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November 30, 2023
Abstract
The introduction of two-parameter ( p , q ) {(p,q)} -calculus and Lie algebras in 1991 has spurred a wave of recent research into ( p , q ) {(p,q)} -special polynomials, including ( p , q ) {(p,q)} -Bernoulli, ( p , q ) {(p,q)} -Euler, ( p , q ) {(p,q)} -Genocchi and ( p , q ) {(p,q)} -Frobenius–Euler polynomials. These investigations have been carried out by numerous researchers in order to uncover a wide range of identities associated with these polynomials and applications. In this article, we aim to introduce ( p , q ) {(p,q)} -sine and ( p , q ) {(p,q)} -cosine based λ-array type polynomials and derive numerous properties of these polynomials such as ( p , q ) {(p,q)} -integral representations, ( p , q ) {(p,q)} -partial derivative formulae and ( p , q ) {(p,q)} -addition formulae. It is worth noting that the utilization of the ( p , q ) {(p,q)} -polynomials introduced in this study, along with other ( p , q ) {(p,q)} -polynomials, can lead to the derivation of various identities that differ from the ones presented here.
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November 30, 2023
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In this paper, a new integral identity is provided. Based on this equality, Simpson-type dual integral inequalities for functions whose first derivatives are s -convex via Riemann–Liouville fractional integrals are established.
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November 30, 2023
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The aim of this article is to construct a ( p , q ) {(p,q)} -analogue of wavelets Kantorovich–Baskakov operators and investigate some statistical approximation properties. We study weighted statistical approximation by means of a Bohman–Korovkin-type theorem, and statistical rate of convergence by means of the weighted modulus of smoothness ω ρ α {\omega_{\rho_{\alpha}}} associated to the space B ρ α ( ℝ + ) {B_{\rho\alpha}(\mathbb{R_{+}})} and Lipschitz-type maximal functions.
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November 30, 2023
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In 2015, Abdeljawad defined the conformable fractional derivative (Grunwald–Letnikov technique) to iterate the conformable fractional integral of order 0 < α ≤ 1 {0<\alpha\leq{1}} (Riemann approach), yielding Hadamard fractional integrals when α = 0 {\alpha=0} . The Gronwall type inequality for generalized operators unifying Riemann–Liouville and Hadamard fractional operators is obtained in this study. We use this inequality to show how the order and initial conditions affect the solution of differential equations with generalized fractional derivatives. More features for generalized fractional operators are established, as well as solutions to initial value problems in several new weighted spaces of functions.
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October 27, 2023
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In this paper, we employ the techniques in [C. Cavaterra, S. Dipierro, Z. Gao and E. Valdinoci, Global gradient estimates for a general type of nonlinear parabolic equations, J. Geom. Anal. 32 2022, 2, Paper No. 65] and the approach in [H. T. Dung and N. T. Dung, Sharp gradient estimates for a heat equation in Riemannian manifolds, Proc. Amer. Math. Soc. 147 2019, 12, 5329–5338] to derive sharp gradient estimates for a positive solution to the heat equation u t = Δ u + a u log u u_{t}=\Delta u+au\log u in a complete noncompact Riemannian manifold (where a is a real constant). This is an extension of the gradient estimates of Dung and Dung.
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October 27, 2023
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The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p -Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations.
