Preface

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this review paper, we recall, in a unifying manner, our recent results concerning the Lie symmetries of nonlinear Fokker-Plank equations, associated to the (weighted) Tsallis and Kaniadakis entropies. The special values of the Tsallis parameters, highlighted by the classification of these symmetries, clearly indicate algebraic and geometric invariants which differentiate the Lie algebras involved. We compare these values with the ones previously obtained by several authors, and we try to establish connections between our theoretical families of entropies and specific entropies arising in several applications found in the literature.
We focus on the discovered correlations, but we do not neglect dissimilarities, which might provide -in the future-deeper details for an improved extended panorama of the Tsallis entropies.
</abstract>

Entropy- A Tale of Ice and Fire

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper we prove among others that, if (<italic>Aj</italic>)<italic>j</italic>=1,...,<italic>m</italic> are positive definite matrices of order <italic>n</italic> ≥ 2 and <italic>qj</italic> ≥ 0, <italic>j</italic> = 1, ..., <italic>m</italic> with <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2023-0003_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M1"><mstyle displaystyle="true"><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><mrow><msub><mi>q</mi><mi>j</mi></msub><mo>=</mo><mn>1</mn></mrow></mstyle></math><tex-math>$$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$</tex-math></alternatives></inline-formula>, then
<disp-formula id="j_awutm-2023-0003_eq_002"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2023-0003_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" id="M2"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mn>0</mn></mtd><mtd columnalign="left"><mrow><mo>≤</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>−</mo><msub><mrow><mi>min</mi></mrow><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></mrow></msub><mrow><mo>{</mo> <mrow><msub><mi>q</mi><mi>i</mi></msub></mrow> <mo>}</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>×</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo> <mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>q</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><munder><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mi>m</mi></mrow></munder><msub><mi>q</mi><mi>i</mi></msub><msub><mi>q</mi><mi>j</mi></msub><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>+</mo><msub><mi>A</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow> <mo>]</mo></mrow></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>≤</mo></mtd><mtd columnalign="left"><mrow><mfrac><mn>1</mn><mrow><msub><mrow><mi>min</mi></mrow><mrow><mi>i</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>}</mo></mrow></msub><mrow><mo>{</mo> <mrow><msub><mi>q</mi><mi>i</mi></msub></mrow> <mo>}</mo></mrow></mrow></mfrac></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mo>×</mo></mtd><mtd columnalign="left"><mrow><mrow><mo>[</mo> <mrow><munderover><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><msub><mi>q</mi><mi>i</mi></msub><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msub><mi>q</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mn>2</mn><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><munder><mrow><msup><mo>∑</mo><mtext>​</mtext></msup></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>&lt;</mo><mi>j</mi><mo>≤</mo><mi>m</mi></mrow></munder><msub><mi>q</mi><mi>i</mi></msub><msub><mi>q</mi><mi>j</mi></msub><msup><mrow><mrow><mo>[</mo> <mrow><mi>det</mi><mrow><mo>(</mo><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>+</mo><msub><mi>A</mi><mi>j</mi></msub></mrow><mo>)</mo></mrow></mrow> <mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow> <mo>]</mo></mrow><mo>.</mo></mrow></mtd></mtr></mtable></math>
<tex-math><?CDATA \begin{eqnarray}0 &#x0026; \le \frac{1}{1-{\min }_{i\in \{1,\ldots ,m\}}\{{q}_{i}\}}\\ \times &#x0026; [\underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}(1-{q}_{i}){[\det ({A}_{i})]}^{-1}-{2}^{n+1}\mathop{{\sum }^{\text{}}}\limits_{1\le i\lt j\le m}{q}_{i}{q}_{j}{[\det ({A}_{i}+{A}_{j})]}^{-1}]\\ \le &#x0026; \underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}{[\det ({A}_{i})]}^{-1}-{[\det (\underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}{A}_{i})]}^{-1}\\ \le &#x0026; \frac{1}{{\min }_{i\in \{1,\ldots ,m\}}\{{q}_{i}\}}\\ \times &#x0026; [\underset{i=1}{\overset{m}{{\sum }^{\text{}}}}{q}_{i}(1-{q}_{i}){[\det ({A}_{i})]}^{-1}-{2}^{n+1}\mathop{{\sum }^{\text{}}}\limits_{1\le i\lt j\le m}{q}_{i}{q}_{j}{[\det ({A}_{i}+{A}_{j})]}^{-1}].\end{eqnarray}?></tex-math></alternatives></disp-formula>
</abstract>

Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

<abstract xmlns="http://www.w3.org/1999/xhtml">

The aim of this paper is to study the existence of the unique mild solution for non-linear fractional integro-differential equations with state-dependent nonlocal condition. The result was obtained by using nonlinear alternative of Granas-Frigon for contraction in Fréchet spaces. To illustrate the result, an example is provided.
</abstract>

Uniqueness Results for Fractional Integro-differential Equations with State-Dependent Nonlocal Conditions in Fréchet Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

Here we examine the multivariate quantitative approximations of Banach space valued continuous multivariate functions on a box or ℝ<italic>N</italic> , <italic>N</italic> ∈ ℕ, by the multivariate normalized, quasi-interpolation, Kantorovich type and quadrature type neural network operators. We treat also the case of approximation by iterated operators of the last four types. These approximations are derived by establishing multidimensional Jackson type inequalities involving the multivariate modulus of continuity of the engaged function or its high order Fréchet derivatives. Our multivariate operators are defined by using a multidimensional density function induced by a hyperbolic tangent like sigmoid function. The approximations are pointwise and uniform. The related feed-forward neural network is with one hidden layer.
</abstract>

