Hyperstability of a logarithmic-type functional equation on restricted domains

Let $X$ be a real normed vector space and $f:(0,\infty )\to X$. In this paper, we prove the hyperstability of the logarithmic functional equation

$f\left(x^y\right)?yf\left(x\right)=0$

on $\Gamma$ of Lebesgue measure zero. More precisely, we prove that if $f:(0,\infty )\to X$ satisfies

$6f\left(x^y\right)?yf\left(x\right)6\le ?(x,y)$

for all $(x,y)\in {\Gamma }^{+}?\{(x,y):y>\alpha \left(x\right)\}[\mathrm{resp.}{\Gamma }^{?}?\{(x,y):0<y<\alpha \left(x\right)\}]$ of Lebesgue measure zero, where $\alpha :(0,\infty )\to (0,\infty )$ is an arbitrary given function and $?:(0,\infty )\times (0,\infty )\to [0,\infty )$ satisfies the condition $?(x,y)∕y\to 0$ as $y\to \infty$ [resp. $y\to 0$], then $f$ satisfies the functional equation $f\left(x^y\right)=yf\left(x\right)$ for all $x>0$ and$y>0$.     

一个Gauss型函数方程

给出了任意两个正实数的几何-调和平均值的一个积分表示式,并由此去探讨了函数方程f(ab,2ab/a+b)=f(a,b),a,b＞0其中f:R+×R+→R是此方程的一个未知函数.     

Recovering the initial distribution for space-fractional diffusion equation by a logarithmic regularization method

In this paper, the backward problem for space-fractional diffusion equation is investigated. We proposed a so-called logarithmic regularization method to solve it. Based on the conditional stability and an a posteriori regularization parameter choice rule, the convergence rate estimates are given under a-priori bound assumption for the exact solution.     

A universal Cauchy functional equation over the positive reals

The functional equation af(xy)+bf(x)f(y)+cf(x+y)+d(f(x)+f(y))=0 whose shape contains all the four well-known forms of Cauchy's functional equation is solved for solutions which are functions having the positive reals as their domain. This complements an earlier work of Dhombres in 1988 where the same functional equation was solved for solutions whose domains contain zero, which leaves out the logarithmic function. Here not only the logarithmic function is recovered but the analysis is entirely different and is based on solving appropriate difference equations.     

The stability of Cauchy's gamma-beta functional equation

We obtain the super stability of Cauchy's gamma-beta functional equation

B(x,y)f(x+y)=f(x)f(y),     

Another logarithmic functional equation

Summary. Let f : ]0,￥[? \Bbb R f :\,]0,\infty[\to \Bbb R be a real valued function on the set of positive reals. The functional equations¶¶f(x + y) - f(x) - f(y) = f(x^-1 + y^-1) f(x + y) - f(x) - f(y) = f(x^{-1} + y^{-1}) ¶and¶f(xy) = f(x) + f(y) f(xy) = f(x) + f(y) ¶are equivalent to each other.     

On the stability of a cubic functional equation

In this paper, we will find out the general solution and investigate the generalized Hyers-Ulam-Rassias stability problem for the following cubic functional equation
2f（x ＋ 2y） ＋ f（2x - y） = 5f（x ＋ y） ＋ 5f（x - y）＋ 15f（y）
in the spirit of Hyers, Ulam, Rassias and Gavruta.     

On the stability of a quartic functional equation

In this paper, we prove the generalized Hyers-Ulam stability for the following quartic functional equation

f(2x+y)+f(2x−y)=4f(x+y)+4f(x−y)+24f(x)−6f(y).     

Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity

We study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, we obtain the existence of global solution and blow-up at +∞ under some suitable conditions. On the other hand, the results for decay estimates of the global solutions are also given. Our result in this paper means that the polynomial nonlinearity is a critical condition of blow-up in finite time for the solutions of semilinear heat equations.     

A functional equation originating from quadratic forms

In this paper, we obtain the general solution and the stability of the 2-variable quadratic functional equation

f(x+y,z+w)+f(x−y,z−w)=2f(x,z)+2f(y,w).     

A functional partial differential equation arising in a cell growth model with dispersion

In this paper we solve an initial‐boundary value problem that involves a pde with a nonlocal term. The problem comes from a cell division model where the growth is assumed to be stochastic. The deterministic version of this problem yields a first‐order pde; the stochastic version yields a second‐order parabolic pde. There are no general methods for solving such problems even for the simplest cases owing to the nonlocal term. Although a solution method was devised for the simplest version of the first‐order case, the analysis does not readily extend to the second‐order case. We develop a method for solving the second‐order case and obtain the exact solution in a form that allows us to study the long time asymptotic behaviour of solutions and the impact of the dispersion term. We establish the existence of a large time attracting solution towards which solutions converge exponentially in time. The dispersion term does not appear in the exponential rate of convergence.     

Stability for the functional equation of cubic type

In this article, we establish the stability of the orthogonally cubic type functional equation (1.2) for all x₁,x₂,x₃ with xi⊥xj(i,j=1,2,3), where ⊥ is the orthogonality in the sense of Rätz, and investigate the stability of the n-dimensional cubic type functional equation (1.3), where n?3 is an integer.     

Solution and stability of a cubic functional equation

In this paper, we investigate the general solution and the stability of a cubic functional equation f（x ＋ ny） ＋ f（x - ny） ＋ f（nx） = n^2 f（x ＋ y） ＋ n^2 f（x - y）＋ （n^3 - 2n^2 ＋ 2）f（x）,where n ≥ 2 is an integer. Furthermore, we prove the stability by the fixed point method.     

On the solvability of a functional equation

This article studies properties of solutions of a functional equation arising in dynamic programming of multistage decision processes. A sufficient condition for the existence, uniqueness and iterative approximation of solutions of the functional equation is established. A few other behaviours of solutions for certain functional equations which are particular cases of the functional equation are discussed. The results presented in this article extend, improve and unify some known results in literature.     

Stability of the quadratic functional equation in Lipschitz spaces

Let G be an Abelian group with a metric d and E a normed space. For any f:G→E we define the quadratic difference of the function f by the formula

Qf(x,y):=2f(x)+2f(y)−f(x+y)−f(x−y)     

A further remark on the logarithmic derivatives of a square matrix

It is well-known that the logarithmic derivativeμ[A] of a square matrixA has its value no less than the maximum real partp of the eigenvalues ofA and there exist matricesA such thatμ[A] >p with respect to any norm inCⁿ. The purpose of this note is to show thatμ[A] can be made equal top for some operator norm inCⁿ if and only if those eigenvalues ofA with maximum real part are simple roots of the minimal polynomials forA.     

Analytic solutions of a second order iterative functional differential equation

This paper is concerned with an iterative functional differential equation

c₁x(z)+c₂x^′(z)+c₃x^″(z)=x(az+bx^′(z))