Additive \(\rho\)--functional inequalities
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Authors
Choonkil Park
- Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea.
Abstract
In this paper, we solve the additive \(\rho\)-functional inequalities
\[\|f(x + y) - f(x) - f(y)\| \leq \| \rho( 2f (\frac{ x + y}{ 2}) - f(x) - f(y) ) \|, \qquad (1)\] ;
\[\|2f (\frac{ x + y}{ 2}) - f(x) - f(y)\| \leq \| \rho(f(x + y) - f(x) - f(y) ) \|, \qquad (2)\] ;
where \(\rho\) is a fixed non-Archimedean number with \(|\rho|<1\) or \(\rho\) is a fixed complex number with \(|\rho|<1\).
Using the direct method, we prove the Hyers-Ulam stability of the additive \(\rho\)-functional inequalities (1)
and (2) in non-Archimedean Banach spaces and in complex Banach spaces, and prove the Hyers-Ulam
stability of additive \(\rho\)-functional equations associated with the additive \(\rho\)-functional inequalities (1) and (2)
in non-Archimedean Banach spaces and in complex Banach spaces.
Share and Cite
ISRP Style
Choonkil Park, Additive \(\rho\)--functional inequalities, Journal of Nonlinear Sciences and Applications, 7 (2014), no. 5, 296--310
AMA Style
Park Choonkil, Additive \(\rho\)--functional inequalities. J. Nonlinear Sci. Appl. (2014); 7(5):296--310
Chicago/Turabian Style
Park, Choonkil. "Additive \(\rho\)--functional inequalities." Journal of Nonlinear Sciences and Applications, 7, no. 5 (2014): 296--310
Keywords
- Hyers-Ulam stability
- additive \(\rho\)-functional equation
- additive \(\rho\)-functional inequality
- non-Archimedean normed space
- Banach space.
MSC
- 46S10
- 39B62
- 39B52
- 47S10
- 12J25
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