A wavelike functional equation of Pexider type (Q1601716)

A function \(f:G\to H\) is said to be a generalized quadratic polynomial if it can be represented in the form \(f(x)=Q^0+ Q^1(x)+Q^2(x)\), where \(Q^0\) is an arbitrary constant, \(Q^1\) is an additive function and \(Q^2\) is a diagonalization of a symmetric bi-additive function. The first result of this paper is as follows: Let \((G,+)\) and \((H,+)\) be abelian groups, and assume that \(2u=v\) is solvable in \(H\). The general solution of the equation \[ f: G\times G\to H,\;f(x+t,y+s) +f(x-t,y-s) =f(x+s,y-t) +f(x-s,y+t) \tag{1} \] is given by \[ f(x+y)= a(x)+b(y) +c(x,y) \] where \(a\) and \(b\) are generalized quadratic polynomials and \(c\) a skew-symmetric bi-additive function. The main result is given in section (3). Let \((G,+)\) and \((H,+)\) be abelian groups such that the equation \(2u=v\) is solvable in both \(G\) and \(H\). It is shown that if \(f_1,f_2, f_3,f_4:G \times G\to H\) satisfy the functional equation \[ f_1(x+ t,y+s)+ f_2(x-t,y-s) =f_3(x+s,y-t) +f_4(x-s,y+t) \] for all \(x,y,s,t\in G\), then \(f_1,f_2,f_3\), and \(f_4\) are given by \(f_1=w+h\), \(f_2=w-h\), \(f_3=w+k\), \(f_4=w-k\) where \(w:G\times G\to H\) is an arbitrary solution of (1) and \(h,k: G\times G\to H\) are arbitrary solutions of \(\Delta^3_{y,t} g(x,y)=0\) and \(\Delta^3_{x,t} g(x, y)=0\) for all \(x,y,s,t\in G\). Here, the partial difference operators \(\Delta_{x, t}\) and \(\Delta_{y,t}\) are defined by \[ \Delta_{x,t} g(x,y)=g(x+t,y) -g(x,y), \Delta_{y,t} g(x,y)= g(x,y+t)- g(x,t). \]