@article{Sun2026, 
author = {Yu Sun and Zhanbing Bai},
title = {Multiplicity results for some higher order iterative systems},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {5},
pages = {14655-14667},
keywords = {iterative system, multiplicity, positive solutions, Green function, fixed-point theorem},
url = {https://www.sciopen.com/article/10.3934/math.2026601},
doi = {10.3934/math.2026601},
abstract = {Our study was concerned with the higher order iterative system where the nonlinearities exhibited dependence on the first-order derivatives         {                                                u            i                          (              n              )                                (          x          )          +                      λ            i                                f            i                    (          x          ,                      u                          i              +              1                                (          x          )          ,                      u                          i              +              1                        ′                    (          x          )          )          =          0          ,                                                        1          ≤          i          ≤          m          ,          x          ∈          [          0          ,          1          ]          ;                                                          u                          m              +              1                                (          x          )          =                      u            1                    (          x          )          ,                          x          ∈          [          0          ,          1          ]          ,                        with two-point boundary conditions         u    i    (  0  )  =      u    i    ′    (  0  )  =  ⋯  =      u    i          (      n      −      2      )        (  0  )  =      u    i          (      r      )        (  1  )  =  0  ,where    m  ≥  2  ,  n  ≥  3  ,  i  ∈  {  1  ,  2  ,  ⋯  ,  m  },    r  ∈  {  1  ,  2  ,  ⋯  ,  n  −  2  } but fixed and        λ    i    ∈      (          0      ,      ∞        )   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.}
}