scholarly journals Additive ρ-functional inequalities in β-homogeneous normed spaces

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1651-1658
Author(s):  
Choonkil Park
Keyword(s):  
Banach Spaces ◽  
Complex Number ◽  
Functional Equations ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Homogeneous Complex ◽  
Fixed Complex

In this paper, we solve the following additive ?-functional inequalities ||f (x + y) - f (x) - f (y)|| ? ???(2f (x+y/2) - f(x) + -f (y))??, (1) where ? is a fixed complex number with |?|<1, and ??2f(x+y/2)-f(x)- f(y)???||?(f(x+y)-f(x)-f(y))||, (2) where ? is a fixed complex number with |?|<1/2 , and prove the Hyers-Ulam stability of the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces and prove the Hyers-Ulam stability of additive ?-functional equations associated with the additive ?-functional inequalities (1) and (2) in ?-homogeneous complex Banach spaces.

scholarly journals Hyers-Ulam stability of quadratic forms in 2-normed spaces

2019 ◽  
Vol 52 (1) ◽  
pp. 496-502
Author(s):  
Won-Gil Park ◽  
Jae-Hyeong Bae
Keyword(s):  
Banach Spaces ◽  
Quadratic Forms ◽  
Functional Equations ◽  
Ulam Stability ◽  
Normed Spaces

AbstractIn this paper, we obtain Hyers-Ulam stability of the functional equationsf (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w),f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w)andf (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)in 2-Banach spaces. The quadratic forms ax2 + bxy + cy2, ax2 + by2 and axy are solutions of the above functional equations, respectively.

scholarly journals Generalized functional inequalities in Banach spaces

2021 ◽  
Vol 7 (2) ◽  
pp. 337-349
Author(s):  
H. Dimou ◽  
Y. Aribou ◽  
S. Kabbaj
Keyword(s):  
Banach Spaces ◽  
Direct Method ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability

Abstract In this paper, we solve and investigate the generalized additive functional inequalities ‖ F ( ∑ i = 1 n x i ) - ∑ i = 1 n F ( x i ) ‖ ≤ ‖ F ( 1 n ∑ i = 1 n x i ) - 1 n ∑ i = 1 n F ( x i ) ‖ \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| and ‖ F ( 1 n ∑ i = 1 n x i ) - 1 n ∑ i = 1 n F ( x i ) ‖ ≤ ‖ F ( ∑ i = 1 n x i ) - ∑ i = 1 n F ( x i ) ‖ . \left\| {F\left( {{1 \over n}\sum\limits_{i = 1}^n {{x_i}} } \right) - {1 \over n}\sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\| \le \left\| {F\left( {\sum\limits_{i = 1}^n {{x_i}} } \right) - \sum\limits_{i = 1}^n {F\left( {{x_i}} \right)} } \right\|. Using the direct method, we prove the Hyers-Ulam stability of the functional inequalities (0.1) in Banach spaces and (0.2) in non-Archimedian Banach spaces.

scholarly journals ADDITIVE FUNCTIONAL INEQUALITIES AND DERIVATIONS ON HILBERT C*-MODULES

2013 ◽  
Vol 55 (2) ◽  
pp. 341-348 ◽  
Author(s):  
FRIDOUN MORADLOU
Keyword(s):  
Banach Spaces ◽  
Functional Inequality ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Formula Group ◽  
Image Position

AbstractIn this paper we investigate the following functional inequality $ \begin{eqnarray*} \| f(x-y-z) - f(x-y+z) + f(y) +f(z)\| \leq \|f(x+y-z) - f(x)\| \end{eqnarray*}$ in Banach spaces, and employing the above inequality we prove the generalized Hyers–Ulam stability of derivations in Hilbert C*-modules.

scholarly journals On the stability problem of quadratic functional equations in 2-Banach spaces

2017 ◽  
pp. 5054-5061
Author(s):  
Seong Sik Kim ◽  
Ga Ya Kim ◽  
Soo Hwan Kim
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Stability Problem ◽  
Functional Equations ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Normed Spaces ◽  
The Stability ◽  
Stability Results

In this paper, we investigate the stability problem in the spirit of Hyers-Ulam, Rassias and G·avruta for the quadratic functional equation:f(2x + y) + f(2x ¡ y) = 2f(x + y) + 2f(x ¡ y) + 4f(x) ¡ 2f(y) in 2-Banach spaces. These results extend the generalized Hyers-Ulam stability results by thequadratic functional equation in normed spaces to 2-Banach spaces.

scholarly journals Ulam Stability of a Functional Equation in Various Normed Spaces

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1119
Author(s):  
Krzysztof Ciepliński
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Functional Equations ◽  
Direct Method ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Quadratic Mappings ◽  
The Stability

In this note, we study the Ulam stability of a general functional equation in four variables. Since its particular case is a known equation characterizing the so-called bi-quadratic mappings (i.e., mappings which are quadratic in each of their both arguments), we get in consequence its stability, too. We deal with the stability of the considered functional equations not only in classical Banach spaces, but also in 2-Banach and complete non-Archimedean normed spaces. To obtain our outcomes, the direct method is applied.

scholarly journals Functional Inequalities Associated with Cauchy Additive Functional Equation in Non-Archimedean Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
A. Ebadian ◽  
N. Ghobadipour ◽  
Th. M. Rassias ◽  
M. Eshaghi Gordji
Keyword(s):  
Functional Equation ◽  
Additive Functional ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Normed Spaces ◽  
Additive Functional Equation ◽  
Rassias Stability

We investigate the generalized Hyers-Ulam stability of the functional inequalities∥f((x+y+z)/4)+f((3x−y−4z)/4)+f((4x+3z)/4)∥≤∥2f(x)∥and∥f((y−x)/3)+f((x−3z)/3)+f((3x+3z−y)/3)∥≤∥f(x)∥in non-Archimedean normed spaces in the spirit of the Th. M. Rassias stability approach.

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.

scholarly journals Hyers-Ulam stability of a finite variable mixed type quadratic-additive functional equation in quasi-Banach spaces

2020 ◽  
Vol 5 (6) ◽  
pp. 5993-6005 ◽  
Author(s):  
K. Tamilvanan ◽  
◽  
Jung Rye Lee ◽  
Choonkil Park ◽  
◽  
...  
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Mixed Type ◽  
Additive Functional ◽  
Ulam Stability ◽  
Additive Functional Equation
Sign in / Sign up

Export Citation Format

Share Document