求解函数解析式的解方程组法

Linear System and a Function Equation

针对函数方程f(x)+bf(g(x))=h(x),其中f(x)为待求函数,b为任意实数,h(x)为已知函数,g(x)满足gn(x)=x.经过多次迭代换元,构建由待求函数构成的线性方程组,运用Cramer法则,可得出该函数方程有唯一解的充要条件为bn≠(-1)n,且此时可解出f(x)=11-(-b)n∑n-1i=0(-b)ih(gi(x)).

This paper studies the function equation f（x） ＋ bf（g（x）） = h（x）, where f（x） is the unknown function, b is a real constant, h（x） is known and the function g（x） satisfies g^n（x） = x. A linear system consisting of the unknown function is constructed and the Cramer＇s rule is applied to conclude that b^n ≠ （- 1）^n is a necessary and sufficient condition for the equation to have aunique solution. Under such a condition f（x）=1/1-（-b）^n n-1∑i=0 （-b）^ih（g^i（x）） is the explicitexpression of the solution.