On the functional equation \(x+f(y+f(x))=y+f(x+f(y))\). II. (English) Zbl 1386.39034
Summary: For an abelian group (\(G,+,0)\) we consider the functional equation
\[
f : G \to G, \quad x + f(y + f(x)) = y + f(x + f(y)) \quad (\forall x, y \in G),{(1)}
\]
together with the condition
\[
f(0) = 0.{(0)}
\]
The main question is that of existence of solutions of \((1)\wedge (0)\), specifically in the case when \(G\) is the direct sum \(\mathbb Z_n^{(J)}\) of copies of a finite or infinite cyclic group (Theorems 3.2 and 4.20).
For Part I, see [the author, ibid. 86, No. 1–2, 187–200 (2013; Zbl 1408.39018)].
For Part I, see [the author, ibid. 86, No. 1–2, 187–200 (2013; Zbl 1408.39018)].
MSC:
| 39B12 | Iteration theory, iterative and composite equations |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
Citations:
Zbl 1408.39018References:
| [1] | Bachmann, P.: Niedere Zahlentheorie. Teubner, Leipzig (1902) |
| [2] | Balcerowski, M.: On the functional equation x + f (y + f(x)) = y + f (x + f(y)). Aequ. Math. 75, 297-303 (2008) · Zbl 1148.39018 · doi:10.1007/s00010-007-2905-7 |
| [3] | Bourbaki, N.: Eléments de Mathématique, Livre II: Algèbre. chapitre 2. Hermann, Paris (1962) · Zbl 0142.00102 |
| [4] | Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Clarendon Press, Oxford (1979) · Zbl 0423.10001 |
| [5] | Hua, L.K.: Introduction to Number Theory. Springer, New York (1982) · Zbl 0483.10001 |
| [6] | Jacobson, N.: Lectures in Abstract Algebra, vol. I. Van Nostrand, Princeton (1966) |
| [7] | Jacobson, N.: Basic Algebra I. Freeman, New York (1985) · Zbl 0557.16001 |
| [8] | Jivulescu, M.A., Napoli, A., Messina, A.: Elementary symmetric functions of two solvents of a quadratic matrix equation. Rep. Math. Phys. 62, 369-387 (2008) · Zbl 1171.15012 · doi:10.1016/S0034-4877(08)80031-6 |
| [9] | Lehmer, D.H.: Computer technology applied to the theory of numbers. In: Studies in Number Theory, pp. 117-151. The Mathematical Association of America (1969) · Zbl 0215.06404 |
| [10] | LeVeque, W.J.: Topics in Number Theory, vol. I. Addison-Wesley, Reading (1965) · Zbl 1408.39018 |
| [11] | MacLane, S., Birkhoff, G.: Algebra. Macmillan, New York (1968) |
| [12] | Pall, G., Taussky, O.: Scalar matrix quadratic residues. Mathematika 12, 94-96 (1965) · Zbl 0125.29503 · doi:10.1112/S0025579300005192 |
| [13] | Rätz, J.: On the homogeneity of additive mappings. Aequ. Math. 14, 67-71 (1976) · Zbl 0329.39002 · doi:10.1007/BF01836207 |
| [14] | Rätz, J.: On the functional equation x + f (y + f(x)) = y + f (x + f(y)), II. Report of meeting. Aequ. Math. 84, 301-302 (2012) |
| [15] | Rätz, J.: On the functional equation x + f (y + f(x)) = y + f (x + f(y)), III. Report of meeting. Aequ. Math. 86, 305 (2013) · Zbl 1408.39018 · doi:10.1007/s00010-013-0188-8 |
| [16] | Rätz, J.: On the functional equation x + f (y + f(x)) = y + f (x + f(y)), IV. Report of meeting. Aequ. Math. (to appear) · Zbl 1408.39018 |
| [17] | Rätz, J.: On the functional equation x + f (y + f(x)) = y + f (x + f(y)). Aequ. Math. 86, 187-200 (2013) · Zbl 1408.39018 · doi:10.1007/s00010-013-0188-8 |
| [18] | Riesel, H.: Prime Numbers and Computer Methods for Factorization. Birkhäuser, Boston (1985) · Zbl 0582.10001 |
| [19] | Sierpiński, W.: Elementary theory of numbers (A. Schinzel, ed.). Polish Scientific Publishers, Warszawa (1987) · Zbl 0638.10001 |
| [20] | Warner, S.: Modern Algebra, vol. I. Prentice-Hall, Englewood Cliffs (1965) · Zbl 0134.24903 |
| [21] | Warner, S.: Modern Algebra, vol. II. Prentice-Hall, Englewood Cliffs (1965) · Zbl 0134.24903 |
| [22] | Wilansky, A.: Functional Analysis. Blaisdell, New York (1964) · Zbl 0136.10603 |
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