Fixed point approach to the Hyers-Ulam-Rassias approximation‎ ‎of homomorphisms and derivations on Non-Archimedean random Lie $C^*$-algebras

Abstract

‎In this paper‎, ‎using fixed point method‎, ‎we prove the generalized Hyers-Ulam stability of‎ <br />‎random homomorphisms in random $C^*$-algebras and random Lie $C^*$-algebras‎ <br />‎and of derivations on Non-Archimedean random C$^*$-algebras and Non-Archimedean random Lie C$^*$-algebras for‎ <br />‎the following $m$-variable additive functional equation:‎ <br />‎$$sum_{i=1}^m f(x_i)=frac{1}{2m}left[sum_{i=1}^mfleft( m x_i‎ + ‎sum_{j=1~,ineq j}^m x_jright)+fleft(sum_{i=1}^m x_iright) right]$$‎ <br />‎The concept of Hyers-Ulam-Rassias stability originated from Th‎. ‎M.‎ <br />‎Rassias� stability theorem that appeared in his paper‎: ‎On the‎ <br />‎stability of the linear mapping in Banach spaces‎, ‎Proc‎. ‎Amer.‎ <br />‎Math‎. ‎Soc‎. ‎72 (1978)‎, ‎297-300.