Analysis in Theory and Applications
type: channel
description: Analysis in Theory and Applications (ATA), used to have the title Approximation Theory and its Applications (from 1984 to 2002), publishes research results in broad areas of approximation including Dynamic Systems, Geometric Analysis, Harmonic Analysis, Numerical Approximation, Signal Analysis, Special Functions, Differential Equations, Wavelets, and Approximation Theory, as well as their applications. The journal is supervised by Office of Education of Jiangsu Province and sponsored by Nanjing University.
publisher: Global Science Press
language: en
publication name: Analysis in Theory and Applications
issn: 1672-4070
items: Weighted Boundedness of Toeplitz Type Operator Associated to General Singular Integral Operator Satisfying a Variant of Hörmander’s Condition
items: Variable Exponent Central Morrey Estimates for Multilinear Bochner-Riesz Operators
items: Liouville Type Theorems for a Class of General Parabolic Hessian Quotient Type Equations
items: A Rigidity Result for the Schiffer Conjecture on Domain with a Hole
items: Separation of Sequences and Multipliers in the Space of Tempered Distributions
items: Estimates for Parabolic Schrödinger Operators with Certain Nonnegative Potentials
items: Double Cosine and Cosine-Sine Fourier Transforms and Generalized Lipschitz Classes in Uniform Metric
items: On Noveley Inequalities for Convex Stochastic Processes on Time Scale Utilizing Quantum Operators
items: Multiplicity of Solutions for an Integro-Differential Equation with Trudinger-Moser Subcritical or Critical Nonlinearities
items: Two-Weight Inequality of Calder\'{o}n-Zygmund Operator on Non-Homogeneous Metric Measure Space
items: On Non-Linearly Coupled Schrodinger Systems
items: Existence of Solutions for Fractional Differential Equations Involving Two Riemann-Liouville Fractional Orders
items: Some Characterizations of Bloch Functions
items: Decomposition Formulas of Kampé de Fériets Double Hypergeometric Functions
items: Erdös Type Inequality for Lorentz Polynomials
items: Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators
items: Domain of Euler Mean in the Space of Absolutely $p$-Summable Double Sequences with $0< p
items: Hopf Bifurcation of a Nonresident Computer Virus Model with Delay
items: An Upwind Multistep Difference with Mixed Finite Volume Element Method for a Positive Semi-Definite Contamination Treatment Problem
items: An Inverse Operator Approach to a Fractional Nonlinear Integro-Differential Equation
items: Quasi-Periodic Solutions of the Generalized KdV Equation
items: On Stationary $p(·)$-Kirchhoff-Type Problem with Mixed Boundary Conditions
items: The Character of Thurston’s Circle Packings with Obtuse Exterior Intersection Angles
items: Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions
items: Distributional Boundary Values of Holomorphic Functions on Tubular Domains
items: On Solutions of Differential-Difference Equations in $\mathbb{C}^n$
items: Author Index to Volume 40
items: Generalization of Certain Hyperbolic Integrals and a Dilogarithm Functional Relation
items: A Certain Class of Equi-Statistical Convergence Based on $(p, q)$-integers via Deferred Nörlund Mean and Related Approximation Theorems
items: Analytic Functions Related to the Strip Domains Involving Generalized Sălăgean Operator

Weighted Boundedness of Toeplitz Type Operator Associated to General Singular Integral Operator Satisfying a Variant of Hörmander’s Condition
type: item
description: In this paper, we establish the weighted sharp maximal function inequalities for the Toeplitz type operator associated to the general singular integral operator satisfying a variant of Hörmander's condition. As an application, we obtain the boundedness of the operator on weighted Lebesgue and Morrey spaces.
subject: Articles
  topic: section
subject: Toeplitz type operator
  topic: keywords
subject: singular integral operator
  topic: keywords
subject: sharp maximal function
  topic: keywords
subject: weighted BMO
  topic: keywords
subject: weighted Lipschitz function.
  topic: keywords
subject: 42B20
  topic: subjects
subject: 42B25
  topic: subjects
creator: Lanzhe Liu
creator: Xiaoqi Zhou
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-29
publication date: 2025-09-29
starting page: 284
ending page: 298
doi: 10.4208/ata.OA-2024-0020

