\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 297, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/297\hfil Convolutions and Green's functions]
{Convolutions and Green's functions
for two families of boundary value problems
for fractional differential equations}

\author[P. W. Eloe, J. T. Neugebauer \hfil EJDE-2016/297\hfilneg]
{Paul W. Eloe, Jeffrey T. Neugebauer}

\address{Paul W. Eloe \newline
Department of Mathematics,
University of Dayton, Dayton, Ohio 45469, USA}
\email{peloe1@udayton.edu}

\address{Jeffrey T. Neugebauer \newline
Department of Mathematics and Statistics,
Eastern Kentucky University,
Richmond, Kentucky  40475, USA}
\email{Jeffrey.Neugebauer@eku.edu}

\thanks{Submitted August 31, 2016 Published November 22, 2016.}
\subjclass[2010]{26A33, 34A08, 34A40, 26D20}
\keywords{Fractional boundary value problem;
 fractional differential inequalities}

\begin{abstract}
 We consider families of two-point boundary value problems for fractional
 differential equations where the fractional derivative is assumed to
 be the Riemann-Liouville fractional derivative.  The problems considered
 are such that appropriate differential operators commute and the problems
 can be constructed as nested boundary value problems for lower order
 fractional differential equations.  Green's functions are then constructed
 as convolutions of lower order Green's functions.  Comparison theorems are
 known for the Green's functions for the lower order problems and so, we
 obtain analogous comparison theorems for the two families of higher order
 equations considered here.  We also pose a related open question for a
 family of Green's functions that do not apparently have convolution
 representations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Sign properties of Green's functions for boundary value problems and
maximum principles for boundary value problems are very closely related
\cite{ecubo, ek, er}.  In recent years, and possibly beginning with \cite{bailu},
many authors have obtained sign properties of a specific Green's function
for a boundary value problem for a fractional differential equation in order
to employ a fixed point theorem to obtain sufficient conditions for the
existence of positive solutions.

In \cite{eln}, the authors initiated the study of families of two-point
boundary value problems for a fractional differential equation and obtained
both sign properties and comparison theorems for the associated Green's functions.
The results in \cite{eln} are motivated by the results for ordinary differential
equations in, say, \cite{eh,ek,er}.  A primary application of this type of
analysis is to the $\mu_{0}-$positivity of operators defined on a cone in a
Banach space.  We refer the reader to the early development the theory of
 $\mu_{0}-$positivity of operators defined on a cone in a Banach space in
the definitive works of Krasnosel'ski\u{i} \cite{k} and Krein and Rutman \cite{kr};
for applications to higher order ordinary differential equations we refer the
reader to \cite{kt} or to \cite{eh}.  That application is not pursued in this
article.

The technique in \cite{eln} is naive; only one boundary condition is stacked
at the right, and thus, Green's functions are constructed explicitly.
Regardless, it serves as a beginning to carry the classical results for ordinary
differential equations over to classes of fractional differential equations.
For example, recently \cite{en1, en2, en3}, the authors have employed comparison
results to compare eigenvalues and characterize principal eigenvalues for
boundary value problems for fractional differential equations.  Such work has
been extended to the Caputo fractional calculus \cite{hk} and the discrete
fractional calculus \cite{hy}.

In this article, we continue to consider families of two-point boundary value
problems for linear fractional differential equations.  The purpose of this
 work is to develop methods to allow for more than one boundary conditions
stacked at the right.  The primary tool is to construct Green's functions
as convolutions of Green's functions for lower order boundary value problems.
We are currently limited to consider problems for which the lower ordered
fractional derivatives commute.  Under the assumption that the lower order
fractional derivatives commute, sign properties and comparison theorems will
be established that are anticipated by the classical results for ordinary
differential equations.  We shall also consider a specific example in which
the convolutions methods proposed here fail and numerical experiments indicate
that anticipated comparison theorems are valid.

The paper is organized as follows: In section 2, we shall provide the basic
definitions and properties of fractional calculus which are employed.
We shall also state two comparison theorems proved in \cite{eln} for the sake
of exposition.  In section 3, we consider a specific problem and introduce
the convolution method.  In section 4, we develop the comparison theorems
for two families of boundary value problems.  We close in section 5
with an example to show that the expected comparison results may continue
to be valid in the case when the appropriate lower order fractional
derivatives do not commute.

\section{Preliminaries}

We recall the definitions of the Riemann-Liouville fractional integral and
the Riemann-Liouville fractional derivative.  For the sake of exposition,
the initial point throughout the article is $a=0$.

\begin{definition} \label{def2.1} \rm
Let $0<\alpha$.  For $t>0$, the $\alpha$-th Riemann-Liouville fractional
integral of a function, $u$, is defined by
$$
I^{\alpha}_{0+} u(t)= \frac{1}{\Gamma(\alpha)} \int^{t}_{0} (t-s)^{\alpha-1} u(s)ds,
$$
provided the right-hand side exists. For $\alpha =0$, define
$I^{\alpha}_{0+}$ to be the identity map.  Moreover, let $n$ denote a positive
integer and assume $n - 1 < \alpha \leq n$.
The $\alpha$-th Riemann-Liouville fractional derivative of the function
$u : [0,\infty) \longrightarrow \mathbb{R}$, denoted
$$
D^{\alpha}_{0+} u(t) = \frac{1}{\Gamma( n- \alpha)}
\frac{d^{n}}{dt^{n}} \int^{t}_{0+} (t - s)^{n -\alpha - 1} u(s) ds
= D^{n} I^{n - \alpha}_{0} u(t),
$$
provided the right-hand side exists.
Again for $\alpha =0$, define $D^{\alpha}_{0+}$ to be the identity map.
\end{definition}

