Solvability of higher order three-point iterative systems

Abstract

In this paper, we consider an iterative system of nonlinear $n^{\text{th}}$ order differential equations: $$ y_i^{(n)}(t)+\lambda_i p_i(t)f_i(y_{i+1}(t))=0,\qquad 1\leq i\leq m,\qquad y_{m+1}(t)= y_1(t),\qquad t\in[0,1], $$ with three-point non-homogeneous boundary conditions $$ \begin{gathered} y_i(0)={y_i}'(0)=\cdots=y_i^{(n-2)}(0)=0, \\ \alpha_iy_i^{(n-2)}(1)-\beta_i y_i^{(n-2)}(\eta)=\mu_i,\qquad 1\leq i\leq m, \end{gathered} $$ where $n\geq 3,$ $\eta\in (0,1)$, $\mu_i\in (0, \infty)$ is a parameter, $f_i:\mathbb{R}^+ \rightarrow \mathbb{R}^+ $ is continuous, $p_i:[0,1] \rightarrow \mathbb{R}^+$ is continuous and $p_i$ does not vanish identically on any closed subinterval of $[0,1]$ for $1\leq i\leq m$. We express the solution of the boundary value problem as a solution of an equivalent integral equation involving kernels and obtain bounds for these kernels. By an application of Guo–Krasnosel'skii fixed point theorem on a cone in a Banach space, we determine intervals of the eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_m$ for which the boundary value problem possesses a positive solution. As applications, we provide examples demonstrating our results.