FUNCTIONAL INEQUALITIES ASSOCIATED WITH BI-CAUCHY ADDITIVE FUNCTIONAL EQUATIONS

In this paper, we prove the Hyers-Ulam stability for the following functional inequalities:parallel to f(x(1), y(1)) + f(x(2), y(2)) + f(x(3), y(3))parallel to <= parallel to f(x(1) + x(2) + x(3), y(1) + y(2) + y(3))parallel to, (1)parallel to f(x(1), y(1)) + f(x(2), y(2)) + f(x(3), y(3))parallel to <= parallel to 2f (x(1) +x(2) + x(3)/2, y(1) + y(2) + y(3)/2)parallel to, (2)parallel to f(x(1,) y(1)) + f(x(2), y(2)) + 2f(x(3), y(3))parallel to <= parallel to 2f (x(1) + x(2)/2 + x(3), y(1) + y(2)/2 + y(3))parallel to (3)in Banach spaces.C. Park was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A2004299). D. Y. Shin was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2010-0021792)