\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 121, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2012/121\hfil Green's functional for a nonlocal problem]
{Green's functional for second-order linear differential equation with
nonlocal conditions}
\author[K. Oru\c{c}o\u glu, K. \"Ozen \hfil EJDE-2012/121\hfilneg]
{Kam\.il Oru\c{c}o\u glu, Kemal \"Ozen} % in alphabetical order
\address{Kam\.il Oru\c{c}o\u glu \newline
Istanbul Technical University, Department of Mathematics, Istanbul,
34469, Turkey}
\email{koruc@itu.edu.tr}
\address{Kemal \"Ozen \newline
Istanbul Technical University, Department of Mathematics, Istanbul,
34469, Turkey,\newline
Nam\i k Kemal University, Department of Mathematics,
Tek\.irda\u g, 59030, Turkey}
\email{ozenke@itu.edu.tr}
\thanks{Submitted February 23, 2012. Published July 19, 2012.}
\subjclass[2000]{34A30, 34B05, 34B10, 34B27, 45A05}
\keywords{Green's function; nonlocal boundary conditions;
\hfill\break\indent nonsmooth coefficient; adjoint problem}
\begin{abstract}
In this work, we present a new constructive technique which is based
on Green's functional concept. According to this technique, a linear
completely nonhomogeneous nonlocal problem for a second-order ordinary
differential equation is reduced to one and only one integral equation
in order to identify the Green's solution. The coefficients of
the equation are assumed to be generally variable nonsmooth functions
satisfying some general properties such as $p$-integrability and boundedness.
A system of three integro-algebraic equations called the special adjoint
system is obtained for this problem. A solution of this special adjoint
system is Green's functional which enables us to determine the Green's
function and the Green's solution for the problem. Some illustrative
applications and comparisons are provided with some known results.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks
\section{Introduction}
Green functions of linear boundary-value problems for ordinary differential
equations with smooth coefficients have been investigated in detail in several
studies \cite{Kre71,Naim69,Shi68,Stak98,TiVaSv80}. In this work, a linear
nonlocal problem is studied for a second-order differential equation. The
coefficients of the equation are assumed to be generally nonsmooth functions
satisfying some general properties such as $p$-integrability and boundedness.
The operator of this equation, in general, does not have a formal adjoint operator,
or any extension of the traditional type for this operator exists only on a space
of distributions \cite{Ho76,Shi68}. In addition, the considered problem does not
have a meaningful traditional type adjoint problem, even for simple
cases of a differential equation and nonlocal conditions. Due to these facts,
some serious difficulties arise in the application of the classical methods
for such a problem. As can be seen from \cite{Kre71}, similar difficulties
arise even for classical type boundary value problems if the coefficients of
the differential equation are, for example, continuous nonsmooth functions.
For this reason, a Green's functional approach is introduced for the investigation
of the considered problem. This approach is based on
\cite{Akh80,Akh84,AkhOruc02,Akh07,Oruc05} and on some methods of functional analysis.
The main idea of this approach is related to the usage of a new concept of the
adjoint problem named adjoint system. Such an adjoint system includes three
integro-algebraic equations with an unknown element $(f_2(\xi),f_1,f_0)$ in
which $f_2(\xi)$ is a function, and $f_j$ for $j=0,1$ are real numbers.
One of these equations is an integral equation with respect to $f_2(\xi)$
and generally includes $f_j$ as parameters. The other two equation can be
considered as a system of algebraic equations with respect to $f_0$ and $f_1$,
and they may include some integral functionals defined on $f_2(\xi)$.
The form of the adjoint system depends on the operators of the equation and
the conditions. The role of the adjoint system is similar to that of the adjoint
operator equation in the general theory of the linear operator equations in
Banach spaces \cite{BrPa70,KaAk82,Kre71}. The integral representation of the
solution is obtained by a concept of the Green functional which is introduced
as a special solution $f(x)=(f_2(\xi,x),f_1(x),f_0(x))$ of the corresponding
adjoint system having a special free term depending on $x$ as a parameter.
