Abstract
In this paper we study estimation, continuous dependence and Hyers–Ulam stability for continuous solutions of a second order iterative equation. First we give an estimate for a bound of its continuous solutions. Then we give a Lipschitz estimation with the Lipschitz conditions (i.e., Hölder \(\hbox {exponent} =1\)) to given functions, which implies a continuous dependence of solutions and encourages a further discussion on the Hyers–Ulam stability. We similarly give a Hölder estimation with the Hölder conditions (i.e., Hölder \(\hbox {exponent} <1\)) to given functions, in a weaker sense than the above Lipschitz one, but it does not imply a continuous dependence. This actually suggests a question: Is the continuous dependence of solutions really critical for the Hölder exponent between \(<1\) and \(=1\)?
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Dedicated to Professor Karol Baron on his 70th birthday.
This work was supported by NSFC # 11401606, NSFC # 11771307, NSFC # 11726623, NSFC # 11521061, PCSIRT IRT 15R53, MYRG2015-00058-L2-FST, FDCT/099/2012/A3, and FDCT/031/2016/A1.
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Kou, K.I., Tang, X. & Zhang, W. Estimation, dependence and stability of solutions of an iterative equation. Aequat. Math. 93, 59–77 (2019). https://doi.org/10.1007/s00010-018-0555-6
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DOI: https://doi.org/10.1007/s00010-018-0555-6