Ikehara's Theorem Research Articles

A simple proof of the Wiener–Ikehara Tauberian Theorem

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series ∑n=1∞a(n)ns. We provide a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method also allows us to derive a version of the Wiener–Ikehara theorem with an error term.

Dynamics on time scales of wave solutions for nonlinear neural networks

A time scale is an arbitrary non-empty closed subset of denoted by . In this work, the traveling wave solutions on time scales for a class of nonlinear delayed dynamical neural networks are investigated. The outcomes of this study are new and have been obtained by inequality technique, Ikehara's theorem, the weighted energy method, Taylor formula, comparison principle and the first integral mean value theorem. This is the first work focused for the wave solution in time space scales for a dynamical delayed neural networks model.

Bicomplex Landau and Ikehara Theorems for the Dirichlet Series

The aim of this paper is to generalize the Landau‐type Tauberian theorem for the bicomplex variables. Our findings extend and improve on previous versions of the Ikehara theorem. Also boundedness result for the bicomplex version of Ikehara–Korevaar theorem is derived. The purpose of this article is to substantially extend the various complex Tauberian theorems for the Dirichlet series to the bicomplex domain.

Novel traveling waves solutions for nonlinear delayed dynamical neural networks with leakage term

In this work, the exponential stability of monotone traveling wave solutions for a class of nonlinear delayed dynamical neural networks with leakage term and distributed delays is investigated. The outcomes of this study are new and has obtained by applying of Ikeharas theorem, the weighted energy method, Taylor formula, comparison principle and the first integral mean value theorem. In addition, the leakage term is considered for the first time in this work to the dynamical model of cellular neural networks.

Asymptotic behavior of traveling waves for non-quasi-monotone system with delay

This paper is concerned with a population dynamic model with delay. In this work, by rewriting the equation and using the Ikehara's theorem, we show the exact asymptotic behavior of the profile as \(\xi\rightarrow\)-\(\infty\) for critical speed.

On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: A constructive approach

We construct explicit counterexamples that show that it is impossible to get any remainder, other than the classical ones, in the Wiener-Ikehara theorem and the Ingham-Karamata theorem under just an additional analytic continuation hypothesis to a half-plane (or even to the whole complex plane).

An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions

We study the family of Fourier-Laplace transformsFα,β(z)=F.p.∫0∞tβexp⁡(itα−izt)dt,Imz<0, for α>1 and β∈C, where Hadamard finite part is used to regularize the integral when Reβ≤−1. We prove that each Fα,β has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems.

Complex Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior

We provide several Tauberian theorems for Laplace transforms with local pseudofunction boundary behavior. Our results generalize and improve various known versions of the Ingham-Fatou-Riesz theorem and the Wiener-Ikehara theorem. Using local pseudofunction boundary behavior enables us to relax boundary requirements to a minimum. Furthermore, we allow possible null sets of boundary singularities and remove unnecessary uniformity conditions occurring in earlier works; to this end, we obtain a useful characterization of local pseudofunctions. Most of our results are proved under one-sided Tauberian hypotheses; in this context, we also establish new boundedness theorems for Laplace transforms with pseudomeasure boundary behavior. As an application, we refine various results related to the Katznelson-Tzafriri theorem for power series.

An Ikehara-type theorem for functions convergent to zero

We establish an analogue of the Ikehara theorem for positive non-increasing functions convergent to zero. In particular, this provides a complete proof of the results formulated in Diekmann & Kaper (1978) [5] and Carr & Chmaj (2004) [1], which are widely used nowadays to prove the uniqueness of traveling waves for various reaction–diffusion equations.

Asymptotics and uniqueness of traveling wavefronts for a delayed model of the Belousov–Zhabotinsky reaction

ABSTRACT This paper is concerned with asymptotics and uniqueness of traveling wavefronts for a delayed model of the Belousov–Zhabotinsky reaction. It is known that this system admits traveling wavefronts for both monostable and bistable types. In this paper, we further study the monostable case. We first establish the precisely asymptotic behavior of traveling wavefronts with the help of Ikehara's Theorem. Then based on the obtained asymptotic behavior, the uniqueness of the traveling wavefronts is proved by the strong comparison principle and the sliding method, when time delay , which complements the uniqueness results obtained by Trofimchuk et al.

Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models

This paper is concerned with the existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. By using Schauder's fixed point theorem and the existence of contracting rectangles, we obtain the existence result. Then we investigate the asymptotic behavior of positive monotone traveling wave solutions by using the modified Ikehara's Theorem. With the help of their asymptotic behavior, we provide a sufficient condition which guarantee that all positive traveling wave solutions of the system are non-monotone. Furthermore, to illustrate our main results, the existence and non-monotonicity of traveling wave solutions of Lotka-Volterra predator-prey model and modified Leslie-Gower predator-prey models with different kinds of functional responses are also discussed.

Note on the absence of remainders in the Wiener-Ikehara theorem

We show that it is impossible to get a better remainder than the classical one in the Wiener-Ikehara theorem even if one assumes analytic continuation of the Mellin transform after subtraction of the pole to a half-plane. We also prove a similar result for the Ingham-Karamata theorem.

Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal

<p style='text-indent:20px;'>This paper is concerned with the traveling waves for a three-species competitive system with nonlocal dispersal. It has been shown by Dong, Li and Wang (DCDS 37 (2017) 6291-6318) that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first investigate the asymptotic behavior of traveling waves at negative infinity by a modified version of Ikehara's Theorem. Then we prove the uniqueness of traveling waves by applying the stronger comparison principle and the sliding method. Finally, we establish the exponential stability of traveling waves with large speeds by the weighted-energy method and the comparison principle, when the initial perturbation around the traveling wavefront decays exponentially as <i>x</i> → -∞, but can be arbitrarily large in other locations.

Entire solution in an ignition nonlocal dispersal equation: Asymmetric kernel

This paper mainly focuses on the front-like entire solution of a classical nonlocal dispersal equation with ignition nonlinearity. Especially, the dispersal kernel function J may not be symmetric here. The asymmetry of J has a great influence on the profile of the traveling waves and the sign of the wave speeds, which further makes the properties of the entire solution more diverse. We first investigate the asymptotic behavior of the traveling wave solutions since it plays an essential role in obtaining the front-like entire solution. Due to the impact of f′(0) = 0, we can no longer use the common method which mainly depends on Ikehara theorem and bilateral Laplace transform to study the asymptotic rates of the nondecreasing traveling wave and the nonincreasing one tending to 0, respectively, so we adopt another method to investigate them. Afterwards, we establish a new entire solution and obtain its qualitative properties by constructing proper supersolution and subsolution and by classifying the sign and size of the wave speeds.

Stability and uniqueness of traveling waves of a non-local dispersal SIR epidemic model

This paper is mainly concerned with the exponential stability and uniqueness of traveling waves of a delayed nonlocal dispersal SIR epidemic model. We first prove the stability of traveling waves by using the weighted energy method, where the traveling waves are allowed to be non-monotone. Next we establish the exact asymptotic behavior of traveling waves at-8 by using Ikehara's theorem. Then the uniqueness of traveling waves is obtained by the stability result. Finally, we discuss how the nonlocal dispersal affects the stability of traveling waves. The conclusion shows that the nonlocal dispersal slows down the convergence rate of the solution to the traveling waves.

Generalization of the Wiener–Ikehara theorem

We study the Wiener–Ikehara theorem under the so-called log-linearly slowly decreasing condition. Moreover, we clarify the connection between two different hypotheses on the Laplace transform occurring in exact forms of the Wiener–Ikehara theorem, that is, in “if and only if” versions of this theorem.

Uniqueness and stability of traveling waves for cellular neural networks with multiple delays

In this paper, we investigate the properties of traveling waves to a class of lattice differential equations for cellular neural networks with multiple delays. Following the previous study [38] on the existence of the traveling waves, here we focus on the uniqueness and the stability of these traveling waves. First of all, by establishing the a priori asymptotic behavior of traveling waves and applying Ikehara's theorem, we prove the uniqueness (up to translation) of traveling waves ϕ(n−ct) with c≤c⁎ for the cellular neural networks with multiple delays, where c⁎<0 is the critical wave speed. Then, by the weighted energy method together with the squeezing technique, we further show the global stability of all non-critical traveling waves for this model, that is, for all monotone waves with the speed c<c⁎, the original lattice solutions converge time-exponentially to the corresponding traveling waves, when the initial perturbations around the monotone traveling waves decay exponentially at far fields, but can be arbitrarily large in other locations.

Stability for Monostable Wave Fronts of Delayed Lattice Differential Equations

This paper is concerned with the stability of traveling wave fronts for delayed monostable lattice differential equations. We first investigate the existence non-existence and uniqueness of traveling wave fronts by using the technique of monotone iteration method and Ikehara theorem. Then we apply the contraction principle to obtain the existence, uniqueness, and positivity of solutions for the Cauchy problem. Next, we study the stability of a traveling wave front by using comparison theorems for the Cauchy problem and initial-boundary value problem of the lattice differential equations, respectively. We show that any solution of the Cauchy problem converges exponentially to a traveling wave front provided that the initial function is a perturbation of the traveling wave front, whose asymptotic behaviour at \(-\infty \) satisfying some restrictions. Our results can apply to many lattice differential equations, for examples, the delayed cellular neural networks model and discrete diffusive Nicholson’s blowflies equation.

A proof of a conjecture of Bateman and Diamond on Beurling generalized primes

Beurling’s generalized prime system is a sequence \(\{p_i\}\) of real numbers \(p_i\) satisfying \(1 3/2\); but \(E(x)=O(\log ^{-3/2}x)\) does not suffice. Bateman and Diamond conjectured the condition \(\int (E(x)\log \,x)^2x^{-1}dx<\infty \). A proof of this conjecture following Kahane’s method is given in this paper. The proof is based on Poission’s summation formula and the Wiener–Ikehara tauberian theorem and applies classical ideas.

Wiener–Ikehara theorems and the Beurling generalized primes

The Wiener–Ikehara theorem is classical. Here we prove a converse of the theorem with “if and only if” conditions. We also extend this theorem to cover upper estimates and lower estimates respectively. Significant applications of these theorems to the study of the Beurling generalized primes are introduced too.