<abstract xmlns="http://www.w3.org/1999/xhtml">We apply the averaging theory of first and second order for studying the limit cycles of generalized polynomial Linard systems of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0001_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>y</mi><mo>-</mo><mn>1</mn><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>y</mi><mo>,</mo><mi> </mi><mi> </mi><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mi>x</mi><mo>-</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>-</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>y</mi><mo>-</mo><mi>h</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mrow><mi>y</mi></mrow><mn>2</mn></msup><mo>,</mo></mrow></math><tex-math>\dot x = y - 1\left( x \right)y,\,\,\dot y = - x - f\left( x \right) - g\left( x \right)y - h\left( x \right){y^2},</tex-math></alternatives></disp-formula>where <italic>l</italic>(<italic>x</italic>) = <italic>∊l</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>l</italic>2(<italic>x</italic>), <italic>f</italic> (<italic>x</italic>) = <italic>∊ f</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>f</italic>2(<italic>x</italic>), <italic>g</italic>(<italic>x</italic>) = <italic>∊g</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>g</italic>2(<italic>x</italic>) and <italic>h</italic>(<italic>x</italic>) = <italic>∊h</italic>1(<italic>x</italic>) + <italic>∊</italic>2<italic>h</italic>2(<italic>x</italic>) where <italic>lk</italic>(<italic>x</italic>) has degree <italic>m</italic> and <italic>fk</italic>(<italic>x</italic>), <italic>gk</italic>(<italic>x</italic>) and <italic>hk</italic>(<italic>x</italic>) have degree <italic>n</italic> for each <italic>k</italic> = 1, 2, and <italic>∊</italic> is a small parameter.</abstract>

Limit cycles of Liénard polynomial systems type by averaging method

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we show the existence of solutions for the nonlinear elliptic equations of the form<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_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><mi> </mi><mi>a</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>u</mi><mo>,</mo><mo>∇</mo><mi>u</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mn>0</mn><mn>1</mn></msubsup><mi>L</mi><mi>ϕ</mi><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mo>∞</mo></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow><mo>,</mo></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\left\{ {\matrix{ { - {\rm{div}}\,a\left( {x,u,\nabla u} \right) = f,} \hfill \cr {u \in W_0^1L\varphi \left( \Omega \right) \cap {L^\infty }\left( \Omega \right),} \hfill \cr } } \right.</tex-math></alternatives></disp-formula>where <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>a</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>ξ</mi></mrow><mo>)</mo></mrow><mo>⋅</mo><mi>ξ</mi><mo>≥</mo><msubsup><mrow><mover><mi>ϕ</mi><mo>¯</mo></mover></mrow><mi>x</mi><mrow><mo>-</mo><mn>1</mn></mrow></msubsup><mrow><mo>(</mo><mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mi>h</mi><mrow><mo>(</mo><mrow><mrow><mo>|</mo><mi>s</mi><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>x</mi><mo>,</mo><mrow><mo>|</mo><mi>ξ</mi><mo>|</mo></mrow></mrow><mo>)</mo></mrow></mrow></math><tex-math>a\left( {x,s,\xi } \right) \cdot \xi \ge \bar \varphi _x^{ - 1}\left( {\varphi \left( {x,h\left( {\left| s \right|} \right)} \right)} \right)\varphi \left( {x,\left| \xi \right|} \right)</tex-math></alternatives></inline-formula> and <italic>h</italic> : ℝ+→]0, 1] is a continuous decreasing function with unbounded primitive. The second term <italic>f</italic> belongs to <italic>LN</italic>(Ω) or to <italic>Lm</italic>(Ω), with <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0002_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>m</mi><mo>=</mo><mfrac><mrow><mi>r</mi><mi>N</mi></mrow><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></math><tex-math>m = {{rN} \over {r + 1}}</tex-math></alternatives></inline-formula> for some <italic>r</italic> > 0 and <italic>φ</italic> is a Musielak function satisfying the Δ2-condition.</abstract>

Department of Mathematics, Faculty of Sciences Dhar El Mahraz

Department of Mathematics, Faculty of Sciences El Jadida

Existence of a weak bounded solution for nonlinear degenerate elliptic equations in Musielak-Orlicz spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we discuss the hypercyclic properties of composition operators on Orlicz function spaces. We give some different conditions under which a composition operator on Orlicz spaces is hyper-cyclic or not. Similarly, multiplication operators are considered. It is shown that there is no hypercyclic multiplication operator on Orlicz spaces.</abstract>

Hypercyclicity of Composition Operators on Orlicz Function Spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">Via Leray-Schauder’s nonlinear alternative, we obtain the existence of a weak solution for a nonlocal problem driven by an operator of elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.</abstract>

Leray-Schauder’s solution for a nonlocal problem in a fractional Orlicz-Sobolev space

