APPROXIMATION OF CUBIC MAPPINGS WITH n-VARIABLES IN β-NORMED LEFT BANACH MODULE ON BANACH ALGEBRAS
APPROXIMATION OF CUBIC MAPPINGS WITH n-VARIABLES IN β-NORMED LEFT BANACH MODULE ON BANACH ALGEBRAS
Let M = {1, 2, <TEX>${\ldots}$</TEX>, n} and let V = {<TEX>$I{\subseteq}M:1{\in}I$</TEX>}. Denote M\I by <TEX>$I^c$</TEX> for <TEX>$I{\in}V$</TEX>. The goal of this paper is to investigate the solution and the stability using the alternative fixed point of generalized cubic functional equation <TEX>$ \sum\limits_{I{\in}V}f(\sum\limits_{i{\in}I}a_ix_i-\sum\limits_{i{\in}I^c}a_ix_i)=2{^{n-2}{a_1}}\sum\limits_{i=2}^na_i^2[f(x_1+x_i)+f(x_1-x_i)]+2{^{n-1}{a_1}(a^2_1-\sum\limits_{i=2}^2a^2_i)f(x_1)$</TEX> in <TEX>${\beta}$</TEX>-Banach modules on Banach algebras, where …