We shall employ a standard notation, $D^{\alpha}_{0+}$,
to denote fractional derivatives for noninteger $\alpha$ and $D^{j}$
to denote classical derivatives in the case, $\alpha = j$ is a nonnegative integer.
 We shall require only a few well known properties in fractional calculus
which we state below.  We refer the reader to the monograph of
 Diethelm \cite{Diethelm} or the recent article \cite{eln} for basic
definitions and properties.
\begin{gather}\label{p1}
I^{\alpha_1}_{0+} I^{\alpha_2}_{0+} u(t)
= I^{\alpha_1+\alpha_2}_{0+} u(t) = I^{\alpha_2}_{0+} I^{\alpha_1}_{0+} u(t),
\quad \text{if } \alpha_1,\alpha_2>0, \\
\label{p2}
D^{\alpha_1}_{0+} I^{\alpha_2}_{0+} u(t)= I^{\alpha_2-\alpha_1}_{0+} u(t),
 \quad \text{if } 0\le \alpha_1\le \alpha_2, \\
\label{p3}
D^{\alpha}_{0+} I^{\alpha}_{0+} u(t) = u(t), \quad \text{if } 0\le \alpha , \\
\label{p4}
I^{\alpha}_{0+} D^{\alpha}_{0+} u(t)= u(t)
+ \sum^{n}_{i=1} c_{i} t^{\alpha-n+(i-1)}, \quad \text{if } 0\le \alpha ,
\end{gather}
We also require the power rules \cite{Diethelm,eln},
\begin{equation}\label{p5}
I_{0+}^{\alpha_2}t^{\alpha_1}
=\frac{\Gamma (\alpha_1 +1)}{\Gamma (\alpha_2 +\alpha_1 +1)}t^{\alpha_2
+\alpha_1},\quad \text{if } \alpha_1 > -1, \; \alpha_2 \ge 0,
\end{equation}
and
\begin{equation}\label{p6}
D_{0+}^{\alpha_2}t^{\alpha_1}
=\frac{\Gamma (\alpha_1 +1)}{\Gamma (\alpha_1+1 -\alpha_2)}t^{\alpha_1-\alpha_2},
\quad \text{if } \alpha_1 > -1, \; \alpha_2 \ge  0.
\end{equation}
In \eqref{p6}, it is assumed that $\alpha_2 -\alpha_1$ is not a positive integer.
If $\alpha_2 -\alpha_1$ is a positive integer, then the right hand side of \eqref{p6}
vanishes.  To see this, one can appeal to the convention that
$\frac{1}{\Gamma (\alpha_1+1 -\alpha_2)}=0$ if $\alpha_2 -\alpha_1$
is a positive integer, or one can perform the calculation on the left hand
side and calculate
$$
D^n t^{n-(\alpha_2-\alpha_1)}=0.
$$

In \cite{eln}, the authors obtained comparison results for boundary value
problems for lower order fractional differential equations.
For the sake of self-containment, we summarize those results here.

Let $2\le n$ denote an integer and let $n-1<\bar{\alpha} \le n$.
For each $0<b$, $0\le \beta\le n-1$, consider a boundary value problem (BVP)
for the fractional differential equation of the form
\begin{equation}\label{2e1}
D^{\bar{\alpha}}_{0+} u+h(t)=0, \quad 0<t<b,
\end{equation}
with two-point boundary conditions of the form
\begin{equation}\label{2e2}
u^{(i)}(0)=0, \quad  i=0,\dots n-2, \quad D_{0+}^{\beta}u(b)=0.
\end{equation}

It is shown in \cite{eln} that the Green's function of \eqref{2e1}--\eqref{2e2}
has the form
\begin{equation}\label{Galphabar}
G(\bar{\alpha} , \beta , b;t,s)
= \begin{cases}
\frac{t^{\bar{\alpha}-1} (b-s)^{\bar{\alpha}-1-\beta}}{b^{\bar{\alpha}-1-\beta}
  \Gamma(\bar{\alpha} )} -\frac{(t-s)^{\bar{\alpha}-1}}{\Gamma(\bar{\alpha} )},
& 0\le s\leq t\le b, \\[4pt]
\frac{t^{\bar{\alpha}-1} (b-s)^{\bar{\alpha}-1-\beta}}{b^{\bar{\alpha}-1-\beta}
\Gamma(\bar{\alpha} )}, & 0\le t\leq s\le b
\end{cases}
\end{equation}
and the solution of \eqref{2e1}, \eqref{2e2} is
$u(t)=\int_{0}^{b}G(\bar{\alpha} ,\beta ,b;t,s)h(s)ds$.

\begin{theorem}[\cite{eln}] \label{newt21}
If $0\le \beta_1<\beta_2\le n-1$, then
\begin{equation}\label{newe23}
0< G(\bar{\alpha} , \beta_1 , b;t,s)< G(\bar{\alpha} , \beta_2 , b;t,s), \quad (t,s)\in (0,b)\times (0,b).
\end{equation}
\end{theorem}

If $n\ge 2$ the proof Theorem \ref{newt21} is readily extended to obtain
the following result.

\begin{theorem}[\cite{eln}] \label{t21}
Let $j\in \{ 0, \dots ,n-2\}$, $i\in \{0,\dots j\}$.
If $0\le \beta_1<\beta_2\le n-1$, then
\begin{equation}\label{e23}
0< \Big( \frac{\partial^{i}}{\partial t^{i}}\Big)G(\bar{\alpha} ,
 \beta_1 , b;t,s)< \Big( \frac{\partial^{i}}{\partial t^{i}}\Big)G(\bar{\alpha} ,
 \beta_2 , b;t,s), \quad (t,s)\in (0,b)\times (0,b).
\end{equation}
\end{theorem}

Theorems \ref{newt21} and \ref{t21} compare Green's functions as functions of
$\beta $. The following theorem compares Green's functions as functions of $b$.

\begin{theorem}[\cite{eln}] \label{t22}
Assume $0<b_1<b_2$.  If $0\le \beta < \bar{\alpha}-1$, then
\begin{equation}\label{e24}
0< G(\bar{\alpha} , \beta , b_1;t,s)< G(\bar{\alpha} , \beta , b_2;t,s), 
 \quad (t,s)\in (0,b_1)\times (0,b_1),
\end{equation}
and if $\bar{\alpha}-1<\beta \le n-1$, then
\begin{equation}\label{e25}
G(\bar{\alpha} , \beta , b_1;t,s)> G(\bar{\alpha} , \beta , b_2;t,s)>0, 
 \quad (t,s)\in (0,b_1)\times (0,b_1).
\end{equation}
If $\beta = \bar{\alpha}-1$, then $G(\bar{\alpha} , \bar{\alpha}-1 , b;t,s)$ 
is independent of $b$ on $(0,b)\times (0,b)$.
\end{theorem}

