<abstract xmlns="http://www.w3.org/1999/xhtml">Inspired by the papers [2, 10] we will study, on 2-divisible groups that need not be abelian, the alienation problem between Drygas’ and the exponential Cauchy functional equations, which is expressed by the equation
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0002_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>f</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mo>)</mo></mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mn>2</mn><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mrow><mo>-</mo><mi>y</mi></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math><tex-math>f\left( {x + y} \right) + g\left( {x + y} \right)g\left( {x - y} \right) = f\left( x \right)f\left( y \right) + 2g\left( x \right) + g\left( y \right) + g\left( { - y} \right).</tex-math></alternatives></disp-formula>We also consider an analogous problem for Drygas’ and the additive Cauchy functional equations as well as for Drygas’ and the logarithmic Cauchy functional equations. Interesting consequences of these results are presented.</abstract>

Department of Mathematics, Faculty of Sciences

Alienation of Drygas’ and Cauchy’s Functional Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">In this work it was proved Matkowski’s fixed point theorem. The consequences of this theorem are also presented.</abstract>

A New Proof and Consequences of the Fixed Point Theorem of Matkowski

<abstract xmlns="http://www.w3.org/1999/xhtml">For a continuous and positive function <italic>ω</italic> (<italic>λ</italic>); <italic>λ</italic>> 0 and <italic>μ</italic> a positive measure on [0; ∞) we consider the following 𝒟-<italic>logarithmic integral transform</italic><disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mi>ℒ</mi><mi>o</mi><mi>g</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>0</mn><mo>∞</mo></msubsup><mrow><mi>w</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mn>1</mn><mtext>n</mtext><mrow><mo>(</mo><mrow><mfrac><mrow><mi>λ</mi><mo>+</mo><mi>T</mi></mrow><mi>λ</mi></mfrac></mrow><mo>)</mo></mrow><mi>d</mi><mi>μ</mi><mrow><mo>(</mo><mi>λ</mi><mo>)</mo></mrow><mo>,</mo></mrow></mrow></mrow></math><tex-math>\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( T \right): = \int_0^\infty {w\left( \lambda \right)1{\rm{n}}\left( {{{\lambda + T} \over \lambda }} \right)d\mu \left( \lambda \right),}</tex-math></alternatives></disp-formula>
where the integral is assumed to exist for <italic>T</italic> a positive operator on a complex Hilbert space <italic>H</italic>.We show among others that, if <italic>A</italic>, <italic>B ></italic> 0 with <italic>BA</italic> + <italic>AB</italic> ≥ 0, then
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>𝒟</mi><mi>ℒ</mi><mi>o</mi><mi>g</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>+</mo><mi>𝒟</mi><mi>ℒ</mi><mi>o</mi><mi>g</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>≥</mo><mi>𝒟</mi><mi>ℒ</mi><mi>o</mi><mi>g</mi><mrow><mo>(</mo><mrow><mi>w</mi><mo>,</mo><mi>μ</mi></mrow><mo>)</mo></mrow><mrow><mo>(</mo><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mo>)</mo></mrow><mo>.</mo></mrow></math><tex-math>\mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( A \right) + \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( B \right) \ge \mathcal{D}\mathcal{L}og\left( {w,\mu } \right)\left( {A + B} \right).</tex-math></alternatives></disp-formula>In particular we have
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_003.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mrow><mi>π</mi></mrow><mn>2</mn></msup><mo>+</mo><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow><mo>)</mo></mrow><mo>≥</mo><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>+</mo><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mi>B</mi><mo>)</mo></mrow><mo>,</mo></mrow></math><tex-math>{1 \over 6}{\pi ^2} + {\rm{di}}\log \left( {A + B} \right) \ge {\rm{di}}\log \left( A \right) + {\rm{di}}\log \left( B \right),</tex-math></alternatives></disp-formula>
where the <italic>dilogarithmic function</italic> dilog : [0; ∞) → ℝ is defined by
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_amsil-2021-0004_eq_004.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mtext>di</mtext><mo>log</mo><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>1</mn><mi>t</mi></msubsup><mrow><mfrac><mrow><mn>1</mn><mi>n</mi><mi>s</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>s</mi></mrow></mfrac><mi>d</mi><mi>s</mi><mo>,</mo></mrow></mrow><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>t</mi><mo>≥</mo><mn>0.</mn></mrow></math><tex-math>{\rm{di}}\log \left( t \right): = \int_1^t {{{1ns} \over {1 - s}}ds,} \,\,\,\,t \ge 0.</tex-math></alternatives></disp-formula>Some examples for integral transform 𝒟𝒧<italic>og</italic> (˙;˙) related to the operator monotone functions are also provided.</abstract>

