rss_2.0Sciendo RSS Feed for Moroccan Journal of Pure and Applied Analysishttps://sciendo.com/journal/MJPAAhttps://www.sciendo.comhttps://sciendo-parsed.s3.eu-central-1.amazonaws.com/64724ba2215d2f6c89dc33dc/cover-image.jpghttps://sciendo.com/journal/MJPAA140216https://sciendo.com/article/10.2478/mjpaa-2023-0025<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we present an existence result of weak solutions for some parabolic equations involving the so-called CEV model with jumps.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00252023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0026<abstract> <title style='display:none'>Abstract</title> <p>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"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mi>B</mml:mi><mml:mo>⊂</mml:mo><mml:mi>B</mml:mi><mml:mo>+</mml:mo><mml:mi>C</mml:mi><mml:mo>⇒</mml:mo><mml:mi>A</mml:mi><mml:mo>⊂</mml:mo><mml:mi>C</mml:mi></mml:mrow></mml: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.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00262023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0027<abstract> <title style='display:none'>Abstract</title> <p>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>.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00272023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0024<abstract> <title style='display:none'>Abstract</title> <p>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.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00242023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0028<abstract> <title style='display:none'>Abstract</title> <p>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"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mrow><mml:mo>{</mml:mo> <mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>M</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>σ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mi>u</mml:mi><mml:mo>-</mml:mo><mml:msub><mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:mi>u</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>λ</mml:mi><mml:mi>ϑ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi></mml:msub><mml:mrow><mml:mfrac><mml:mrow><mml:mi>ϑ</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mi>d</mml:mi><mml:mi>x</mml:mi></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>r</mml:mi></mml:msup><mml:mi> </mml:mi><mml:mtext>in</mml:mtext><mml:mi> </mml:mi><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mi>u</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>.</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msubsup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mn>0</mml:mn><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mo>.</mml:mo><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow> </mml:mrow></mml:mrow></mml: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"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:mi>σ</mml:mi><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mo>∫</mml:mo><mml:mi mathvariant="normal">Ω</mml:mi></mml:msub><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mrow><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>u</mml:mi></mml:mrow> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mo>+</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mrow><mml:mo>∇</mml:mo><mml:mi>u</mml:mi></mml:mrow> <mml:mo>|</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac><mml:mi>d</mml:mi><mml:mi>x</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml: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.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00282023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0021<abstract> <title style='display:none'>Abstract</title> <p>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.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00212023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0020<abstract> <title style='display:none'>Abstract</title> <p>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.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00202023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0023<abstract> <title style='display:none'>Abstract</title> <p>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.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00232023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0022<abstract> <title style='display:none'>Abstract</title> <p>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.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00222023-09-29T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0019<abstract> <title style='display:none'>Abstract</title> <p>For a Bloch function <italic>f</italic> in the unit ball in ℂ<italic>ⁿ</italic>, we study the maximal locus of the Bloch norm of <italic>f</italic>; namely, the set <italic>L_f</italic> where the Bergman length of the gradient vector field of <italic>f</italic> attains its maximum. We prove that for <italic>n</italic> ≥<italic>,</italic> the set <italic>L_f</italic> consists of a finite union of real analytic sets with dimensions at most 2<italic>n −</italic> 2. This is not the case for <italic>n</italic> = 1 as was proved earlier by Cima and Wogen. We also give some rigidity properties of the set <italic>L_f</italic>. In particular, we give some sufficient criteria for constructing extreme functions in the Little Bloch ball.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00192023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0014<abstract> <title style='display:none'>Abstract</title> <p>There exist two real valued periodic functions on the real line such that, for every <italic>x</italic> ∈ ℝ, <italic>f</italic>₁(<italic>x</italic>) + <italic>f</italic>₂(<italic>x</italic>) = <italic>x</italic>, but it is impossible to find two real valued periodic functions on the real line such that, for every <italic>x</italic> ∈ ℝ, <italic>f</italic>₁(<italic>x</italic>) + <italic>f</italic>₂(<italic>x</italic>) = <italic>x</italic>². The purpose of this note is to prove this result and also to study the possibility of decomposing more general polynomials into sum of periodic functions.