Variable Exponent Central Morrey Estimates for Multilinear Bochner-Riesz Operators
type: item
description: We prove the boundedness of multilinear Bochner-Riesz operator on central Morrey spaces with variable exponents. Moreover, we also show that the variable exponent Lipschitz commutator of Bochner-Riesz operator are bounded on above spaces. As applications, we also obtain the bounds of the commutators generated by $\lambda$-central BMO functions and Bochner-Riesz operators on central Morrey spaces with variable exponents.
subject: Articles
  topic: section
subject: Multilinear Bochner-Riesz operator
  topic: keywords
subject: commutator
  topic: keywords
subject: central Morrey space with variable exponent
  topic: keywords
subject: boundedness.
  topic: keywords
subject: 42B15
  topic: subjects
subject: 42B20
  topic: subjects
subject: 42B35
  topic: subjects
creator: Yanqi Yang
creator: Lulu Yang
creator: Shuangping Tao
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-29
publication date: 2025-09-29
starting page: 259
ending page: 283
doi: 10.4208/ata.OA-2024-0022

Liouville Type Theorems for a Class of General Parabolic Hessian Quotient Type Equations
type: item
description: We first consider the a priori estimates to a class of general parabolic $(k,l)-$ Hessian quotient type equations of the form with $0{\le}1<k{\le}n.$ We derive that any $k-$admissible-monotone solution to or has interior gradient estimates and Pogorelov type estimates. As an application, we prove Liouville type theorems for these equations.
subject: Articles
  topic: section
subject: Parabolic $(k
  topic: keywords
subject: l)$-Hessian quotient type equations
  topic: keywords
subject: Pogorelov type estimates
  topic: keywords
subject: Liouville type theorems.
  topic: keywords
subject: 35K55
  topic: subjects
subject: 35B45
  topic: subjects
subject: 35B08
  topic: subjects
creator: Hui Liu
creator: Ni Xiang
creator: Lina Zheng
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-29
publication date: 2025-09-29
starting page: 238
ending page: 258
doi: 10.4208/ata.OA-2025-0006

A Rigidity Result for the Schiffer Conjecture on Domain with a Hole
type: item
description: Let $\Omega$ be a domain with a hole containing the origin in $\mathbb{R}^2$ and $u$ be a solution to the problem where $\partial^{\pm}\Omega$ represents the outer and inner boundaries of $\Omega,$ respectively, $c$ is a constant. Let ${\mu}_k$ denote the $k{\rm th}$ Neumann eigenvalue of the Laplacian on $\Omega$ and${\Omega}_h$ is the hole. We establish that if $\mu< {\mu}_8,$ then $\Omega$ is an annulus.
subject: Articles
  topic: section
subject: Schiffer conjecture
  topic: keywords
subject: overdetermined problem
  topic: keywords
subject: symmetry.
  topic: keywords
subject: 35J25
  topic: subjects
subject: 35N05
  topic: subjects
creator: Yingxin Sun
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-29
publication date: 2025-09-29
starting page: 229
ending page: 237
doi: 10.4208/ata.OA-2024-0023

Separation of Sequences and Multipliers in the Space of Tempered Distributions
type: item
description: We consider the notions of $v$-separation and $(N,v)$-separation for increasing sequences that tend to infinity. We study several of the connections between the properties of a multiplier in the space $S'(\mathbb{R})$ and in other related spaces and the separation properties of the sequence of its zeros. We also prove that a distributional division problem $$Fh=f,$$always has tempered solutions $h$ for any tempered data $f$ if and only if the non integrable function $1/F$ admits regularizations that are tempered, and that this holds if and only if the pseudofunction $\mathcal{P} f (1/F)$ is tempered.
subject: Articles
  topic: section
subject: Tempered distributions
  topic: keywords
subject: division problems
  topic: keywords
subject: separation of sequences.
  topic: keywords
subject: 46F10
  topic: subjects
creator: Ricardo Estrada
creator: Kevin Kellinsky-Gonzalez
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-29
publication date: 2025-09-29
starting page: 208
ending page: 228
doi: 10.4208/ata.OA-2024-0015

Estimates for Parabolic Schrödinger Operators with Certain Nonnegative Potentials
type: item
description: In this paper, the parabolic Schrödinger operator $\mathcal{P}={\partial}_t-\triangle+V(x)$ on $\mathbb{R}^{n+1}$ is considered, where $n\ge3,$ nonnegative potential $V$ belongs to the reverse Hölder class $RH_q$ with $q \ge n /2$. The $L^p$ boundedness of operators $V^{\alpha}\mathcal{P}^{-\beta},$ $V^{\alpha}{\nabla}\mathcal{P}^{-\beta}$ and their adjoint operators are established.
subject: Articles
  topic: section
subject: $L^p$ estimate
  topic: keywords
subject: parabolic Schrödinger operator
  topic: keywords
subject: reverse Hölder class.
  topic: keywords
subject: 42B25
  topic: subjects
subject: 35J10
  topic: subjects
subject: 42B37
  topic: subjects
creator: Yanhui Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-29
publication date: 2025-09-29
starting page: 197
ending page: 207
doi: 10.4208/ata.OA-2023-0009

