Fixed points and quadratic rhofunctional equations

In this paper, we solve the quadratic ρ -functional equations f ( x + y ) + f ( x − y ) − 2 f ( x ) − 2 f ( y ) = ρ ( 2 f ( x + y /2 ) + 2 f ( x − y/ 2 ) − f ( x ) − f ( y ) ) , ( 1 ) where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ≠ 1 ; 2 , and 2 f ( x + y/ 2 ) + 2 f ( x − y/ 2 ) − f ( x ) − f ( y ) = ρ ( f ( x + y ) + f ( x − y ) − 2 f ( x ) − 2 f ( y ) ) ; ( 2 ) where ρ is a fixed non-Archimedean number or a fixed real or complex number with ρ ≠ 1 ; − 1/ 2 . Using the fixed point method, we prove the Hyers-Ulam stability of the quadratic ρ -functional equations (1) and (2) in non-Archimedean Banach spaces and in Banach spaces.