<abstract xmlns="http://www.w3.org/1999/xhtml">In this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.</abstract>

Weighted Variable Exponent Sobolev spaces on metric measure spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">We prove the existence of weak solutions to a generalized p-Laplacian systems in degenerate form. The techniques of Young measure for elliptic systems are used to prove the existence result.</abstract>

Weak solutions for generalized p-Laplacian systems via Young measures

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article, we introduce the concept of rough subgroupoid of a groupoid as a generalization of a rough subgroup and give some features about the lower and the upper approximations in a groupoid [1]. We give some of the characterization of them.</abstract>

The Upper and Lower Approximations in Rough Subgroupoid of a Groupoid

<abstract xmlns="http://www.w3.org/1999/xhtml">In this work, we construct a new general two-point quadrature rules for the Riemann–Stieltjes integral <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0010_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><mrow><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mi> </mi><mi>d</mi><mi>u</mi><mi> </mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></math><tex-math>$\int_a^b {f(t)} \,du\,(t)$</tex-math></alternatives></inline-formula>, where the integrand <italic>f</italic> is assumed to be satisfied with the Hölder condition on [<italic>a</italic>, <italic>b</italic>] and the integrator <italic>u</italic> is of bounded variation on [<italic>a</italic>, <italic>b</italic>]. The dual formulas under the same assumption are proved. Some sharp error <italic>Lp</italic>–Error estimates for the proposed quadrature rules are also obtained.</abstract>

Two-Point Quadrature Rules for Riemann–Stieltjes Integrals with <italic>Lp</italic>–error estimates

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we establish some Opial type inequalities for Riemann-Stieltjes integrals of functions with two variables. The obtained inequalities generalize those previously demonstrated (see [2])</abstract>

Opial type inequalities for double Riemann-Stieltjes integrals

<abstract xmlns="http://www.w3.org/1999/xhtml">This paper reviews, from different points of view, results on Bernoulli numbers and polynomials, the distribution of prime numbers in connexion with the Riemann hypothesis. We give an account on the theorem of G. Robin, as formulated by J. Lagarias. The other parts are devoted to the series <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0012_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>𝒨</mi><msub><mrow><mi>i</mi></mrow><mi>s</mi></msub><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><mrow><mfrac><mrow><mi>μ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>n</mi></mrow><mi>s</mi></msup></mrow></mfrac><msup><mrow><mi>z</mi></mrow><mi>n</mi></msup></mrow></mrow></math><tex-math>$\mathcal{M}{i_s}(z) = \sum\limits_{n = 1}^\infty {{{\mu (n)} \over {{n^s}}}{z^n}} $</tex-math></alternatives></inline-formula>. A significant result is that the real part <italic>f</italic> of<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0012_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>∑</mo><mrow><mfrac><mrow><mi>μ</mi><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mi>n</mi></mfrac><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>i</mi><mi>n</mi><mi>π</mi><mi>θ</mi></mrow></msup></mrow></mrow></math><tex-math>$$\sum {{{\mu (n)} \over n}{e^{2in\pi \theta }}}$$</tex-math></alternatives></disp-formula>is an example of a non-trivial real-valued continuous function <italic>f</italic> on the real line which is 1-periodic, is not odd and has the property <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0012_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mo>∑</mo><mrow><mi>h</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><mrow><mi>f</mi><mo>(</mo><mi>h</mi><mo>/</mo><mi>k</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></mrow></math><tex-math>$\sum\nolimits_{h = 1}^n {f(h/k) = 0}$</tex-math></alternatives></inline-formula> for every positive integer <italic>k</italic>.</abstract>

Harmonic numbers, harmonic series and zeta function

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we present our results (see our papers), which concern the existence of the renormalized solutions for equations of the type:<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0013_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><mo>∂</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mtext>div</mtext><mrow><mo>(</mo><mrow><mi>a</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>-</mo><mtext>div</mtext><mrow><mo>(</mo><mrow><mi>Φ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mi> </mi><mi> </mi><mi> </mi><mtext>in</mtext><mi> </mi><mi>Q</mi><mo>=</mo><mi>Ω</mi><mo>×</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>)</mo><mo>,</mo></mrow></math><tex-math>$${{\partial b(x,u)} \over {\partial t}} - {\rm{div}}\left( {a(x,t,u,\nabla u)} \right) - {\rm{div}}\left( {\Phi \left( {x,t,u} \right)} \right) = f\,\,\,{\rm{in}}\,Q = \Omega \times (0,T),$$</tex-math></alternatives></disp-formula>where <italic>b</italic>(<italic>x</italic>, ·) is a strictly increasing <italic>C</italic>1-function for any <italic>x</italic> ∈ Δ, <italic>a</italic>(<italic>x</italic>, <italic>t</italic>, <italic>s</italic>, ξ) and Φ(<italic>x</italic>, <italic>t</italic>, <italic>s</italic>) are a Carathéodory functions. The function <italic>f</italic> is in <italic>L</italic>1(<italic>Q</italic>).</abstract>

