Abstract
Let be a function, where is a group and is an abelian group. In this paper, the following third order Cauchy difference of = − − − − + + + + + + − − − − ( ), is studied. We first give some special solutions of on free groups. Then sufficient and necessary conditions on finite cyclic groups and symmetric groups are also obtained. MSC:39B52, 39A70.
Highlights
It is well known from [ ] that Jensen’s functional equation f (x + y) + f (x – y) = f (x), ( . )with the additional condition f ( ) =, is equivalent to Cauchy’s equation f (x + y) = f (x) + f (y) on the real line
3 Solution on a free group we study the solutions on a free group
Proof Note that C( )f (·, y, z) is a homomorphism and H is an abelian group, which yields
Summary
Let f : G → H be a function, where (G, ·) is a group and (H, +) is an abelian group. We consider the following functional equation: f (x x x x ) – f (x x x ) – f (x x x ) – f (x x x ) – f (x x x )
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