{
  "id": "2102.01789",
  "version": "v1",
  "published": "2021-02-02T22:45:44.000Z",
  "updated": "2021-02-02T22:45:44.000Z",
  "title": "A new type of functional equations on semigroups with involutions",
  "authors": [
    "Iz-iddine El-Fassi"
  ],
  "categories": [
    "math.FA"
  ],
  "abstract": "Let $S$ be a commutative semigroup, $K$ a quadratically closed commutative field of characteristic different from $2$, $G$ a $2$-cancellative abelian group and $H$ an abelian group uniquely divisible by $2$. The aim of this paper is to determine the general solution $f:S^2\\to K$ of the d'Alembert type equation: $$ f(x+y,z+w)+f(x+\\sigma(y),z+\\tau(w)) =2f(x,z)f(y,w),\\quad\\quad (x,y,z,w\\in S) $$ the general solution $f:S^2\\to G$ of the Jensen type equation: $$ f(x+y,z+w)+f(x+\\sigma(y),z+\\tau(w)) =2f(x,z),\\quad\\quad (x,y,z,w\\in S) $$ the general solution $f:S^2\\to H$ of the quadratic type equation quation: $$ f(x+y,z+w)+f(x+\\sigma(y),z+\\tau(w)) =2f(x,z)+2f(y,w),\\quad\\quad (x,y,z,w\\in S) $$ where $\\sigma,\\tau: S\\to S$ are two involutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-02T22:45:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "39B52",
      "65Q20"
    ],
    "keywords": [
      "functional equations",
      "general solution",
      "involutions",
      "quadratic type equation quation",
      "jensen type equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}