scholarly journals On Ulam's Type Stability of the Cauchy Additive Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Janusz Brzdęk
Keyword(s):  
Functional Equation ◽  
General Result ◽  
Commutative Group ◽  
Ulam’S Type Stability ◽  
Additive Equation ◽  
Class Of Functions

We prove a general result on Ulam's type stability of the functional equationfx+y=fx+fy, in the class of functions mapping a commutative group into a commutative group. As a consequence, we deduce from it some hyperstability outcomes. Moreover, we also show how to use that result to improve some earlier stability estimations given by Isaac and Rassias.

scholarly journals An explicit representation for disappointment aversion and other betweenness preferences

2020 ◽  
Vol 15 (4) ◽  
pp. 1509-1546
Author(s):  
Simone Cerreia-Vioglio ◽  
David Dillenberger ◽  
Pietro Ortoleva
Keyword(s):  
Functional Equation ◽  
Expected Utility ◽  
General Result ◽  
Explicit Representation ◽  
Implicit Representation ◽  
Utility Representation ◽  
Disappointment Aversion

One of the most well known models of non‐expected utility is Gul's (1991) model of disappointment aversion. This model, however, is defined implicitly, as the solution to a functional equation; its explicit utility representation is unknown, which may limit its applicability. We show that an explicit representation can be easily constructed, using solely the components of the implicit representation. We also provide a more general result: an explicit representation for preferences in the betweenness class that also satisfy negative certainty independence (Dillenberger 2010) or its counterpart. We show how our approach gives a simple way to identify the parameters of the representation behaviorally and to study the consequences of disappointment aversion in a variety of applications.

On a functional equation for general branching processes

1973 ◽  
Vol 10 (1) ◽  
pp. 198-205 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Population Size ◽  
General Class ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variable ◽  
Slow Variation ◽  
Age Dependent ◽  
Class Of Functions

If Z(t) denotes the population size in a Bellman-Harris age-dependent branching process such that a non-denenerate random variable W, then it is known that E(W) = 1 and that ϕ (u) = E(e–uW) satisfies a well-known integral equation. In this situation Athreya [1] has recently found a NASC for E(W |log W| y) <∞, for γ > 0. This paper generalizes Athreya's results in two directions. Firstly a more general class of branching processes is considered; secondly conditions are found for E(W 1 + βL(W)) < ∞ for 0 β < 1, where L is one of a class of functions of slow variation.

scholarly journals The solution of the functional equation of D'Alembert's type for commutative groups

1982 ◽  
Vol 5 (2) ◽  
pp. 315-335 ◽  
Author(s):  
A. L. Rukhin
Keyword(s):  
Functional Equation ◽  
Commutative Group ◽  
Matrix Elements ◽  
Dimensional Representation ◽  
Finite Dimensional Representation ◽  
Finite Dimensional

A functional equation of the formϕ1(x+y)+ϕ2(x−y)=∑inαi(x)βi(y), where functionsϕ1,ϕ2,αi,βi,i=1,…,nare defined on a commutative group, is solved. We also obtain conditions for the solutions of this equation to be matrix elements of a finite dimensional representation of the group.

scholarly journals On Koenigs' ratios for iterates of real functions

1969 ◽  
Vol 10 (1-2) ◽  
pp. 207-213 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Functional Equation ◽  
General Result ◽  
Real Function ◽  
Real Solutions ◽  
Real Functions ◽  
Image Position ◽  
Schröder Functional Equation

In a recent note, M. Kuczama [5] has obtained a general result concerning real solutions φ(x) on the interval 0 ≦ x < a ≦∞ of the Schröder functional equation providing the known real function satisfies the following (quite weak) conditions: f(x) is continuous and strictly increasing in ([0 a);(0) = 0 and 0 <f(x) <x for x ∈ (0, a); limx→0+ {f(x)/x} = s; and f(x)/x is monotonic in (0, a).

scholarly journals On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Jaeyoung Chung ◽  
Bogeun Lee ◽  
Misuk Ha
Keyword(s):  
Functional Equation ◽  
Euclidean Space ◽  
Functional Equations ◽  
Direct Consequence ◽  
Functional Inequality ◽  
Commutative Group ◽  
Bounded Functions

LetGbe a uniquely2-divisible commutative group and letf,g:G→Candσ:G→Gbe an involution. In this paper, generalizing the superstability of Lobačevskiǐ’s functional equation, we considerf(x+σy)/22-g(x)f(y)≤ψ(x)orψ(y)for allx,y∈G, whereψ:G→R+. As a direct consequence, we find a weaker condition for the functionsfsatisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.

