Zygfryd Kominek, a Mathematician, a Teacher, a Friend

<abstract xmlns="http://www.w3.org/1999/xhtml">We consider the following families of real-valued functions defined on 𝕉2: feebly continuous functions (FC), very feebly continuous functions (VFC), and two-feebly continuous functions (TFC). It is known that the inclusions FC ⊂ VFC ⊂ TFC are proper. We study pointwise and uniform limits of sequences with terms taken from these families.</abstract>

Faculty of Mathematics, Physics, and Informatics

Limits of Sequences of Feebly-Type Continuous Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">The paper consists of two parts. At first, assuming that (Ω, <italic>A, P</italic>) is a probability space and (<italic>X, ϱ</italic>) is a complete and separable metric space with the σ-algebra 𝒝 of all its Borel subsets we consider the set 𝒭<italic>c</italic> of all 𝒝 ⊗ 𝒜<italic>-</italic>measurable and contractive in mean functions <italic>f</italic> : <italic>X ×</italic> Ω → <italic>X</italic> with finite integral ∫ Ω<italic>ϱ</italic> (<italic>f</italic>(<italic>x, ω</italic>), <italic>x</italic>) <italic>P</italic> (<italic>dω</italic>) for <italic>x</italic> ∈ <italic>X</italic>, the weak limit π <italic>f</italic> of the sequence of <italic>iterates</italic> of <italic>f</italic> ∈ 𝒭<italic>c</italic>, and investigate continuity-like property of the function <italic>f ↦</italic> π<italic>f, f</italic> ∈ 𝒭<italic>c</italic>, and Lipschitz solutions <italic>φ</italic> that take values in a separable Banach space of the equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0015_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mrow><msub><mo>∫</mo><mi>Ω</mi></msub><mrow><mi>φ</mi><mrow><mo>(</mo><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>ω</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mi>P</mi><mrow><mo>(</mo><mrow><mi>d</mi><mi>ω</mi></mrow><mo>)</mo></mrow></mrow></mrow><mo>+</mo><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>.</mo></mrow></math><tex-math>\varphi \left( x \right) = \int_\Omega {\varphi \left( {f\left( {x,\omega } \right)} \right)P\left( {d\omega } \right)} + F\left( x \right).</tex-math></alternatives></disp-formula>Next, assuming that <italic>X</italic> is a real separable Hilbert space, Λ: <italic>X</italic> → <italic>X</italic> is linear and continuous with ||Λ || <italic>&lt;</italic> 1, and <italic>µ</italic> is a probability Borel measure on <italic>X</italic> with finite first moment we examine continuous at zero solutions <italic>φ</italic> : <italic>X</italic> → 𝔺 of the equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2019-0015_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>φ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mover><mi>μ</mi><mo>⌢</mo></mover><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>φ</mi><mrow><mo>(</mo><mrow><mi>Λ</mi><mi>x</mi></mrow><mo>)</mo></mrow></mrow></math><tex-math>\varphi \left( x \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over \mu } \left( x \right)\varphi \left( {\Lambda x} \right)</tex-math></alternatives></disp-formula>which characterizes the limit distribution π <italic>f </italic>for some special <italic>f</italic> ∈ 𝒭<italic>c</italic>.</abstract>

Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">We show that a large class of summable ideals can be defined using a certain kind of “sparseness” of subsets of the line near zero, but it is still an open question whether this gives a characterization of the whole class.</abstract>

Some Kinds of Sparseness on the Real Line and Ideals on <italic>ω</italic>

<abstract xmlns="http://www.w3.org/1999/xhtml">We present three different types of bijective functions <italic>f</italic> : <italic>I → I</italic> on a compact interval <italic>I</italic> with finitely many discontinuities where certain iterates of these functions will be continuous. All these examples are strongly related to permutations, in particular to derangements in the first case, and permutations with a certain number of successions (or small ascents) in the second case. All functions of type III form a direct product of a symmetric group with a wreath product. It will be shown that any iterative root <italic>F</italic> : <italic>J → J</italic> of the identity of order <italic>k</italic> on a compact interval <italic>J</italic> with finitely many discontinuities is conjugate to a function <italic>f</italic> of type III, i.e., <italic>F</italic> = <italic>φ−</italic>1 ∘ <italic>f</italic> ∘ <italic>φ</italic> where <italic>φ</italic> is a continuous, bijective, and increasing mapping between <italic>J</italic> and [0, <italic>n</italic>] for some integer <italic>n</italic>.</abstract>

