GENERAL STABILITY OF THE EXPONENTIAL AND LOBAČEVSKIǏ FUNCTIONAL EQUATIONS

2016 ◽  
Vol 94 (2) ◽  
pp. 278-285 ◽  
Author(s):  
JAEYOUNG CHUNG
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Mixed Variables ◽  
General Stability ◽  
Exponential Functional ◽  
Image Position

Let $S$ be a semigroup possibly with no identity and $f:S\rightarrow \mathbb{C}$. We consider the general superstability of the exponential functional equation with a perturbation $\unicode[STIX]{x1D713}$ of mixed variables $$\begin{eqnarray}\displaystyle |f(x+y)-f(x)f(y)|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$ In particular, if $S$ is a uniquely $2$-divisible semigroup with an identity, we obtain the general superstability of Lobačevskiǐ’s functional equation with perturbation $\unicode[STIX]{x1D713}$$$\begin{eqnarray}\displaystyle \biggl|f\biggl(\frac{x+y}{2}\biggr)^{2}-f(x)f(y)\biggr|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$

An Analogue of the Wave Equation and Certain Related Functional Equations

1969 ◽  
Vol 12 (6) ◽  
pp. 837-846 ◽  
Author(s):  
John A. Baker
Keyword(s):  
Functional Equation ◽  
Wave Equation ◽  
Functional Equations ◽  
Geometric Interpretation ◽  
Continuous Solutions ◽  
Difference Analogue ◽  
Image Position

Consider the functional equation(1)assumed valid for all real x, y and h. Notice that (1) can be written(2)a difference analogue of the wave equation, if we interpret etc., (i. e. symmetric h differences), and that (1) has an interesting geometric interpretation. The continuous solutions of (1) were found by Sakovič [5].

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

Graphs in the plane invariant under an area preserving linear map and general continuous solutions of certain quadratic functional equations

1985 ◽  
Vol 97 (2) ◽  
pp. 261-278 ◽  
Author(s):  
P. J. McCarthy ◽  
M. Crampin ◽  
W. Stephenson
Keyword(s):  
Functional Equation ◽  
Arbitrary Function ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Continuous Solutions ◽  
Linear Map ◽  
Area Preserving Maps ◽  
Image Position ◽  
Area Preserving

AbstractThe requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the formfor all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

Solutions of the Functional Equation (f(x))2 - f(x2) = h(x)

1960 ◽  
Vol 3 (2) ◽  
pp. 113-120 ◽  
Author(s):  
I. N. Baker
Keyword(s):  
Functional Equation ◽  
Number Theory ◽  
Functional Equations ◽  
Image Position

In a recent paper [2] Lambek and Moser have introduced the functional equations1and2in connection with some problems of number theory, in particular in dealing with the sums by pairs of sets of integers. The second may be put into the same form as (1) by the substitutions x = In z, f(ln z) = F(z), h(ln z) = H(z).

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART II

2012 ◽  
Vol 85 (2) ◽  
pp. 202-216 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
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AbstractWe investigate the Pexider-type functional equation where f, g, h are real functions defined on an abelian group G. We solve this equation under the assumptions G=ℝ and f is continuous.

scholarly journals Uniqueness theorems for a general class of functional equations

1970 ◽  
Vol 11 (3) ◽  
pp. 362-366 ◽  
Author(s):  
C. T. Ng
Keyword(s):  
Functional Equation ◽  
General Class ◽  
Functional Equations ◽  
Initial Conditions ◽  
Infinite Interval ◽  
Uniqueness Theorems ◽  
Image Position

In a previous paper [1] J. Aczél has shown the following Theorem 1. If in the (closed, half-closed or open, finite or infinite) interval 〈A, B〉and there f, F are there, f F are continuous, F intern (the value F(x, y) lies strictly between x and y) and u → H(u, v, x, y) or v → H(u, v, x, y) are injective (i.e. or , then the functional equation (*) with the initial conditionshas at most one solution.

scholarly journals ON SOME PEXIDER-TYPE FUNCTIONAL EQUATIONS CONNECTED WITH THE ABSOLUTE VALUE OF ADDITIVE FUNCTIONS. PART I

2012 ◽  
Vol 85 (2) ◽  
pp. 191-201 ◽  
Author(s):  
BARBARA PRZEBIERACZ
Keyword(s):  
Functional Equation ◽  
Abelian Group ◽  
Functional Equations ◽  
Additive Functions ◽  
Absolute Value ◽  
Real Functions ◽  
The Absolute ◽  
Image Position

AbstractWe investigate the Pexider-type functional equation where f,g,h are real functions defined on an abelian group G.

