Functional Equations Research Articles

Alienation of the quadratic, exponential and d’Alembert functional equations

AbstractLet $$(S,+,0)$$ ( S , + , 0 ) be a commutative monoid, $$\sigma :S\rightarrow S$$ σ : S → S be an endomorphism with $$\sigma ^2=id$$ σ 2 = i d and let K be a field of characteristic different from 2. We study the solutions $$f,g,h:S\rightarrow K$$ f , g , h : S → K of the Pexider type functional equation $$\begin{aligned} f(x+y)+f(x+\sigma y)+g(x+y)=2f(x)+2f(y)+g(x)g(y) \end{aligned}$$ f ( x + y ) + f ( x + σ y ) + g ( x + y ) = 2 f ( x ) + 2 f ( y ) + g ( x ) g ( y ) resulting from summing up the well known quadratic and exponential functional equations side by side. We show that under some additional assumptions the above equation forces f and g to split back into the system of two equations $$\begin{aligned} \left\{ \begin{array}{ll}f(x+y)+f(x+\sigma y)=2f(x)+2f(y)\\ g(x+y)=g(x)g(y)\end{array}\right. \end{aligned}$$ f ( x + y ) + f ( x + σ y ) = 2 f ( x ) + 2 f ( y ) g ( x + y ) = g ( x ) g ( y ) for all $$x,y\in S$$ x , y ∈ S (alienation phenomenon). We also consider an analogous problem for the quadratic and d’Alembert functional equations as well as for the quadratic, exponential and d’Alembert functional equations.

Analyzing the Effect of Tethered Cable on the Stability of Tethered UAVs Based on Lyapunov Exponents

In the working process of the tethered unmanned aerial vehicle (UAV), there is interference from the tethered cable, which can easily lead to the instability of the UAV. To solve the above problems, a method based on the Lyapunov exponent is proposed to analyze the stability of tethered cables for tethered UAVs. The dynamics equation of the UAV platform is established using the Euler–Poincare equation. The tension formula of the tethered cable is derived from the catenary theory and the principle of micro-segment equilibrium. Based on the Lyapunov exponential method, the stability changes of the tethered UAV in the takeoff, hovering, and landing stages are simulated and analyzed in a MATLAB environment. Prototype tests are carried out to prove the correctness of the simulation model and calculation conclusions. The results show that with an increase in the density of the tethered cable, the stability of the tethered UAV tends to decrease. At the same time, stability is affected by the density of the tethered cable more often during takeoff than during landing.

A NEW PROGRAM FOR THE ENTIRE FUNCTIONS IN NUMBER THEORY

In this paper, we propose a new program for introducing the sign of the functional equation to present the entire functions of order one in number theory. We suggest some open problems for the zeros of these entire functions related to the completed Dedekind zeta function, completed quadratic Dirichlet L-functions, completed Ramanujan zeta function and completed automorphic L-function. These lead to the contribution to giving the deep understanding for diffusion equations associated with the entire functions of order one in number theory.

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

On an alternative additive-quadratic functional equation

On an alternative additive-quadratic functional equation

Weyl symmetry for curve counting invariants via spherical twists

We study the curve counting invariants of Calabi–Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande–Thomas invariants. When the divisor arises as resolution of a curve of A 1 A_1 -singularities, our results match the rationality of the associated Calabi–Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossing formulas and generalized Donaldson–Thomas invariants for surface-like objects.

ON A TEMPERED XI FUNCTION ASSOCIATED WITH THE RIEMANN XI FUNCTION

In this paper, we propose a tempered xi function obtained by the recombination of the decomposable functions for the Riemann xi function for the first time. We first obtain its functional equation and series representation. We then suggest three equivalent open problems for the zeros for it. We finally consider its behaviors on the critical line.

A note on generalized exponential stability of impulsive stochastic functional differential equations

This brief considers the generalized exponential stability of impulsive stochastic functional differential equations (ISFDEs). Initially, An extension of Halanay inequality is derived. Subsequently, by employing this generalized Halanay inequality, we establish the generalized exponential stability result for the given systems.

