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.

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).

scholarly journals The new investigation of the stability of mixed type additive-quartic functional equations in non-Archimedean spaces

2020 ◽  
Vol 53 (1) ◽  
pp. 174-192
Author(s):  
Anurak Thanyacharoen ◽  
Wutiphol Sintunavarat
Keyword(s):  
Functional Equation ◽  
Normed Space ◽  
Functional Equations ◽  
Mixed Type ◽  
Additive Group ◽  
Ulam Stability ◽  
Quartic Functional Equation ◽  
The Stability

AbstractIn this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where f maps from an additive group to a complete non-Archimedean normed space.

scholarly journals Some remarks on the stability of the Cauchy equation and completeness

2021 ◽  
Author(s):  
Harald Fripertinger ◽  
Jens Schwaiger
Keyword(s):  
Normed Space ◽  
Additive Function ◽  
Functional Equations ◽  
Constant Function ◽  
Cauchy Equation ◽  
Cauchy Difference ◽  
International Conference ◽  
The Difference ◽  
The Stability ◽  
The Given

AbstractIt was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

scholarly journals On the stability of set-valued functional equations with the fixed point alternative

2012 ◽  
Vol 2012 (1) ◽  
pp. 81 ◽  
Author(s):  
Hassan Kenary ◽  
Hamid Rezaei ◽  
Yousof Gheisari ◽  
Choonkil Park
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
The Stability

scholarly journals A General Theorem on the Stability of a Class of Functional Equations Including Quartic-Cubic-Quadratic-Additive Equations

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 282
Author(s):  
Yang-Hi Lee ◽  
Soon-Mo Jung
Keyword(s):  
Functional Equations ◽  
Direct Method ◽  
General Theorem ◽  
Stability Problems ◽  
General Stability ◽  
Stability Theorems ◽  
Additive Type ◽  
The Stability

We prove general stability theorems for n-dimensional quartic-cubic-quadratic-additive type functional equations of the form by applying the direct method. These stability theorems can save us the trouble of proving the stability of relevant solutions repeatedly appearing in the stability problems for various functional equations.

On an Exponential Functional Inequality and its Distributional Version

2015 ◽  
Vol 58 (1) ◽  
pp. 30-43 ◽  
Author(s):  
Jaeyoung Chung
Keyword(s):  
Functional Equation ◽  
Tensor Product ◽  
Functional Inequality ◽  
Schwartz Distribution ◽  
Exponential Functional ◽  
The Stability

AbstractLet G be a group and 𝕂 = ℂ or ℝ. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions f : G → 𝕂 satisfying the inequalityWhere ϕ: Gn-1 → [0,∞]. Also as a a distributional version of the above inequality we consider the stability of the functional equationwhere u is a Schwartz distribution or Gelfand hyperfunction, o and ⊗ are the pullback and tensor product of distributions, respectively, and S(x1, ..., xn) = x1 + · · · + xn.

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