On the quadratic functional equation modulo a subgroup
Let \(X\) be a topological vector space, \(Y\) a topological abelian group and \(K\) a discrete subgroup of \(Y\). The author considers mappings \(f:X\to Y\) continuous at the origin and satisfying the quadratic equation modulo \(K\): \[ f(x+y)+f(x-y)-2f(x)-2f(y)\in K,\qquad x,y\in X. \] He proves that for each such a mapping \(f\) there exists a quadratic function \(Q:X\to Y\) continuous at the origin an such that \[ f(x)-f(0)-Q(x)\in K,\qquad x\in X. \] Also the ``pexiderized'' congruence: \[ f_1(x+y)+f_2(x-y)-f_3(x)-f_4(y)\in K,\qquad x,y\in X \] and its particular form: \[ f(x+y)+f(x-y)-g(x)-g(y)\in K,\qquad x,y\in X \] are investigated.
scientific article; zbMATH DE number 5017249