A HYPERSTABILITY RESULT FOR THE CAUCHY EQUATION

2013 ◽  
Vol 89 (1) ◽  
pp. 33-40 ◽  
Author(s):  
JANUSZ BRZDĘK
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Normed Space ◽  
Functional Equations ◽  
Cauchy Equation ◽  
Cauchy Functional Equation ◽  
Real Numbers ◽  
Formula One ◽  
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AbstractWe prove a hyperstability result for the Cauchy functional equation$f(x+ y)= f(x)+ f(y)$, which complements some earlier stability outcomes of J. M. Rassias. As a consequence, we obtain the slightly surprising corollary that for every function$f$, mapping a normed space${E}_{1} $into a normed space${E}_{2} $, and for all real numbers$r, s$with$r+ s\gt 0$one of the following two conditions must be valid:$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = \infty , &&\displaystyle\end{eqnarray*}$$$$\begin{eqnarray*}\displaystyle \sup _{x, y\in E_{1}}\Vert f(x+ y)- f(x)- f(y)\Vert \hspace{0.167em} \mathop{\Vert x\Vert }\nolimits ^{r} \hspace{0.167em} \mathop{\Vert y\Vert }\nolimits ^{s} = 0. &&\displaystyle\end{eqnarray*}$$In particular, we present a new method for proving stability for functional equations, based on a fixed point theorem.

scholarly journals On some functional equations from additive and non-additive measures-I

1980 ◽  
Vol 23 (2) ◽  
pp. 145-150 ◽  
Author(s):  
PL Kannappan
Keyword(s):  
Functional Equation ◽  
Shannon Entropy ◽  
Functional Equations ◽  
Statistical Thermodynamics ◽  
Rényi Entropies ◽  
Algebraic Properties ◽  
Image Position ◽  
Additive Measures ◽  
Measure Entropy ◽  
Renyi Entropies

It is known that while the Shannon and the Rényi entropies are additive, the measure entropy of degree β proposed by Havrda and Charvat (7) is non-additive. Ever since Chaundy and McLeod (4) considered the following functional equationwhich arose in statistical thermodynamics, (1.1) has been extensively studied (1, 5, 6, 8). From the algebraic properties of symmetry, expansibility and branching of the entropy (viz. Shannon entropy Hn, etc.) one obtains the sum representationwhich with the property of additivity yields the functional equation (1.1), (9, 10).

Functional Equations, Distributions and Approximate Identities

1990 ◽  
Vol 42 (4) ◽  
pp. 696-708 ◽  
Author(s):  
John A. Baker
Keyword(s):  
Functional Equation ◽  
Lebesgue Measure ◽  
Functional Equations ◽  
Schwartz Distributions ◽  
Integrable Functions ◽  
Approximate Identities ◽  
The Subject ◽  
Image Position ◽  
Complex Valued

The subject of this paper is the use of the theory of Schwartz distributions and approximate identities in studying the functional equationThe aj’s and b are complex-valued functions defined on a neighbourhood, U, of 0 in Rm, hj. U → Rn with hj(0) = 0 and fj, g: Rn → C for 1 ≦ j ≦ N. In most of what follows the aj's and hj's are assumed smooth and may be thought of as given. The fj‘s, b and g may be thought of as the unknowns. Typically we are concerned with locally integrable functions f1, … , fN such that, for each s in U, (1) holds for a.e. (almost every) x ∈ Rn, in the sense of Lebesgue measure.

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