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In this study, we introduce the following additive functional equation: g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y ) g\left(\lambda u+v+2y)=\lambda g\left(u)+g\left(v)+2g(y) for all λ ∈ C \lambda \in {\mathbb{C}} , all unitary elements u , v u,v in a unital Poisson C * {C}^{* } -algebra P P , and all y ∈ P y\in P . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the aforementioned additive functional equation in unital Poisson C * {C}^{* } -algebras. Furthermore, we apply to study Poisson C * {C}^{* } -algebra homomorphisms and Poisson C * {C}^{* } -algebra derivations in unital Poisson C * {C}^{* } -algebras.
In this study, we introduce the following additive functional equation: g ( λ u + v + 2 y ) = λ g ( u ) + g ( v ) + 2 g ( y ) g\left(\lambda u+v+2y)=\lambda g\left(u)+g\left(v)+2g(y) for all λ ∈ C \lambda \in {\mathbb{C}} , all unitary elements u , v u,v in a unital Poisson C * {C}^{* } -algebra P P , and all y ∈ P y\in P . Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the aforementioned additive functional equation in unital Poisson C * {C}^{* } -algebras. Furthermore, we apply to study Poisson C * {C}^{* } -algebra homomorphisms and Poisson C * {C}^{* } -algebra derivations in unital Poisson C * {C}^{* } -algebras.
In this study, we solve the system of additive functional equations: h ( x + y ) = h ( x ) + h ( y ) , g ( x + y ) = f ( x ) + f ( y ) , 2 f x + y 2 = g ( x ) + g ( y ) , \left\{\begin{array}{l}h\left(x+y)=h\left(x)+h(y),\\ g\left(x+y)=f\left(x)+f(y),\\ 2f\left(\frac{x+y}{2}\right)=g\left(x)+g(y),\end{array}\right. and we investigate the stability of (homomorphism, derivation)-systems in Banach algebras.
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 \phi \left(x+y,z+w)+\phi \left(x-y,z-w)-2\phi \left(x,z)-2\phi \left(x,w)=0 in ρ \rho -complete convex modular spaces satisfying Fatou property or Δ 2 {\Delta }_{2} -condition.
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