The first component $f_2(\xi,x)$ of Green functional $f(x)$ is corresponded
to Green's function for the problem. The other two components $f_j(x)$ for
$j=0,1$ correspond to the unit effects of the conditions. If the homogeneous
problem has a nontrivial solution, then the Green functional does not exist.
In summary, this approach is principally different from the classical methods
used for constructing Green functions\cite{Stak98}.
\section{Statement of the problem}
Let $\mathbb{R}$ be the set of real numbers. Let $G=(x_0,x_1)$ be a bounded
open interval in $\mathbb{R}$. Let $L_p(G)$ with $1\leq p<\infty$ be the
space of $p$-integrable functions on $G$. Let $L_{\infty}(G)$ be the space
of measurable and essentially bounded functions on $G$, and let $W_p^{(2)}(G)$
with $1\leq p\leq \infty$ be the space of all functions $u=u(x)\in L_p(G)$
having derivatives $d^ku/dx^k \in L_p(G)$, where $k=1,2$.
The norm on the space $W_p^{(2)}(G)$ is defined as
\[
\| {u}\|_{W_p^{(2)}(G)}=\sum_{k=0}^{2}\| {d^ku\over dx^k}\|_{L_p(G)}\, .
\]
We consider the second-order boundary value problem
\begin{equation} \label{e2.1}
(V_2u)(x)\equiv u''(x)+A_1(x)u'(x)+A_0(x)u(x)=z_2(x),\quad x\in G,
\end{equation}
subject to the nonlocal boundary conditions
\begin{equation} \label{e2.2}
\begin{gathered}
V_1u \equiv a_1u(x_0)+b_1u'(x_0)+\int_{x_0}^{x_1}g_1(\xi)u''(\xi)d\xi=z_1,\\
V_0u \equiv a_0u(x_0)+b_0u'(x_0)+\int_{x_0}^{x_1}g_0(\xi)u''(\xi)d\xi=z_0,
\end{gathered}
\end{equation}
which are more general conditions than the ones in \cite{Akh07}.
We investigate a solution to the problem in the space $W_p=W_p^{(2)}(G)$.
Furthermore, we assume that the following conditions are satisfied:
$A_i\in L_p(G)$ and $g_i\in L_q(G)$ for $i=0,1$ are given functions; $a_i,b_i$
for $i=0,1$ are given real numbers; $z_2\in L_p(G)$ is a given function
and $z_i$ for $i=0,1$ are given real numbers.
Problem \eqref{e2.1}-\eqref{e2.2} is a linear completely nonhomogeneous problem
which can be considered as an operator equation:
\begin{equation} \label{e2.3}
Vu=z,
\end{equation}
with the linear operator $V=(V_2,V_1,V_0)$ and $z=(z_2(x),z_1,z_0)$.
The assumptions considered above guarantee that $V$ is bounded from $W_p$
to the Banach space $E_p\equiv L_p(G)\times\mathbb{R}\times\mathbb{R}$
consisting of element $z=(z_2(x),z_1,z_0)$ with
\[
\| z\|_{E_p}=\| z_2\|_{L_p(G)}+| z_1| +| z_0|, \quad 1\le p\le \infty.
\]
If, for a given $z\in E_p$, the problem \eqref{e2.1}-\eqref{e2.2} has
a unique solution $u\in W_p$ with $\| u\|_{W_p}\leq c_0\| z\|_{E_p}$,
then this problem is called a well-posed problem, where $c_0$ is a
constant independent of $z$. Problem \eqref{e2.1}-\eqref{e2.2} is well-posed
if and only if $V:W_p\to E_p$ is a (linear) homeomorphism.
\section{Adjoint space of the solution space}
Problem \eqref{e2.1}-\eqref{e2.2} is investigated by means of a new concept of
the adjoint problem. This concept is introduced in the
papers~\cite{Akh84,AkhOruc02} by the adjoint operator $V^{*}$ of $V$.
Some isomorphic decompositions of the space $W_p$ of solutions and its
adjoint space $W_p^{*}$ are employed. Any function $u\in W_p$ can be
represented as
\begin{equation} \label{e3.1}
u(x)=u(\alpha)+u'(\alpha)(x-\alpha)+\int_\alpha ^x (x-\xi)u''(\xi)d\xi
\end{equation}
where $\alpha$ is a given point in $\overline{G}$ which is the set of
closure points for $G$. Furthermore, the trace or value operators
$D_0u=u(\gamma),D_1u=u'(\gamma)$ are bounded and surjective from $W_p$
onto $\mathbb{R}$ for a given point $\gamma$ of $\overline{G}$.
