scholarly journals Stability of Functional Inequalities with Cauchy-Jensen Additive Mappings

2007 ◽  
Vol 2007 ◽  
pp. 1-13 ◽  
Author(s):  
Young-Sun Cho ◽  
Hark-Mahn Kim
Keyword(s):  
Additive Mapping ◽  
Functional Inequalities ◽  
Ulam Stability ◽  
Additive Mappings

We investigate the generalized Hyers-Ulam stability of the functional inequalities associated with Cauchy-Jensen additive mappings. As a result, we obtain that if a mapping satisfies the functional inequalities with perturbation which satisfies certain conditions, then there exists a Cauchy-Jensen additive mapping near the mapping.

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 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 Functional Inequalities Associated with Additive Mappings

2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Jaiok Roh ◽  
Ick-Soon Chang
Keyword(s):  
Banach Space ◽  
Functional Inequality ◽  
Functional Inequalities ◽  
Stability Theorems ◽  
Group Divisible ◽  
Additive Mappings

The functional inequality‖f(x)+2f(y)+2f(z)‖≤‖2f(x/2+y+z)‖+ϕ  (x,y,z) (x,y,z∈G)is investigated, whereGis a group divisible by2,f:G→Xandϕ:G3→[0,∞)are mappings, andXis a Banach space. The main result of the paper states that the assumptions above together with (1)ϕ(2x,−x,0)=0=ϕ(0,x,−x) (x∈G)and (2)limn→∞(1/2n)ϕ(2n+1x,2ny,2nz)=0, orlimn→∞2nϕ(x/2n−1,y/2n,z/2n)=0  (x,y,z∈G), imply thatfis additive. In addition, some stability theorems are proved.

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.

ON APPROXIMATELY ADDITIVE MAPPINGS IN 2-BANACH SPACES

2019 ◽  
Vol 101 (2) ◽  
pp. 299-310 ◽  
Author(s):  
JANUSZ BRZDĘK ◽  
EL-SAYED EL-HADY
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Banach Spaces ◽  
Point Theorem ◽  
Stability Result ◽  
Main Tool ◽  
Ulam Stability ◽  
Cauchy Equation ◽  
Stability Issues ◽  
Additive Mappings

We show how some Ulam stability issues can be approached for functions taking values in 2-Banach spaces. We use the example of the well-known Cauchy equation $f(x+y)=f(x)+f(y)$, but we believe that this method can be applied for many other equations. In particular we provide an extension of an earlier stability result that has been motivated by a problem of Th. M. Rassias. The main tool is a recent fixed point theorem in some spaces of functions with values in 2-Banach spaces.

scholarly journals On the Hyers-Ulam Stability of ψ-Additive Mappings

1993 ◽  
Vol 72 (2) ◽  
pp. 131-137 ◽  
Author(s):  
G. Isac ◽  
T.M. Rassias
Keyword(s):  
Ulam Stability ◽  
Additive Mappings

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 Stability ofψ-additive mappings: applications to nonlinear analysis

1996 ◽  
Vol 19 (2) ◽  
pp. 219-228 ◽  
Author(s):  
George Isac ◽  
Themistocles M. Rassias
Keyword(s):  
Fixed Point ◽  
Nonlinear Analysis ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Ulam Stability ◽  
Mathematical Problems ◽  
Stability Of Mappings ◽  
Additive Mappings

The Hyers-Ulam stability of mappings is in development and several authors have remarked interesting applications of this theory to various mathematical problems. In this paper some applications in nonlinear analysis are presented, especially in fixed point theory. These kinds of applications seem not to have ever been remarked before by other authors.

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