scholarly journals Banach space techniques underpinning a theory for nearly additive mappings

2002 ◽  
Vol 404 ◽  
pp. 1-73 ◽  
Author(s):  
Félix Cabello Sánchez ◽  
Jesús M. F. Castillo
Keyword(s):  
Banach Space ◽  
Additive Mappings

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.

ON HYPERSTABILITY OF ADDITIVE MAPPINGS ONTO BANACH SPACES

2014 ◽  
Vol 91 (2) ◽  
pp. 278-285 ◽  
Author(s):  
YUNBAI DONG ◽  
BENTUO ZHENG
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Banach Spaces ◽  
Studia Math ◽  
Additive Map ◽  
Image Position ◽  
Additive Mappings

AbstractLet$(X,+)$be an Abelian group and$E$be a Banach space. Suppose that$f:X\rightarrow E$is a surjective map satisfying the inequality$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(x-y)\Vert \,|\leq {\it\varepsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(x-y)\Vert ^{p}\}\end{eqnarray}$$for some${\it\varepsilon}>0$,$p>1$and for all$x,y\in X$. We prove that$f$is an additive map. However, this result does not hold for$0<p\leq 1$. As an application, we show that if$f$is a surjective map from a Banach space$E$onto a Banach space$F$so that for some${\it\epsilon}>0$and$p>1$$$\begin{eqnarray}|\,\Vert f(x)-f(y)\Vert -\Vert f(u)-f(v)\Vert \,|\leq {\it\epsilon}\min \{\Vert f(x)-f(y)\Vert ^{p},\Vert f(u)-f(v)\Vert ^{p}\}\end{eqnarray}$$whenever$\Vert x-y\Vert =\Vert u-v\Vert$, then$f$preserves equality of distance. Moreover, if$\dim E\geq 2$, there exists a constant$K\neq 0$such that$Kf$is an affine isometry. This improves a result of Vogt [‘Maps which preserve equality of distance’,Studia Math.45(1973) 43–48].

On Traces of n-Additive Mappings on Semiprime Ring

2017 ◽  
Vol 4 (1) ◽  
pp. 1-10
Author(s):  
Najat Muthana ◽  
◽  
Asma Ali ◽  
Kapil Kumar
Keyword(s):  
Semiprime Ring ◽  
Additive Mappings

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

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