\section{Green's functions as convolutions}

We begin with a specific family of two-point boundary value problems.  
Let $b>0$. Assume $3<\alpha \le 4$ and assume $0\le \beta \le 1 $. 
Consider a two-point BVP for a fractional differential equation of the form
\begin{gather}\label{3e1}
D^{\alpha}_{0+} u(t)+h(t)=0, \quad  0<t<b, \\
\label{3e2}
 u(0)= 0, \quad D^{\beta}_{0+}u(b)=0, \quad D^{\alpha -2}_{0+} u(0)= 0,  \quad  
 D^{\alpha -2}_{0+}  u(b)=0.
\end{gather}
We shall construct the Green's function, $G(\alpha, \beta ,b; t,s)$, 
with two strategies, one by direct computation and another by a convolution 
of lower order Green's functions.  We shall also produce a calculation to 
verify the two constructions are equivalent.

To produce a direct computation, apply the operator, $I^{\alpha}_{0+}$ to 
\eqref{3e1}, employ \eqref{p4} and
\begin{equation}\label{3firstformofu}
u(t)+c_1 t^{\alpha -4} +c_2 t^{\alpha -3} + c_{3} t^{\alpha -2} 
+ c_4 t^{\alpha -1} + I^{\alpha }_{0+} h(t) =0.
\end{equation}
Since $u(0)=0$, $c_1 =0$ because of the singularity of $t^{\alpha -4}$ at $t=0$, 
and $D^{\alpha -2}_{0} u(0)=0$ implies $c_{3}=0$.
Thus,
\begin{equation*}
u(t)+ c_2 t^{\alpha -3} +  c_4 t^{\alpha -1} + I^{\alpha }_{0+} h(t) =0.
\end{equation*}

Apply the boundary conditions, $D^{\beta}_{0+}u(b)=0$ and $D^{\alpha -2}_{0+}u(b)=0$ 
(and the power rule) to obtain the system of equations
\begin{gather*}
c_2 b^{\alpha -3-\beta}\frac{\Gamma (\alpha -2)}{\Gamma (\alpha -2-\beta )} 
+  c_4 b^{\alpha -1-\beta }\frac{\Gamma (\alpha )}{\Gamma (\alpha -\beta )} 
+ I^{\alpha -\beta}_{0+} h(b)=0, \\
c_4 \Gamma (\alpha )b+ I^2_{0+} h(b)=0.
\end{gather*}
Thus, $c_2$ and $c_4$ are explicitly obtained and
\begin{equation} \label{3coefficients}
\begin{gathered}
c_2= \frac{\Gamma (\alpha -2-\beta )}{b^{\alpha - 3- \beta}
\Gamma (\alpha -2 )}
\Big( \frac{b^{\alpha -2-\beta}}{\Gamma (\alpha -\beta )}
I_{0+}^2h(b)-I_{0+}^{\alpha -\beta}h(b)\Big),\\
c_4= -\frac{1}{\Gamma (\alpha )b}I^2_{0+} h(b),
\end{gathered}
\end{equation}
and \eqref{3firstformofu} reduces to
\begin{equation*}
u(t)+ c_2 t^{\alpha -3} +  c_4 t^{\alpha -1} + I^{\alpha }_{0} h(t) =0,
\end{equation*}
where $c_2$ and $c_4$ are given in \eqref{3coefficients}.
Using \eqref{3coefficients} we  define
$$
g(\alpha ,\beta ,b;t,s)
=\frac{t^{\alpha -1}(b-s)}{\Gamma (\alpha )b}
+\frac{\left(-b^{\alpha -2-\beta}(b-s)+(b-s)^{\alpha -1-\beta}\right)
\Gamma (\alpha -2-\beta )t^{\alpha -3}}{b^{\alpha -3-\beta}
\Gamma (\alpha -2 )\Gamma (\alpha -\beta)}.
$$
Then the Green's function of BVP \eqref{3e1}, \eqref{3e2}, is
\begin{equation}\label{3Gconstruction}
G(\alpha , \beta ,b;t,s)
=\begin{cases}
g(\alpha ,\beta ,b;t,s), & 0\le t<s\le b,\\
g(\alpha ,\beta ,b;t,s)-\frac{(t-s)^{\alpha -1}}{\Gamma(\alpha )},
& 0\le s<t\le b,
\end{cases}
\end{equation}
and the solution $u$ of  \eqref{3e1}, \eqref{3e2}, has the form
$$
u(t)=\int_{0}^{1}G(\alpha , \beta, b; t,s)h(s)ds.
$$

We now construct $G(\alpha , \beta, b; t,s)$ as a convolution of two Green's 
functions of lower order BVPs.
Consider a change of variable, $v(t)=D^{\alpha -2}_{0+} u(t) $. 
Note that $3<\alpha \le 4$ implies $1<\alpha -2 \le 2$ and
$$
D^2 v(t)= D^2D^{\alpha -2}_{0+} u(t)
=D^2D^2I_{0+}^{4-\alpha }u(t)=D^4I_{0+}^{4-\alpha }u(t)
= D^{\alpha}_{0+} u(t).
$$
Using only the boundary conditions 
$D^{\alpha -2}_{0+} u(0)= 0,  D^{\alpha -2}_{0+}  u(b)=0$ 
from \eqref{3e2}, it is the case that $v$ satisfies a conjugate or Dirichlet 
boundary value problem for an ordinary differential equation,
\begin{gather}\label{2ode}
v''(t)+h(t)=0, \quad  0<t<b, \\
\label{2conj}
 v(0)= 0, \quad v(b)=0.
\end{gather}
Thus,
\begin{equation}\label{vrep}
v(t)=\int_{0}^{b}G_{\rm conj}(b;t,s)h(s)ds, \quad 0\le t\le b,
\end{equation}
where $G_{\rm conj}(b;t,s)$ is well known and has the form
\begin{equation}\label{Gconj}
G_{\rm conj} (b;t,s) =\begin{cases} \frac{t(b-s)}{b}, & 0\le t<s \le b,\\[4pt]
\frac{s(b-t)}{b}, & 0\le s<t\le b,
\end{cases}
\end{equation}

