On a cubic–quadratic equation relative to elliptic curves

Abstract

Abstract We investigate the Hyers–Ulam stability of the following cubic–quadratic functional equation relative to elliptic curves f ( x + y + z , u + v + w ) + f ( x + y − z , u + v + w ) + 2 f ( x , u − w ) + 2 f ( y , v − w ) = f ( x + y , u + w ) + f ( x + y , v + w ) + f ( x + z , u + w ) + f ( x − z , u + v − w ) + f ( y + z , v + w ) + f ( y − z , u + v − w ) $f(x+y+z,u+v+w)+f(x+y-z,u+v+w)+2f(x,u-w)+2f(y,v-w) =f(x+y,u+w)+f(x+y,v+w)+f(x+z,u+w)+f(x-z,u+v-w)+f(y+z,v+w)+f(y-z,u+v-w)$ . The function f ( x , y ) = x 3 + a x + b − y 2 $$ f(x,y)=x^{3}+ax+b-y^{2}$$ having level curves as elliptic curves is a solution of the above functional equation.

Keywords

• 2-Banach space
• Elliptic curve
• Hyers–Ulam stability