Hyers–Ulam stability and hyperstability of a Jensen-type functional equation on 2-Banach spaces

Abstract

Abstract The main aim of this paper is to establish the Hyers–Ulam stability and hyperstability of a Jensen-type quadratic mapping in 2-Banach spaces. That is, we prove the various types of Hyers–Ulam stability and hyperstability of the Jensen-type quadratic functional equation of the form g ( x + y 2 + z ) + g ( x + y 2 − z ) + g ( x − y 2 + z ) + g ( x − y 2 − z ) = g ( x ) + g ( y ) + 4 g ( z ) , $$ g \biggl( \frac{x+y}{2} + z \biggr) + g \biggl( \frac{x+y}{2} - z \biggr) + g \biggl( \frac{x-y}{2} + z \biggr) + g \biggl( \frac{x-y}{2} - z \biggr) = g(x) + g(y) + 4 g(z), $$ in 2-Banach spaces by using the Hyers direct method.

Keywords

• Hyers–Ulam stability
• Hyperstability
• Jensen-type function
• Quadratic function
• 2-Banach space
• Direct method