Operator Subadditivity of the 𝒟-Logarithmic Integral Transform for Positive Operators in Hilbert Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known recursive sequences, as the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, etc. We establish some general limit formulas, where the product of the first <italic>n</italic> terms of these sequences appears. Furthermore, we prove some general limits that connect these sequences to the number <italic>e</italic>(≈ 2:71828:::).</abstract>

A Note on Two Fundamental Recursive Sequences

<abstract xmlns="http://www.w3.org/1999/xhtml">We introduce the notion of an (<italic>α, β, γ</italic>) triple system, which generalizes the familiar generalized Jordan triple system related to a construction of simple Lie algebras. We then discuss its realization by considering some bilinear algebras and vice versa. Next, as a new concept, we study triality relations (a triality group and a triality derivation) associated with these triple systems; the relations are a generalization of the automorphisms and derivations of the triple systems. Also, we provide examples of several involutive triple systems with a tripotent element.</abstract>

Triality Groups Associated with Triple Systems and their Homotope Algebras

<abstract xmlns="http://www.w3.org/1999/xhtml">We consider the MHD system in a bounded domain Ω ⊂ ℝ<italic>N</italic>, <italic>N</italic> = 2; 3, with Dirichlet boundary conditions. Using Dan Henry’s semigroup approach and Giga–Miyakawa estimates we construct global in time, unique solutions to fractional approximations of the MHD system in the base space (<italic>L</italic>2(Ω
))<italic>N</italic> × (<italic>L</italic>2(Ω
))<italic>N</italic>. Solutions to MHD system are obtained next as a limits of that fractional approximations.</abstract>

MHD Equations in a Bounded Domain

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper our considerations are focused on some Markov chain associated with certain piecewise-deterministic Markov process with a statedependent jump intensity for which the exponential ergodicity was obtained in [4]. Using the results from [3] we show that the law of iterated logarithm holds for such a model.</abstract>

The Law of the Iterated Logarithm for Random Dynamical System with Jumps and State-Dependent Jump Intensity

<abstract xmlns="http://www.w3.org/1999/xhtml">We establish necessary and sufficient conditions allowing separation of pair of real functions by an <italic>m</italic>-convex and by an <italic>m</italic>-affine function. Some examples and a geometric interpretation of <italic>m</italic>-convexity of a function is exhibited, as well as a Jensen’s inequality for this kind of function.</abstract>

Sandwich Type Results For <italic>m</italic>-Convex Real Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.</abstract>

Departamento de Análisis Matemático, Instituto de Matemáticas

Department of Mathematics and Informatics

The African University Ahmed Draia of Adrar

Some Existence Results for Systems of Impulsive Stochastic Differential Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">Ring properties of amalgamated products are investigated. We offer new, elementary arguments which extend results from [5] and [12] to noncommutative setting and also give new properties of amalgamated rings.</abstract>

A Note on Amalgamated Rings Along an Ideal

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we introduce some new types of contractive mappings by combining Caristi contraction, Ćirić-quasi contraction and weak contraction in the framework of a metric space. We prove some fixed point theorems for such type of mappings over complete metric spaces with the help of <italic>φ</italic>-diminishing property. Some examples are given in strengthening the hypothesis of our established theorems.</abstract>

Some Fixed Point Theorems Via Combination of Weak Contraction and Caristi Contractive Mapping

<abstract xmlns="http://www.w3.org/1999/xhtml">The convergence of sequences and non-unique fixed points are established in ℳ-orbitally complete cone metric spaces over the strongly minihedral cone, and scalar weighted cone assuming the cone to be strongly minihedral. Appropriate examples and applications validate the established theory. Further, we provide one more answer to the question of the existence of the contractive condition in Cone metric spaces so that the fixed point is at the point of discontinuity of a map. Also, we provide a negative answer to a natural question of whether the contractive conditions in the obtained results can be replaced by its metric versions.</abstract>

Results in Strongly Minihedral Cone and Scalar Weighted Cone Metric Spaces and Applications

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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 