The function $u$ satisfies a two-point BVP for a fractional differential 
equation of the form
\begin{gather}\label{2fracde}
D^{\alpha -2}_{0+} u(t)= v(t), \quad  0<t<b, \\
\label{temporaryconditions}
u(0)= 0, \quad D_{0+}^{\beta}u(b)=0.
\end{gather}
 The Green's function, $G(\alpha-2, \beta ,b;t,s)$,  
(with $\bar{\alpha}=\alpha -2$) is given by \eqref{Galphabar} for 
 \eqref{2fracde}, \eqref{temporaryconditions} and has the form
\begin{equation}\label{Galpha-2}
G(\alpha -2, \beta , b;t,s) 
= \begin{cases} 
\frac{t^{\alpha-3} (b-s)^{\alpha-3-\beta}}{b^{\alpha-3-\beta} \Gamma(\alpha -2)} 
-\frac{(t-s)^{\alpha-3}}{\Gamma(\alpha -2)}, & 0\le s\leq t\le b, \\[4pt]
\frac{t^{\alpha-3} (b-s)^{\alpha-3-\beta}}{b^{\alpha-3-\beta} \Gamma(\alpha -2)},
 & 0\le t\leq s\le b. 
\end{cases}
\end{equation}
Thus,
$$
u(t)=\int_{0}^{b}G(\alpha -2, \beta ,b;t,s)(-v(s))ds, \quad 0\le t\le b.
$$
Since $v$ has the form \eqref{vrep},
\begin{align*}
u(t) &=\int_{0}^{b}G(\alpha -2, \beta ,b;t,s)
 \Big(-\int_{0}^{b}G_{\rm conj}(b;s,r)h(r)dr\Big)ds, \quad 0\le t\le b, \\
    &=\int_{0}^{b}\Big(-\int_{0}^{b}G(\alpha -2, \beta ,b;t,s)G_{\rm conj}
 (b;s,r)ds\Big)h(r)dr
\end{align*}
and
\begin{equation}\label{urep}
u(t)=\int_{0}^{b}G(\alpha , \beta ,b;t,s)h(s)ds, \quad 0\le t\le b,
\end{equation}
where $G(\alpha , \beta ,b;t,s)$ is the Green's function for the original 
$\alpha$ fractional order BVP \eqref{3e1}, \eqref{3e2}; in particular,
\begin{equation}\label{2Galpha}
G(\alpha , \beta ,b;t,s) = -\int_{0}^{b}G(\alpha -2 , \beta ,b;t,r)
G_{\rm conj}(b;r,s)dr, \quad (t,s)\in [0,b]\times [0,b].
\end{equation}

To gain confidence that the two constructions are valid, we show the equivalence 
of \eqref{3Gconstruction} and \eqref{2Galpha} in the case that $\beta =0$.  
To that end, recall a calculation that employs the special beta function.  
Make a change of variable, $t=\tau +s(x-\tau)$ to calculate
\begin{align*}
&\int_{\tau}^{x}(x-t)^{m-1}(t-\tau)^{n-1}dt \\
&=(x-\tau)^{m+n-1}\int_{0}^{1}(1-s)^{m-1}s^{n-1}ds=(x-\tau)^{m+n-1}B(m,n)
\end{align*}
where $B(m,n)$ denotes the special beta function.  Thus,
$$
\int_{\tau}^{x}(x-t)^{m-1}(t-\tau)^{n-1}dt 
= (x-\tau)^{m+n-1}\frac{\Gamma (m)\Gamma (n)}{\Gamma (m+n)}.
$$
Rewrite $G_{\rm conj} (b;t,s)$ in the form
\begin{equation*}
G_{\rm conj} (b;t,s) 
=\begin{cases} 
\frac{t(b-s)}{b}, & 0\le t<s \le b,\\
\frac{t(b-s)}{b}-(t-s), & 0\le s<t\le b,
\end{cases}
\end{equation*}
Then for $t<s$, write $-\int_{0}^{b}G(\alpha -2 ,0 ,b;t,r)G_{\rm conj}(b;r,s)dr$ as
\begin{align*}
&-\int_{0}^{b}G(\alpha -2 ,0,b;t,r)G_{\rm conj}(b;r,s)dr \\
&=-\frac{t^{\alpha -3}}{b^{\alpha -3}\Gamma (\alpha -2 )}
 \int_{0}^{b}(b-r)^{\alpha -3}rdr \frac{b-s}{b}\\
&\quad +\frac{1}{\Gamma (\alpha -2)}\int_{0}^{t}(t-r)^{\alpha -3}rdr
 \frac{b-s}{b}+\frac{t^{\alpha -3}}{b^{\alpha -3}\Gamma (\alpha -2 )}
 \int_{s}^{b}(b-r)^{\alpha -3}(r-s)dr\\
&=\frac{-1}{\Gamma (\alpha )}\Big(\Big(b(b-s)
 -\frac{(b-s)^{\alpha -1}}{b^{\alpha -3}}\Big)t^{\alpha -3}
 -\frac{(b-s)}{b}t^{\alpha -1}\Big),
\end{align*}
and note that the three terms produced here match the three terms 
that are summed to produce $g(\alpha, 0, b; t,s)$ for $t<s$.

