Generalization of a result of Aczél, Ghermänescu and Hosszú

D. Ž., Djokovič (1964) Generalization of a result of Aczél, Ghermänescu and Hosszú. A MAGYAR TUDOMÁNYOS AKADÉMIA MATEMATIKAI KUTATÓ INTÉZETÉNEK KÖZLEMÉNYEI, 9 (1-2). pp. 51-59.

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Abstract

J . Aczél, M. Ghermänescu and M. Hosszú in their paper [1] have considered the basic cyclic functional equation (1.1) F(X₁, X₂, . . . , Xₙ ) + F(X₂, X₃, .. ., Xₙ, X₁) + ... + F(Xₙ, X₁, . . ., Xₙ_₁) = 0 and the derived equation (1.2) F(X₁, X₂, ..., Xₚ) + F(X₂, X₃ Xₚ+₁) + ... + (Xₙ_ₚ+₁, Xₙ_ₚ+₂, . . . , Xₙ) + F(Xₙ_ₚ+₂, Xₙ_ₚ+₂, .. ., Xₙ, X₁) + ... + F(Xₙ, X₁, . . ., Xₚ_₁) = 0 , where p and n (> p) are two arbitrary positive integers. In the mentioned paper they have formulated three theorems, the second being: The most general solution of (1.2) in the case when n ≥ 2p — 1 is given by (1.3) F(X₁, X₂, . . ., Xₚ) = f(X₁, X₂, . . ., Xₚ_₁) f(X₂, X₃, ..., Xₚ) where f is an arbitrary function. This theorem (and two others) are proved under the following assumptions: 1° Xᵢ ∈ S, where S is an arbitrary non-empty set; 2° The values of the function F lie in an additive abelian group M; 3° The group M is such that the equation mX = A (X, A ∈ M) has a unique solution X = A/m for every m ≤ n (m ∈ N). I n the proof of first and second theorems it is sufficient to take m — n in 3°. The idea of the proof of the third theorem, concerning equation (1.2) in case p < n < 2p — 1, is shown in [1] for two particular values of n and p. For details of the proof in the general case and for similar equations of. M. Hosszú [2] (also there the same assumptions 1°, 2° and 3° are made). In the second paragraph of this paper we shall solve the equation (1.2) for n ≥ 2p — 1, under the assumptions 1° and 2° only. In the third paragraph we shall solve the generalized equation (1.2) when all functions are taken to be different. Finally, we shall investigate the equation (1.2) with 1° and 2° in the case p < n < 2p — 1 for some special values of difference 2p — 1 — n.

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