scholarly journals Stability of the Exponential Functional Equation in Riesz Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Bogdan Batko
Keyword(s):  
Functional Equation ◽  
Representation Theorem ◽  
Spectral Representation ◽  
Cauchy Functional Equation ◽  
Exponential Functional ◽  
The Stability ◽  
Class Of Functions

We deal with the stability of the exponential Cauchy functional equationF(x+y)=F(x)F(y)in the class of functionsF:G→Lmapping a group (G, +) into a Riesz algebraL. The main aim of this paper is to prove that the exponential Cauchy functional equation is stable in the sense of Hyers-Ulam and is not superstable in the sense of Baker. To prove the stability we use the Yosida Spectral Representation Theorem.

scholarly journals ON THE COMPLEXITY OF COMPUTING OPTIMAL SOLUTIONS

1991 ◽  
Vol 02 (03) ◽  
pp. 207-220 ◽  
Author(s):  
ZHI-ZHONG CHEN ◽  
SEINOSUKE TODA
Keyword(s):  
Polynomial Time ◽  
General Result ◽  
Optimization Problems ◽  
Randomized Algorithm ◽  
Optimal Solution ◽  
Maximum Clique ◽  
Optimal Solutions ◽  
Np Optimization ◽  
Parallel Queries ◽  
Class Of Functions

We study the computational complexity of computing optimal solutions (the solutions themselves, not just their cost) for NP optimization problems where the costs of feasible solutions are bounded above by a polynomial in the length of their instances (we simply denote by NPOP such an NP optimization problem). It is of particular interest to find a computational structure (or equivalently, a complexity class) which. captures that complexity, if we consider the problems of computing optimal solutions for NPOP’s as a class of functions giving those optimal solutions. In this paper, we will observe that [Formula: see text] the class of functions computable in polynomial-time with one free evaluation of unbounded parallel queries to NP oracle sets, captures that complexity. We first show that for any NPOP Π, there exists a polynomial-time bounded randomized algorithm which, given an instance of Π, uses one free evaluation of parallel queries to an NP oracle set and outputs some optimal solution of the instance with very high probability. We then show that for several natural NPOP’s, any function giving those optimal solutions is at least as computationally hard as all functions in [Formula: see text]. To show the hardness results, we introduce a property of NPOP’s, called paddability, and we show a general result that if Π is a paddable NPOP and its associated decision problem is NP-hard, then all functions in [Formula: see text] are computable in polynomial-time with one free evaluation of an arbitrary function giving optimal solutions for instances of Π. The hardness results are applications of this general result. Among the NPOP’s, we include MAXIMUM CLIQUE, MINIMUM COLORING, LONGEST PATH, LONGEST CYCLE, 0–1 TRAVELING SALESPERSON, and 0–1 INTEGER PROGRAMMING.

scholarly journals Ulam’s Type Stability and Generalized Norms

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 502
Author(s):  
Laura Manolescu
Keyword(s):  
Functional Equation ◽  
Symmetric Spaces ◽  
Ulam Stability ◽  
Cauchy Equation ◽  
Cauchy Functional Equation ◽  
Type V ◽  
Ulam’S Type Stability

A symmetric functional equation is one whose form is the same regardless of the order of the arguments. A remarkable example is the Cauchy functional equation: f ( x + y ) = f ( x ) + f ( y ) . Interesting results in the study of the rigidity of quasi-isometries for symmetric spaces were obtained by B. Kleiner and B. Leeb, using the Hyers-Ulam stability of a Cauchy equation. In this paper, some results on the Ulam’s type stability of the Cauchy functional equation are provided by extending the traditional norm estimations to ther measurements called generalized norm of convex type (v-norm) and generalized norm of subadditive type (s-norm).

scholarly journals Stability of generalized additive Cauchy equations

2000 ◽  
Vol 24 (11) ◽  
pp. 721-727 ◽  
Author(s):  
Soon-Mo Jung ◽  
Ki-Suk Lee
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Cauchy Equation ◽  
Class Of Functions ◽  
Rassias Stability

A familiar functional equationf(ax+b)=cf(x)will be solved in the class of functionsf:ℝ→ℝ. Applying this result we will investigate the Hyers-Ulam-Rassias stability problem of the generalized additive Cauchy equationf(a1x1+⋯+amxm+x0)=∑i=1mbif(ai1x1+⋯+aimxm)in connection with the question of Rassias and Tabor.

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