<abstract xmlns="http://www.w3.org/1999/xhtml">In this research we lay the concept of log <italic>m</italic>-convex functions defined on real intervals containing the origin, some algebraic properties are exhibit, in the same token discrete Jensen type inequalities and integral inequalities are set and shown.</abstract>

Log <italic>m</italic>-Convex Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we study the existence of positive weak solutions for a quasilinear elliptic system involving weighted (<italic>p</italic>(.), <italic>q</italic>(.))−Laplacian operators. The approach is based on sub-supersolutions method and on Schauder’s fixed point theorem.</abstract>

On positive weak solutions for a class of weighted (<italic>p</italic>(.), <italic>q</italic>(.))−Laplacian systems

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article, inspired by the concepts of extended <italic>b</italic>-metric spaces, we introduce the notion of complex valued extended <italic>b</italic>-metric spaces. Using this new idea, some fixed point theorems involving rational contractive inequalities are proved. The established results herein augment several significant work in the comparable literature.</abstract>

Department of Mathematics, Faculty of Physical Sciences

Fixed Point Theorems in Complex Valued Extended <italic>b</italic>-Metric Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we are interested in the existence of solutions for Dirichlet problem associated with the degenerate nonlinear elliptic equations<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2019-0012_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>-</mo><mtext>div</mtext><mrow><mo>[</mo><mrow><mi>𝒜</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow><msub><mrow><mi>ω</mi></mrow><mn>1</mn></msub><mo>+</mo><mi>𝒝</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow><msub><mrow><mi>ω</mi></mrow><mn>2</mn></msub></mrow><mo>]</mo></mrow><mo>=</mo><msub><mrow><mi>f</mi></mrow><mn>0</mn></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mrow><msub><mrow><mi>D</mi></mrow><mi>j</mi></msub><msub><mrow><mi>f</mi></mrow><mi>j</mi></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi> </mi><mi> </mi><mtext>in</mtext></mrow><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mtext>on</mtext><mi> </mi><mi> </mi><mi> </mi><mi> </mi><mo>∂</mo><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\left\{ {\matrix{ { - {\rm{div}}\left[ {\mathcal{A}\left( {x,\nabla u} \right){\omega _1} + \mathcal{B}\left( {x,u,\nabla u} \right){\omega _2}} \right] = {f_0}\left( x \right) - \sum\limits_{j = 1}^n {{D_j}{f_j}\left( x \right)\,\,{\rm{in}}} \,\,\,\,\,\Omega ,} \hfill \cr {u\left( x \right) = 0\,\,\,\,{\rm{on}}\,\,\,\,\partial \Omega {\rm{,}}} \hfill \cr } } \right.</tex-math></alternatives></disp-formula>in the setting of the weighted Sobolev spaces.</abstract>

Existence Results For A Class Of Nonlinear Degenerate Elliptic Equations

<abstract xmlns="http://www.w3.org/1999/xhtml">A new proof of Euler’s formular for calculating ζ(2<italic>k</italic>) is given. Some new inequalities and identities for ζ(2<italic>k</italic> + 1) have also been given. The Riemann’s functional equation together with trigonometric identities were used to establish the results.</abstract>

Some Identities and inequalities related to the Riemann zeta function

<abstract xmlns="http://www.w3.org/1999/xhtml">A mathematical model for transmission dynamics of tuberculosis among healthcare workers is formulated. Tuberculosis is an airborne disease caused by Mycobacterium tuberculosis bacteria that affect the lungs of a host. Previous research had concentrated on mathematical modeling of transmission dynamics of tuberculosis without considering the impact of compliance rate to particulate respirator by healthcare workers on the transmission. Therefore, how compliance rate to particulate respirator reduces the transmission of tuberculosis is an active question, and we develop a new system of ordinary differential equations that explicitly explores the impact of compliance rate to particulate respirator by healthcare workers upon transmission. Rigorous analysis of the model shows that the disease-free equilibrium point is locally asymptotically stable when the basic reproduction number, <italic>Ro</italic> &lt; 1. This is established through the analysis of characteristic equation. Basic reproduction, <italic>Ro</italic> is the number of new cases that an existing case generates on average over the infectious period in a susceptible population. We also show that the endemic equilibrium point is locally asymptotically stable for <italic>Ro</italic> > 1, by using Routh-Hurwitz criteria for stability. Sensitivity analysis is carried out to determine the relative importance of the model parameters to the disease transmission. The result of the sensitivity analysis shows that the most sensitive parameter is β (Human-to-human transmission rate), followed by Λ (Human recruitment rate). Also, the result shows that increase in <italic>ψ</italic> (compliance rate to particulate respirator by healthcare workers) leads to decrease in <italic>Ro</italic> which reduces tuberculosis spread among healthcare workers.</abstract>