A few results on some nonlinear parabolic problems in Orlicz-Sobolev spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we prove the existence of entropy solutions for anisotropic elliptic unilateral problem associated to the equations of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0014_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><msub><mrow><mo>∂</mo></mrow><mi>i</mi></msub><msub><mrow><mi>a</mi></mrow><mi>i</mi></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi><mo>)</mo><mo>-</mo></mrow><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mrow><msub><mrow><mo>∂</mo></mrow><mi>i</mi></msub><msub><mrow><mi>φ</mi></mrow><mi>i</mi></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>,</mo></mrow></mrow></math><tex-math>$$ - \sum\limits_{i = 1}^N {{\partial _i}{a_i}(x,u,\nabla u) - } \sum\limits_{i = 1}^N {{\partial _i}{\phi _i}(u) = f,} $$</tex-math></alternatives></disp-formula>where the right hand side <italic>f</italic> belongs to <italic>L</italic>1(Ω). The operator <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0014_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mo>-</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></msubsup><mrow><msub><mrow><mo>∂</mo></mrow><mi>i</mi></msub><msub><mrow><mi>a</mi></mrow><mi>i</mi></msub><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow></mrow></math><tex-math>$- \sum\nolimits_{i = 1}^N {{\partial _i}{a_i}\left( {x,u,\nabla u} \right)} $</tex-math></alternatives></inline-formula> is a Leray-Lions anisotropic operator and <italic>ϕi</italic> ∈ <italic>C</italic>0(ℝ,ℝ).</abstract>

Existence of Solutions for Some Nonlinear Elliptic Anisotropic Unilateral Problems with Lower Order Terms

<abstract xmlns="http://www.w3.org/1999/xhtml">We prove existence of entropy solutions to general class of unilateral nonlinear parabolic equation in inhomogeneous Musielak-Orlicz spaces avoiding ceorcivity restrictions on the second lower order term. Namely, we consider<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2018-0015_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mtable><mtr><mtd><mrow><mi>u</mi><mo>≥</mo><mi>ψ</mi></mrow></mtd><mtd><mrow><mtext>in</mtext></mrow></mtd><mtd><mrow><msub><mrow><mi>Q</mi></mrow><mi>T</mi></msub><mo>,</mo></mrow></mtd></mtr></mtable></mtd></mtr><mtr><mtd><mfrac><mrow><mo>∂</mo><mi>b</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow><mo>(</mo><mrow><mi>a</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>+</mo><mi>d</mi><mi>i</mi><mi>v</mi><mrow><mo>(</mo><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>t</mi><mo>,</mo><mi>u</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mn>1</mn></msup><mrow><mo>(</mo><mrow><msub><mrow><mi>Q</mi></mrow><mi>T</mi></msub></mrow><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mrow></math><tex-math>$$\left\{ \matrix{ \matrix{ {u \ge \psi } \hfill &amp; {{\rm{in}}} \hfill &amp; {{Q_T},} \hfill \cr } \hfill \cr {{\partial b(x,u)} \over {\partial t}} - div\left( {a\left( {x,t,u,\nabla u} \right)} \right) = f + div\left( {g\left( {x,t,u} \right)} \right) \in {L^1}\left( {{Q_T}} \right). \hfill \cr} \right.$$</tex-math></alternatives></disp-formula>The growths of the monotone vector field <italic>a</italic>(<italic>x</italic>, <italic>t</italic>, <italic>u</italic>, ᐁ<italic>u</italic>) and the non-coercive vector field <italic>g</italic>(<italic>x</italic>, <italic>t</italic>, <italic>u</italic>) are controlled by a generalized nonhomogeneous <italic>N</italic>- function <italic>M</italic> (see (3.3)-(3.6)). The approach does not require any particular type of growth of <italic>M</italic> (neither Δ2 nor ᐁ2). The proof is based on penalization method.</abstract>

Département de Mathématiques, Laboratoire LAMA. Faculté des Sciences Dhar-Mahrez

Obstacle parabolic equations in non-reflexive Musielak-Orlicz spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">By using some analytical techniques, we prove a complete monotonicity property of a certain function involving the (<italic>p</italic>, <italic>k</italic>)-digamma function. Subsequently, we derive some inequalities for the (<italic>p</italic>, <italic>k</italic>)- digamma function. As special cases of the established results, we deduce some new results concerning the <italic>p</italic>-digamma and the <italic>k</italic>-digamma functions. Our results are extensions of some previous results due to Qiu and Vuorinen, Mortici, and Merovci.</abstract>

Extension of complete monotonicity results involving the digamma function

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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 