Double Cosine and Cosine-Sine Fourier Transforms and Generalized Lipschitz Classes in Uniform Metric
type: item
description: For complex-valued functions $f \in L^1(\mathbb{R}^2_+)$, where $\mathbb{R}_+ := [0,\infty)$ we give sufficient conditions under which the double cosine or cosine-sine Fourier transform of $f$ belongs to a generalized Lipschitz class defined by the mixed modulus of smoothness of orders $m,n \in \mathbb{N} = \{1,2,\cdots \}$ in uniform metric. The sharpness of these conditions is established under some restriction for non-negative functions.
subject: Articles
  topic: section
creator: Sergey Volosivets
creator: Yulia Krotova
rights: Copyright (c) 2025 Analysis in Theory and Applications
date: 2025-09-15
publication date: 2025-09-15
doi: 10.4208/ata.OA-2022-0018

On Noveley Inequalities for Convex Stochastic Processes on Time Scale Utilizing Quantum Operators
type: item
description: We aim to present a innovation extension of the mean-square stochastic integral of a process on time scale based on a q-operator and fractional q-operator. By presenting the q-convex definition for a stochastic process and using quantum operators, we were able to prove some well-known inequalities such as the HermitHadamardand the Jensen for the processes.
subject: Articles
  topic: section
creator: Azizollah Babakhani
creator: Rosineide F. da Paz
rights: Copyright (c) 2025 Analysis in Theory and Applications
date: 2025-09-15
publication date: 2025-09-15
doi: 10.4208/ata.OA-2024-0048

Multiplicity of Solutions for an Integro-Differential Equation with Trudinger-Moser Subcritical or Critical Nonlinearities
type: item
description: This paper study a general nonlocal problem characterized by the equation: \[ -\mathcal{L}_K u + V(x)u = P(x)h(u) \] in $\mathbb{R}$. Here, $\mathcal{L}_K$ represents a nonlocal integro-differential operator, and $h$ is a nonlinear term displaying subcritical or critical growth of Trudinger-Moser type. We initially prove the existence of a ground state solution by employing variational methods and innovative analytical approaches. Furthermore, through the application of constrained variational methods, Brouwer degree theory, and a quantitative deformation lemma, we establish the existence of a sign-changing solution with minimal energy, surpassing the energy of the ground state solution.
subject: Articles
  topic: section
creator: Ying Cai
creator: Guangze Gu
creator: Zhipeng Yang
rights: Copyright (c) 2025 Analysis in Theory and Applications
date: 2025-09-15
publication date: 2025-09-15
doi: 10.4208/ata.OA-2025-0002

Two-Weight Inequality of Calder\'{o}n-Zygmund Operator on Non-Homogeneous Metric Measure Space
type: item
description: Let $(\mathcal{X}, d, \mu)$ be a non-homogeneous metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, we study weighted inequalities of the Calder\'{o}n-Zygmund operator on $(\mathcal{X}, d, \mu)$. Specifically, for $1 < p < \infty$, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted $L^p$ inequality holds. We deal with this problem by developing a vector-valued theory for Calder\'{o}n-Zygmund operators on the non-homogeneous metric measure spaces which is interesting in its own right.
subject: Articles
  topic: section
creator: Yaoyao Han
rights: Copyright (c) 2025 Analysis in Theory and Applications
date: 2025-09-15
publication date: 2025-09-15
doi: 10.4208/ata.OA-2024-0019

On Non-Linearly Coupled Schrodinger Systems
type: item
description: Abstract. In this paper we study the following nonlinear Schr\"{o}dinger system: \[ \begin{cases} -\Delta u + a(x) u = \lambda(x) f(u, v), & x \in \mathbb{R}^N, \\ -\Delta v + b(x) v = \lambda(x) g(u, v), & x \in \mathbb{R}^N, \\ u(x) \to 0, & v(x) \to 0 \end{cases} \] as $ |x| \to \infty $. Here, $ a, b, \lambda \in C(\mathbb{R}^N, \mathbb{R}) $ are all non-periodic in $ x_i $ for $ i = 1, \cdots, N, N \geq 3, f, g \in C(\mathbb{R}^2, \mathbb{R}) $. We show that this system has infinitely many solutions with small negative energies and infinitely many large-energy solutions. To the best of our knowledge, there is no corresponding result about such a Schr\"{o}dinger system.
subject: Articles
  topic: section
creator: Jing Zhang
rights: Copyright (c) 2025 Analysis in Theory and Applications
date: 2025-09-15
publication date: 2025-09-15
doi: 10.4208/ata.OA-2025-0014

