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A Survey on Max-Type Difference Equations

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Advances in Discrete Dynamical Systems, Difference Equations and Applications (ICDEA 2021)

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Abstract

In this paper we present a survey on max-type difference equations. Mainly, we focus on different generalizations of the Lyness’ max-type difference equation, \(x_{n+1} = \frac{\max \{ x_n^k, A\}}{x_n^{\ell }x_{n-1}}\), where A is a positive real number, k and \(\ell \) are integer numbers and the initial conditions are positive; and on the reciprocal difference equation with maximum, \(x_{n+1} = \max \left\{ \frac{A_0}{x_n}, \dots , \frac{A_k}{x_{n-k}} \right\} \), where \(A_j\), \(j=0,\ldots ,k\), and the initial conditions are positive. We also collect some interesting results about several variations of those equations considering powers, periodic coefficients or changes of variables. Furthermore, we deal with a generalization of the idea of max-type difference equations, namely, rank-type difference equations. Finally, we show some applications and propose open problems related to the topic.

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Notes

  1. 1.

    Let \(\varOmega \) represent a measurable Lebesgue subset of \(\mathbb {R}^n\) of non-zero measure; and let \(\mu \) be a Lebesgue measure. Then \(L_{\infty }\) is the set of measurable functions, \(f:\varOmega \rightarrow \mathbb {C}\), such that there exists \(\varOmega _0 \subset \varOmega \), measurable with \(\mu (\varOmega \setminus \varOmega _0)=0\), and \(f|_{\varOmega _0}\) being bounded. See [57].

  2. 2.

    Let X be a metric space and let \(f:X\rightarrow X\) be a continuous function. The \(\omega \)-limit set of \(x\in X\) is \(\omega (x,f)= \bigcap _{n\in \mathbb {N}}\overline{\{f^k(x):k>n\}}\), where the bar denotes the closure of a set. For further information on \(\omega \)-limit sets of metric spaces, consult [9].

  3. 3.

    A function f is contractive if there exists \(0\le \alpha < 1\) and a real number r such that \(|f(x)-r|\le \alpha |x-r|\) for all \(x\in \mathbb {R}\). Notice that r is a globally attractor fixed point.

  4. 4.

    Let \(G:\mathbb {R}^n\rightarrow \mathbb {R}^m\). Assume there exists \(L\ge 0\) such that \(||G(x)-G(y)||_{\infty }\le L||x-y||_{\infty }\) for all \(x,y\in \mathbb {R}^n\), where \(||\cdot ||_{\infty }\) is the sup-norm. If \(L<1\), we will call G sup-contractive. Moreover, if \(L=1\), we call G sup-non-expansive.

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Acknowledgements

This work has been supported by the grant MTM2017-84079-P funded by MCIN/AEI/10.13039/501100011033 and by ERDF “A way of making Europe”, by the European Union.

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  1. Departamento de Matemáticas, Universidad de Murcia, Murcia, Spain

    Antonio Linero-Bas & Daniel Nieves-Roldán

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  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. University of Rhode Island, Kingston, RI, USA

    Mustafa R. S. Kulenović

  3. University of Sarajevo, Sarajevo, Bosnia and Herzegovina

    Senada Kalabušić

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Linero-Bas, A., Nieves-Roldán, D. (2023). A Survey on Max-Type Difference Equations. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_6

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