In addition, the values $u(\alpha),u'(\alpha)$ and the derivative $u''(x)$
are unrelated elements of the function $u\in W_p$ such that for any real
numbers $\nu_0,\nu_1$ and any function $\nu_2\in L_p(G)$, there exists one
and only one $u\in W_p$ such that $u(\alpha)=\nu_0,u'(\alpha)=\nu_1$
and $u''(x)=\nu_2(x)$. Therefore, there exists a linear homeomorphism between
$W_p$ and $E_p$. In other words, the space $W_p$ has the isomorphic
decomposition $W_p=L_p(G)\times\mathbb{R}\times\mathbb{R}$.
\begin{theorem}[\cite{Akh07}] \label{thm3.1}
If $1\leq p<\infty$, then any linear bounded functional $F\in W_p^{*}$
can be represented as
\begin{equation} \label{e3.2}
F(u)=\int_{x_0}^{x_1}u''(x)\varphi_2(x)dx+u'(x_0)\varphi_1+u(x_0)\varphi_0
\end{equation}
with a unique element $\varphi=(\varphi_2(x),\varphi_1,\varphi_0)\in E_q$ where $p+q=pq$. Any linear bounded functional $F\in W_\infty^{*}$ can be represented as
\begin{equation} \label{e3.3}
F(u)=\int_{x_0}^{x_1}u''(x)d\varphi_2+u'(x_0)\varphi_1+u(x_0)\varphi_0
\end{equation}
with a unique element
$\varphi=(\varphi_2(e),\varphi_1,\varphi_0)\in \widehat{E_1}
=(BA(\sum,\mu))\times\mathbb{R}\times\mathbb{R}$ where $\mu$ is
the Lebesgue measure on $\mathbb{R}$, $\sum$ is $\sigma$-algebra of the
$\mu$-measurable subsets $e\subset G$ and $BA(\sum,\mu)$ is the space of
all bounded additive functions $\varphi_2(e)$ defined on $\sum$ with
$\varphi_2(e)=0$ when $\mu(e)=0$~\cite{KaAk82}. The inverse is also valid; that is,
if $\varphi\in E_q$, then \eqref{e3.2} is bounded on $W_p$ for $1\leq p<\infty$
and $p+q=pq$. If $\varphi\in \widehat{E_1}$, then \eqref{e3.3} is bounded on
$W_\infty$.
\end{theorem}
\begin{proof}
\cite{Akh07} The operator $Nu\equiv(u''(x),u'(x_0),u(x_0)):W_p\to E_p$ is bounded
and has a bounded inverse $N^{-1}$ represented by
\begin{equation} \label{e3.4}
\begin{gathered}
u(x)=(N^{-1}h)(x) \equiv \int_{x_0}^{x}(x-\xi)h_2(\xi)d\xi+h_1(x-x_0)+h_0,\\
h=(h_2(x),h_1,h_0)\in E_p.
\end{gathered}
\end{equation}
The kernel $\ker N$ of $N$ is trivial and the image $\operatorname{Im}N$ of $N$
is equal to $E_p$. Hence, there exists a bounded adjoint operator
$N^{*}:E_p^{*}\to W_p^{*}$ with $\ker N^{*}=\{0\}$ and
$\operatorname{Im}N^{*}=W_p^{*}$. In other words, for a given
$F\in W_p^{*}$ there exists a unique $\psi\in E_p^{*}$ such that
\begin{equation} \label{e3.5}
F=N^{*}\psi \quad \text{or}\quad F(u)=\psi(Nu),\quad u\in W_p.
\end{equation}
If $1\leq p<\infty$, then $E_p^{*}=E_q$ in the sense of an isomorphism~\cite{KaAk82}. Therefore, the functional $\psi$ can be represented by
\begin{equation} \label{e3.6}
\psi(h)=\int_{x_0}^{x_1}\varphi_2(x)h_2(x)dx+\varphi_1h_1+\varphi_0h_0,
\quad h\in E_p,
\end{equation}
with a unique element $\varphi=(\varphi_2(x),\varphi_1,\varphi_0)\in E_q$.
