Approximate \(C^{\ast}\)-algebra homomorphisms associated to an Apollonius-Jensen type additive mapping: a fixed point approach.

*(English)*Zbl 1315.39019
Pardalos, Panos M. (ed.) et al., Nonlinear analysis. Stability, approximation, and inequalities. In honor of Themistocles M. Rassias on the occasion of his 60th birthday. New York, NY: Springer (ISBN 978-1-4614-3497-9/hbk; 978-1-4614-3498-6/ebook). Springer Optimization and Its Applications 68, 457-470 (2012).

The stability problem of functional equations originated from a question of Ulam in 1940, concerning the stability of group homomorphisms. In 1941, D. H. Hyers gave a first affirmative answer to the question for Banach spaces.

In 2003, Cǎdariu and Radu applied the fixed point method to investigate the Jensen functional equation [L. Cǎdariu and V. Radu, JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 4, 7 p. (2003; Zbl 1043.39010)]. They were able to present a short and a simple proof (different of the “direct method”, initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of the Jensen functional equation.

In an inner product space, the equality \[ \|z-x\|^2+\|z-y\|^2=\frac{1}{2}\|x-y\|^2+2\|z-\frac{x+y}{2}\|^2 \] holds, it is called the Apollonius’ identity. If the following functional equation, which is motivated by the above equation, namely \[ Q(z-x)+Q(z-y)=\frac{1}{2}Q(x-y)+2Q(z-\frac{x+y}{2}), \] holds, then it is called quadratic. For this reason, the above functional equation is called a quadratic functional equation of Apollonius type, and each solution of the functional equation is said to be a quadratic mapping of Apollonius type.

Recently, C. Park [Math. Nachr. 281, No. 3, 402–411 (2008; Zbl 1142.39023)] introduced and investigated the following functional equation \[ f((\sum_{i=1}^nz_i)-(\sum_{i=1}^nx_i))+f((\sum_{i=1}^nz_i)-(\sum_{i=1}^ny_i))=2f((\sum_{i=1}^nz_i)-\frac{(\sum_{i=1}^nx_i)+(\sum_{i=1}^ny_i)}{2}), \] which is called the generalized Apollonius-Jensen-type additive functional equation and whose solution is said to be a generalized Apollonius-Jensen-type additive mapping.

In the present paper, the authors adopt the idea of Cǎdariu and Radu [loc. cit.] to prove the generalized Hyers-Ulam stability result of \(C^*\)-algebra homomorphisms as well as to prove the generalized Ulam-Hyers stability of generalized derivations on \(C^*\)-algebra for additive functional equations of \(n\)-Apollonius type.

For the entire collection see [Zbl 1242.00056].

In 2003, Cǎdariu and Radu applied the fixed point method to investigate the Jensen functional equation [L. Cǎdariu and V. Radu, JIPAM, J. Inequal. Pure Appl. Math. 4, No. 1, Paper No. 4, 7 p. (2003; Zbl 1043.39010)]. They were able to present a short and a simple proof (different of the “direct method”, initiated by Hyers in 1941) for the generalized Hyers-Ulam stability of the Jensen functional equation.

In an inner product space, the equality \[ \|z-x\|^2+\|z-y\|^2=\frac{1}{2}\|x-y\|^2+2\|z-\frac{x+y}{2}\|^2 \] holds, it is called the Apollonius’ identity. If the following functional equation, which is motivated by the above equation, namely \[ Q(z-x)+Q(z-y)=\frac{1}{2}Q(x-y)+2Q(z-\frac{x+y}{2}), \] holds, then it is called quadratic. For this reason, the above functional equation is called a quadratic functional equation of Apollonius type, and each solution of the functional equation is said to be a quadratic mapping of Apollonius type.

Recently, C. Park [Math. Nachr. 281, No. 3, 402–411 (2008; Zbl 1142.39023)] introduced and investigated the following functional equation \[ f((\sum_{i=1}^nz_i)-(\sum_{i=1}^nx_i))+f((\sum_{i=1}^nz_i)-(\sum_{i=1}^ny_i))=2f((\sum_{i=1}^nz_i)-\frac{(\sum_{i=1}^nx_i)+(\sum_{i=1}^ny_i)}{2}), \] which is called the generalized Apollonius-Jensen-type additive functional equation and whose solution is said to be a generalized Apollonius-Jensen-type additive mapping.

In the present paper, the authors adopt the idea of Cǎdariu and Radu [loc. cit.] to prove the generalized Hyers-Ulam stability result of \(C^*\)-algebra homomorphisms as well as to prove the generalized Ulam-Hyers stability of generalized derivations on \(C^*\)-algebra for additive functional equations of \(n\)-Apollonius type.

For the entire collection see [Zbl 1242.00056].

Reviewer: Nasrin Eghbali (Ardabil)

##### MSC:

39B82 | Stability, separation, extension, and related topics for functional equations |

46L05 | General theory of \(C^*\)-algebras |

46B03 | Isomorphic theory (including renorming) of Banach spaces |

39B52 | Functional equations for functions with more general domains and/or ranges |

46L57 | Derivations, dissipations and positive semigroups in \(C^*\)-algebras |