文摘
Schauder’s fixed point theorem and the Banach contraction principle are used to study the polynomial-like iterative functional equation $$\begin{aligned} \lambda _1f(x)+\lambda _2f^2(x)+\cdots +\lambda _n f^n(x)=F(x). \end{aligned}$$We give sufficient conditions for the existence, uniqueness, and stability of the periodic and continuous solutions. We examine the monotonicity, convexity, and differentiability of the solutions of the family \(2f(x)+\lambda f^2(x)=\sin (x)\), (\(\lambda \in [0,1]\)).KeywordsIterative functional equationperiodic solutionsfixed point theoremThis work was partially supported by the National Natural Science Foundation of China (Grant No. 11326120, 11501069), Foundation of Chongqing Municipal Education Commission (Grant No. KJ1400528, KJ1600320).