By expressions \eqref{e3.5} and \eqref{e3.6}, any $F\in W_p^{*}$ can
uniquely be represented by \eqref{e3.2}. For a given $\varphi\in E_q$,
the functional $F$ represented by \eqref{e3.2} is bounded on $W_p$.
Hence, \eqref{e3.2} is a general form for the functional $F\in W_p^{*}$.
The proof is complete due to that the case $p=\infty$ can also be
shown \cite{Akh07}.
\end{proof}
Theorem \ref{thm3.1} guarantees that $W_p^{*}=E_q$ for all $1\leq p<\infty$,
and $W_{\infty}^{*}=E_{\infty}^{*}=\widehat{E_1}$. The space $E_1$ can
also be considered as a subspace of the space $\widehat{E_1}$
(see \cite{AkhOruc02,Akh07}).
\section{Adjoint operator and adjoint system of the integro-algebraic equations}
Investigating an explicit form for the adjoint operator $V^{*}$ of $V$
is taken into consideration in this section. To this end, any
$f=(f_2(x),f_1,f_0)\in E_q$ is taken as a linear bounded functional on
$E_p$ and also
\begin{equation} \label{e4.1}
f(Vu)\equiv\int_{x_0}^{x_1}f_2(x)(V_2u)(x)dx+f_1(V_1u)+f_0(V_0u),\quad
u\in W_p,
\end{equation}
can be assumed. By substituting expressions \eqref{e2.1} and \eqref{e2.2},
and expression \eqref{e3.1} (for $\alpha=x_0$) of $u\in W_p$ into \eqref{e4.1},
we have
\begin{equation} \label{e4.2}
\begin{split}
f(Vu)&\equiv\int_{x_0}^{x_1}f_2(x)[u''(x)+A_1(x)\{u'(x_0)
+\int_{x_0}^x u''(\xi)d\xi\}\\
&\quad +A_0(x)\{u(x_0)+u'(x_0)(x-x_0)+\int_{x_0}^x (x-\xi)u''(\xi)d\xi\}]dx\\
&\quad +f_1\{a_1u(x_0)+b_1u'(x_0)+\int_{x_0}^{x_1}g_1(\xi)u''(\xi)d\xi\}\\
&\quad +f_0\{a_0u(x_0)+b_0u'(x_0)+\int_{x_0}^{x_1}g_0(\xi)u''(\xi)d\xi\}.
\end{split}
\end{equation}
After some calculations, we can obtain
\begin{equation} \label{e4.3}
\begin{split}
f(Vu)&\equiv \int_{x_0}^{x_1}f_2(x)(V_2u)(x)dx+\sum_{i=0}^{1}f_i(V_iu)\\
&=\int_{x_0}^{x_1}(w_2f)(\xi)u''(\xi)d\xi+(w_1f)u'(x_0)+(w_0f)u(x_0)\\
&\equiv (wf)(u),\quad \forall f\in E_q,\quad \forall u\in W_p,\quad 1\leq p\leq\infty
\end{split}
\end{equation}
where
\begin{equation} \label{e4.4}
\begin{split}
(w_2f)(\xi)
&= f_2(\xi)+f_1g_1(\xi)+f_0g_0(\xi)+\int_{\xi}^{x_1}f_2(s)\{A_1(s)
+A_0(s)(s-\xi)\}ds,\\
w_1f&= b_1f_1+b_0f_0+\int_{x_0}^{x_1}f_2(s)\{A_1(s)+A_0(s)(s-x_0)\}ds\\
w_0f&= a_1f_1+a_0f_0+\int_{x_0}^{x_1}f_2(s)A_0(s)ds.
