Existence and uniqueness of solutions for multi-order fractional differential equations with integral boundary conditions

Abstract

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