In this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions \( \textstyle\begin{cases} ({}^{C}D_{0+}^{\alpha}u)(t)+\sum_{i=1}^{m}\lambda _{i}(t)({}^{C}D_{0+}^{\alpha _{i}}u)(t)+ \sum_{j=1}^{n}\mu _{j}(t)({}^{C}D_{0+}^{\beta _{j}}u)(t)\\ \quad{}+\sum_{k=1}^{p}\xi _{k}(t)({}^{C}D_{0+}^{\gamma _{k}}u)(t)+\sum_{l=1}^{q}\omega _{l}(t)({}^{C}D_{0+}^{\delta _{l}}u)(t)\\ \quad{}+\sigma (t)u(t)+f(t,u(t))=0,\quad t\in [0,1],\\ u^{\prime \prime}(0)=u^{\prime \prime \prime}(0)=0,\qquad u^{\prime}(0)=\eta _{1}\int _{0}^{1}u(s)\,ds,\qquad u(1)=\eta _{2}\int _{0}^{1}u(s)\,ds, \end{cases} \) where $0<\delta _{1}<\delta _{2}<\cdots <\delta _{q}<1<\gamma _{1}<\gamma _{2}<\cdots <\gamma _{p}<2<\beta _{1}<\beta _{2}<\cdots <\beta _{n}<3<\alpha _{1}<\alpha _{2}<\cdots <\alpha _{m}<\alpha <4$ and $\eta _{1}+2(1-\eta _{2})\neq 0$ . Using a fixed point theorem and Banach contractive mapping principle, we obtain some existence and uniqueness results.