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August 31, 2023
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This paper studies the inhomogeneous defocusing coupled Schrödinger system i u ˙ j + Δ u j = | x | - ρ ( ∑ 1 ≤ k ≤ m a j k | u k | p ) | u j | p - 2 u j , ρ > 0 , j ∈ [ 1 , m ] . i\dot{u}_{j}+\Delta u_{j}=\lvert x\rvert^{-\rho}\bigg{(}\sum_{1\leq k\leq m}a_% {jk}\lvert u_{k}\rvert^{p}\biggr{)}\lvert u_{j}\rvert^{p-2}u_{j},\quad\rho>0,% \,j\in[1,m]. The goal of this work is to prove the scattering of energy global solutions in the conformal space made up of f ∈ H 1 ( ℝ N ) {f\in H^{1}(\mathbb{R}^{N})} such that x f ∈ L 2 ( ℝ N ) {xf\in L^{2}(\mathbb{R}^{N})} . The present paper is a complement of the previous work by the first author and Ghanmi [T. Saanouni and R. Ghanmi, Inhomogeneous coupled non-linear Schrödinger systems, J. Math. Phys. 62 2021, 10, Paper No. 101508]. Indeed, the supplementary assumption x u 0 ∈ L 2 {xu_{0}\in L^{2}} enables us to get the scattering in the mass-sub-critical regime p 0 < p ≤ 2 - ρ N + 1 {p_{0}<p\leq\frac{2-\rho}{N}+1} , where p 0 {p_{0}} is the Strauss exponent. The proof is based on the decay of global solutions coupled with some non-linear estimates of the source term in Strichartz norms and some standard conformal transformations. Precisely, one gets | t | α ∥ u ( t ) ∥ L r ( ℝ N ) ≲ 1 \lvert t\rvert^{\alpha}\lVert u(t)\rVert_{L^{r}(\mathbb{R}^{N})}\lesssim 1 for some α > 0 {\alpha>0} and a range of Lebesgue norms. The decay rate in the mass super-critical regime is the same one as of e i ⋅ Δ u 0 {e^{i\cdot\Delta}u_{0}} . This rate is different in the mass sub-critical regime, which requires some extra assumptions. The novelty here is the scattering of global solutions in the weighted conformal space for the class of source terms p 0 < p < 2 - ρ N - 2 + 1 {p_{0}<p<\frac{2-\rho}{N-2}+1} . This helps to better understand the asymptotic behavior of the energy solutions. Indeed, the source term has a negligible effect for large time and the above non-linear Schrödinger problem behaves like the associated linear one. In order to avoid a singular source term, one assumes that p ≥ 2 {p\geq 2} , which restricts the space dimensions to N ≤ 3 {N\leq 3} . In a paper in progress, the authors treat the same problem in the complementary case ρ < 0 {\rho<0} .
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August 31, 2023
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In the current investigation, we offer the generalized version of q 1 q 2 {q_{1}q_{2}} -Simpson’s type inequalities via ( α , m ) {(\alpha,m)} -coordinated convex functions. To validate their generalized behavior, we demonstrate the link between our outcomes and the already derived ones. Moreover, we provide some application to special means of positive real numbers to support our findings. The principal outcomes raised in this investigation are extensions and generalizations of the comparable results in the history on Simpson’s inequalities for coordinated convex functions.
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August 9, 2023
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This article presents a proof of the bounded convergence theorem for Riemann integrals. An effort has been made to keep the exposition concise and self-contained.
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August 1, 2023
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The objective of this work is to study the asymptotic justification of the two-dimensional model of viscoelastic von Kármán membrane shells. More precisely, we consider a three-dimensional model for a nonlinearly viscoelastic membrane shell with a specific class of boundary conditions of von Kármán type. Using techniques from formal asymptotic analysis with the thickness of the shell as a small parameter, we show that the scaled three-dimensional solution still leads to the two-dimensional model of viscoelastic von Kármán membrane shells.
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July 25, 2023
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We investigate the exterior differential operator on quasi-Kähler manifolds and show some relations of its components for smooth functions.
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July 25, 2023
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In this paper, we introduce the concepts of pseudo- n -multiplier and pseudo- n -Jordan multiplier on Banach algebras and compare those with the classical notions of n -multiplier and n -Jordan multiplier. We show that under special hypotheses every pseudo- ( n + 1 ) {(n+1)} -Jordan multiplier is a pseudo- n -Jordan multiplier and vice versa.
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June 27, 2023
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In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform.
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June 1, 2023
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In this paper we prove a new variant of q -Hermite–Hadamard–Mercer-type inequality for the functions that satisfy the Jensen–Mercer inequality (JMI). Moreover, we establish some new midpoint- and trapezoidal-type inequalities for differentiable functions using the JMI. The newly developed inequalities are also shown to be extensions of preexisting inequalities in the literature.
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June 1, 2023
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Many authors investigated the characteristics of the Bell, Euler, Bernoulli, and Genocchi polynomials because of their numerous uses in statistics, number theory, and other branches of science. A generating function for mixed-type Apostol–Euler polynomials of order η related with Bell polynomials is presented in this study. We also construct essential identities of Apostol–Euler polynomials of order η connected with Bell polynomials, such as the correlation formula, the implicit summation formula, the derivative formula, some association with Stirling numbers, and specific cases. Also we establish certain symmetric identities and their known consequences. In addition, we investigate several interesting relationships related to umbral calculus.
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June 1, 2023
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In this work, we propose some unified integral formulas for the Mittag-Leffler confluent hypergeometric function (MLCHF), and our findings are assessed in terms of generalized special functions. Additionally, certain unique cases of confluent hypergeometric function have been corollarily presented.
