Additive ρ-functional inequalities

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 nonArchimedean number with \(|\rho| lt;1\) or \(\rho\) is a fixed complex number with \(|\rho| lt;1\). Using the direct method, we prove the HyersUlam stability of the additive \(\rho\)functional inequalities (1) and (2) in nonArchimedean Banach spaces and in complex Banach spaces, and prove the HyersUlam stability of additive \(\rho\)functional equations associated with the additive \(\rho\)functional inequalities (1) and (2) in nonArchimedean Banach spaces and in complex Banach spaces.