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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].
##### 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
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