\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{graphicx} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 152, pp. 1--19.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/152\hfil Non-existence of solutions] {Non-existence of solutions for two-point fractional and third-order\\ boundary-value problems} \author[G. L. Karakostas \hfil EJDE-2013/152\hfilneg] {George L. Karakostas} % in alphabetical order \address{George L. Karakostas \newline Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece} \email{gkarako@uoi.gr} \thanks{Submitted October 3, 2012. Published June 28, 2013.} \subjclass[2000]{34B15, 34A10, 34B27, 34B99} \keywords{Third order differential equation; two-point boundary-value problem; \hfill\break\indent fractional boundary condition; nonexistence of solutions} \begin{abstract} In this article, we provide sufficient conditions for the non-existence of solutions of the boundary-value problems with fractional derivative of order $\alpha\in(2,3)$ in the Riemann-Liouville sense \begin{gather*} D_{0+}^{\alpha}x(t)+\lambda a(t)f(x(t))=0,\quad t\in(0,1),\\ x(0)=x'(0)=x'(1)=0, \end{gather*} and in the Caputo sense \begin{gather*} ^CD^{\alpha}x(t)+f(t,x(t))=0,\quad t\in(0,1),\\ x(0)=x'(0)=0, \quad x(1)=\lambda\int_0^1x(s)ds; \end{gather*} and for the third-order differential equation $$x'''(t)+(Fx)(t)=0, \quad \text{a.e. }t\in [0,1],$$ associated with three among the following six conditions $$x(0)=0,\quad x(1)=0,\quad x'(0)=0, \quad x'(1)=0, \quad x''(0)=0, \quad x''(1)=0.$$ Thus, fourteen boundary-value problems at resonance and six boundary-value problems at non-resonanse are studied. Applications of the results are, also, given. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{application}[theorem]{Application} \allowdisplaybreaks \section{Introduction} The problem of existence of solutions of fractional differential equations and general third-order differential equations, satisfying two-point boundary conditions, has been extensively discussed in the literature, see, e.g., \cite{3, CAB, F, FL, MG, inp, LKJ, 4, MA, 5, OR, 6,TROY, yao, yao1, 7, 8} and the references therein. In this work we are dealing with non-existence of solutions and this, because the two problems (namely, existence and non-existence) are equally important in the theory of differential equations. To our knowledge, the problem of non-existence for such boundary-value problems has not been studied sufficiently. Indeed, only a few results have been given on it, and not in a systematic research. See, e.g., \cite{MES, glk, wcz, yang, zwg}. Let $I$ be the interval $[0,1]$. We are going to give sufficient conditions for the non-existence of solutions of the two boundary-value problems \begin{gather}\label{f1} D_{0+}^{\alpha}x(t)+\lambda a(t)f(x(t))=0,\quad 0 0$, and$f:[0,+\infty)\to [0,+\infty)$is continuous. The symbol$D_{0+}^{\alpha}u$represents the Riemann-Liouville fractional derivative of the continuous function$u:I\to\mathbb{R}$of order$\alpha\in(2,3)$, (see, e.g., \cite{WANG}), i.e. the quantity defined by $$D_{0+}^{\alpha}u(t) =\frac{1}{\Gamma(n-\alpha)}\Big(\frac{d}{dt}\Big)^n \int_0^t\frac{u(s)}{(t-s)^{\alpha-n+1}}ds,\quad n=\lfloor{\alpha}\rfloor+1.