Stability of an additive-quadratic functional equation in modular spaces

Abstract

Using the direct method, we prove the Hyers-Ulam-Rassias stability of the following functional equation: phi ( x + y , z + w ) + phi ( x - y , z - w ) - 2 phi ( x , z ) - 2 phi ( x , w ) = 0 \phi \left(x+y,z+w)+\phi \left(x-y,z-w)-2\phi \left(x,z)-2\phi \left(x,w)=0 in rho \rho -complete convex modular spaces satisfying Fatou property or Delta 2 {\Delta }_{2} -condition.

Using the direct method, we prove the Hyers-Ulam-Rassias stability of the following functional equation: ϕ( x + y , z + w ) + ϕ( x - y , z - w ) - 2 ϕ( x , z ) - 2 ϕ( x , w ) = 0 in ρ-complete convex modular spaces satisfying Fatou property or Δ2-condition.