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00142023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0016<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we study the cyclicity problem with respect to the forward shift operator <italic>S_b</italic> acting on the de Branges–Rovnyak space ℋ (<italic>b</italic>) associated to a function <italic>b</italic> in the closed unit ball of <italic>H</italic>^∞ and satisfying log(1− |<italic>b</italic>| ∈ <italic>L</italic>¹(𝕋). We present a characterisation of cyclic vectors for <italic>S_b</italic> when <italic>b</italic> is a rational function which is not a finite Blaschke product. This characterisation can be derived from the description, given in [22], of invariant subspaces of <italic>S_b</italic> in this case, but we provide here an elementary proof. We also study the situation where <italic>b</italic> has the form <italic>b</italic> = (1+ <italic>I</italic>)/2, where <italic>I</italic> is a non-constant inner function such that the associated model space <italic>K</italic>_I = ℋ (<italic>I</italic>) has an orthonormal basis of reproducing kernels.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00162023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0012<abstract> <title style='display:none'>Abstract</title> <p>We consider a Nevanlinna–Pick interpolation problem on finite sequences of the unit disc, constrained by Beurling–Sobolev norms. We find sharp asymptotics of the corresponding interpolation quantities, thereby improving the known estimates. On our way we obtain a S. M. Nikolskii type inequality for rational functions whose poles lie outside of the unit disc. It shows that the embedding of the Hardy space <italic>H</italic>² into the Wiener algebra of absolutely convergent Fourier/Taylor series is invertible on the subset of rational functions of a given degree, whose poles remain at a given distance from the unit circle.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00122023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0017<abstract> <title style='display:none'>Abstract</title> <p>The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice <italic>X</italic>, including weighted <italic>ℓ</italic>^p spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier–Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>X</mml:mi><mml:mo>∞</mml:mo></mml:msubsup></mml:mrow></mml:math> <tex-math>\mathbb{D}_X^\infty</tex-math> </alternatives> </inline-formula> . Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_002.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>X</mml:mi><mml:mo>∞</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:mo>|</mml:mo><mml:msup><mml:mrow><mml:mi>ℂ</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup></mml:mrow></mml:math> <tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N}</tex-math> </alternatives> </inline-formula> and the <italic>ℓ</italic>^p-unit balls <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_003.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>𝔻</mml:mi></mml:mrow><mml:mi>X</mml:mi><mml:mo>∞</mml:mo></mml:msubsup><mml:mo>,</mml:mo><mml:mo>|</mml:mo><mml:msup><mml:mrow><mml:mi>ℂ</mml:mi></mml:mrow><mml:mi>N</mml:mi></mml:msup><mml:mo>=</mml:mo><mml:msubsup><mml:mrow><mml:mi>𝔹</mml:mi></mml:mrow><mml:mi>p</mml:mi><mml:mi>N</mml:mi></mml:msubsup></mml:mrow></mml:math> <tex-math>\mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N</tex-math> </alternatives> </inline-formula> , in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_004.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ℤ</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mi>N</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>α</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> <tex-math>X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right))</tex-math> </alternatives> </inline-formula>, <italic>s</italic> = (<italic>s</italic>₁, <italic>s</italic>₂, … ) and <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0017_ieq_005.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>X</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>ℓ</mml:mi></mml:mrow><mml:mi>p</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mi>ℤ</mml:mi></mml:mrow><mml:mo>+</mml:mo><mml:mi>N</mml:mi></mml:msubsup><mml:mo>,</mml:mo><mml:mi> </mml:mi><mml:mi> </mml:mi><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>α</mml:mi><mml:mo>!</mml:mo></mml:mrow><mml:mrow><mml:mrow><mml:mo>|</mml:mo> <mml:mi>α</mml:mi> <mml:mo>|</mml:mo></mml:mrow><mml:mo>!</mml:mo></mml:mrow></mml:mfrac></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>t</mml:mi></mml:msup><mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mrow><mml:mo>|</mml:mo> <mml:mi>α</mml:mi> <mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> <tex-math>X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha \right|} \right)}^s}} \right)</tex-math> </alternatives> </inline-formula>, <italic>s,t</italic> ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00172023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0011<abstract> <title style='display:none'>Abstract</title> <p>This note is dedicated to recalling the virtues and the important contributions in mathematics of mohamed zarabi who passed a way on mid december 2021.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00112023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0015<abstract> <title style='display:none'>Abstract</title> <p>Zarrabi proved in 1993 that if the spectrum of a contraction <italic>T</italic> on a Banach space is a countable subset of the unit circle 𝕋, and if <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_001.