Multiplicity results for some higher order iterative systems

Yu Sun; Zhanbing Bai; Yu Sun; Zhanbing Bai

doi:10.3934/math.2026601

AIMS Mathematics

2026, Volume 11, Issue 5: 14655-14667. doi: 10.3934/math.2026601

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Multiplicity results for some higher order iterative systems

Yu Sun ,
Zhanbing Bai ^,

College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China

Received: 06 February 2026 Revised: 01 May 2026 Accepted: 19 May 2026 Published: 25 May 2026
MSC : 34B18

Our study was concerned with the higher order iterative system where the nonlinearities exhibited dependence on the first-order derivatives
$ \left\{ \begin{aligned} &u_i^{(n)}(x) + \lambda_i f_i(x, u_{i+1}(x), u'_{i+1}(x)) = 0, \quad\quad \quad &1 \leq i \leq m, x \in [0, 1]; \\ &u_{m+1}(x) = u_1(x), &x \in [0, 1], \end{aligned}\right. $
with two-point boundary conditions
$ u_i(0) = u'_i(0) = \cdots = u_i^{(n-2)}(0) = u^{(r)}_i(1) = 0, $
where $ m\geq2, n\geq3, i\in\{1, 2, \cdots, m \} $, $ r\in \{1, 2, \cdots, n-2\} $ but fixed and $ \lambda_i \in \left(0, \infty\right) $ were constants. By giving the corresponding Green's function and its related properties, the form of the solution of the system could be obtained. Combined with the fixed-point theorem of functional type on cones due to Bai and Ge, we established novel conclusions on the existence of multiple positive solutions for higher order iterative systems. A notable feature was the involvement of first derivatives in the nonlinear terms.
- iterative system,
- multiplicity,
- positive solutions,
- Green function,
- fixed-point theorem
Citation: Yu Sun, Zhanbing Bai. Multiplicity results for some higher order iterative systems[J]. AIMS Mathematics, 2026, 11(5): 14655-14667. doi: 10.3934/math.2026601

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Abstract

Our study was concerned with the higher order iterative system where the nonlinearities exhibited dependence on the first-order derivatives

$ \left\{ \begin{aligned} &u_i^{(n)}(x) + \lambda_i f_i(x, u_{i+1}(x), u'_{i+1}(x)) = 0, \quad\quad \quad &1 \leq i \leq m, x \in [0, 1]; \\ &u_{m+1}(x) = u_1(x), &x \in [0, 1], \end{aligned}\right. $

with two-point boundary conditions

$ u_i(0) = u'_i(0) = \cdots = u_i^{(n-2)}(0) = u^{(r)}_i(1) = 0, $

where $ m\geq2, n\geq3, i\in\{1, 2, \cdots, m \} $, $ r\in \{1, 2, \cdots, n-2\} $ but fixed and $ \lambda_i \in \left(0, \infty\right) $ were constants. By giving the corresponding Green's function and its related properties, the form of the solution of the system could be obtained. Combined with the fixed-point theorem of functional type on cones due to Bai and Ge, we established novel conclusions on the existence of multiple positive solutions for higher order iterative systems. A notable feature was the involvement of first derivatives in the nonlinear terms.

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