Now we use the convolution representation \eqref{2Galpha} and extend 
comparison theorems for Green's functions in \cite{eln} 
(see Theorems \ref{newt21} and \ref{t22}) for boundary value problems 
with precisely one boundary condition specified at the right.

\begin{theorem}\label{3t1}
If $0\le \beta_1 <\beta_2\le 1$, then
\begin{equation}\label{comparison}
0>G(\alpha , \beta_1, b;t,s)>G(\alpha ,\beta_2, b;t,s), \quad 
(t,s)\in (0,b)\times (0,b).
\end{equation}
\end{theorem}

\begin{proof}
Using \eqref{newe23}, if $0\le \beta_1 <\beta_2\le 1$, then
$$
0<G(\alpha -2, \beta_1, b;t,s)<G(\alpha -2,\beta_2, b;t,s), \quad 
(t,s)\in (0,b)\times (0,b).
$$
Since
$$
0<G_{\rm conj}(b;t,s), \quad (t,s)\in (0,b)\times (0,b),
$$
the result follows immediately from the representation in \eqref{2Galpha}.
\end{proof}

\begin{theorem}\label{3t2}
Assume $0<b_1<b_2$. If $0\le \beta<\alpha -3$, then
\begin{equation}\label{bmonup}
0>G(\alpha ,\beta ,b_1;t,s)>G(\alpha , \beta ,b_2;t,s), \quad 
(t,s)\in (0,b_1)\times (0,b_1).
\end{equation}
\end{theorem}

\begin{proof}
Applying  \eqref{e24}, if $0\le \beta<\alpha -3$, then
\begin{equation*}
0<G(\alpha -2,\beta ,b_1;t,s)<G(\alpha -2, \beta ,b_2;t,s), \quad 
(t,s)\in (0,b_1)\times (0,b_1).
\end{equation*}
Since $0<G_{\rm conj}(b;t,s)$ and 
$0<\frac{\partial}{\partial b}G_{\rm conj}(b;t,s)$ for 
$(t,s)\in (0,b_1)\times (0,b_1)$ the comparison in Theorem \eqref{3t2} follows.
\end{proof}

\section{Two families of boundary conditions}

In this section, we shall employ the convolution construction and consider 
two families of boundary value problems for a higher order fractional 
differential equation.  The first family we consider is motivated by the 
two-point Lidstone boundary value problem for ordinary differential equations 
\cite{aw, lidstone}.

Assume $n\in\mathbb{N}$, $2n-1<\alpha\le 2n$, $0\le \beta\le 1$ and consider the BVP
\begin{gather}\label{5e1}
D^{\alpha}_{0+}u(t) +h(t)=0, \quad 0<t<b, \\
\label{5e2}
\begin{gathered}
 u(0)= 0, \quad D^{\beta}_{0+}u(b)=0, \quad
 D^{\alpha -2l}_{0+} u(0)= 0,  \\
   D^{\alpha -2l}_{0+}  u(b)=0, \quad  l=1,\dots,n-1.
\end{gathered}
\end{gather}
Denote by $G_n(\alpha ,\beta ,b;t,s)$ the Green's function for 
\eqref{5e1}, \eqref{5e2}. And so, if $3<\alpha \le 4$, then
$$
G_2(\alpha ,\beta ,b;t,s)=G(\alpha ,\beta ,b;t,s)
$$
is given by \eqref{2Galpha}.

Inductively, we construct functions $G_{n-k}(\alpha -2k,\beta ,b;t,s)$ by
\begin{equation}\label{inductiveconstruction}
G_{n-k}(\alpha -2k, \beta , b;t,s)
=-\int^b_0 G_{n-(k+1)}(\alpha-2(k+1), \beta ,b;t,r)G_{\rm conj}(b;r,s)ds,
\end{equation}
$n-k=3, \dots ,n-1$. It then follows that the Green's function 
$G_n(\alpha ,\beta ,b;t,s)$ the Green's function for \eqref{5e1}, \eqref{5e2} 
is of the form
$$
G_n(\alpha, \beta , b;t,s)=-\int^b_0 G_{n-1}(\alpha-2, \beta ,b;t,r)
G_{\rm conj}(b;r,s)ds,
$$
where $G_{n-1}(\alpha -2,\beta ,b;t,s)$ is the Green's function for the BVP
\begin{gather*}
D^{\alpha -2}_{0+}u(t) +h(t)=0, \quad 0<t<b, \\
u(0)= 0, \quad D^{\beta}_{0+}u(b)=0, \quad D^{\alpha -2l}_{0+} u(0)= 0, 
 \quad   D^{\alpha -2l}_{0+}  u(b)=0, \ l=1,\dots,n-2.
\end{gather*}

To see this, we make the change of variable $v(t)=D^{\alpha-2}_{0+} u(t)$. 
 Then 
$$
D^2 v(t)=D^2D^{\alpha-2}_{0+} u(t)=D^\alpha_{0+} u(t)=-h(t).
$$ 
Since $v(0)=D^{\alpha-2}_{0+} u(0)=0$ and $v(b)=D^{\alpha-2}_{0+} u(b)=0$, 
$v$ satisfies the Dirichlet BVP
\begin{gather*}
v''+h(t)=0,\quad 0<t<b, \\
v(0)=0,\quad v(b)=0.
\end{gather*}
 Also, $u$ satisfies a lower order BVP
\begin{gather*}
 D^{\alpha-2}_{0+}u(t)=v(t), \quad 0<t<b, \\
 u(0)= 0, \quad D^{\beta}_{0+}u(b)=0, \quad D^{\alpha-2l}_{0+}u(0)=0, 
\quad D^{\alpha-2l}_{0+}u(b)=0, \quad l=2,\dots,k,
\end{gather*}
and by the induction hypothesis,
\begin{align*}
u(t)
&=\int^b_0 G_{n-1}(\alpha-2, \beta ,b;t,s)(-v(s))ds\\
&=\int^b_0\Big(-\int^b_0 G_{n-1}(\alpha-2, \beta ,b;t,s)
 G_{\rm conj}(b;s,r)ds\Big)h(r)dr\\
&=\int^b_0 G_{n}(\alpha ,\beta ,b;t,s)h(s)ds,
\end{align*}
where $G_{n}(\alpha, \beta ,b;t,s)
=-\int^b_0 G_{n-1}(\alpha-2, \beta ,b;t,r)G_{\rm conj}(b;r,s)ds$.