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>lim</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>‖</mml:mo> <mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow> <mml:mo>‖</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msqrt><mml:mi>n</mml:mi></mml:msqrt></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> <tex-math>{\lim _{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {\sqrt n }} = 0</tex-math> </alternatives> </inline-formula>, then <italic>T</italic> is an isometry, so that ‖<italic>Tⁿ</italic>‖ = 1 for every <italic>n</italic> ∈ ℤ. It is also known that if <italic>C</italic> is the usual triadic Cantor set then every contraction <italic>T</italic> on a Banach space such that <italic>Spec</italic>(<italic>T</italic> ) ⊂ 𝒞 satisfying <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_002.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>lim</mml:mo><mml:mi> </mml:mi><mml:mi>s</mml:mi><mml:mi>u</mml:mi><mml:msub><mml:mrow><mml:mi>p</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>→</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:msub><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mo>‖</mml:mo> <mml:mrow><mml:msup><mml:mrow><mml:mi>T</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:msup></mml:mrow> <mml:mo>‖</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mi>α</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mo>&lt;</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo></mml:mrow></mml:math> <tex-math>\lim \,su{p_{n \to + \infty }}{{\log \left( {\left\| {{T^{ - n}}} \right\|} \right)} \over {{n^\alpha }}} &lt; + \infty</tex-math> </alternatives> </inline-formula> for some <inline-formula> <alternatives> <inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2023-0015_ieq_003.png"/> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:mo>&lt;</mml:mo><mml:mfrac><mml:mrow><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi> </mml:mi><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>3</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mo>−</mml:mo><mml:mo>log</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mfrac></mml:mrow></mml:math> <tex-math>\alpha &lt; {{\log \left( 3 \right) - \log \left( 2 \right)} \over {2\,\log \left( 3 \right) - \log \left( 2 \right)}}</tex-math> </alternatives> </inline-formula> is an isometry.</p> <p>In the other direction an easy refinement of known results shows that if a closed <italic>E</italic> ⊂ 𝕋 is not a “strong <italic>AA</italic>⁺-set” then for every sequence (<italic>u_n)_n</italic>_≥1 of positive real numbers such that lim inf<italic>_n</italic>_→+∞<italic>u_n</italic> = + ∞ there exists a contraction <italic>T</italic> on some Banach space such that <italic>Spec</italic>(<italic>T</italic> )⊂ <italic>E</italic>, ‖<italic>T</italic>^{−<italic>n</italic>}‖ = <italic>O</italic>(<italic>u</italic>_n) as <italic>n</italic> → + ∞ and sup_n_≥1 ‖<italic>T</italic>⁻ⁿ‖ = + ∞.</p> <p>We show conversely that if <italic>E</italic> ⊂ 𝕋 is a strong <italic>AA</italic>⁺-set then there exists a nondecreasing unbounded sequence (<italic>u</italic>_n)_n_≥1 such that for every contraction <italic>T</italic> on a Banach space satsfying <italic>Spec</italic>(<italic>T</italic>) ⊂ <italic>E</italic> and ‖<italic>T</italic>⁻ⁿ ‖ = <italic>O</italic>(<italic>u</italic>_n) as <italic>n</italic> → + ∞ we have <italic>sup_n</italic>_&gt;0 ‖<italic>T</italic>⁻ⁿ ‖ ≤ <italic>K</italic>, where <italic>K &lt;</italic> + ∞ denotes the “<italic>AA</italic>⁺-constant” of <italic>E</italic> (closed countanble subsets of 𝕋 and the triadic Cantor set are strong <italic>AA</italic>⁺-sets of constant 1).</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00152023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0018<abstract> <title style='display:none'>Abstract</title> <p>In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00182023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0013<abstract> <title style='display:none'>Abstract</title> <p>We extend Carleson’s formula to radially polynomially weighted Dirichlet spaces.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00132023-06-07T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0008<abstract> <title style='display:none'>Abstract</title> <p>New copulas, based on perturbation theory, are introduced to clarify a <italic>symmetrization</italic> procedure for asymmetric copulas. We give also some properties of the <italic>symmetrized</italic> copula mainly conservation of concordance. Finally, we examine some copulas with a prescribed symmetrized part. The start point of the treatment is the independence copula and the last one will be an arbitrary member of Farlie-Gumbel-Morgenstein family. By the way, we study topologically, the set of all symmetric copulas and give some of its classical and new properties.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00082023-02-01T00:00:00.000+00:00https://sciendo.com/article/10.2478/mjpaa-2023-0007<abstract> <title style='display:none'>Abstract</title> <p>To have a more realistic model, in this paper, This manuscript is devoted to investigating a fractional-order mathematical model of Kouidere et al. That describes the dynamics of spread of African swine fever virus (ASFV). The aim of this work is to protect susceptible pigs from the virus, In our model, by including three controls which represent: the iron fencing and spraying pesticides and get rid.</p> <p>The aims of this paper is to reduce the number of infected pigs and ticks by using optimal control strategy and fractinal order derivation.</p> <p>Pontryagin’s maximal principle is used to describe optimal controls with Caputo time-fractional derivative and the optimal system is resolved in an iterative manner. Numerical simulations are presented based on the presented method. We finished tis article with a conclusion.</p> </abstract>ARTICLEtruehttps://sciendo.com/article/10.2478/mjpaa-2023-00072023-02-01T00:00:00.000+00:00en-us-1