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article, we provide sufficient conditions for the existence of periodic solutions of the eighth-order differential equation
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_mjpaa-2020-0005_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>8</mn><mo>)</mo></mrow></mrow></msup><mo>-</mo><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mi>λ</mi></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mi>μ</mi></mrow><mn>2</mn></msup></mrow><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow></msup><mo>+</mo><mi>A</mi><mover><mi>x</mi><mo>⃜</mo></mover><mo>+</mo><mi>B</mi><mover><mi>x</mi><mo>¨</mo></mover><mo>+</mo><msup><mrow><mi>p</mi></mrow><mn>2</mn></msup><msup><mrow><mi>λ</mi></mrow><mn>2</mn></msup><msup><mrow><mi>μ</mi></mrow><mn>2</mn></msup><mi>x</mi><mo>=</mo><mi>ɛ</mi><mi>F</mi><mrow><mo>(</mo><mrow><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mover><mi>x</mi><mo>˙</mo></mover><mo>,</mo><mover><mi>x</mi><mo>¨</mo></mover><mo>,</mo><mover><mi>x</mi><mo>⃛</mo></mover><mo>,</mo><mover><mi>x</mi><mo>⃜</mo></mover><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow></msup><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mrow><mo>(</mo><mn>7</mn><mo>)</mo></mrow></mrow></msup></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math><tex-math>{x^{\left( 8 \right)}} - \left( {1 + {p^2} + {\lambda ^2} + {\mu ^2}} \right){x^{\left( 6 \right)}} + A\ddddot x + B\ddot x + {p^2}{\lambda ^2}{\mu ^2}x = \varepsilon F\left( {t,x,\dot x,\ddot x,\dddot x,\ddddot x,{x^{\left( 5 \right)}},{x^{\left( 6 \right)}}{x^{\left( 7 \right)}}} \right),</tex-math></alternatives></disp-formula>
where <italic>A</italic> = <italic>p</italic>2λ2 + <italic>p</italic>2<italic>µ</italic>2 + λ2<italic>µ</italic>2 + <italic>p</italic>2 + λ2 + <italic>µ</italic>2, <italic>B</italic> = <italic>p</italic>2 λ2 + <italic>p</italic>2<italic>µ</italic>2 + λ2<italic>µ</italic>2 + <italic>p</italic>2λ2<italic>µ</italic>2, with λ, <italic>µ</italic> and <italic>p</italic> are rational numbers different from −1, 0, 1, and <italic>p</italic> ≠ ±λ, <italic>p</italic> ≠±<italic>µ</italic>, λ ≠±<italic>µ</italic>, ɛ is sufficiently small and <italic>F</italic> is a nonlinear non-autonomous periodic function. Moreover we provide some applications.</abstract>

On the limit cycles for a class of eighth-order differential equations

<abstract xmlns="http://www.w3.org/1999/xhtml">For <italic>r</italic> ∈ (1, 2], the authors establish sufficient conditions for the existence of solutions for a class of boundary value problem for <italic>r</italic>th order Caputo-Hadamard fractional differential inclusions satisfying nonlinear integral conditions. Both cases of convex and nonconvex valued right hand sides are considered.</abstract>

Laboratoire des Mathematiques Appliqués et Pures

Boundary value problems for Caputo-Hadamard fractional differential inclusions with Integral Conditions

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we establish several inequalities involving certain generalizations of the hyperbolic functions. The established results serve as generalizations of some known results in the literature. Among other analytical techniques, the procedure makes use of l’Hospital rule for monotonicity.</abstract>

Department of Mathematics, Faculty of Mathematical Sciences

Some Inequalities for Generalized Hyperbolic Functions

<abstract xmlns="http://www.w3.org/1999/xhtml">We prove the existence of renormalized solutions to a class of nonlinear evolution equations, supplemented with initial and Dirichlet condition in the framework of generalized Sobolev spaces. The data are assumed merely integrable.</abstract>

On some nonlinear parabolic equations with variable exponents and measure data

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we continue studying the properties of weak soft axioms discussed and studied in [8]. We initiate and explore soft semi-<italic>R</italic>0 spaces at soft point in terms of soft semi-open sets and study its characterizations and properties. It is interesting to mention that this soft contains the soft semi-closure of each of its soft point singletons. We also define soft semi-<italic>R</italic>1 spaces at soft point and discuss some of its characterizations.</abstract>

More On Weak Soft Axioms

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we introduce the notion of (<italic>q</italic>, <italic>p</italic>)-mixing operators from the injective tensor product space <italic>E</italic> ̂⊗∈<italic>F</italic> into a Banach space <italic>G</italic> which we call (<italic>q</italic>, <italic>p</italic>, <italic>F</italic>)-mixing. In particular, we extend the notion of (<italic>q</italic>, <italic>p</italic>, <italic>E</italic>)-summing operators which is a special case of (<italic>q</italic>, <italic>p</italic>, <italic>F</italic>)-mixing operators to Lipschitz case by studying their properties and showing some results for this notion.</abstract>

Lipschitz (<italic>q</italic>, <italic>p</italic>, <italic>E</italic>)-summing operators on injective Lipschitz tensor products of spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">In this work, we establish some new (<italic>k</italic>, <italic>s</italic>) −fractional integral inequalities of continuous random variables by using the (<italic>k</italic>, <italic>s</italic>) −Riemann-Liouville fractional integral operator.</abstract>

Some estimations on continuous random variables for (<italic>k</italic>, <italic>s</italic>) −fractional integral 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 