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

In this paper, we prove the following generalization of the classical Darbo fixed point principle : Let <italic>X</italic> be a Banach space and <italic>µ</italic> be a montone measure of noncompactness on <italic>X</italic> which satisfies the generalized Cantor intersection property. Let <italic>C</italic> be a nonempty bounded closed convex subset of <italic>X</italic> and <italic>T : C → C</italic> be a continuous mapping such that for any countable set Ω ⊂ <italic>C</italic>, we have <italic>µ</italic>(<italic>T</italic>(Ω)) ≤ <italic>kµ</italic>(Ω), where <italic>k</italic> is a constant, 0 ≤ <italic>k</italic> &lt; 1. Then <italic>T</italic> has at least one fixed point in <italic>C</italic>. The proof is based on a combined use of topological methods and partial ordering techniques and relies on the Schauder and the Knaster-Tarski fixed point principles.
</abstract>

On Darbo’s fixed point principle

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

In this paper, we study the solvability to the left of the positive infimum of all eigenvalues for some non-resonant quasilinear elliptic problems involving variable exponents. We first prove the existence of at least a weak solution for some non-variational systems by using a surjectivity result for pseudomonotone operators. Furthermore, under additional conditions, we show that the solution is unique and provide examples. Second, we deal with non-resonant gradient-type systems and obtain existence by using a variational approach.
</abstract>

Solvability of Parametric Elliptic Systems with Variable Exponents

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

The purpose of the present article is to provide a new parabolic Sobolev embedding map between a parabolic weighted Sobolev space and the space of square-integrable functions on a cylinder. Furthermore, the embedding constant is furnished explicitly.
</abstract>

On a New Parabolic Sobolev Embedding Map

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

This paper explores the concept of normal structure by introducing a generalized form known as <italic>P</italic>-proximal normal structure. Focusing on the framework of Hausdorff locally convex spaces, the study establishes optimal proximity outcomes for both cyclic and noncyclic relatively <italic>P</italic>-nonexpansive mappings. Furthermore, the paper presents in the realm of probabilistic normed spaces, considered as instances of Hausdorff locally convex spaces, some theorems that address the existence of best proximity points within this context.
</abstract>

Hassan II University of Casablanca, Faculty of Sciences Ben M’Sik

Laboratory of Analyse Modelisation and Simulation

On best proximity point theorems for relatively nonexpansive mappings in locally convex spaces

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

Initial conditions can have a substantial impact on the behavior of iterative root-finding techniques for nonlinear equations. By allowing complex starting points and complex roots, it is possible to examine the basins of attraction in the complex plane in order to compare the performance of various iterative techniques. In this paper, a one-parameter family of third-order root-finding methods is studied by varying its parameter <italic>A</italic> within −2.0 and 2.4 and applying it to a polynomial equation of high degree (degree 25). This family includes the Euler–Chebyshev’s (<italic>A</italic> = 0), Halley’s (<italic>A</italic> = 1) and BSC (<italic>A</italic> = 2) techniques. According to the results, the one-parameter family provides the best performance for values near <italic>A</italic> = 1, which equals to the Halley’s method.
</abstract>

Basins of attraction of a one-parameter family of root-finding techniques

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

In this paper, we present an existence result of weak solutions for some parabolic equations involving the so-called CEV model with jumps.
</abstract>

Laboratoire LAR2A, Department of Mathematics, Faculty of Sciences Tétouan

Department of Mathematics, Faculty of Sciences Tétouan

An existence result for two-dimensional parabolic integro-differential equations involving CEV model

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

In this paper we generalize a result of R. Urbański from paper [7] which states that for subsets <italic>A</italic>, <italic>B</italic>, <italic>C</italic> of topological vector space <italic>X</italic> the following implication holds 
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0026_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>⊂</mo><mi>B</mi><mo>+</mo><mi>C</mi><mo>⇒</mo><mi>A</mi><mo>⊂</mo><mi>C</mi></mrow></math>
<tex-math>A + B \subset B + C \Rightarrow A \subset C</tex-math>
</alternatives>
</disp-formula>
 provided that <italic>B</italic> is bounded and <italic>C</italic> is closed and convex. The generalization is given in Theorem 5 where we prove this result for <italic>k</italic>- convex subsets of a topological vector space. Also we introduce a notion of some abstract like-closure operation for subsets of linear space and we study its connections to order cancellation law.
</abstract>

On some generalization of order cancellation law for subsets of topological vector space

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

In this paper, we deal with a nonlinear elliptic problems that incorporate a Hardy potential and a nonlinear convection term. We establish the existence and regularity of solutions under various assumptions concerning the summability of the source term <italic>f</italic>.
</abstract>

