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Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between \(C^*\)-algebras. (English) Zbl 1125.39027
The stability of the following quadratic functional equation of Euler-Lagrange type \[ \sum_{i=1}^nr_iQ\left(\sum_{j=1}^nr_j(x_i-x_j)\right)+\left(\sum_{i=1}^nr_i\right)Q \left(\sum_{i=1}^nr_ix_i\right)= \left(\sum_{i=1}^nr_i\right)^2\sum_{i=1}^nr_iQ(x_i). \] was established by K.-W. Jun and H.-M. Kim [Nonlinear Anal., Theory Methods Appl. 62, No. 6 (A), 975–987 (2005; Zbl 1081.39027)].
The author of the paper under review proves the Hyers-Ulam-Rassias stability of the following functional equation in Banach modules over a unital \(C^*\)-algebra \(A\) \[ \sum_{i=1}^nr_iL\left(\sum_{j=1}^nr_j(x_i-x_j)\right)+\left(\sum_{i=1}^nr_i\right)L \left(\sum_{i=1}^nr_ix_i\right)=\left (\sum_{i=1}^nr_i\right)\sum_{i=1}^nr_iL(x_i), \] where \(r_1,\dots,r_n\in (0,\infty)\) and \(L\) is a mapping between Banach \(A\)-modules.

39B82 Stability, separation, extension, and related topics for functional equations
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
46L05 General theory of \(C^*\)-algebras
39B52 Functional equations for functions with more general domains and/or ranges