On Iteration of Bijective Functions with Discontinuities

<abstract xmlns="http://www.w3.org/1999/xhtml">Some differentiability results from the paper of D.Ş. Marinescu &amp; M. Monea [7] on delta-convex mappings, obtained for real functions, are extended for mappings with values in a normed linear space. In this way, we are nearing the completion of studies established in papers [2], [5] and [7].</abstract>

Delta-Convexity With Given Weights

<abstract xmlns="http://www.w3.org/1999/xhtml">Using the Borel classification of set-valued maps, we present here some new results on set-valued maps which are similar to some of the well known theorems on functions due to Lebesgue and Kuratowski. We consider set-valued maps of two variables in perfectly normal topological spaces. It was proved in [11] that a set-valued map lower semicontinuous (i.e. of lower Borel class 0) in the first and upper semicontinuous (i.e. of upper Borel class 0) in the second variable is of upper Borel class 1 and also (with stronger assumptions) of lower Borel class 1. This result cannot be generalized into higher Borel classes. In this paper we show that a set-valued map of the upper (resp. lower) Borel class α in the first and lower semicontinuous and upper quasicontinuous (upper semicontinuous and lower quasicontinuous) in the second variable is of the lower (resp. upper) Borel class α + 1. Also other cases are considered.</abstract>

On the Borel Classes of Set-Valued Maps of Two Variables

<abstract xmlns="http://www.w3.org/1999/xhtml">Given a continuous strictly monotonic real-valued function α, defined on an interval <italic>I</italic>, and a function <italic>ω</italic> : <italic>I →</italic> (0, +∞) we denote by <italic>B</italic>α<italic>ω</italic> the Bajraktarević mean generated by α and weighted by <italic>ω</italic>:<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0012_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>B</mi><mi>ω</mi><mi>α</mi></msubsup><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><msup><mi>α</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mrow><mfrac><mrow><mi>ω</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>ω</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>ω</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></mfrac><mi>α</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mfrac><mrow><mi>ω</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mrow><mi>ω</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>ω</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></mfrac><mi>α</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>I</mi><mo>.</mo></mrow></math><tex-math>B_\omega ^\alpha \left({x,y} \right) = {\alpha ^{- 1}}\left({{{\omega \left(x \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(x \right) + {{\omega \left(y \right)} \over {\omega \left(x \right) + \omega \left(y \right)}}\alpha \left(y \right)} \right),\,\,\,x,y \in I.</tex-math></alternatives></disp-formula>We find a necessary integral formula for all possible three times differentiable solutions (<italic>φ, ψ</italic>) of the functional equation<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0012_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msubsup><mi>B</mi><mi>s</mi><mi>ϕ</mi></msubsup><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>r</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><msubsup><mi>B</mi><mi>t</mi><mi>ψ</mi></msubsup><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>x</mi><mo>+</mo><mi>r</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mi>y</mi><mo>,</mo></mrow></math><tex-math>r\left(x \right)B_s^\varphi \left({x,y} \right) + r\left(y \right)B_t^\psi \left({x,y} \right) = r\left(x \right)x + r\left(y \right)y,</tex-math></alternatives></disp-formula>where <italic>r, s, t</italic> : <italic>I →</italic> (0, +∞) are three times differentiable functions and the first derivatives of <italic>φ, ψ</italic> and <italic>r</italic> do not vanish. However, we show that not every pair (<italic>φ, ψ</italic>) given by the found formula actually satisfies the above equation.</abstract>

