scholarly journals On superstability of exponential functional equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Batool Noori ◽  
M. B. Moghimi ◽  
Abbas Najati ◽  
Choonkil Park ◽  
Jung Rye Lee
Keyword(s):  
Functional Equations ◽  
Exponential Functional

AbstractThe aim of this paper is to prove the superstability of the following functional equations: $$\begin{aligned}& f \bigl(P(x,y) \bigr)= g(x)h(y), \\& f(x+y)=g(x)h(y). \end{aligned}$$ f ( P ( x , y ) ) = g ( x ) h ( y ) , f ( x + y ) = g ( x ) h ( y ) .

scholarly journals Stability of Exponential Functional Equations with Involutions

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jaeyoung Chung ◽  
Soon-Yeong Chung
Keyword(s):  
Commutative Semigroup ◽  
Functional Equations ◽  
Exponential Functional ◽  
Not Otherwise Specified ◽  
Stability Of Functional Equations ◽  
The Stability

LetSbe a commutative semigroup if not otherwise specified andf:S→ℝ. In this paper we consider the stability of exponential functional equations|f(x+σ(y))-g(x)f(y)|≤ϕ(x)or ϕ(y),|f(x+σ(y))-f(x)g(y)|≤ϕ(x)orϕ(y)for allx,y∈Sand whereσ:S→Sis an involution. As main results we investigate bounded and unbounded functions satisfying the above inequalities. As consequences of our results we obtain the Ulam-Hyers stability of functional equations (Chung and Chang (in press); Chávez and Sahoo (2011); Houston and Sahoo (2008); Jung and Bae (2003)) and a generalized result of Albert and Baker (1982).

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}$$$\begin{eqnarray}\displaystyle \biggl|f\biggl(\frac{x+y}{2}\biggr)^{2}-f(x)f(y)\biggr|\leq \unicode[STIX]{x1D713}(x,y)\quad \text{for all }x,y\in S. & & \displaystyle \nonumber\end{eqnarray}$$

scholarly journals Alienation of the logarithmic and exponential functional equations

2015 ◽  
Vol 90 (1) ◽  
pp. 107-121 ◽  
Author(s):  
Zygfryd Kominek ◽  
Justyna Sikorska
Keyword(s):  
Functional Equations ◽  
Exponential Functional

scholarly journals Ulam’s Type Stability of Involutional-Exponential Functional Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jaeyoung Chung
Keyword(s):  
Growth Condition ◽  
Commutative Semigroup ◽  
Functional Equations ◽  
Integrable Function ◽  
Exponential Functional ◽  
Locally Integrable Function ◽  
The Stability ◽  
Ulam’S Type Stability

LetSbe a commutative semigroup,f,g:S→Candσ:S→San involution. In this paper we consider the stability of involution-exponential functional equationsfx+σy-gxfy≤ϕxresp.,  ϕy, |f(x+σy)-f(x)g(y)|≤ϕ(x) [resp.,  ϕ(y)]for allx,y∈S, whereϕ:S→R+satisfies the growth condition: there existsC>1such thatlimk→∞C-kϕ(kx)=0for eachx∈S. We also consider the stability ofL∞-version|f(x+σy)-f(x)f(y)|L∞(R2n)≤ϵ,wheref:Rn→Cis a locally integrable function.

On some functional equations related to derivations and bicircular projections in rings

2014 ◽  
Vol 49 (2) ◽  
pp. 313-331
Author(s):  
Maja Fošner ◽  
◽  
Benjamin Marcen ◽  
Nejc Širovnik ◽  
Joso Vukman ◽  
...  
Keyword(s):  
Functional Equations

Functional equations related to weightable quasi-metrics

2015 ◽  
Vol 4 (1047) ◽  
Author(s):  
M.J. Campion ◽  
E. Indurain ◽  
G. Ochoa
Keyword(s):  
Functional Equations

Solutions and Stability of Generalized Mixed Type QCA-Functional Equations in Random Normed Spaces

2013 ◽  
Vol 59 (2) ◽  
pp. 299-320
Author(s):  
M. Eshaghi Gordji ◽  
Y.J. Cho ◽  
H. Khodaei ◽  
M. Ghanifard
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Functional Equations ◽  
Mixed Type ◽  
Additive Functional ◽  
Normed Spaces ◽  
Additive Functional Equation ◽  
Probabilistic Normed Spaces ◽  
Random Normed Spaces ◽  
Generalized Stability

Abstract In this paper, we investigate the general solution and the generalized stability for the quartic, cubic and additive functional equation (briefly, QCA-functional equation) for any k∈ℤ-{0,±1} in Menger probabilistic normed spaces.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

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