Existence of Solutions for Fractional Differential Equations Involving Two Riemann-Liouville Fractional Orders
type: item
description: In this work, we study existence and uniqueness of solutions for multi-point boundary value problem of nonlinear fractional differential equations with two fractional derivatives. By using the variety of fixed point theorems, such as Banach\u2019s fixed point theorem, Leray-Schauder\u2019s nonlinear alternative and Leray-Schauder\u2019s degree theory, the existence of solutions is obtained. At the end, some illustrative examples are discussed.
subject: Articles
  topic: section
subject: Riemann-Liouville integral
  topic: keywords
subject: existence
  topic: keywords
subject: fixed point theorem
  topic: keywords
subject: Leray-Schauder alternative.
  topic: keywords
subject: 26A33
  topic: subjects
subject: 34A12
  topic: subjects
subject: 34A08
  topic: subjects
creator: Mohamed Houas
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2018-11-14
publication date: 2018-11-14
starting page: 253
ending page: 274
doi: 10.4208/ata.OA-2017-0021

Some Characterizations of Bloch Functions
type: item
description: We define Bloch-type functions of $\mathcal{C}^1(\mathbb{D})$ on the unit disk of complex plane $\mathbb{C}$ and characterize them in terms of weighted Lipschitz functions. We also discuss the boundedness of a composition operator $C_{\phi}$\u00a0acting between two Bloch-type spaces. These obtained results generalize the corresponding known ones to the setting of $\mathcal{C}^1(\mathbb{D})$.
subject: Articles
  topic: section
subject: Bloch space
  topic: keywords
subject: majorant
  topic: keywords
subject: composition operator.
  topic: keywords
subject: 11K70
  topic: subjects
subject: 32A70
  topic: subjects
creator: Guanghua He
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2018-11-14
publication date: 2018-11-14
starting page: 225
ending page: 231
doi: 10.4208/ata.OA-2017-0059

Decomposition Formulas of Kampé de Fériets Double Hypergeometric Functions
type: item
description: This paper is sequel to the authors paper [18]. By using the generalized Burchnall-Chaundy operator method, the authors are aiming at deriving certain decomposition formulas for some interesting special cases of Kamp\u00e9 de F\u00e9riets series of double hypergeometric series $F^{p:q;k}_{l:m;n}$.
subject: Articles
  topic: section
subject: Generalized hypergeometric function
  topic: keywords
subject: Kampé de Fériets functions
  topic: keywords
subject: decomposition formulas
  topic: keywords
subject: inverse pairs of symbolic operators.
  topic: keywords
subject: 35K65
  topic: subjects
subject: 33C20
  topic: subjects
subject: 33C65
  topic: subjects
creator: Maged G. Bin-Saad, Anvar Hasanov & Mamasali Turaev
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2018-11-14
publication date: 2018-11-14
starting page: 275
ending page: 292
doi: 10.4208/ata.OA-2017-0077

Erdös Type Inequality for Lorentz Polynomials
type: item
description: An elementary, but very useful tool for proving inequalities for polynomials with restricted zeros is the Bernstein or Lorentz representation of polynomials. In the present paper, we give two classes of Lorentz polynomials, for which the Erd\u00f6s-type inequality holds.
subject: Articles
  topic: section
subject: Lorentz representation of polynomials
  topic: keywords
subject: constrained polynomials
  topic: keywords
subject: Morkov-type inequalities
  topic: keywords
subject: Erdös-type inequality.
  topic: keywords
subject: 11C08
  topic: subjects
subject: 41A17
  topic: subjects
creator: Laiyi Zhu, Dapeng Zhou & Zhiyong Huang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2018-11-14
publication date: 2018-11-14
starting page: 232
ending page: 240
doi: 10.4208/ata.OA-2017-0002

Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators
type: item
description: $[b,T]$ denotes the commutator of generalized Calder\u00f3n-Zygmund operators $T$ with Lipschitz function $b$, where $b \u2208 \rm{Lip}_\u03b2(R^n)$, $(0<\u03b2\u22641)$ and $T$ is a $\u03b8(t)$\u2212type Calder\u00f3n-Zygmund operator. The commutator $[b,T]$ generated by $b$ and $T$ is defined by\u00a0 $$[b,T] f(x)=b(x)Tf(x)\u2212T(bf)(x)=\u222bk(x,y)(b(x)\u2212b(y))f(y)dy.$$ In this paper, the authors discuss the boundedness of the commutator $[b,T]$ on weighted Hardy spaces and weighted Herz type Hardy spaces and prove that $[b,T]$ is bounded from $H^p(\u03c9^p)$ to $L^q(\u03c9^q)$, and from $H\dot{K}^{\u03b1,p}_{q_1}(\u03c9_1,\u03c9^{q_1}_2)$ to $\dot{K}^{\u03b1,p}_{q_2}(\u03c9_1,\u03c9^{q_2}_2)$. The results extend and generalize the well-known ones in [7].
subject: Articles
  topic: section
subject: Commutator
  topic: keywords
subject: Lipschitz function
  topic: keywords
subject: weighted hardy space
  topic: keywords
subject: Herz space.
  topic: keywords
subject: 42B20
  topic: subjects
subject: 42B25
  topic: subjects
creator: Cuilan Wu, Yunjie Wang & Lisheng Shu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2018-11-14
publication date: 2018-11-14
starting page: 209
ending page: 224
doi: 10.4208/ata.OA-2017-0050

Domain of Euler Mean in the Space of Absolutely $p$-Summable Double Sequences with $0< p
type: item
description: In this study, as the domain of four dimensional Euler mean $E(r,s)$ of orders $r$, $s$ in the space $\mathcal{L}_p$ for $0<p<1$, we examine the double sequence space $\varepsilon^{r,s}_p$ and some properties of four dimensional Euler mean. We determine the $\u03b1$- and $\u03b2(bp)$-duals of the space $\varepsilon^{r,s}_p$, and characterize the classes $(\varepsilon^{r,s}_p:\mathcal{M}_u)$,\u00a0$(\varepsilon^{r,s}_p:\mathcal{C}_{bp})$\u00a0and $(\varepsilon^{r,s}_p:\mathcal{L}_q)$ of four dimensional matrix transformations, where $1\u2264q<\u221e$. Finally, we shortly emphasize on the Euler spaces of single and double sequences, and note some further suggestions.
subject: Articles
  topic: section
subject: Summability theory
  topic: keywords
subject: double sequences
  topic: keywords
subject: double series
  topic: keywords
subject: alpha-
  topic: keywords
subject: beta- and gamma-duals
  topic: keywords
subject: matrix domain of 4-dimensional matrices
  topic: keywords
subject: matrix transformations.
  topic: keywords
subject: 46A45
  topic: subjects
subject: 40C05
  topic: subjects
creator: Medine Ye\u015filkayagil & Feyzi Ba\u015far
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2018-11-14
publication date: 2018-11-14
starting page: 241
ending page: 252
doi: 10.4208/ata.OA-2017-0056

Hopf Bifurcation of a Nonresident Computer Virus Model with Delay
type: item
description: In this paper, a delayed nonresident computer virus model with graded infection rate is considered in which the following assumption is imposed: latent computers have lower infection ability than infectious computers. With the aid of the bifurcation theory, sufficient conditions for stability of the infected equilibrium of the model and existence of the Hopf bifurcation are established. In particular, explicit formulae which determine direction and stability of the Hopf bifurcation are derived by means of the normal form theory and the center manifold reduction for functional differential equations. Finally, a numerical example is given in order to show the feasibility of the obtained theoretical findings.
subject: Articles
  topic: section
subject: Computer virus
  topic: keywords
subject: delay
  topic: keywords
subject: Hopf bifurcation
  topic: keywords
subject: SLA model
  topic: keywords
subject: Periodic solution.
  topic: keywords
subject: 34C15
  topic: subjects
subject: 34C23
  topic: subjects
subject: 37G15
  topic: subjects
subject: 37N25
  topic: subjects
creator: Zizhen Zhang, Yougang Wang & Massimiliano Ferrara
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2018-11-14
publication date: 2018-11-14
starting page: 199
ending page: 208
doi: 10.4208/ata.OA-2016-0035

