scholarly journals Approximate Multi-Jensen, Multi-Euler-Lagrange Additive and Quadratic Mappings in -Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Tian Zhou Xu
Keyword(s):  
Banach Spaces ◽  
Direct Method ◽  
Ulam Stability ◽  
Quadratic Mappings

We prove the generalized Hyers-Ulam stability of multi-Jensen, multi-Euler-Lagrange additive, and quadratic mappings in -Banach spaces, using the socalled direct method. The corollaries from our main results correct some outcomes from Park (2011).

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 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 Approximate Quartic and Quadratic Mappings in Quasi-Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-18
Author(s):  
M. Eshaghi Gordji ◽  
H. Khodaei ◽  
Hark-Mahn Kim
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Banach Spaces ◽  
Mixed Type ◽  
Quadratic Mapping ◽  
Ulam Stability ◽  
Linear Spaces ◽  
Quadratic Mappings

we establish the general solution for a mixed type functional equation of aquartic and a quadratic mapping in linear spaces. In addition, we investigate the generalized Hyers-Ulam stability inp-Banach spaces.

scholarly journals Approximate multi-Jensen and multi-quadratic mappings in 2-Banach spaces

2013 ◽  
Vol 29 (2) ◽  
pp. 159-166
Author(s):  
KRZYSZTOF CIEPLINSKI ◽  
◽  
TIAN ZHOU XU ◽  
Keyword(s):  
Banach Spaces ◽  
Ulam Stability ◽  
Quadratic Mappings ◽  
Additive Mappings

In this paper we prove the generalized Hyers-Ulam stability of multi-Jensen and multi-quadratic mappings in 2-Banach spaces. The corollaries from our main results correct some outcomes from [Park, W.-G., Approximate additive mappings in 2-Banach spaces and related topics, J. Math. Anal. Appl., 376 (2011) 193–202].

scholarly journals Solution and Hyers-Ulam-Rassias Stability of Generalized Mixed Type Additive-Quadratic Functional Equations in Fuzzy Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
M. Eshaghi Gordji ◽  
H. Azadi Kenary ◽  
H. Rezaei ◽  
Y. W. Lee ◽  
G. H. Kim
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Functional Equations ◽  
Mixed Type ◽  
Direct Method ◽  
Quadratic Functional ◽  
Ulam Stability ◽  
Fixed Point Methods ◽  
Rassias Stability

By using fixed point methods and direct method, we establish the generalized Hyers-Ulam stability of the following additive-quadratic functional equationf(x+ky)+f(x−ky)=f(x+y)+f(x−y)+(2(k+1)/k)f(ky)−2(k+1)f(y)for fixed integerskwithk≠0,±1in fuzzy Banach spaces.

scholarly journals On the Hyers-Ulam Stability of a General Mixed Additive and Cubic Functional Equation inn-Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Tian Zhou Xu ◽  
John Michael Rassias
Keyword(s):  
Functional Equation ◽  
Banach Spaces ◽  
Direct Method ◽  
Ulam Stability ◽  
Cubic Functional Equation ◽  
Cubic Function

The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation inn-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.

STABILITY ANALYSIS IN BANACH SPACES FOR A GENERAL ADDITIVE FUNCTIONAL EQUATION

2020 ◽  
Vol 9 (11) ◽  
pp. 9179-9186
Author(s):  
P. Agilan ◽  
J.M. Rassias ◽  
V. Vallinayagam
Keyword(s):  
Functional Equation ◽  
Stability Analysis ◽  
Banach Spaces ◽  
Direct Method ◽  
Additive Functional ◽  
Ulam Stability ◽  
General Control ◽  
Additive Functional Equation ◽  
Control Functions ◽  
Stability Results

In this paper, we present the Hyers-Ulam stability of generalized additive functional equation in Banach spaces and stability results have been obtained by a classical direct method by various general control functions.

scholarly journals Generalized Hyers-Ulam stability of cubic functional inequality

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1969-1978
Author(s):  
Hark-Mahn Kim ◽  
Eunyoung Son
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Direct Method ◽  
Functional Inequality ◽  
Fixed Point Method ◽  
Point Method ◽  
Ulam Stability

In this article, we investigate the generalized Hyers-Ulam stability of a cubic functional inequality in Banach spaces and in non-Archimedean Banach spaces by using fixed point method and direct method, respectively.

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 Existence and Ulam stability for implicit fractional q-difference equations

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Saïd Abbas ◽  
Mouffak Benchohra ◽  
Nadjet Laledj ◽  
Yong Zhou
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Difference Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Ulam Stability ◽  
Uniqueness Of Solutions ◽  
Stability Results ◽  
Rassias Stability

AbstractThis paper deals with some existence, uniqueness and Ulam–Hyers–Rassias stability results for a class of implicit fractional q-difference equations. Some applications are made of some fixed point theorems in Banach spaces for the existence and uniqueness of solutions, next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Two illustrative examples are given in the last section.

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