Kannappan–Wilson and Van Vleck–Wilson functional equations on semigroups

Kannappan–Wilson and Van Vleck–Wilson functional equations on semigroups

Stability analysis of a delayed SEIRQ epidemic model with diffusion

In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SEIRQ epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SEIRQ model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value $R_0$ of the delayed SEIRQ model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that the unique disease-free equilibrium is asymptotically stable , what is proven by the numericals scchema. Moreover,we search under their condition the endemic equilibrium is asymptotically stable .

Decompositions of Unit Hypercubes and the Reversion of a Generalized Möbius Series

Let $s_d(n)$ be the number of distinct decompositions of the $d$-dimensional hypercube with $n$ rectangular regions that can be obtained via a sequence of splitting operations. We prove that the generating series $y = \sum_{n \geq 1} s_d(n)x^n$ satisfies the functional equation $x = \sum_{n\geq 1} \mu_d(n)y^n$, where $\mu_d(n)$ is the $d$-fold Dirichlet convolution of the Möbius function. This generalizes a recent result by Goulden et al., and shows that $s_1(n)$ also gives the number of natural exact covering systems of $\mathbb{Z}$ with $n$ residual classes. We also prove an asymptotic formula for $s_d(n)$ and describe a bijection between $1$-dimensional decompositions and natural exact covering systems.

The Generalized Eta Transformation Formulas as the Hecke Modular Relation

The transformation formula under the action of a general linear fractional transformation for a generalized Dedekind eta function has been the subject of intensive study since the works of Rademacher, Dieter, Meyer, and Schoenberg et al. However, the (Hecke) modular relation structure was not recognized until the work of Goldstein-de la Torre, where the modular relations mean equivalent assertions to the functional equation for the relevant zeta functions. The Hecke modular relation is a special case of this, with a single gamma factor and the corresponding modular form (or in the form of Lambert series). This has been the strongest motivation for research in the theory of modular forms since Hecke’s work in the 1930s. Our main aim is to restore the fundamental work of Rademacher (1932) by locating the functional equation hidden in the argument and to reveal the Hecke correspondence in all subsequent works (which depend on the method of Rademacher) as well as in the work of Rademacher. By our elucidation many of the subsequent works will be made clear and put in their proper positions.

P-Laplacian functional differential equations with neutral delay arguments: new oscillation theorems

p-Laplacian functional differential equations with neutral delay arguments: new oscillation theorems

Existence and uniqueness of continuous solutions for iterative functional differential equations in Banach algebras

Existence and uniqueness of continuous solutions for iterative functional differential equations in Banach algebras

Adomian Decomposition Method and Block by Block Method for Solving Nonlinear Functional Volterra Integral Equation in Two-dimensions

This work proposes a new defintion of the nonlinear functional Volterra integral equation ib 2D of the second kind with continuous kernel (NT-DFVIE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions are obtained by two powerful methods Adomian Decomposition Method (ADM) and Block by block method (BBM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

Existence of a Class of Doubly Perturbed Stochastic Functional Differential Equations with Poisson Jumps

In this paper, we use successive approximations and Picard iterative method to establish the existence and uniqueness of mild solution for a class of doubly perturbed impulsive neutral stochastic functional differential equations with Poisson jumps in Hilbert spaces. An example of a doubly perturbed stochastic differential equation with delays is given to illustrate our main results.

Report of Meeting: The Twenty-third Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 – February 3, 2024

Report of Meeting: The Twenty-third Katowice–Debrecen Winter Seminar on Functional Equations and Inequalities Brenna (Poland), January 31 – February 3, 2024

All solutions to a Schröder type functional equation

All solutions to a Schröder type functional equation

A Kannappan-Cosine Functional Equation on Semigroups

Abstract In this paper we determine the complex-valued solutions of the Kannappan-cosine functional equation g(xyz 0) = g(x)g(y) − f(x)f(y), x, y ∈ S, where S is a semigroup and z 0 is a fixed element in S.

Nonlocal boundary value problems for functional hybrid differential equations involving generalized w-Caputo fractional operator

In this manuscript , we establish the existence and uniqueness of solutions for boundary value problems of nonlinear hybrid fractional differential equations involving generalized $\omega-$Caputo fractional derivatives. The proofs are based on Krasnoselskii fixed point theorem and some basic concept of $\omega-$Caputo fractional analysis. As application, an nontrivial example is given in the last part of this paper to illustrate our theoretical results.