Fuzzy stability of quadratic-cubic functional equations

In this paper, the direct method and the fixed point alternative method are implemented to give Hyers-Ulam-Rassias stability of the functional equation

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.     

Generalized Hyers-Ulam Stability for a General Mixed Functional Equation in Quasi-��-normed Spaces

In this paper, we establish the general solution and investigate the generalized Hyers-Ulam stability of the following mixed additive and quadratic functional equation

On some composite functional inequalities

The aim of the paper is to deal with the following composite functional inequalities

Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces

In this paper, we establish a general solution and the generalized Hyers-Ulam-Rassias stability of the following general mixed additive-cubic functional equation

Bounded solutions of the Golab—Schinzel equation

Summary. Let \Bbb K {\Bbb K} be either the field of reals or the field of complex numbers, X be an F-space (i.e. a Fréchet space) over \Bbb K {\Bbb K} n be a positive integer, and f : X ? \Bbb K f : X \to {\Bbb K} be a solution of the functional equation¶¶f(x + f(x)ⁿ y) = f(x) f(y) f(x + f(x)^n y) = f(x) f(y) .¶We prove that, if there is a real positive a such that the set { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} contains a subset of second category and with the Baire property, then f is continuous or { x ? X : |f(x)| ? (0, a)} \{ x \in X : |f(x)| \in (0, a)\} for every x ? X x \in X . As a consequence of this we obtain the following fact: Every Baire measurable solution f : X ? \Bbb K f : X \to {\Bbb K} of the equation is continuous or equal zero almost everywhere (i.e., there is a first category set A ì X A \subset X with f(X \A) = { 0 }) f(X \backslash A) = \{ 0 \}) .     

On the functional equation f(x+g(y))-f(y+g(y))=f(x)-f(y) on groups

We solve the equation

Quantitative form of the Luzin <Emphasis Type="Italic">C</Emphasis>-property

We prove the following statement, which is a quantitative form of the Luzin theorem on C-property: Let (X, d, μ) be a bounded metric space with metric d and regular Borel measure μ that are related to one another by the doubling condition. Then, for any function f measurable on X, there exist a positive increasing function η ∈ Ω (η(+0) = 0 and η(t)t ^−a decreases for a certain a > 0), a nonnegative function g measurable on X, and a set E ⊂ X, μE = 0 , for which

Numerical verification of condition for approximately midconvex functions

Let X be a normed space and V be a convex subset of X. Let a\colon \mathbbR₊ ? \mathbbR₊{\alpha \colon \mathbb{R}_+ \to \mathbb{R}_+}. A function f \colon V ? \mathbbR{f \colon V \to \mathbb{R}} is called α-midconvex if

D'Alembert's functional equation on restricted domains

Summary In the class of functionalsf:X , whereX is an inner product space with dimX 3, we study the D'Alembert functional equationf(x + y) + f(x – y) = 2f(x)f(y) (1) on the restricted domainsX ₁ = {(x, y) X ²/x, y = 0} andX ₂ = {(x, y) X ²/x = y}. In this paper we prove that the equation (1) restricted toX ₁ is not equivalent to (1) on the whole spaceX. We also succeed in characterizing all common solutions if we add the conditionf(2x) = 2f²(x) – 1. Using this result, we prove the equivalence between (1) restricted toX ₂ and (1) on the whole spaceX. This research follows similar previous studies concerning the additive, exponential and quadratic functional equations.     

Solution and stability of generalized mixed type additive and quadratic functional equation in non-Archimedean spaces

In this paper, we achieve the general solution and the generalized Hyres–Ulam–Rassias stability of the following additive–quadratic functional equation

Generalized weighted quasi-arithmetic means

Under some natural assumptions on real functions f and g defined on a real interval I, we show that a two variable function M _f,g : I ² → I defined by

On functions with the cauchy difference bounded by a functional. III

We are going to discuss special cases of a conditional functional inequality

Quotient mean,its invariance with respect to a quasi-arithmetic mean-type mapping,and some applications

Under some conditions on the functions f and g defined in a real interval I the function

On quasimonotone increasing homeomorphisms

Let f ? C(\Bbb Rⁿ,\Bbb Rⁿ) f\in C(\Bbb R^n,\Bbb R^n) be quasimonotone increasing such that Y(f(y)-f(x)) ￡ -c Y(y-x) (x << y) \Psi (f(y)-f(x)) \!\le -c \Psi (y-x) (x\ll y) for a linear and strictly positive functional Y \Psi and c > 0. We prove that f is a homeomorphism with decreasing and Lipschitz continuous inverse and we prove the global asymptotic stability of the equilibrium solution of x￠=f(x) x'=f(x) .     

An algorithm for semi-infinite polynomial optimization

We consider the semi-infinite optimization problem:

From Chaos to Global Convergence

The main purpose of this paper is to investigate dynamical systems F : \mathbbR² ? \mathbbR²{F : \mathbb{R}^2 \rightarrow \mathbb{R}^2} of the form F(x, y) = (f(x, y), x). We assume that f : \mathbbR² ? \mathbbR{f : \mathbb{R}^2 \rightarrow \mathbb{R}} is continuous and satisfies a condition that holds when f is non decreasing with respect to the second variable. We show that for every initial condition x₀ = (x ₀, y ₀), such that the orbit

Approximation of several dimensional functions by trigonometric polynomials

Let f(x, y) be a periodic function defined on the region D

Diameter preserving linear maps and isometries

We study linear bijections of C(X) which preserve the diameter of the range, that is, the seminorm r(f)=sup{|f(x)-f(y)| : x, y ? X}\varrho (f)={\rm sup}\{|f(x)-f(y)| : x, y\in X\}.     

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper: