Linear difference inequalities with constant coefficients with the sum equal to zero

Stevo Stević; Stevo Stević

doi:10.3934/math.20251176

AIMS Mathematics

2025, Volume 10, Issue 11: 26744-26766. doi: 10.3934/math.20251176

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Linear difference inequalities with constant coefficients with the sum equal to zero

Stevo Stević ^{1,2,3
,
, ,}

1.
Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/Ⅲ, 11000 Beograd, Serbia
2.
Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11000 Beograd, Serbia
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Received: 26 September 2025 Revised: 07 November 2025 Accepted: 12 November 2025 Published: 18 November 2025
MSC : primary 39B99, 39B62; secondary 39A06

Many difference equations of the form
$ x_{n+k} = f(x_{n+k-1}, \ldots, x_n), \quad n\in {\mathbb N}, $
where $ k\in {\mathbb N} $, model some phenomena in nature and society. The most interesting cases usually occur when the function $ f $ satisfies the condition $ f(x, \ldots, x) = x $ on its domain of definition. Because of this the difference inequalities $ x_{n+k}\le f(x_{n+k-1}, \ldots, x_n) $ and $ x_{n+k}\ge f(x_{n+k-1}, \ldots, x_n) $ are of some interest. If $ f $ is a smooth function, then it can be approximated by a linear function. Motivated by some concrete examples, here we mostly consider the sequences that satisfy the linear difference inequality
$ \begin{align*} \sum\limits_{j = 1}^ka_jx_{n+l-j}\ge 0, \quad n\in {\mathbb N}_0, \end{align*}$
where $ k\in {\mathbb N}_2 $, $ l\in {\mathbb N}_0, $ and the coefficients $ a_j\in {\mathbb R}, $ $ j = \overline{2, k-1} $, $ a_1, a_k\in {\mathbb R}\setminus\{0\}, $ satisfy the condition $ \sum_{j = 1}^ka_j = 0. $
- linear difference inequalities,
- constant coefficients,
- difference equations,
- bounded sequences
Citation: Stevo Stević. Linear difference inequalities with constant coefficients with the sum equal to zero[J]. AIMS Mathematics, 2025, 10(11): 26744-26766. doi: 10.3934/math.20251176

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Abstract

Many difference equations of the form

$ x_{n+k} = f(x_{n+k-1}, \ldots, x_n), \quad n\in {\mathbb N}, $

where $ k\in {\mathbb N} $, model some phenomena in nature and society. The most interesting cases usually occur when the function $ f $ satisfies the condition $ f(x, \ldots, x) = x $ on its domain of definition. Because of this the difference inequalities $ x_{n+k}\le f(x_{n+k-1}, \ldots, x_n) $ and $ x_{n+k}\ge f(x_{n+k-1}, \ldots, x_n) $ are of some interest. If $ f $ is a smooth function, then it can be approximated by a linear function. Motivated by some concrete examples, here we mostly consider the sequences that satisfy the linear difference inequality

$ \begin{align*} \sum\limits_{j = 1}^ka_jx_{n+l-j}\ge 0, \quad n\in {\mathbb N}_0, \end{align*}$

where $ k\in {\mathbb N}_2 $, $ l\in {\mathbb N}_0, $ and the coefficients $ a_j\in {\mathbb R}, $ $ j = \overline{2, k-1} $, $ a_1, a_k\in {\mathbb R}\setminus\{0\}, $ satisfy the condition $ \sum_{j = 1}^ka_j = 0. $

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