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Extended stability problem for alternative Cauchy-Jensen mappings. (English) Zbl 1141.39028
By using the direct method, the authors investigate the stability property of the Euler-Lagrange additive functional equation of the form \[ f (ax + by) + f (bx + ay) + (a + b)[f (-x) + f (-y)] = 0. \] Also, by using the fixed point method, they prove the stability property of the Euler-Lagrange functional equation of the form \[ f (ax + by) + f (ax - by) + 2af (-x) = 0. \]

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
46B03 Isomorphic theory (including renorming) of Banach spaces
46L05 General theory of \(C^*\)-algebras
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