On a Functional Equation Appearing on the Margins of a Mean Invariance Problem

<abstract xmlns="http://www.w3.org/1999/xhtml">Under some simple conditions on the real functions <italic>f</italic> and <italic>g</italic> defined on an interval <italic>I</italic> ⊂ (0, <italic>∞</italic>), the two-place functions <italic>Af</italic> (<italic>x, y</italic>) = <italic>f</italic> (<italic>x</italic>) + <italic>y − f</italic> (<italic>y</italic>) and <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2020-0015_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>G</mi><mi>g</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></mfrac><mi>y</mi></mrow></math><tex-math>{G_g}\left({x,y} \right) = {{g\left(x \right)} \over {g\left(y \right)}}y</tex-math></alternatives></inline-formula> generalize, respectively, <italic>A</italic> and <italic>G</italic>, the classical weighted arithmetic and geometric means. In this note, basing on the invariance identity <italic>G</italic> ∘ (<italic>H, A</italic>) = <italic>G</italic> (equivalent to the Pythagorean harmony proportion), a suitable weighted extension <italic>Hf,g</italic> of the classical harmonic mean <italic>H</italic> is introduced. An open problem concerning the symmetry of <italic>Hf,g</italic> is proposed. As an application a method of effective solving of some functional equations involving means is presented.</abstract>

Generalization of the Harmonic Weighted Mean Via Pythagorean Invariance Identity and Application

<abstract xmlns="http://www.w3.org/1999/xhtml">Counterparts of the Ohlin and Levin–Stečkin theorems for strongly convex functions are proved. An application of these results to obtain some known inequalities related with strongly convex functions in an alternative and unified way is presented.</abstract>

Ohlin and Levin–Stečkin-Type Results for Strongly Convex Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">In the present paper we establish necessary and sufficient conditions under which two functions can be separated by a delta-convex function. This separation will be understood as a separation with respect to the partial order generated by the Lorentz cone. An application to a stability problem for delta-convexity is also given.</abstract>

On a Separation Theorem for Delta-Convex Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">The main goal of this paper is to give a completely elementary proof for the decomposition theorem of Wright convex functions which was discovered by C. T. Ng in 1987. In the proof, we do not use transfinite tools, i.e., variants of Rodé’s theorem, or de Bruijn’s theorem related to functions with continuous differences.</abstract>

An Elementary Proof for the Decomposition Theorem of Wright Convex Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">In [12] a close connection between stability results for the Cauchy equation and the completion of a normed space over the rationals endowed with the usual absolute value has been investigated. Here similar results are presented when the valuation of the rationals is a <italic>p</italic>-adic valuation. Moreover a result by Z<sc>ygfryd</sc> K<sc>ominek</sc> ([5]) on the stability of the Pexider equation is formulated and proved in the context of Banach spaces over the field of <italic>p</italic>-adic numbers.</abstract>

Connections Between the Completion of Normed Spaces Over Non-Archimedean Fields and the Stability of the Cauchy Equation

<abstract xmlns="http://www.w3.org/1999/xhtml">W. Orlicz in 1951 has observed that if {<italic>fn</italic>(<italic>·, y</italic>)}<italic>n</italic>∈<italic>N</italic> converges in measure to <italic>f</italic>(<italic>·, y</italic>) for each <italic>y</italic> ∈ [0, 1], then {<italic>fn</italic>}<italic>n</italic>∈<italic>N</italic> converges in measure to <italic>f</italic> on [0, 1] <italic>×</italic> [0, 1]. The situation is different for the convergence in category even if we assume the convergence in category of sequences {<italic>fn</italic>(<italic>·, y</italic>)}<italic>n</italic>∈<italic>N</italic> for each <italic>y</italic> ∈ [0, 1] and <italic>{fn</italic>(<italic>x, ·</italic>)<italic>}n</italic>∈<italic>N</italic> for each <italic>x</italic> ∈ [0, 1].</abstract>

Convergence in Measure and in Category

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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 