<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we consider the hypo-<italic>q</italic>-operator norm and hypo-<italic>q-</italic>numerical radius on a Cartesian product of algebras of bounded linear operators on Banach spaces. A representation of these norms in terms of semi-inner products, the equivalence with the <italic>q</italic>-norms on a Cartesian product and some reverse inequalities obtained via the scalar reverses of Cauchy-Buniakowski-Schwarz inequality are also given.</abstract>

Hypo-<italic>q</italic>-Norms on a Cartesian Product of Algebras of Operators on Banach Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>{rn}n</italic>∈𝕅 be a strictly increasing sequence of nonnegative real numbers satisfying the asymptotic formula <italic>rn ~ αβn</italic>, where <italic>α, β</italic> are real numbers with <italic>α ></italic> 0 and <italic>β ></italic> 1. In this note we prove some limits that connect this sequence to the number <italic>e</italic>. We also establish some asymptotic formulae and limits for the counting function of this sequence. All of the results are applied to some well-known sequences in mathematics.</abstract>

Notes on a General Sequence

<abstract xmlns="http://www.w3.org/1999/xhtml">This work presents a numerical comparison between two efficient methods namely the fractional natural variational iteration method (FNVIM) and the fractional natural homotopy perturbation method (FNHPM) to solve a certain type of nonlinear Caputo time-fractional partial differential equations in particular, nonlinear Caputo time-fractional wave-like equations with variable coefficients. These two methods provided an accurate and efficient tool for solving this type of equations. To show the efficiency and capability of the proposed methods we have solved some numerical examples. The results show that there is an excellent agreement between the series solutions obtained by these two methods. However, the FNVIM has an advantage over FNHPM because it takes less time to solve this type of nonlinear problems without using He’s polynomials. In addition, the FNVIM enables us to overcome the diffi-culties arising in identifying the general Lagrange multiplier and it may be considered as an added advantage of this technique compared to the FNHPM.</abstract>

Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">In this study, we give two sequences <italic>{L</italic>+<italic>n}n≥</italic>1 and <italic>{L−n}n≥</italic>1 derived by altering the Lucas numbers with {<italic>±</italic>1, <italic>±</italic>3}, terms of which are called as altered Lucas numbers. We give relations connected with the Fibonacci <italic>Fn</italic> and Lucas <italic>Ln</italic> numbers, and construct recurrence relations and Binet’s like formulas of the <italic>L</italic>+<italic>n</italic> and <italic>L−n</italic> numbers. It is seen that the altered Lucas numbers have two distinct factors from the Fibonacci and Lucas sequences. Thus, we work out the greatest common divisor (<italic>GCD</italic>) of <italic>r</italic>-consecutive altered Lucas numbers. We obtain <italic>r</italic>-consecutive <italic>GCD</italic> sequences according to the altered Lucas numbers, and show that their <italic>GCD</italic> sequences are unbounded or periodic in terms of values <italic>r</italic>.</abstract>

The GCD Sequences of the Altered Lucas Sequences

<abstract xmlns="http://www.w3.org/1999/xhtml">In this research we introduce the concept of <italic>m</italic>-convex function of higher order by means of the so called <italic>m</italic>-divided difference; elementary properties of this type of functions are exhibited and some examples are provided.</abstract>

<italic>m</italic>-Convex Functions of Higher Order

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article, we will define Padovan’s hybrid numbers, based on the new noncommutative numbering system studied by Özdemir ([7]). Such a system that is a set involving complex, hyperbolic and dual numbers. In addition, Padovan’s hybrid numbers are created by combining this set, satisfying the relation <italic>ih</italic> = <italic>−hi</italic> = <italic>ɛ</italic> + <italic>i</italic>. Given this, some properties and identities are shown for these numbers, such as Binet’s formula, generating matrix, characteristic equation, norm, and generating function. In addition, these numbers are extended to the integer field and some identities are made.</abstract>

University of Trás-os-Montes and Alto Douro

The Hybrid Numbers of Padovan and Some Identities

<abstract xmlns="http://www.w3.org/1999/xhtml">A general fixed point theorem for two pairs of absorbing mappings satisfying a new type of implicit relation ([37]), without weak compatibility in <italic>Gp</italic>-metric spaces is proved. As applications, new results for mappings satisfying contractive conditions of integral type and for <italic>ϕ</italic>-contractive mappings are obtained.</abstract>

A General Fixed Point Theorem for Two Pairs of Absorbing Mappings in <italic>Gp</italic> -Metric Spaces

Report of Meeting. The Twentieth Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), January 29–February 1, 2020

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Annales Mathematicae Silesianae

Annales Mathematicae Silesianae publishes significant research and survey papers from all branches of pure and applied mathematics, and reports of meetings. It welcomes contributed papers that develop important, new mathematical ideas and results or solve outstanding problems.  Article processing chargesThe journal does not have article processing charges (APCs) nor article submission charges.  Archiving  Sciendo archives the contents of this journal in Portico - digital long-term preservation service of scholarly books, journals and collections.  Plagiarism Policy  The editorial board is participating in a growing community of Similarity Check System's users in order to ensure that the content published is original and trustworthy. Similarity Check is a medium that allows for comprehensive manuscripts screening, aimed to eliminate plagiarism and provide a high standard and quality peer-review process.  Peer Review process  Annales Mathematicae Silesianae is a single-blind peer reviewed journal.Normally the papers are peer reviewed by two independent referees who are experts in the appropriate branch of mathematics. According to their opinions the manuscripts are either accepted or rejected. In the case their opinions contradict each other we send the paper to the third reviewer.Reviewers are asked to provide detailed, constructive comments that will help both the editors to make a decision on the publication and the author(s) to improve their manuscript. Preparing the review the reviewers may use the following scheme: review_scheme.pdf. Reviewers are also asked to declare that there is no conflict of interests between them and the authors.  Target Group  - researchers in all branches of pure and applied mathematics 

Annales Mathematicae Silesianae

Volume 34 (2020): Issue 2 (September 2020)

Hypo-q-Norms on a Cartesian Product of Algebras of Operators on Banach Spaces

Notes on a General Sequence

Numerical Comparison of FNVIM and FNHPM for Solving a Certain Type of Nonlinear Caputo Time-Fractional Partial Differential Equations

The GCD Sequences of the Altered Lucas Sequences

m-Convex Functions of Higher Order

The Hybrid Numbers of Padovan and Some Identities

A General Fixed Point Theorem for Two Pairs of Absorbing Mappings in G_p -Metric Spaces

Report of Meeting. The Twentieth Debrecen–Katowice Winter Seminar on Functional Equations and Inequalities Hajdúszoboszló (Hungary), January 29–February 1, 2020

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