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Periodic and continuous solutions of a polynomial-like iterative equation

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Abstract

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 \(2f(x)+\lambda f^2(x)=\sin (x)\), (\(\lambda \in [0,1]\)).

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

    Che Tat Ng

  2. School of Mathematics, Chongqing Normal University, Chongqing, 401331, People’s Republic of China

    Hou Yu Zhao

Authors
  1. Che Tat Ng
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  2. Hou Yu Zhao
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Corresponding author

Correspondence to Hou Yu Zhao.

Additional information

This 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).

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Ng, C.T., Zhao, H.Y. Periodic and continuous solutions of a polynomial-like iterative equation. Aequat. Math. 91, 185–200 (2017). https://doi.org/10.1007/s00010-016-0456-5

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  • DOI: https://doi.org/10.1007/s00010-016-0456-5

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