Hyperbolic Tangent Like Relied Banach Space Valued Neural Network Multivariate Approximations

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper a new method to compute approximate analytical solutions for differential equations of fractional order is presented. The proposed computational method, called the Piecewise Polynomial Least Squares Method (PWPLSM), is a combination of the Polynomial Least Squares Method and of piecewise-defined functions. Numerical results for differential equations of different fractional orders are discussed. The approximate solutions obtained with PWPLSM are compared with other existing analytical and numerical solutions. The tables and figures included demonstrate the accuracy of the new method.
</abstract>

Piecewise Polynomial Least Squares Method for Differential Equations of Fractional Order

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper, we study the following fractional differential equation involving the Atangana-Baleanu-Caputo fractional derivative: <disp-formula id="j_awutm-2023-0007_eq_001"><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_awutm-2023-0007_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" id="M1"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mi>B</mi><msub><mi>C</mi><mi>a</mi></msub><msubsup><mi>D</mi><mi>τ</mi><mi>θ</mi></msubsup><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo>,</mo><mi>x</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mi>G</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo>,</mo><mi>x</mi><mo stretchy="false">(</mo><mi>ϑ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mi>ϑ</mi><mo>∈</mo><mi>J</mi><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">]</mo><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>x</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>φ</mi><mi>a</mi></msub><mo>∈</mo><mi>ℝ</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math><tex-math>$$\left\{ {\matrix{ {AB{C_a}D_\tau ^\theta [x(\vartheta ) - F(\vartheta ,x(\vartheta ))] = G(\vartheta ,x(\vartheta )),\;\;\;{\kern 1pt} \vartheta \in J: = [a,b],} \hfill \cr {x(a) = {\varphi _a} \in .} \hfill \cr } } \right.$$</tex-math></alternatives></disp-formula>
The result is based on a Dhage fixed point theorem. Further, an example is provided for the justification of our main result.
</abstract>

Lakshmikantham Monotone Iterative Principle for Hybrid Atangana-Baleanu-Caputo Fractional Differential Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this paper we consider three concepts of uniform dichotomy in mean for stochastic skew-evolution semiflows: uniform exponential dichotomy in mean, uniform polynomial dichotomy in mean and uniform <italic>h</italic>- dichotomy in mean. Some characterizations of these notions and connections between these concepts are given. The obtained results can be considered generalizations to the dichotomy case of the results from [29].
</abstract>

On uniform dichotomy in mean of stochastic skew-evolution semiflows in Banach spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

In this article, our aim is to study analytic functions related with Salagean operator and associated with the right half of the lemniscate of Bernoulli. We find the estimates of the third Hankel determinant for new family of analytic functions. It is important to mention that our results generalize a number of existence results in the literature.
</abstract>

Estimate of third Hankel determinant for a subfamily of analytic functions

<abstract xmlns="http://www.w3.org/1999/xhtml">

The paper studies a Cournot game involving <italic>n</italic> private firms, where the firms’ interactions are analyzed in a discrete-time setting with time delay. The past production levels of other private firms have an impact on the production levels of each private firm. We performed a local stability analysis on the nonlinear system associated with the model and found that it has only one equilibrium point, which is asymptotically stable if and only if some conditions involving the number of firms, the degree of product di erentiation and the time delay are fulfilled. In addition, numerical examples are presented in this paper, which illustrate the theoretical results obtained.
</abstract>

Stability properties of a Cournot-type oligopoly model with time delay

<abstract xmlns="http://www.w3.org/1999/xhtml">

This paper deals with the existence and uniqueness of common fixed points in metric and dislocated metric spaces. In addition, an application and some illustrative examples are given in order to justify the validity and credibility of our results, also, their superiority over some similar theorems existing in unique common fixed points theory’s domain.
</abstract>

Fixed Point Results in Incomplete Metric and Dislocated Metric Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">

Along this paper, we study an evolution inclusion governed by the so-called sweeping process. The right side of the inclusion contains a set-valued perturbation, supposed to be the external forces exercised on the system. We prove existence and relaxation results under weak assumptions on the perturbation by taking a truncated Lipschitz condition. These perturbations have non-convex and unbounded values without any compactness condition; we just assume a linear growth assumption on the element of minimal norm. The approach is based on the construction of approximate solutions. The relaxation is obtained by proving the density of the solution set of the original problem in a closure of the solution set of the relaxed one.
</abstract>

Laboratory of Pure and Applied Mathematics, Faculty of Exact Sciences and Computer

Non-convex perturbation to evolution problems involving Moreau’s sweeping process

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Annals of West University of Timisoara - Mathematics and Computer Science

 Annals of West University of Timisoara - Mathematics and Computer Science, first issued in 1963, publishes peer-reviewed original and quality papers in all branches of Mathematics and Computer Science. The works devoted to interdisciplinary topics and relevant applications are also considered. The journal publishes research papers, expository papers and also short notes. Generally, the survey papers are only invited articles. The papers must be new, correct and of interest relative to the recent advances in the fields of Mathematics and Computer Science. Proposals for special issues are welcome and should be sent to the Editor-in-Chief. From volume 2023 Annals of West University of Timisoara - Mathematics and Computer Science publishes in a continuous model.     Why subscribe and read   It publishes original papers in all areas of Mathematics and Computer Science   Why submit    Fast and constructive peer-review   It publishes research papers, surveys and book reviews in any topic belonging to Mathematics or Computer Science      Rejection Rate    30%      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. Target Group    - researchers in all branches of mathematics and computer science 