$$ \begin{theorem} Assume that$f$satisfies the condition $$\label{g} L_f:=\sup_{|u|>0}\frac{|f(u)|}{|u|}< \frac{\Gamma(\alpha)}{\lambda\int_0^1s(1-s)^{\alpha-2}a(s)ds}.$$ Then problem \eqref{f1}-\eqref{f2} does not admit solutions. \end{theorem} \begin{proof} According to \cite{MES}, the problem is equivalent to the operator equation \eqref{oe}, where the operator$T$has the integral form \eqref{v} with the Green's function$G$being defined by the type $$G(t,s):=\begin{cases} \frac{(1-s)^{\alpha-2}t^{\alpha-1}}{\Gamma(\alpha)}a(s), \quad 0\leq t\leq s\leq 1,\\ \Big[\frac{(1-s)^{\alpha-2}t^{\alpha-1}}{\Gamma(\alpha)}-\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\Big]a(s), \quad 0\leq s\leq t\leq 1. \end{cases}$$ Here$\Gamma(\alpha)$is the gamma function at$\alpha$. It is not hard to see that$G$takes positive values and it satisfies $$G(t,s)\leq G(1,s), \quad (t,s)\in [0,1]\times[0,1].$$ Therefore, the result will follow from Theorem \ref{t1} and inequality \eqref{l}, when we know that the quantity in the right side of relation \eqref{g} is less than or equal to$1/K_{\eqref{f1}-\eqref{f2}}. So, we have to prove this fact. Indeed, we have \begin{align*} K_{\eqref{f1}-\eqref{f2}} &=\lambda \sup_{t\in I}\int_0^1G(t,s)a(s)ds\\ &=\lambda \int_0^1G(1,s)a(s)ds \\ &=\lambda\frac{1}{\Gamma(\alpha)} \int_0^1\Big[(1-s)^{\alpha-2} -(1-s)^{\alpha-1}\Big]a(s)ds\\ & = \lambda\frac{1}{\Gamma(\alpha)} \int_0^1s(1-s)^{\alpha-2}a(s)ds, \end{align*} from which the result follows. \end{proof} \begin{application}\rm Consider the values $$\alpha=2.7, \quad a(t):=2t+3, \quad f(u):=\frac{8u^2+u}{u+1}(4+\sin(u)),$$ as they appear in \cite{MES}. Then we have \begin{align*} \int_0^1s(1-s)^{\alpha-2}a(s)ds &=\int_0^1(1-s)^{\alpha-2}(2s^2+3s)ds\\ &=\int_0^1s^{\alpha-2}(2(1-s)^2+3(1-s))ds\\ &=\int_0^1s^{\alpha-2}(5-7s+2s^2)ds =\frac{3\alpha+7}{(\alpha-1)\alpha(\alpha+1)}. \end{align*} and therefore it follows that $$\frac{\Gamma(\alpha)}{\lambda\int_0^1s(1-s)^{\alpha-2}a(s)ds} =\frac{(\alpha-1)\Gamma(\alpha+2)}{\lambda(3\alpha+7)} \approx\frac{26.23339972}{15.1\lambda}.$$ Since it holds $$\frac{|f(u)|}{|u|}=\frac{8u+1}{u+1}(4+\sin(u))\leq 40,$$ we have, also, $$\frac{\|f(u(\cdot))\|}{\|u(\cdot)\|}\leq 40,$$ which, due to Theorem \ref{t1}, shows that, if the parameter\lambda$is chosen such that $$\lambda<\frac{26.23339972}{40\times 15.1}=\frac{26.23339972}{604}=0.04343278,$$ then the problem has no (any, and not necessarily positive) solution. This upper bound of the parameter$\lambda$agrees with the value of the parameter suggested in \cite{MES}. \end{application} \section{Nonexistence for the fractional BVP \eqref{f3}-\eqref{f4}}\label{fr2} Here we discuss non-existence for the fractional boundary-value problem \eqref{f3}-\eqref{f4}, studied in \cite{CW}, where, again, (the Krasnoselskii's fixed point theorem in cones is applied and) the existence of solutions is investigated. It is assumed that$^CD^{\alpha}u$is the Caputo fractional derivative of the continuous function$u:I\to\mathbb{R}$at the real number$\alpha\in(2,3)$defined by $$^CD^{\alpha}u(t):=\frac{1}{\Gamma(n-\alpha)}\int_0^t(t-s)^{(\alpha-1)}u(s)ds, \quad n=\lfloor{\alpha}\rfloor+1.