Nonlinear elliptic boundary value problems with convection term and Hardy potential

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

We investigate the existence of non-trivial weak solutions for the following <italic>p</italic>(<italic>x</italic>)-Kirchhoff bi-nonlocal elliptic problem driven by both <italic>p</italic>(<italic>x</italic>)-Laplacian and <italic>p</italic>(<italic>x</italic>)-Biharmonic operators
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0028_eq_001.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mrow><mo>{</mo> <mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>M</mi><mrow><mo>(</mo><mi>σ</mi><mo>)</mo></mrow><mrow><mo>(</mo><mrow><msubsup><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mn>2</mn></msubsup><mi>u</mi><mo>-</mo><msub><mrow><mi mathvariant="normal">Δ</mi></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msub><mi>u</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>λ</mi><mi>ϑ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><msup><mrow><mrow><mrow><mo>(</mo><mrow><mrow><msub><mo>∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow><mfrac><mrow><mi>ϑ</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><msup><mrow><mrow><mrow><mo>|</mo> <mi>u</mi> <mo>|</mo></mrow></mrow></mrow><mrow><mi>q</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msup><mi>d</mi><mi>x</mi></mrow></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>r</mi></msup><mi> </mi><mtext>in</mtext><mi> </mi><mi mathvariant="normal">Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>2</mn><mo>,</mo><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></msup><mrow><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>)</mo></mrow><mo>∩</mo><msubsup><mrow><mi>W</mi></mrow><mn>0</mn><mrow><mn>1</mn><mo>,</mo><mi>p</mi><mrow><mo>(</mo><mo>.</mo><mo>)</mo></mrow></mrow></msubsup><mrow><mo>(</mo><mi mathvariant="normal">Ω</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow> </mrow></mrow></math>
<tex-math>\left\{ {\matrix{ {M\left( \sigma \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill \cr {u \in {W^{2,p\left( . \right)}}\left( \Omega \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right),} \hfill \cr } } \right.</tex-math>
</alternatives>
</disp-formula>
under some suitable conditions on the continuous functions <italic>p</italic>, <italic>q</italic>, the non-negative function <italic>ϑ</italic> and <italic>M</italic>(<italic>σ</italic>), where
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0028_eq_002.png"></graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mi>σ</mi><mo>:</mo><mo>=</mo><mrow><msub><mo>∫</mo><mi mathvariant="normal">Ω</mi></msub><mrow><mfrac><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mi mathvariant="normal">Δ</mi><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msup></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mrow><mrow><mo>|</mo> <mrow><mo>∇</mo><mi>u</mi></mrow> <mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></msup></mrow><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></mfrac><mi>d</mi><mi>x</mi><mo>.</mo></mrow></mrow></mrow></math>
<tex-math>\sigma : = \int_\Omega {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.}</tex-math>
</alternatives>
</disp-formula>
Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.
</abstract>

AMNEA Group, Laboratory MAIS, Department of Mathematics, FSTE, BP 509, Boutalamine Errachidia

<italic>p</italic>(<italic>x</italic>)-Kirchhoff bi-nonlocal elliptic problem driven by both <italic>p</italic>(<italic>x</italic>)-Laplacian and <italic>p</italic>(<italic>x</italic>)-Biharmonic operators

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Moroccan Journal of Pure and Applied Analysis

 The Moroccan Journal of Pure and Applied Analysis "MJPAA" (Print ISSN: 2605-6364) is a peer-reviewed open access journal published by the "Université Sidi Mohamed Ben Abdellah" (Sidi Mohamed Ben Abdallah University), covering all main-stream branches of pure and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, numerical analysis, optimization theory, and control theory.   MJPAA is published by Sidi Mohamed Ben Abdallah University in three issues in January, May and September.   Why subscribe and read    The journal covers a wide range of branches of mathematical analysis.   Pure and applied analysis topics are intensely studied by researchers throughout the world especially in Africa, the Arab and European countries.   An edition committee of high-level and international renown.   Editorial board representing different regions of the world and various research teams.      Why submit    No publication charges for the authors or their institutions.   The journal is available to be indexed in the most important databases.   The journal is among the best in its field in Africa and the Arab world.      Rejection Rate    45%      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.   Publisher:Sidi Mohamed Ben Abdallah University (Route Imouzzer, BP 2626 Fez 30000, Morocco, Tel. : 0212 5 35 60 96 60/ 61)   Contact:Professor Youssef Akdim, Sidi Mohamed Ben Abdellah Universitymjpaa.edition1@usmba.ac.ma 