# Additive $\rho$--functional inequalities

Volume 7, Issue 5, pp 296--310
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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.

•  46S10
•  39B62
•  39B52
•  47S10
•  12J25

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