Abstract
Abstract In this paper, we investigate the orthogonal stability of functional equations in orthogonality modules over a unital Banach algebra. Using a fixed point method, we prove the Hyers-Ulam stability of the orthogonally Jensen additive functional equation 2 f ( x + y 2 ) = f ( x ) + f ( y ) , the orthogonally Jensen quadratic functional equation 2 f ( x + y 2 ) + 2 f ( x − y 2 ) = f ( x ) + f ( y ) , the orthogonally cubic functional equation f ( 2 x + y ) + f ( 2 x − y ) = 2 f ( x + y ) + 2 f ( x − y ) + 12 f ( x ) , and the orthogonally quartic functional equation f ( 2 x + y ) + f ( 2 x − y ) = 4 f ( x + y ) + 4 f ( x − y ) + 24 f ( x ) − 6 f ( y ) for all x, y with x ⊥ y , where ⊥ is the orthogonality in the sense of Rätz. MSC:39B55, 47H10, 39B52, 46H25.
Highlights
Ger and Sikorska [ ] investigated the orthogonal stability of the Cauchy functional equation f (x + y) = f (x) + f (y), namely they showed that, if f is a mapping from an orthogonality space X into a real Banach space Y and f (x + y) – f (x) – f (y) ≤ ε for all x, y ∈ X with x ⊥ y and for some ε > , there exists exactly one orthogonally additive mapping g : X → Y such that f (x) – g(x)
Introduction and preliminaries Assume thatX is a real inner product space and f : X → R is a solution of the orthogonal Cauchy functional equation f (x + y) = f (x) + f (y), where x, y =
In Section, we prove the Hyers-Ulam stability of the orthogonally Jensen additive functional equation in orthogonality modules over a unital Banach algebra
Summary
Ger and Sikorska [ ] investigated the orthogonal stability of the Cauchy functional equation f (x + y) = f (x) + f (y), namely they showed that, if f is a mapping from an orthogonality space X into a real Banach space Y and f (x + y) – f (x) – f (y) ≤ ε for all x, y ∈ X with x ⊥ y and for some ε > , there exists exactly one orthogonally additive mapping g : X → Y such that f (x) – g(x) The first author treating the stability of the quadratic equation was Skof [ ] by proving that, if f is a mapping from a normed space X into a Banach space Y satisfying f (x + y) + f (x – y) – f (x) – f (y) ≤ ε for some ε > , there is a unique quadratic mapping g : X → Y such that f (x) –
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