let $\phi \:\vartheta (R)\rightarrow \vartheta (R)\cup \{\emptyset \}$ be a function, where $\vartheta (R)$ denotes the set of all ideals of $R$. A proper ideal $I\in \vartheta (R)$ is called $\phi$-prime ideal if $ra\in I-\phi(I)$ implies $r\in I$ or $a\in I$. An element $a\in R$ is called $\phi $-prime to $I$ if $ra\in I-\phi (I)$ (with $r\in R$) implies that $r\in I$. We denote by $p(I)$ the set of all elements of $R$ that are not $\phi$-prime to $I$. $I$ is called a $\phi$-primal ideal of $R$ if the set $P=p(I)\cup \phi(I)$ forms an ideal of $R$. Throughout this work, we define almost primal and $\phi$-primal ideals, and we also show that they enjoy many of the properties of primal ideals.

Keywords: Primal ideal; $\phi$-prime ideal; weakly primal ideal; $\phi$-primal ideal.

MSC: 13A15, 16Y60

Abstract

The property ($gR$), introduced in [Aiena, P., Guillen, J. and Pe\~{n}a, P., {\it Property ($gR$) and perturbations}, to appear in Acta Sci. Math. (Szeged), 2012], is an extension to the context of B-Fredholm theory, of property ($R$), introduced in [Aiena, P., Guillen, J. and Pe\~{n}a, P., {\it Property ($R$) for bounded linear operators}, Mediterr. J. Math. {\bf 8} (4), 491-508, 2011]. In this paper we continue the study of property ($gR$) and we consider its preservation under perturbations by finite rank and nilpotent operators. We also prove that if $T$ is left polaroid (resp\. right polaroid) and $N$ is a nilpotent operator which commutes with $T$ then $T+N$ is also left polaroid (resp\. right polaroid).

Keywords: Property ($gR$); semi B-Fredholm operator; perturbation.

MSC: 47A10, 47A11, 47A53, 47A55

Abstract

Let $P(z)$ be a polynomial and $P'(z)$ its derivative. In this paper, we shall obtain certain compact generalizations and sharp refinements of some results of Govil, Malik, Turán and others concerning the maximum modulus of $P(z)$ and $P'(z)$ on the unit circle $|z|=1$, which also yields a number of other interesting results for various choices of parameters.

Keywords: Inequalities; zeros of polynomials; derivative; maximum modulus.

MSC: 30C10, 30C15

Abstract

In the present study, a new theorem on the degree of approximation of function $\tilde{f}$, conjugate to a periodic function $f$ belonging to weighted $W(L_r,\xi(t))$-class using semi-monotonicity on the generating sequence $\{p_n\}$ has been established.

Keywords: Generalized Lipschitz $W(L_r,\xi(t))$-class of functions; conjugate Fourier series; degree of approximation; $C^1$ means; $N_p$ means; product summability $C^{1}\cdot N_{p}$ transform.

MSC: 40G05, 42B05, 42B08

Abstract

In this paper we extend the notion of equal convergence of Császár and Laczkovich with the help of ideals of the set of positive integers and introduce the ideas of $\Cal{I}$ and $\Cal{I}^*$-equal convergence and prove certain properties. Throughout the investigation two classes of ideals, one satisfying ``Chain Condition'' and another called $P$-ideals play a very important role. We also introduce certain related notions of convergence and prove an Egoroff-type theorem for $\Cal{I}^*$-equal convergence.

Keywords: Ideal; filter; $\Cal{I}$ and $\Cal{I}^*$-equal convergence; $P$-ideal; Chain condition; $\Cal{I}^*$-uniform equal convergence; $\Cal{I}^*$-almost uniform equal convergence; $\Cal{I}^*$-quasi vanishing restriction; Egoroff Theorem.

MSC: 40G15, 40A99, 46A99

Abstract

The purpose of this paper is to introduce the set-valued Prešić-Ćirić type contraction in 0-complete partial metric spaces and to prove some coincidence and common fixed point theorems for such mappings in product spaces, in partial metric case. Results of this paper extend, generalize and unify several known results in metric and partial metric spaces. An example shows how the results of this paper can be used while the existing one cannot.

Keywords: Set-valued mapping; partial metric space; fixed point; Prešić type mapping.

MSC: 47H10, 54H25

Abstract

T. G. Bhaskar and V. Lakshmikantham [Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006) 1379--1393], V. Lakshmikantham and Lj\. B. Ćirić [Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70 (2009) 4341--4349] introduced the concept of a coupled coincidence point of a mapping $F$ from $X\times X$ into $X$ and a mapping $g$ from $X$ into $X$. In this paper we prove a coupled coincidence fixed point theorem in the setting of a generalized $b$-metric space. Three examples are presented to verify the effectiveness and applicability of our main result.

Keywords: Common fixed point; coupled coincidence fixed point; $b$-metric space; $G$-metric space; generalized $b$-metric space.

MSC: 54H25, 47H10, 54E50

Abstract

In this paper we derive a new companion of Grüss' type inequality for Riemann-Stieltjes integral. Applications to the approximation problem of the Riemann-Stieltjes are also pointed out.

Keywords: Ostrowski's inequality; bounded variation; Riemann-Stieltjes integral.

MSC: 26D15, 26D20, 41A55

Abstract

In this paper, we study the growth and the oscillation of solutions of linear difference equations with meromorphic coefficients. Also, we investigate the growth of difference polynomials generated by meromorphic solutions of some difference equations. We improve and generalize some results due to Z. X. Chen, I. Laine and C. C. Yang.

Keywords: Linear difference equations; meromorphic function; order of growth; type of meromorphic function.

MSC: 39A10, 30D35, 39A12

Abstract

We introduce a new iteration process and prove that it is faster than all of Picard, Mann and Agarwal et al\. processes. We support analytic proof by a numerical example. Our process is independent of all three processes just mentioned. We also prove some weak and strong convergence theorems for two nonexpansive mappings. Moreover, we apply our results to find solutions of constrained minimization problems and feasibility problems.

Keywords: Nonexpansive mapping; weak convergence; strong convergence; rate of convergence; iteration process.

MSC: 47H05, 49M05