\end{split}
\end{equation}
The operators $w_2,w_1,w_0$ are linear and bounded from the space $E_q$
of the triples $f=(f_2(x),f_1,f_0)$ into the spaces
$L_q(G),\mathbb{R},\mathbb{R}$ respectively. Therefore, the operator
$w=(w_2,w_1,w_0):E_q\to E_q$ represented by $wf=(w_2f,w_1f,w_0f)$
is linear and bounded. By \eqref{e4.3} and Theorem \ref{thm3.1},
the operator $w$ is an adjoint operator for the operator $V$ when
$1\leq p<\infty$, in other words, $V^{*}=w$. When $p=\infty$,
$w:E_1\to E_1$ is bounded; in this case, the operator $w$ is the restriction
of the adjoint operator $V^{*}:E_{\infty}^{*}\to W_{\infty}^{*}$ of $V$
onto $E_1\subset E_{\infty}^{*}$.
Equation \eqref{e2.3} can be transformed into the following equivalent equation
\begin{equation} \label{e4.5}
VSh=z,
\end{equation}
with an unknown $h=(h_2,h_1,h_0)\in E_p$ by the transformation $u=Sh$ where $S=N^{-1}$. If $u=Sh$, then $u''(x)=h_2(x)$, $u'(x_0)=h_1$, $u(x_0)=h_0$. Hence, equation \eqref{e4.3} can be rewritten as
\begin{equation} \label{e4.6}
\begin{split}
f(VSh)&\equiv \int_{x_0}^{x_1}f_2(x)(V_2Sh)(x)dx+\sum_{i=0}^{1}f_i(V_iSh)\\
&= \int_{x_0}^{x_1}(w_2f)(\xi)h_2(\xi)d\xi+(w_1f)h_1+(w_0f)h_0\\
&\equiv (wf)(h),\quad \forall f\in E_q,\quad \forall h\in E_p,\quad
1\leq p\leq \infty.
\end{split}
\end{equation}
Therefore, one of the operators $VS$ and $w$ becomes an adjoint operator for the other one. Consequently, the equation
\begin{equation} \label{e4.7}
wf=\varphi,
\end{equation}
with an unknown function $f=(f_2(x),f_1,f_0)\in E_q$ and a given function
$\varphi=(\varphi_2(x),\varphi_1,\varphi_0)\in E_q$ can be considered as
an adjoint equation of \eqref{e4.5}(or of \eqref{e2.3}) for all $1\leq p\leq \infty$.
Equation \eqref{e4.7} can be written in explicit form as the system of equations
\begin{equation} \label{e4.8}
\begin{gathered}
(w_2f)(\xi) = \varphi_2(\xi),\quad \xi\in G,\\
w_1f = \varphi_1,\\
w_0f = \varphi_0.
\end{gathered}
\end{equation}
By the expressions \eqref{e4.4}, the first equation in \eqref{e4.8}
is an integral equation for $f_2(\xi)$ and includes $f_1$ and $f_0$ as parameters;
on the other hand, the other equations in \eqref{e4.8} constitute a system of
two algebraic equations for the unknowns $f_1$ and $f_0$ and they include some
integral functionals defined on $f_2(\xi)$. In other words, \eqref{e4.8}
is a system of three integro-algebraic equations. This system called the adjoint
system for \eqref{e4.5}(or \eqref{e2.3}) is constructed by using \eqref{e4.3}
which is actually a formula of integration by parts in a nonclassical form.
The traditional type of an adjoint problem is defined by the classical Green's
formula of integration by parts~\cite{Stak98}, therefore, has a sense only for
some restricted class of problems.
\section{Solvability conditions for completely nonhomogeneous problem}
The operator $Q=w-I_q$ is considered where $I_q$ is the identity operator
on $E_q$; i.e., $I_qf=f$ for all $f\in E_q$. This operator can also be
defined as $Q=(Q_2,Q_1,Q_0)$ with
\begin{equation} \label{e5.1}
\begin{gathered}
(Q_2f)(\xi) = (w_2f)(\xi)-f_2(\xi),\quad \xi\in G,\\
Q_if = w_if-f_i,\quad i=0,1.
\end{gathered}
\end{equation}
By the expressions \eqref{e4.4} and the conditions imposed on $A_i$ and
$g_i$ for $i=0,1$, $Q_{m}:E_q\to L_q(G)$ is a compact operator, and also
$Q_i:E_q\to \mathbb{R}$ for $i=0,1$ are compact operators where $1