Since $0<G_{\rm conj}(b;t,s)$ and $0<\frac{\partial}{\partial b}G_{\rm conj}(b;t,s)$ 
for $(t,s)\in (0,b)\times (0,b)$ the following comparison results are immediate 
from Theorems \ref{3t1} and \ref{3t2} and the inductive construction 
in \eqref{inductiveconstruction}.

\begin{theorem}\label{4t1}
If $0\le \beta_1 <\beta_2\le 1$, then
\begin{equation} \label{4comparison}
\begin{aligned}
0&> (-1)^{n-k}\Big(\frac{d^{2k}}{dt^{2k}}\Big)G_{n}(\alpha , \beta_1, b;t,s)\\
&> (-1)^{n-k}\Big(\frac{d^{2k}}{dt^{2k}}\Big)G_{n}(\alpha ,\beta_2, b;t,s),
\quad (t,s)\in (0,b)\times (0,b),
\end{aligned}
\end{equation}
for $k=0,\dots ,n-1$.
\end{theorem}

\begin{theorem}\label{new4t2} 
Assume $0<b_1<b_2$ and assume $0\le \beta<\alpha -(n-1)$. 
Then for $n\in\mathbb{N}$, $2n-1<\alpha\le 2n$,
\begin{equation} \label{4bmonup}
\begin{aligned}
0&>(-1)^{n-k}\Big(\frac{d^{2k}}{dt^{2k}}\Big) G_n(\alpha, \beta ,b_1;t,s)\\
&> (-1)^{n-k}\Big(\frac{d^{2k}}{dt^{2k}}\Big)G(\alpha, \beta ,b_2;t,s), \quad
(t,s)\in(0,b_1)\times(0,b_1),
\end{aligned}
\end{equation}
for $k=0,\dots ,n-1$.
\end{theorem}


For a second family we begin with a low order analogue of a right focal 
boundary value problem.  Let $b>0$.
Assume $3<\alpha \le 4$ and $0\le \beta \le 1 $. Consider a two-point BVP
 for a fractional differential equation of the form
\begin{gather}\label{4e1}
D^{\alpha}_{0+} u(t)+h(t)=0, \quad  0<t<b, \\
\label{4e2}  u(0)= 0, \quad D^{\beta}_{0+}u(b)=0, 
\quad D^{\alpha -2}_{0+} u(0)= 0,  \quad   D^{\alpha -1}_{0+}  u(b)=0.
\end{gather}
Again, we set $v(t)=D^{\alpha -2}_{0+} u(t) $. Then $v$ satisfies a right 
focal BVP for an ordinary differential equation,
\begin{gather*}
v''(t)+h(t)=0, \quad  0<t<b, \\
 v(0)= 0, \quad v'(b)=0, \\
v(t)=\int_{0}^{b}G_{\rm foc}(b;t,s)h(s)ds, \quad 0\le t\le b,
\end{gather*}
where $G_{\rm foc}(b;t,s)$ is well known and has the form
\begin{equation*}
G_{\rm foc} (b;t,s) 
=\begin{cases} t, & 0\le t<s \le b,\\
s=t-(t-s), & 0\le s<t\le b,
\end{cases}
\end{equation*}
Then $u$, a solution of \eqref{4e1}, \eqref{4e2}, also satisfies the BVP,
\begin{equation*}
D^{\alpha -2}_{0+} u(t)= v(t), \quad  0<t<b,
\end{equation*}
along with boundary conditions $u(0)= 0, D_{0+}^{\beta}u(b)=0$.
Thus, a solution $u$ of \eqref{4e1}, \eqref{4e2} has the form
\begin{equation*}
u(t)=\int_{0}^{b}\mathcal{G}(\alpha , \beta ,b;t,s)h(s)ds, \quad 0\le t\le b,
\end{equation*}
where $\mathcal{G}(\alpha , \beta ,b;t,s)$ is the convolution
\begin{equation}\label{2Halpha}
\mathcal{G}(\alpha , \beta ,b;t,s) 
= -\int_{0}^{b}G(\alpha -2 , \beta ,b;t,r)G_{\rm foc}(b;r,s)dr, \quad 
(t,s)\in [0,b]\times [0,b],
\end{equation}
and $G(\alpha -2 , \beta ,b;t,s)$ is given by \eqref{Galphabar} with 
$\bar{\alpha}=\alpha -2$.

Since $G_{\rm foc}(b;t,s)>0$ on $(0,b)\times (0,b)$ and 
$\frac{\partial}{\partial b}G_{\rm foc}(b;t,s)\equiv 0$ on $(0,b)\times (0,b)$,
the following results follow from the known comparison results for 
$G(\alpha -2 , \beta ,b;t,s)$.

\begin{theorem}\label{4t21}
If $0\le \beta_1 <\beta_2\le 1$, then
\begin{equation*}
0>\mathcal{G}(\alpha , \beta_1, b;t,s)>\mathcal{G}(\alpha ,\beta_2, b;t,s), 
\quad (t,s)\in (0,b)\times (0,b).
\end{equation*}
\end{theorem}

\begin{theorem}\label{4t2}
Assume $0<b_1<b_2$. If $0\le \beta<\alpha -3$, then
\begin{equation*}
0>\mathcal{G}(\alpha ,\beta ,b_1;t,s)>\mathcal{G}(\alpha , \beta ,b_2;t,s), 
\quad (t,s)\in (0,b_1)\times (0,b_1),
\end{equation*}
and if $\alpha -3< \beta \le 1$, then
\begin{equation}\label{e45}
0>\mathcal{G}(\alpha ,\beta ,b_2;t,s)>\mathcal{G}(\alpha , \beta ,b_1;t,s), 
\quad (t,s)\in (0,b_1)\times (0,b_1).
\end{equation}
\end{theorem}

Note that Theorem \ref{4t2} includes the case $\alpha -3< \beta \le 1$, 
and Theorem \ref{3t2} does not.  Since
$$
\frac{\partial}{\partial b}G_{\rm foc}(b;t,s)\equiv 0 \quad\text{on } 
(0,b)\times (0,b),
$$
Inequality \eqref{e25} can be applied to obtain \eqref{e45}.

We also point out that a further type of comparison result is valid since
$$
0<G_{\rm conj}(b;t,s)<G_{\rm foc}(b;t,s) \quad \text{on } (0,b)\times (0,b).
$$

\begin{theorem}\label{4t3}
If $0\le \beta\le 1$, then
\begin{equation*}
0>G(\alpha , \beta, b;t,s)>\mathcal{G}(\alpha ,\beta, b;t,s), \quad 
(t,s)\in (0,b)\times (0,b),
\end{equation*}
where $G(\alpha , \beta, b;t,s)$ is given by \eqref{2Galpha}.
\end{theorem}

The ideas produced here can be extended inductively.  
We do so but modify on the higher order boundary conditions.  
In \eqref{4e2}, $v$ satisfies right focal boundary conditions; 
for simplicity, to proceed inductively, $v$ will satisfy initial conditions.

