<abstract><title style='display:none'>Abstract</title>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><title style='display:none'>Abstract</title>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><title style='display:none'>Abstract</title>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"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>𝒨</mml:mi><mml:msub><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msub><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>∞</mml:mo></mml:munderover><mml:mrow><mml:mfrac><mml:mrow><mml:mi>μ</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mi>s</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:msup><mml:mrow><mml:mi>z</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:mrow></mml:mrow></mml: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"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>∑</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:mi>μ</mml:mi><mml:mo>(</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:mfrac><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>i</mml:mi><mml:mi>n</mml:mi><mml:mi>π</mml:mi><mml:mi>θ</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml: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"/><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>h</mml:mi><mml:mo>/</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:mrow></mml: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>
<title style='display:none'>Abstract</title>
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.
</abstract>

Sums and products of periodic functions

<abstract>
<title style='display:none'>Abstract</title>
This paper deals with a new class of mixed fractional differential equations with integral boundary conditions. We show an important equivalence result between our problem and nonlinear integral Fredholm equation of the second kind. The existence and uniqueness of a positive solution are proved using Guo-Krasnoselskii’s fixed point theorem and Banach’s contraction mapping principle. Different types of Ulam-Hyers stability are discussed. Three examples are also given to show the applicability of our results.
</abstract>

A new class of mixed fractional differential equations with integral boundary conditions

<abstract>
<title style='display:none'>Abstract</title>
In the history of option pricing, Black-Scholes model is one of the most significant models. In this paper, we present a new numerical strategy for valuing American option pricing problems governed by Black-Scholes model (BSM). Numerical computations are carried out to show the efficiency and robustness of the proposed method. We compare our numerical solution with the ones based on Finite Element Method (FEM) and the Enriched Finite Element Method (PUFEM). Our result shows the efficiency of the proposed strategy. In addition, that approach can be used to treat nonlinear evolutionary problems.
</abstract>

About the valuation of American option under Black-Scholes model : a numerical study

<abstract>
<title style='display:none'>Abstract</title>
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>
<title style='display:none'>Abstract</title>
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>

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

<abstract>
<title style='display:none'>Abstract</title>
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>
<title style='display:none'>Abstract</title>
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>
<title style='display:none'>Abstract</title>
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.
</abstract>

<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

<abstract>
<title style='display:none'>Abstract</title>
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><title style='display:none'>Abstract</title> In this paper, we establish sufficient conditions for the existence and stability of solutions for a class of boundary value problem for implicit fractional differential equations with Caputo fractional derivative. The arguments are based upon the Banach contraction principle. Two examples are included to show the applicability of our results. </abstract>

Laboratory of Mathematics, University of Sidi Bel-Abbes, P.O. Box 89 Sidi Bel Abbes 22000, Algeria

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589,

Laboratory of Mathematics, University of Sidi Bel-Abbes, P.O. Box 89 Sidi Bel Abbes 22000,

Existence and Stability Results for Nonlinear Boundary Value Problem for Implicit Differential Equations of Fractional Order

<abstract><title style='display:none'>Abstract</title>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>

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

<abstract><title style='display:none'>Abstract</title> In this paper, we present Hermite-Hadamard inequality for p-convex functions in fractional integral forms. we obtain an integral equality and some Hermite-Hadamard type integral inequalities for p-convex functions in fractional integral forms. We give some Hermite-Hadamard type inequalities for convex, harmonically convex and p-convex functions. Some results presented in this paper for p-convex functions, provide extensions of others given in earlier works for convex, harmonically convex and p-convex functions.</abstract>

Hermite-Hadamard type inequalities for p-convex functions via fractional integrals

<abstract><title style='display:none'>Abstract</title> Some new inequalities of Hermite-Hadamard type for HA-convex functions defined on positive intervals are given.</abstract>

Inequalities of Hermite-Hadamard Type for HA-Convex Functions

<abstract><title style='display:none'>Abstract</title> In this paper, we introduce a new two-parameter deformation of the classical Gamma function, which we call a (p,k)-analogue of the Gamma function. We also provide some identities generalizing those satisfied by the classical Gamma function. Furthermore, we establish some inequalities involving this new function. </abstract>

Department of Mathematics, University for Development Studies, Navrongo Campus, P. O. Box 24, Navrongo, UE/R,

Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi,

On a (p,k)-analogue of the Gamma function and some associated Inequalities

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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...

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 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...

Moroccan Journal of Pure and Applied Analysis

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