Department of Mathematics, African Institute for Mathematical Sciences

Modeling Tuberculosis Among Healthcare Workers

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we prove some common coupled fixed point theorems for contractive mappings in Menger metric spaces under geometrically convergent t-norms. Also, we prove common fixed point theorems for pairs of weakly compatible mappings, which generalize the results of Jian-Zhong Xiao and al (2011). The main results is supported by a suitable examples.</abstract>

Common coupled fixed point theorems for two pairs of weakly compatible mappings in Menger metric spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">We present in this paper a survey of recent results on the controllability of the parabolic system governed by bilinear control. We first discuss the problem of global controllability which corresponds to the question of whether the solution of the system can be driven to a given state at a some finite time by means of a control. We give some results on the global controllability of bilinear and semilinear reaction-diffusion equations. After this we introduce the case of partial controllability with the control acting on a subregion of the domain. Illustrative examples are also provided.</abstract>

Controllability of some bilinear and semilinear parabolic problems

<abstract xmlns="http://www.w3.org/1999/xhtml">In the present paper, an homogeneous and non-homogeneous inverse problem constrained by the stationary problem in electrocardiology representing the heart, lungs surfaces, and torso model is investigated. Our goal is to reconstruct the electrical potentials on the surface of the heart from the information obtained non invasively on the torso surface. The existence and uniqueness of solution for the heart-torso problem and the related inverse problem is assessed, and the primal and dual problems are discretized using a finite element method. We present some preliminary numerical experiments using an efficient implementation of the proposed scheme in homogeneous and non-homogeneous cases. Finally, we demonstrate the effect of the non-homogeneity on the reconstructed epicardial potential and show that the inverse ECG problem cannot be solved by the classical BEM (boundary element method).</abstract>

Effect of Torso Non-Homogeneities in the quasi-static inverse problems arising in electrocardiology

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we present fixed point results for Boyd and Wong type [3] generalized contractive condition in partial metric spaces. In particular, we generalize the fixed point results due to Akkouchi [1] in complete partial metric spaces in which the continuity requirement for a mapping is relaxed to obtain the results. In addition to that we present a common fixed point theorem for a pair of maps. An illustrative example is also constructed to exhibit the results.</abstract>

Boyd and Wong Type Fixed Point Theorems in Partial Metric Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we propose a novel hybrid model for restoration of images corrupted by multiplicative noise. Using a MAP estimator, we can derive a functional whose minimizer corresponds to the denoised image we want to recover. The energies studied here are inspired by image restoration with non linear variable exponent [1, 2], and it is a combination of fast growth with respect to low gradient and slow growth when the gradient is large. We study a mathematical framework to prove the well posedness of the minimizer problem and we introduce the associated evolution problem, for which we derive numerical approaches. At last, compared experimental results distinctly demonstrate the superiority of the proposed model, in term of removing some muliplicative noise while preserving the edges and reducing the staircase effect.</abstract>

An Efficient Algorithm for Reconstruction Images Corrupted by Some Multiplicative Noises

<abstract xmlns="http://www.w3.org/1999/xhtml">We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2019-0020_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><mo>-</mo><msub><mrow><mi>Δ</mi></mrow><mi>p</mi></msub><mi>u</mi><mo>=</mo><mi>λ</mi><msup><mrow><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mi> </mi><mi> </mi><mi> </mi><mi>i</mi><mi>n</mi><mi> </mi><mi> </mi><mi>Ω</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><msup><mrow><mrow><mrow><mo>|</mo><mrow><mo>∇</mo><mi>u</mi></mrow><mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>.</mo><mi>v</mi><mo>+</mo><msup><mrow><mrow><mrow><mo>|</mo><mi>u</mi><mo>|</mo></mrow></mrow></mrow><mrow><mi>p</mi><mo>-</mo><mn>2</mn></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mi>o</mi><mi>n</mi><mi> </mi><mi> </mi><mi>Γ</mi><mo>.</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\left\{ {\matrix{ { - {\Delta _p}u = \lambda {{\left| u \right|}^{p - 2}}u\,\,\,in\,\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u.v + {{\left| u \right|}^{p - 2}}u = 0\,\,\,on\,\,\Gamma .} \hfill \cr } } \right.</tex-math></alternatives></disp-formula>We successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λ<italic>n</italic>↗∞. To this end we endow <italic>W</italic>1,<italic>p</italic>(Ω) with a norm invoking the trace and use the duality mapping on <italic>W</italic>1,<italic>p</italic> (Ω) to apply mini-max arguments on <italic>C</italic>1-manifold.</abstract>

On the spectrum of Robin boundary <italic>p</italic>-Laplacian problem

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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 

Moroccan Journal of Pure and Applied Analysis

Volume 5 (2019): Issue 2 (December 2019)

Log m-Convex Functions

On positive weak solutions for a class of weighted (p(.), q(.))−Laplacian systems

Fixed Point Theorems in Complex Valued Extended b-Metric Spaces