The kernel of the second order Cauchy difference on semigroups

TY - JOUR

T1 - The kernel of the second order Cauchy difference on semigroups

AU - Stetkær, Henrik

PY - 2017/4/1

Y1 - 2017/4/1

N2 - Let S be a semigroup, H a 2-torsion free, abelian group and C 2f the second order Cauchy difference of a function f: S→ H. Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of C 2f= 0 are the functions of the form f(x) = j(x) + B(x, x) , where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of C 2f= 0 to Fréchet’s functional equation and to polynomials of degree less than or equal to 2.

AB - Let S be a semigroup, H a 2-torsion free, abelian group and C 2f the second order Cauchy difference of a function f: S→ H. Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of C 2f= 0 are the functions of the form f(x) = j(x) + B(x, x) , where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of C 2f= 0 to Fréchet’s functional equation and to polynomials of degree less than or equal to 2.

KW - Cauchy difference

KW - Fréchet

KW - Functional equation

KW - Second order Cauchy difference

KW - Whitehead

U2 - 10.1007/s00010-016-0453-8

DO - 10.1007/s00010-016-0453-8

M3 - Journal article

AN - SCOPUS:85006380608

SN - 0001-9054

VL - 91

SP - 279

EP - 288

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

IS - 2

ER -