Solvability of iterative system of fractional order differential equations with nonhomogeneous boundary conditions (Q6086834)

In this paper, the authors study the following iterative system of fractional order differential equations \[ \begin{cases} D_{0+}^{\alpha}y_i(t)+\lambda_i\phi_i(t)f_i(y_{i+1}(t))=0, ~~ 1\le i\le m, ~~ t\in [0,1],\\ y_{m+1}(t)=y_1(t), ~~ t\in [0,1] \end{cases}\tag{1} \] satisfying three-point boundary conditions with a nonhomogeneous term \[ \begin{cases} y_i(0)=y'_i(0)=\cdots=y_i^{(n-2)}(0)=0,\\ a_iD_{0+}^{\beta}y_i(1)-b_iD_{0+}^{\beta}y_i(\eta)=\xi_i, ~~ 1\le i\le m, \end{cases}\tag{2} \] where \(D_{0+}^{\alpha}, D_{0+}^{\beta}\) represent the Riemann-Liouville fractional order derivatives of orders \(\alpha\) and \(\beta\) respectively, \(\alpha\in (n-1,n],\) \(\beta\in (n-3, n-2],\) \(n\ge 3,\) \(f_i\in C(\mathbb{R}^+,\mathbb{R}^+),\) \(\phi_i\in C([0,1],\mathbb{R}^+),\) \(a_i, b_i\) are real constants, \(\eta\in (0,1)\) and \(\xi_i\in (0,\infty),\) ~\(1\le i\le m.\) By using the Guo-Krasnosel'skiĭ fixed point theorem, the eigenvalue intervals are determined for the existence of positive solutions for the iterative fractional system (1)--(2).