We determine the general solution of the functional equation f(x + ky) + f(x-ky) = g(x + y) + g(x-y) + h(x) + h(y) for fixed integers with k ≠ 0; ±1 without assuming any regularity conditions for the unknown functions f, g, h, and0020\( \tilde{h} \). The method used for solving these functional equations is elementary but it exploits an important result due to Hosszú. The solution of this functional equation can also be obtained in groups of certain type by using two important results due to Székelyhidi.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 3, pp. 399–415, March, 2011.
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Xu, T.Z., Rassias, J.M. & Xu, W.X. A generalized mixed type of quartic–cubic–quadratic–additive functional equations. Ukr Math J 63, 461–479 (2011). https://doi.org/10.1007/s11253-011-0515-y
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DOI: https://doi.org/10.1007/s11253-011-0515-y