Let $b>0$. Let $n\ge 3$ denote an integer and assume $n-1<\alpha \le n$. 
 Let $k\in \{1, \ldots , n-1 \}$ denote an integer and assume $0\le \beta \le k$. 
Consider a two-point BVP for a fractional differential equation of the form
\begin{gather}\label{6e1}
D^{\alpha}_{0+} u(t)+h(t)=0, \quad  0<t<b, \\
\label{6e2}
\begin{gathered}
 u^{(i)}(0)= 0, \quad i=0, \ldots , k-1, \quad
 D^{\beta}_{0+} u(b)= 0,  \\
   D^{\alpha -j}_{0+}  u(b)=0, \quad j=1, \ldots n-k-1 .
\end{gathered}
\end{gather}
Denote the Green's function, if it exists, of \eqref{6e1}, \eqref{6e2} as 
$\mathcal{G}(\alpha, \beta, k, b; t,s)$.  And from what follows, 
$\mathcal{G}(\alpha, \beta, k, b; t,s)$ will exist as a convolution of Green's 
functions of lower order problems.

To treat this boundary value problem as a nested family of problems, we make 
the substitution $v=D^{\alpha -(n-k-1)}_{0+} u$. Then $v$ satisfies the
 initial value problem,
$$
v^{(n-k-1)}(t)+h(t)=0, \quad 0<t<b, \quad v^{(j)}(b)=0, \quad j=0,\ldots ,n-k-2, 
$$
and  a solution $u$ of \eqref{6e1}, \eqref{6e2} satisfies the BVP
\begin{gather}\label{6e3}
D^{\alpha -(n-k-1)}_{0+} u(t)=v(t), \quad  0<t<b, \\
\label{6e4}
u^{(i)}(0)= 0, \quad i=0, \ldots , k-1, \quad D^{\beta}_{0+} u(b)= 0.
\end{gather}
We write
$$
v(t)=\int_{b}^{t}\frac{(t-s)^{n-k-2}}{\Gamma (n-k-1) }(-h(s))ds
 =\int_{0}^{b}G_{ivp}(b; t,s)h(s)ds,
$$
where
\begin{equation*}
G_{ivp}(b;t,s) 
= \begin{cases} 
0, & 0\le s\leq t\le b, \\
\frac{(t-s)^{n-k-2}}{\Gamma (n-k-1)}, & 0\le t\leq s\le b. 
\end{cases}
\end{equation*}
The Green's function for the BVP \eqref{6e3}, \eqref{6e4} has been constructed 
in \cite{eln} (see \eqref{Galphabar}) and has the form
\begin{equation}\label{Galpha-1}
\begin{aligned}
&G(\alpha -(n-k-1), \beta , b;t,s) \\
&= \begin{cases} 
\frac{t^{\alpha-(n-k-2)} (b-s)^{\alpha-(n-k-2)-\beta}}{b^{\alpha-(n-k-2)-\beta} 
\Gamma(\alpha -(n-k-1))} -\frac{(t-s)^{\alpha-(n-k-2)}}{\Gamma(\alpha -(n-k-1))}, 
& 0\le s\leq t\le b, \\[4pt]
\frac{t^{\alpha-(n-k-2)} (b-s)^{\alpha-(n-k-2)-\beta}}{b^{\alpha-(n-k-2)
-\beta} \Gamma(\alpha -(n-k-1))}, 
& 0\le t\leq s\le b. 
\end{cases}
\end{aligned}
\end{equation}
Thus, $\mathcal{G}(\alpha, \beta, k, b; t,s)$ exists and can be written as 
the convolution
\begin{equation}\label{IVPconvolution}
\mathcal{G}(\alpha, \beta, k, b; t,s)
=-\int_{0}^{b}G(\alpha -(n-k-1), \beta , b;t,r)G_{ivp}(b;r,s) dr.
\end{equation}

\begin{theorem}\label{6t1}
If $j\in \{0, \ldots ,k-1 \}$, and if $j\le \beta_1 <\beta_2\le k$, then, 
for $i=0, \ldots ,j$,
\begin{equation}\label{6comparison}
0<(-1)^{n-k} \big( \frac{\partial^{i}}{\partial t^{i}} \big) 
\mathcal{G}(\alpha , \beta_1, k, b;t,s)
<(-1)^{n-k}\big(\frac{\partial^{i}}{\partial t^{i}}\big)
\mathcal{G}(\alpha ,\beta_2, k, b;t,s),
\end{equation}
for $(t,s)\in (0,b)\times (0,b)$.
\end{theorem}

\begin{proof}
Applying Theorem \ref{t21}, if $j\in \{0, \ldots ,k-1 \}$, and 
if $j\le \beta_1 <\beta_2\le k$, then, for $i=0, \ldots ,j$, and 
$(t,s)\in (0,b)\times (0,b)$, we have
$$
0<\big(\frac{\partial^{i}}{\partial t^{i}}\big)
G(\alpha -(n-k-1), \beta_1, k, b;t,s)
<\big(\frac{\partial^{i}}{\partial t^{i}}\big)G(\alpha -(n-k-1),\beta_2, k, b;t,s) .
$$
The parity of $G_{ivp}$ is $(-1)^{n-k}$ and so, again, the result follows 
immediately from the representation in \eqref{IVPconvolution}.
\end{proof}

Theorem \ref{t22} implies the following comparisons.

\begin{theorem}\label{6t2}
Assume $0<b_1<b_2$. If $0\le \beta<\alpha -(n-k)$, then
\begin{equation}\label{6bmonup}
0>(-1)^{n-k}\mathcal{G}(\alpha ,\beta ,k ,b_1;t,s)>(-1)^{n-k}
\mathcal{G}(\alpha , \beta ,k, b_2;t,s), 
\end{equation}
for $(t,s)\in (0,b_1)\times (0,b_1)$,
and if $\alpha -(n-k)< \beta \le k$, then
\begin{equation}\label{6bmondown}
0>(-1)^{n-k}G(\alpha ,\beta ,k, b_2;t,s)>(-1)^{n-k}G(\alpha ,
 \beta ,k, b_1;t,s), 
\end{equation}
for  $(t,s)\in (0,b_1)\times (0,b_1)$.
\end{theorem}

\section{An open question}

In the preceding, we have employed convolutions of Green's functions 
to obtain the expected comparison theorems for two point boundary value 
problems for higher order fractional equations.  
If the Green's function is not constructed as a convolution, it is open 
as to whether there exist families of two-point problems that maintain 
the validity of the expected comparison theorems. 
 We introduce a specific family here to frame the open question.