An Upwind Multistep Difference with Mixed Finite Volume Element Method for a Positive Semi-Definite Contamination Treatment Problem
type: item
description: A positive semi-definite problem of three-dimensional incompressible contamination treatment from nuclear waste in porous media is discussed in this paper. The mathematical model is defined by a nonlinear initial-boundary system consisting of partial differential equations. Four important equations (an elliptic equation, two convection-diffusion equations and a heat conductor equation) determine the physical features. Considering the physical natures and computational efficiency, the authors introduce the conservative mixed finite volume element, upwind approximation and multistep difference to solve this system. The pressure and Darcy velocity are computed by a mixed finite volume element. The concentrations and temperature are solved by a combination of upwind approximation, multistep difference and mixed finite volume element. A multistep difference is used for approximating the partial derivative with respect to time. Mixed finite volume element and upwind differences are given for solving the convection-diffusions equations. Numerical dispersion and nonphysical oscillations could be eliminated, and the computational efficiency is improved by using a large time step. Furthermore, a conservative law is preserved and error estimates in $L^2$-norm is obtained. Finally, two numerical experiments are given to show the efficiency and possible applications.
subject: Articles
  topic: section
subject: Positive semi-definite problem
  topic: keywords
subject: upwind multistep difference with mixed finite volume element
  topic: keywords
subject: local conservation
  topic: keywords
subject: convergence analysis
  topic: keywords
subject: numerical experiments.
  topic: keywords
subject: 65M15
  topic: subjects
subject: 65N30
  topic: subjects
subject: 65N12
  topic: subjects
subject: 76S05
  topic: subjects
creator: Changfeng Li
creator: Yirang Yuan
creator: Huailing Song
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-07-19
publication date: 2025-07-19
starting page: 111
ending page: 139
doi: 10.4208/ata.OA-2022-0024

An Inverse Operator Approach to a Fractional Nonlinear Integro-Differential Equation
type: item
description: This paper studies the existence, uniqueness and stability for a new fractional nonlinear integro-differential equation with an integral boundary condition using several well-known fixed point theorems in a Banach space. The method used is to convert the equation into an equivalent implicit integral equation based on a bounded inverse operator which is an infinite series and uniformly convergent. Furthermore, we compute approximate values of a few multivariate Mittag-Leffler functions by our Python codes in illustrative examples demonstrating the use of key theorems derived. These investigations have a wide range of applications as existence, uniqueness and stability often appear in various pure and applied research areas.
subject: Articles
  topic: section
subject: Fractional nonlinear integro-differential equation
  topic: keywords
subject: uniqueness and existence
  topic: keywords
subject: fixed point theory
  topic: keywords
subject: multivariate Mittag-Leffler function
  topic: keywords
subject: inverse operator.
  topic: keywords
subject: 34B15
  topic: subjects
subject: 34A12
  topic: subjects
subject: 34K20
  topic: subjects
subject: 26A33
  topic: subjects
creator: Chenkuan Li
creator: Wenyuan Liao
creator: Reza Saadati
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-07-19
publication date: 2025-07-19
starting page: 172
ending page: 196
doi: 10.4208/ata.OA-2024-0043

Quasi-Periodic Solutions of the Generalized KdV Equation
type: item
description: This paper concerns the existence of real analytic quasi-periodic solutions close to the constant function 1 of the generalized KdV equation. The proof is based on an abstract KAM theorem for infinite dimensional Hamiltonian systems.
subject: Articles
  topic: section
subject: KAM theorem
  topic: keywords
subject: generalized KdV equation
  topic: keywords
subject: normal form
  topic: keywords
subject: quasi-periodic solution.
  topic: keywords
subject: 37K55
  topic: subjects
subject: 35B15
  topic: subjects
creator: Zhaowei Lou
creator: Yingnan Sun
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-07-19
publication date: 2025-07-19
starting page: 99
ending page: 110
doi: 10.4208/ata.OA-2024-0031

On Stationary $p(·)$-Kirchhoff-Type Problem with Mixed Boundary Conditions
type: item
description: In this paper, we consider the stationary Kirchhoff-type equation with mixed boundary conditions in a variable exponent Sobolev space. We show the existence of a weak solution. Moreover, we prove the existence of a non-trivial weak solution and infinitely many weak solutions under some hypotheses by variational methods.
subject: Articles
  topic: section
subject: Mixed boundary value problem
  topic: keywords
subject: $p(·)$-Kirchhoff-type problem
  topic: keywords
subject: existence of a weak solution
  topic: keywords
subject: variational methods.
  topic: keywords
subject: 35J65
  topic: subjects
subject: 35J20
  topic: subjects
subject: 47J10
  topic: subjects
creator: Junichi Aramaki
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-07-19
publication date: 2025-07-19
starting page: 140
ending page: 171
doi: 10.4208/ata.OA-2022-0010

