Additive \(\rho\)--functional inequalities


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.


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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


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