Let $b>0$. Assume $3<\alpha \le 4$ and assume $\alpha -2 < \gamma <\alpha -1$. 
Consider a two-point BVP for a fractional differential equation of the form
\begin{gather}\label{e1}
D^{\alpha}_{0+} u(t)+h(t)=0, \quad  0<t<b, \\
\label{e2}
 u(0)= 0, \quad u(b)=0, \quad D^{\alpha -2}_{0+} u(0)= 0,  \quad   
D^{\gamma}_{0+}  u(b)=0.
\end{gather}
The technique to set $v=D^{\alpha -2}_{0+} u$ is now fruitless, 
since we do not know how to transform $D^{\gamma}_{0+}  u$ to $v$.  
So apply \eqref{p4} directly to  \eqref{e1}, \eqref{e2}, and a solution $u$ 
of \eqref{e1}, \eqref{e2} has the form
\begin{equation*}
u(t)+c_1 t^{\alpha -4} +c_2 t^{\alpha -3} + c_{3} t^{\alpha -2}
 + c_4 t^{\alpha -1} + I^{\alpha }_{0} h(t) =0.
\end{equation*}
Then apply the boundary conditions to obtain the system of equations
\begin{gather*}
c_2 b^{\alpha -3} +  c_4 b^{\alpha -1} + I^{\alpha }_{0} h(b)=0, \\
c_2 \frac{\Gamma (\alpha -2)}{\Gamma (\alpha -2-\gamma )}
b^{\alpha -3-\gamma}+c_4 \frac{\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}
b^{\alpha -1-\gamma}+ I^{\alpha -\gamma}_{0} h(b)=0.
\end{gather*}
Thus, $c_2$ and $c_4$ are explicitly obtained and
\begin{gather*}
c_2= \frac{1}{\Delta}\Big(b^{\alpha -1}I^{\alpha -\gamma}_{0} h(b)
 -\frac{\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}
 b^{\alpha -1-\gamma}I^{\alpha}_{0} h(b)\Big), \\
c_4= \frac{1}{\Delta}\Big(\frac{\Gamma (\alpha -2)}
 {\Gamma (\alpha -2-\gamma )}b^{\alpha - 3 -\gamma}
 I^{\alpha}_{0} h(b)-b^{\alpha -3}I^{\alpha -\gamma}_{0} h(b)\Big),
\end{gather*}
where
$$
\Delta = \Big(\frac{\Gamma (\alpha )}{\Gamma (\alpha -\gamma )}
 -\frac{\Gamma (\alpha -2)}{\Gamma (\alpha -2-\gamma )}
\Big)b^{2\alpha -\gamma -4}.
$$
Let
\begin{align*}
g(\alpha ,\gamma ,b;t,s) 
&=\frac{-1}{\Delta}\left(b^{\alpha -1}(b-s)^{\alpha -\gamma -1}-b^{\alpha -1-\gamma}(b-s)^{\alpha -1}\right)\frac{t^{\alpha -3}}{\Gamma (\alpha -\gamma)}\\
&\quad +\frac{-1}{\Delta}\Big(\frac{\Gamma (\alpha -2)}
{\Gamma (\alpha -2-\gamma )}b^{\alpha - 3 -\gamma}\frac{(b-s)^{\alpha -1}}
{\Gamma (\alpha )}-b^{\alpha -3}
\frac{(b-s)^{\alpha -\gamma -1}}{\Gamma (\alpha -\gamma )}\Big)t^{\alpha -1}.
\end{align*}
Then the Green's function is
\begin{equation}
G(\alpha , \gamma ,b;t,s) 
=\begin{cases} g(\alpha ,\gamma ,b;t,s), & 0\le t<s\le b,\\
g(\alpha ,\gamma ,b;t,s)-\frac{(t-s)^{\alpha -1}}{\Gamma(\alpha )}, 
& 0\le s<t\le b,
\end{cases}
\end{equation}
and the solution $u$ has the form
$$
u(t)=\int_{0}^{1}G(\alpha , \gamma, b; t,s)h(s)ds.
$$
Note that at $\gamma =\alpha -2$,
$$
G(\alpha , \gamma, b; t,s)=G(\alpha , 0, b; t,s)
$$
where $G(\alpha , 0, b; t,s)$ is given by \eqref{2Galpha} at $\beta =0$ 
and at $\gamma =\alpha -1$,
$$
G(\alpha , \gamma, b; t,s)=\mathcal{G}(\alpha , 0, b; t,s)
$$
where $\mathcal{G}(\alpha , 0, b; t,s)$ is given by \eqref{2Halpha} at $\beta =0$.

A natural question to ask in the context of Theorem \ref{4t3} is the following:
Let $b>0$. Assume $3<\alpha \le 4$ and assume
 $\alpha -2 < \gamma_1<\gamma_2 <\alpha -1$. 
Let $G(\alpha, \gamma_{i},b;t,s)$, $i=1,2$ denote the Green's function for
  \eqref{e1}, \eqref{e2} for $\gamma =\gamma_{i}$, $i=1,2$ respectively.  
Let $G(\alpha , 0, b; t,s)$ be given by \eqref{2Galpha} and 
$\mathcal{G}(\alpha , 0, b; t,s)$ be given by \eqref{2Halpha}.  Is
\[
0>G(\alpha , 0, b;t,s)>G(\alpha , \gamma_1, b;t,s)>G(\alpha , 
\gamma_2, b;t,s)>\mathcal{G}(\alpha ,0, b;t,s), 
\]
for $(t,s)\in (0,b)\times (0,b)$, valid?

Preliminary numerical experiments, not produced here, indicate that the answer 
to this particular question is yes.

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\end{document}