The Character of Thurston’s Circle Packings with Obtuse Exterior Intersection Angles
type: item
description: We study the character of Thurston's circle packings with obtuse exterior intersection angles, and we get some simple criteria for the existence of hyperbolic circle packings. Moreover, the compactness theorem of combinatorial Ricci flows in hyperbolic background geometry with the weight function $Φ ∈ [0, π)$ is obtained. As a consequence, We generalize G. Lin's result [12] from acute exterior intersection angles case to obtuse exterior intersection angles case.
subject: Articles
  topic: section
subject: Character
  topic: keywords
subject: Thurston’s circle packing
  topic: keywords
subject: combinatorial Ricci flow
  topic: keywords
subject: obtuse exterior intersection angles.
  topic: keywords
subject: 52C26
  topic: subjects
subject: 53A70
  topic: subjects
subject: 53E99
  topic: subjects
subject: 57Q15
  topic: subjects
creator: Chunlei Liu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-11
publication date: 2025-04-11
starting page: 80
ending page: 98
doi: 10.4208/ata.OA-2024-0046

Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions
type: item
description: We prove that, for some suitable smooth bounded domain, there exists a solution to the following Neumann problem for the Lane-Emden system: where $Ω$ is some smooth bounded domain in $\mathbb{R}^N,$ $N ≥ 4,$ $\mu > 0,$ $α > 0,$ $β > 0$ are constants and $ε\ne 0$ is a small number. We show that there exists a solution to the slightly supercritical problem for $ε > 0,$ and for $ε < 0,$ there also exists a solution to the slightly subcritical problem if the domain is not convex. Comparing with the single elliptic equations, the challenges and novelty are manifested in the construction of good approximate solutions characterizing the boundary behavior under Neumann boundary conditions, in which process, the selection of the range of nonlinear coupling exponents and the weighted Sobolev spaces requires elaborate discussion.
subject: Articles
  topic: section
subject: Lane-Emden system
  topic: keywords
subject: Neumann problem
  topic: keywords
subject: blow up solutions
  topic: keywords
subject: reduction method.
  topic: keywords
subject: 35J57
  topic: subjects
subject: 35B25
  topic: subjects
subject: 35B38
  topic: subjects
creator: Qing Guo
creator: Junyuan Liu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-11
publication date: 2025-04-11
starting page: 1
ending page: 34
doi: 10.4208/ata.OA-0026

Distributional Boundary Values of Holomorphic Functions on Tubular Domains
type: item
description: The main purpose of this paper is to establish the distributional boundary values of functions in the weighted Hardy space, which improves the results of Carmichael in [4] and [8], where the weight function is linear. As our main result, we will prove that $f(z)$ in $H(ψ, Γ)$ has the $\mathcal{Z}'$ boundary value and can be expressed by the inverse Fourier transform of a distribution. Next, we will establish the $S'$ boundary value under stronger assumptions and give more precise expression if $f(z)$ also converges to $U ∈ D'_{L^p}(\mathbb{R}^n),$ where $1 ≤ p ≤ 2.$ In addition, we will also study the inverse result, in which we will prove that $f(z)$ is holomorphic on $T_Γ.$
subject: Articles
  topic: section
subject: The weighted Hardy space
  topic: keywords
subject: distributional boundary values
  topic: keywords
subject: tubular domains.
  topic: keywords
subject: 32A07
  topic: subjects
subject: 32A40
  topic: subjects
subject: 42B25
  topic: subjects
subject: 42B30
  topic: subjects
creator: Guantie Deng
creator: Weiwei Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-11
publication date: 2025-04-11
starting page: 35
ending page: 51
doi: 10.4208/ata.OA-2022-0017

On Solutions of Differential-Difference Equations in $\mathbb{C}^n$
type: item
description: In this paper, we mainly explore the existence of entire solutions of the quadratic trinomial partial differential-difference equation $$af^2(z)+2\omega f(z)(a_0f(z)+L^{k+s}_{1,2}(f(z)))+b(a_0f(z)+L^{k+s}_{1,2}(f(z)))^2=e^{g(z)}$$ by utilizing Nevanlinna’s theory in several complex variables, where $g(z)$ is entire functions in $\mathbb{C}^n,$ $ω\ne 0$ and $a, b, ω ∈ \mathbb{C}.$ Furthermore, we get the exact froms of solutions of the above differential-difference equation when $ω = 0.$ Our results are generalizations of previous results. In addition, some examples are given to illustrate the accuracy of the results.
subject: Articles
  topic: section
subject: Differential-difference equations
  topic: keywords
subject: Nevanlinna theory
  topic: keywords
subject: finite order
  topic: keywords
subject: entire solutions.
  topic: keywords
subject: 39A45
  topic: subjects
subject: 30D35
  topic: subjects
subject: 39A14
  topic: subjects
subject: 32H30
  topic: subjects
subject: 35A20
  topic: subjects
creator: Ling Yang
creator: Lu Chen
creator: Shimei Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-04-11
publication date: 2025-04-11
starting page: 52
ending page: 79
doi: 10.4208/ata.OA-2024-0044

