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.
Weighted Boundedness of Toeplitz Type Operator Associated to General Singular Integral Operator Satisfying a Variant of Hörmander’s Condition
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.
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Variable Exponent Central Morrey Estimates for Multilinear Bochner-Riesz Operators
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.
central Morrey space with variable exponent
Liouville Type Theorems for a Class of General Parabolic Hessian Quotient Type Equations
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.
A Rigidity Result for the Schiffer Conjecture on Domain with a Hole
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.
Separation of Sequences and Multipliers in the Space of Tempered Distributions
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.
Estimates for Parabolic Schrödinger Operators with Certain Nonnegative Potentials
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.
Double Cosine and Cosine-Sine Fourier Transforms and Generalized Lipschitz Classes in Uniform Metric
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.
Copyright (c) 2025 Analysis in Theory and Applications
On Noveley Inequalities for Convex Stochastic Processes on Time Scale Utilizing Quantum Operators
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.
Multiplicity of Solutions for an Integro-Differential Equation with Trudinger-Moser Subcritical or Critical Nonlinearities
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.
Two-Weight Inequality of Calder\'{o}n-Zygmund Operator on Non-Homogeneous Metric Measure Space
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.
On Non-Linearly Coupled Schrodinger Systems
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.
Existence of Solutions for Fractional Differential Equations Involving Two Riemann-Liouville Fractional Orders
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.
Some Characterizations of Bloch Functions
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 $\mathcal{C}^1(\mathbb{D})$.
Decomposition Formulas of Kampé de Fériets Double Hypergeometric Functions
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 $F^{p:q;k}_{l:m;n}$.
Maged G. Bin-Saad, Anvar Hasanov & Mamasali Turaev
Erdös Type Inequality for Lorentz Polynomials
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.
Weighted Boundedness of Commutators of Generalized Calderόn-Zygmund Operators
$[b,T]$ 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].
Domain of Euler Mean in the Space of Absolutely $p$-Summable Double Sequences with $0< p
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.
Medine Ye\u015filkayagil & Feyzi Ba\u015far
Hopf Bifurcation of a Nonresident Computer Virus Model with Delay
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.
Zizhen Zhang, Yougang Wang & Massimiliano Ferrara
An Upwind Multistep Difference with Mixed Finite Volume Element Method for a Positive Semi-Definite Contamination Treatment Problem
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.
upwind multistep difference with mixed finite volume element
An Inverse Operator Approach to a Fractional Nonlinear Integro-Differential Equation
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.
Fractional nonlinear integro-differential equation
Quasi-Periodic Solutions of the Generalized KdV Equation
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.
On Stationary $p(·)$-Kirchhoff-Type Problem with Mixed Boundary Conditions
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.
The Character of Thurston’s Circle Packings with Obtuse Exterior Intersection Angles
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.
Blowing Up Solutions to Slightly Sub- or Super-Critical Lane-Emden Systems with Neumann Boundary Conditions
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.
Distributional Boundary Values of Holomorphic Functions on Tubular Domains
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 $T_Γ.$
On Solutions of Differential-Difference Equations in $\mathbb{C}^n$
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.
Generalization of Certain Hyperbolic Integrals and a Dilogarithm Functional Relation
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.
A Certain Class of Equi-Statistical Convergence Based on $(p, q)$-integers via Deferred Nörlund Mean and Related Approximation Theorems
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.
rate of convergence and Korovkin type approximation theorems.
Analytic Functions Related to the Strip Domains Involving Generalized Sălăgean Operator
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.