Multiplicity results for some higher order iterative systems

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

$$\{\begin{array}{rlr}& u_i^{(n)}(x)+{\lambda }_i{f}_i(x,u_{i+1}(x),u_{i+1}^{\prime }(x))=0,& 1\le i\le m,x\in [0,1];\\ & u_{m+1}(x)=u_1(x),& x\in [0,1],\end{array}$$

with two-point boundary conditions

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

where $m\ge 2,n\ge 3,i\in \{1,2,\cdots ,m\}$, $r\in \{1,2,\cdots ,n-2\}$ but fixed and ${\lambda }_i\in \left(0,\mathrm{\infty }\right)$ 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.