Author Index to Volume 40
type: item
subject: Articles
  topic: section
subject:
  topic: subjects
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-02-27
publication date: 2025-02-27
starting page: 435
ending page: 435

Generalization of Certain Hyperbolic Integrals and a Dilogarithm Functional Relation
type: item
description: In a previous work [Indag. Math., 23(1) (2012)], I did employ a hyperbolic version of the Beukers, Calabi, and Kolk change of variables to solve $$\int_0^1\int_0^1(1-x^2y^2)^{-1}dxdy,$$ which yielded exact closed-form expressions for some definite integrals and, from one of them, I proved a two-term dilogarithm identity. Here in this note, I derive closed-form expressions for $$\int_0^b[{\rm sinh}^{-1}({\rm cosh} \ x)-x]dx, b\ge 0 \ {\rm and} \ \int_{\alpha/2}^{\beta/2} {\rm ln}({\rm tanh} \ x)dx, \ \ b\in \mathbb{R},$$ where $\alpha := {\rm sinh}^{−1} (1)$ and $β := b + {\rm sinh}^{−1} ({\rm cosh} \ b).$ From these general results, I derive a dilogarithm functional relation.
subject: Articles
  topic: section
subject: Hyperbolic integrals
  topic: keywords
subject: dilogarithm function
  topic: keywords
subject: dilogarithm relations.
  topic: keywords
subject: 40C10
  topic: subjects
subject: 11M06
  topic: subjects
subject: 33B30
  topic: subjects
creator: F. M. S. Lima
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-02-27
publication date: 2025-02-27
starting page: 422
ending page: 434
doi: 10.4208/ata.OA-2018-0011

A Certain Class of Equi-Statistical Convergence Based on $(p, q)$-integers via Deferred Nörlund Mean and Related Approximation Theorems
type: item
description: The concept of equi-statistical convergence is more general than that of the well-established statistical uniform convergence. In this paper, we have introduced the idea of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence under the difference operator including $(p, q)$-integers via deferred Nörlund statistical convergence so as to build up a few inclusion relations between them. We have likewise presented the notion of the deferred weighted (Nörlund type) equi-statistical convergence (presumably new) in view of difference sequence of order $r$ based on $(p, q)$-integers to demonstrate a Korovkin type approximation theorem and proved that our theorem is a generalization (non-trivial) of some well-established Korovkin type approximation theorems which were demonstrated by earlier authors. Eventually, we set up various fascinating examples in connection with our definitions and results.
subject: Articles
  topic: section
subject: Statistical convergence
  topic: keywords
subject: $(p
  topic: keywords
subject: q)$-integers
  topic: keywords
subject: deferred Nörlund summability
  topic: keywords
subject: $\varphi^{p
  topic: keywords
subject: q}_n$-equi-statistical convergence
  topic: keywords
subject: rate of convergence and Korovkin type approximation theorems.
  topic: keywords
subject: 40A05
  topic: subjects
subject: 41A36
  topic: subjects
subject: 40G15
  topic: subjects
creator: A. A. Das
creator: Vishnu Narayan Mishra
creator: S. K. Paikray
creator: P. Parida
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-02-27
publication date: 2025-02-27
starting page: 381
ending page: 404
doi: 10.4208/ata.OA-2018-0018

Analytic Functions Related to the Strip Domains Involving Generalized Sălăgean Operator
type: item
description: In the paper, the authors introduce a new subclass of univalent functions associated with the strip domains by using the generalized Sălăgean operator. The bounds of coefficients and Fekete-Szegö inequality for functions in this class are obtained. The results presented here extend some of the earlier results.
subject: Articles
  topic: section
subject: Analytic functions
  topic: keywords
subject: starlike functions
  topic: keywords
subject: strip domain
  topic: keywords
subject: differential subordination
  topic: keywords
subject: Sălăgean operator.
  topic: keywords
subject: 30C45
  topic: subjects
subject: 30C80
  topic: subjects
creator: Xiaoyuan Wang
creator: Zhiren Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-02-27
publication date: 2025-02-27
starting page: 405
ending page: 421
doi: 10.4208/ata.OA-2018-0002