$$ Also, the function$f :[0,1]\times[0,\to[0,\infty)$is continuous and$0<\lambda<2$. \begin{theorem} Assume that$f$satisfies the condition $$\label{g1} L_f:=\sup_{t\in{I}}\sup_{|u|>0}\frac{|f(t,u)|}{|u|}< \frac{(2-\lambda)(\alpha-2)\Gamma(\alpha+2)}{2\alpha(\alpha-1)}.$$ Then problem \eqref{f3}-\eqref{f4} does not admit solutions. \end{theorem} \begin{proof} According to \cite{CW}, the problem is equivalent to the operator equation \eqref{oe}, where$T$has the integral form \eqref{v}, with the Green's function$G$being given by the type $$G(t,s):=\begin{cases} \frac{2t(1-s)^{\alpha-1}(\alpha-\lambda+\lambda s)-(2-\lambda)\alpha(t-s)^{\alpha-1}}{(2-\lambda)\Gamma(\alpha+1)}, \quad 0\leq s\leq t\leq 1,\\ \frac{2t(1-s)^{\alpha-1}(\alpha-\lambda+\lambda s)}{(2-\lambda)\Gamma(\alpha+1)}, \quad 0\leq t\leq s\leq 1. \end{cases}$$ Due to \cite[Lemmas 2.3 and 2.4]{CW}, the kernel$G$is a nonnegative function and it satisfies the inequality $$G(t,s)\leq\frac{2\alpha}{\lambda(\alpha-2)}G(1,s),$$ for all$s, t, \in[0,1]$and$\lambda\in(0,2)$. Hence, the result will follow from Theorem \ref{t1} and inequality \eqref{l}, when we know that the quantity in the right side of relation \eqref{g1} is less than or equal to the$1/K_{\eqref{f3}-\eqref{f4}}. To prove this fact observe that \begin{align*} {K_{\eqref{f3}-\eqref{f4}}} &=\sup_{t\in I}\int_0^1G(t,s)ds\\ &\leq \frac{2\alpha}{\lambda(\alpha-2)}\int_0^1G(1,s)ds =\frac{2\alpha(\alpha-1)}{(2-\lambda)(\alpha-2)\Gamma(\alpha+2)}. \end{align*} This completes the proof of the theorem. \end{proof} \section{Nonexistence for third-order BVPs at non-resonance}\label{nres} In this section we give information about non-existence for the third-order differential equation \eqref{eq} subject to one of the boundary conditions \eqref{p1}, \eqref{p2},\cdots$, \eqref{p14}. \begin{theorem} The boundary-value problems (at non-resonance) \eqref{eq}-{\eqref{p1}}, \eqref{eq}-{\eqref{p2}}, \dots ,\eqref{eq}-{\eqref{p14}}, do not have solutions provided that \eqref{l} is satisfied, where the number$L_F$is such as the following tables shows: \begin{center} \begin{tabular}{|c|c|c|c|c|c|} \hline &\eqref{eq}-{\eqref{p1}}&\eqref{eq}-{\eqref{p2}}&\eqref{eq}-{\eqref{p3}} &\eqref{eq}-{\eqref{p4}}\\ \hline$0\delta$such that $$\label{a} |f(u)|\leq L_f|u|\quad\text{and}\quad |f'(u)|\geq\delta,\quad u\in\mathbb{R}.$$ \end{itemize} Clearly, the latter implies that either$f'(u)\geq \delta$, for all$u$, or$f'(u)\leq -\delta$, for all$u$. Our main theorem reads as follows. \begin{theorem}\label{t2} The boundary-value problems \eqref{qq}-{\eqref{p15}}, \dots , \eqref{qq}-{\eqref{p20}} (at resonance) do not admit solutions when the parameters$L, \delta$satisfy the relations as in the following tables: \begin{center} \begin{tabular}{|c|c|c|c|} \hline &\hskip .2in \eqref{qq}-{\eqref{p15}}& \eqref{qq}-{\eqref{p16}}&\eqref{qq}-{\eqref{p17}} \\ \hline$\delta