Affiliation:
1. Faculty of Mathematics and Natural Sciences, University of Rzeszów, Pigonia 1, 35-310 Rzeszów, Poland
Abstract
Abstract
Let (S, +) be a commutative semigroup, σ : S → S be an endomorphism with σ2 = id and let K be a field of characteristic different from 2. Inspired by the problem of strong alienation of the Jensen equation and the exponential Cauchy equation, we study the solutions f, g : S → K of the functional equation
f
(
x
+
y
)
+
f
(
x
+
σ
(
y
)
)
+
g
(
x
+
y
)
=
2
f
(
x
)
+
g
(
x
)
g
(
y
)
for
x
,
y
∈
S
.
$$f(x + y) + f(x + \sigma (y)) + g(x + y) = 2f(x) + g(x)g(y)\;\;\;\;{\rm for}\;\;x,y \in S.$$
We also consider an analogous problem for the Jensen and the d’Alembert equations as well as for the d’Alembert and the exponential Cauchy equations.
Reference10 articles.
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3. [3] Ger R., Additivity and exponentiality are alien to each other, Aequationes Math. 80 (2010), 111–118.
4. [4] Ger R., The alienation phenomenon and associative rational operations, Ann. Math. Sil. 27 (2013), 75–88.
5. [5] Ger R., Alienation of additive and logarithmic equations, Ann. Univ. Sci. Budapest. Sect. Comput. 40 (2013), 269–274.
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