<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>In this paper, we introduced a generalization of the raised cosine distribution. We also provided its several distributional properties and characterizations, including percentiles and some applications.</abstract>

On a Generalized Raised Cosine Distribution: Some Properties, Characterizations and Applications

<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>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>A modified system of nonlinear fractional-order differential equations become used to classify humans of various personalities and different Impact Factors of Memory (IFM); with unique set of model parameters. The model was used to interpret and predict the functions of the union of various people with external circumstance(s) and adapted to neighborhood environment in which the statistics collections were achieved to analyze numerous measures affecting marriages, unique challenges in marriage and associated occasions were investigated through the use of questionnaire. Data had been analyzed and the outcomes have been carried out as parameters to validate the model. Adams Predictor-Corrector Method was used to test the chaoticity of the system and it was confirmed via numerical simulations.Numerical simulation outcomes had been presented to reveal the effectiveness of the model and the accuracy of the statements established. The mathematical information implied by the model unveil an underlying mechanism which can give an explanation for couple disruption in relationships that were initially deliberated to remain all the time. Despite the terrible aspects of relationships, some human beings were still satisfied in their intimate members of the family. The study was addressed on a field survey (use of questionnaires) and with the aid of interrogating the members one on one. A feasible path for future work is the choice to attain balance through modelling and to validate the results with the aid of numerical simulations.</abstract>

A Modified Fractional Differential Love Model

<abstract><title style='display:none'>Abstract</title>The major emphasis of this work is the derivation of a posteriori error estimates for the mixed finite volume discretization of second-order elliptic equations. The estimates are established for meshes consisting of simplices on unstructured grids. We consider diffusion problems with nonhomogeneous diffusion coefficients. The error estimates are of residual types and are formulated in the energy semi-norm for a locally postprocessed approximate solutions. The estimates are fully computable and locally efficient that they can serve as indicators for adaptive refinement and for the actual control of the error. Numerical results are shown for two test examples in two space dimensions. It is found that the proposed adaptive mixed finite volume method offers a robust and accurate approach for solving second-order elliptic equations, even when highly nonhomogeneous diffusion coefficients are used in the simulations.</abstract>

A posteriori error estimates for mixed finite volume solution of elliptic boundary value problems

<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>
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 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>
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>
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>
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> 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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