一类多元线性函数方程(Ⅰ)(英文)

On a Class of Linear Functional Equations for Functions of Several Variables (Ⅰ)

设函数 f(x1,x2 ,… ,xn)对xn 有连续二阶偏导数 ,我们寻求函数方程 ni=1(- 1) i- 1[f(x1,… ,xi+xi+1,… ,xn+1) + f(x1,… ,xi-xi+1,… ,xn+1) ]+ (- 1) n2 f(x1,x2 ,… ,xn) =0的一般解 .首先 ,给出了方程 ni=1(- 1) i- 1[F(x1,… ,xi+xi+1,… ,xn+1) +F(x1,… ,xi-xi+1,… ,xn+1) ]=0的一般解 ,其次 ,上述第 1式对xn+1两次微分 ,并简化得到形如第 2式的方程 .第 1个函数方程的一般解为f(x1,x2 ,… ,xn) = n-1i=1(- 1) i- 1[A(x1,… ,xi+xi+1,… ,xn) +A(x1,… ,xi-xi+1,… ,xn) ]+ (- 1) n- 12A(x1,x2 ,… ,xn- 1) .其中A(x1,x2 ,… ,xn- 1)是对xn- 1具有连续二阶导数的任意函数 .

By letting the function f(x 1,x 2,...,x n) have continuous partial derivatives of second order with respect to x n ,the functional equation  n i=1 (-1) i-1 [f(x 1,...,x i+x i+1 ,...,x n+1 )+f(x 1,...,x i-x i+1 ,...,x n+1 )]+(-1) n2f(x 1,x 2,...,x n)=0 is considered.First,the general solution of the equation  n i=1 (-1) i-1 [F(x 1,...,x i+x i+1 ,...,x n+1 )+F(x 1,...,x i-x i+1 ,...,x n+1 )]=0 was presented.Then,the first functional equation was twice differentiated with respect to x n+1 and reduced to an equation of the aforementioned type.It is found that the general solution of the first functional equation is f(x 1,x 2,...,x n)= n-1 i=1 (-1) i-1 [A(x 1,...,x i+x i+1 ,...,x n)+A(x 1,...,x i-x i+1 ,...,x n)]+ (-1) n-1 2A(x 1,x 2,...,x n-1 ). Where A(x 1,x 2,...x n-1 ) is an arbitrary twice continuous differentiable with respect to x n-1 .