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 Approximate Solutions of Functional Equations in Vector Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Bogdan Batko
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Spectral Representation ◽  
General Theorem ◽  
Approximate Solutions ◽  
Quadratic Functional ◽  
Cauchy Equation ◽  
Riesz Spaces ◽  
The Stability ◽  
Class Of Functions

We provide a method of approximation of approximate solutions of functional equations in the class of functions acting into a Riesz space (algebra). The main aim of the paper is to provide a general theorem that can act as a tool applicable to a possibly wide class of functional equations. The idea is based on the use of the Spectral Representation Theory for Riesz spaces. The main result will be applied to prove the stability of an alternative Cauchy functional equationF(x+y)+F(x)+F(y)≠0⇒F(x+y)=F(x)+F(y)in Riesz spaces, the Cauchy equation with squaresF(x+y)2=(F(x)+F(y))2inf-algebras, and the quadratic functional equationF(x+y)+F(x-y)=2F(x)+2F(y)in Riesz spaces.

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.

scholarly journals Stability results and separations theorems

2021 ◽  
Author(s):  
Roman Badora
Keyword(s):  
Functional Equation ◽  
Separation Theorems ◽  
Cauchy Functional Equation ◽  
Additive Map ◽  
Different Types ◽  
The Stability ◽  
Stability Results

AbstractThe presented work summarizes the relationships between stability results and separation theorems. We prove the equivalence between different types of theorems on separation by an additive map and different types of stability results concerning the stability of the Cauchy functional equation.

scholarly journals Probabilistic (Quasi)metric Versions for a Stability Result of Baker

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Dorel Miheţ ◽  
Claudia Zaharia
Keyword(s):  
Functional Equation ◽  
Stability Result ◽  
Fixed Point Method ◽  
Normed Spaces ◽  
Cauchy Functional Equation ◽  
Triangular Norms ◽  
Random Normed Spaces ◽  
The Stability ◽  
Probabilistic Metric ◽  
Additive Cauchy Functional Equation

By using the fixed point method, we obtain a version of a stability result of Baker in probabilistic metric and quasimetric spaces under triangular norms of Hadžić type. As an application, we prove a theorem regarding the stability of the additive Cauchy functional equation in random normed spaces.

scholarly journals The Stability of Some Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
M. M. Pourpasha ◽  
Th. M. Rassias ◽  
R. Saadati ◽  
S. M. Vaezpour
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Fixed Point ◽  
Differential Equations ◽  
Vector Space ◽  
Fixed Point Method ◽  
Point Method ◽  
The Stability ◽  
Class Of Functions

We generalize the results obtained by Jun and Min (2009) and use fixed point method to obtain the stability of the functional equationf(x+σ(y))=F[f(x),f(y)], for a class of functions of a vector space into a Banach space whereσis an involution. Then we obtain the stability of the differential equations of the formy′=F[q(x),P(x)y(x)].

scholarly journals On the Stability Problem in Fuzzy Banach Space

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
G. Zamani Eskandani ◽  
P. Găvruţa ◽  
Gwang Hui Kim
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Stability Problem ◽  
Cauchy Functional Equation ◽  
Open Problems ◽  
The Stability ◽  
Fuzzy Banach Space

We investigate the generalized Ulam-Hyers stability of the Cauchy functional equation and pose two open problems in fuzzy Banach space.

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