Blowing-up solutions of multi-order fractional differential equations with the periodic boundary condition

In this paper, we analyze the boundary value problem of a class of multi-order fractional differential equations involving the standard Caputo fractional derivative with the general periodic boundary conditions: {L(D)u(t)=f(t,u(t)),t∈[0,T],T>0,u(0)=u(T)>0,u′(0)=u′(T)>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} L(D)u(t) = f(t,u(t)),\quad t\in[0,T], T>0, \\ u(0) = u(T)>0,\qquad u'(0)=u'(T)>0, \end{cases} $$\end{document} where L(D)=∑i=0naiDSi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L(D)=\sum^{n}_{i=0}a_{i}D^{S_{i}}$\end{document}, 1≤S0<⋯<Sn−1<Sn<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1\leq S_{0}<\cdots<S_{n-1}<S_{n}<2$\end{document}, ai∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{i}\in\mathbb{R}$\end{document}, an≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{n}\neq0$\end{document}, and f:[0,T]×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}$\end{document} is a continuous operation. We get the Green’s function in terms of the Laplace transform. We obtain the existence and uniqueness of solution for the class of multi-order fractional differential equations. We investigate the blowing-up solutions to the special case f(t,u(t))=|u(t)|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(t,u(t))=|u(t)|^{p}$\end{document}, ai≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{i}\geq0$\end{document}, and give an upper bound on the blow-up time Tmax\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$T_{\mathrm{max}}$\end{document}.