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

Abstract

In this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions {(CD0+αu)(t)+∑i=1mλi(t)(CD0+αiu)(t)+∑j=1nμj(t)(CD0+βju)(t)+∑k=1pξk(t)(CD0+γku)(t)+∑l=1qωl(t)(CD0+δlu)(t)+σ(t)u(t)+f(t,u(t))=0,t∈[0,1],u″(0)=u‴(0)=0,u′(0)=η1∫01u(s)ds,u(1)=η2∫01u(s)ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} ({}^{C}D_{0+}^{\\alpha}u)(t)+\\sum_{i=1}^{m}\\lambda _{i}(t)({}^{C}D_{0+}^{\\alpha _{i}}u)(t)+ \\sum_{j=1}^{n}\\mu _{j}(t)({}^{C}D_{0+}^{\\beta _{j}}u)(t)\\\\ \\quad{}+\\sum_{k=1}^{p}\\xi _{k}(t)({}^{C}D_{0+}^{\\gamma _{k}}u)(t)+\\sum_{l=1}^{q}\\omega _{l}(t)({}^{C}D_{0+}^{\\delta _{l}}u)(t)\\\\ \\quad{}+\\sigma (t)u(t)+f(t,u(t))=0,\\quad t\\in [0,1],\\\\ u^{\\prime \\prime}(0)=u^{\\prime \\prime \\prime}(0)=0,\\qquad u^{\\prime}(0)=\\eta _{1}\\int _{0}^{1}u(s)\\,ds,\\qquad u(1)=\\eta _{2}\\int _{0}^{1}u(s)\\,ds, \\end{cases} $$\\end{document} 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.