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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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