\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 $1<p<\infty$.
That is, $Q:E_q\to E_q$ is a compact operator, and therefore has a compact
adjoint operator $Q^{*}:E_p\to E_p$. Since $w=Q+I_q$ and $VS=Q^{*}+I_p$,
 where $I_p=I_q^{*}$, \eqref{e4.5} and \eqref{e4.7} are canonical Fredholm
type equations, and $S$ is a right regularizer of \eqref{e2.3} \cite{Kre71}.
Consequently, we have the following theorem.

\begin{theorem}[\cite{Akh07}] \label{thm5.1}
If $1<p<\infty$, then $Vu=0$ has either only the trivial solution or a finite
 number of linearly independent solutions in $W_p$:\\
\textbf{(1)} If $Vu=0$ has only the trivial solution in $W_p$, then also 
$wf=0$ has only the trivial solution in $E_q$. Then, the operators
 $V:W_p\to E_p$ and $w:E_q\to E_q$ become linear homeomorphisms.\\
\textbf{(2)} If $Vu=0$ has $m$ linearly independent solutions $u_1,u_2,\dots,u_{m}$ 
in $W_p$, then $wf=0$ also has $m$ linearly independent solutions
\[
f^{\star 1\star}=(f_2^{\star 1\star}(x),f_1^{\star 1\star},f_0^{\star 1\star}),
\dots,f^{\star m\star}=(f_2^{\star m\star}(x),f_1^{\star m\star},f_0^{\star m\star})
\]
in $E_q$. In this case, \eqref{e2.3} and \eqref{e4.7} have solutions
 $u\in W_p$ and $f\in E_q$ for given $z\in E_p$ and $\varphi \in E_q$
if and only if the conditions
\begin{equation} \label{e5.2}
\int_{x_0}^{x_1}f_2^{\star i\star}(\xi)z_2(\xi)d\xi+f_1^{\star i\star}z_1
+f_0^{\star i\star}z_0=0,\quad i=1,2,\dots,m,
\end{equation}
and
\begin{equation} \label{e5.3}
\int_{x_0}^{x_1}\varphi_2(\xi)u_i''(\xi)d\xi+\varphi_1u_i'(x_0)+\varphi_0u_i(x_0)=0,
\quad i=1,2,\dots,m,
\end{equation}
are satisfied, respectively.
\end{theorem}

\section{Green's Functional}

Consider the following equation given in the form of a functional identity
\begin{equation} \label{e6.1}
(wf)(u)=u(x),\quad \forall u\in W_p,
\end{equation}
where $f=(f_2(\xi),f_1,f_0)\in E_q$ is an unknown triple and $x\in \overline{G}$ 
is a parameter.

\noindent\textbf{Definition.}
\cite{Akh07} Suppose that $f(x)=(f_2(\xi,x),f_1(x),f_0(x))\in E_q$ 
is a triple with parameter $x\in \overline{G}$. 
If $f=f(x)$ is a solution of \eqref{e6.1} for a given $x\in \overline{G}$, 
then $f(x)$ is called a Green's functional of $V$ (or of \eqref{e2.3}).

Due to that the operator $I_{W_p,C}$ of the imbedding of $W_p$ into the 
space $C(\overline{G})$ of continuous functions on $\overline{G}$ is bounded, 
the linear functional $\theta(x)$ defined by $\theta(x)(u)=u(x)$ is bounded 
on $W_p$ for a given $x\in \overline{G}$. On the other hand, $(wf)(u)=(V^{*}f)(u)$. 
Thus, \eqref{e6.1} can also be written as \cite{Akh84,AkhOruc02}
\begin{equation*}
(V^{*}f)=\theta(x).
\end{equation*}
In other words, \eqref{e6.1} can be considered as a special case of the 
adjoint equation $V^{*}f=\psi$ for some $\psi=\theta(x)$.

By substituting $\alpha=x_0$ into \eqref{e3.1} and using \eqref{e4.3},
 we can rewrite \eqref{e6.1} as
\begin{equation} \label{e6.2}
\begin{split}
&\int_{x_0}^{x_1}(w_2f)(\xi)u''(\xi)d\xi+(w_1f)u'(x_0)+(w_0f)u(x_0)\\
&=\int_{x_0}^{x}(x-\xi)u''(\xi)d\xi
+u'(x_0)(x-x_0)+u(x_0),\quad \forall f\in E_q,\quad \forall u\in W_p.
\end{split}
\end{equation}
The elements $u''(\xi)\in L_p(G),u'(x_0)\in \mathbb{R}$ and $u(x_0)\in \mathbb{R}$
of the function $u\in W_p$ are unrelated. Then, we can construct the
system
\begin{equation} \label{e6.3}
\begin{gathered}
(w_2f)(\xi)= (x-\xi)H(x-\xi),\quad \xi\in G,\\
(w_1f)= (x-x_0),\\
(w_0f)= 1,
\end{gathered}
\end{equation}
where $H(x-\xi)$ is a Heaviside function on $\mathbb{R}$.

Equation \eqref{e6.1} is equivalent to the system \eqref{e6.3} which is a 
special case for the adjoint system \eqref{e4.8} when
 $\varphi_2(\xi)=(x-\xi)H(x-\xi)$, $\varphi_1=x-x_0$ and $\varphi_0=1$. 
Therefore, $f(x)$ is a Green's functional if and only if $f(x)$ is a solution 
of  system \eqref{e6.3} for an arbitrary $x\in \overline{G}$. 
For a solution $u\in W_p$ of \eqref{e2.3} and a Green's functional $f(x)$,
 we can rewrite \eqref{e4.3} as
\begin{equation} \label{e6.4}
\begin{split}
&\int_{x_0}^{x_1}f_2(\xi,x)z_2(\xi)d\xi+f_1(x)z_1+f_0(x)z_0\\
&= \int_{x_0}^{x_1}(x-\xi)H(x-\xi)u''(\xi)d\xi + u'(x_0)(x-x_0)+u(x_0).
\end{split}
\end{equation}
The right hand side of \eqref{e6.4} is equal to $u(x)$. Therefore, we can
state the following theorem.

\begin{theorem}[\cite{Akh07}]  \label{thm6.1}
If \eqref{e2.3} has at least one Green's functional $f(x)$, then any 
solution $u\in W_p$ of \eqref{e2.3} can be represented by
\begin{equation} \label{e6.5}
u(x)=\int_{x_0}^{x_1}f_2(\xi,x)z_2(\xi)d\xi+f_1(x)z_1+f_0(x)z_0.
\end{equation}
Additionally, $Vu=0$ has only the trivial solution.
\end{theorem}

Since one of the operators $V:W_p\to E_p$ and $w:E_q\to E_q$ is a homeomorphism, 
so is the other, and, there exists a unique Green's functional, 
where $1\leq p\leq \infty$. Necessary and sufficient conditions for the existence 
of a Green's functional can be given in the following theorem for $1<p<\infty$.

\begin{theorem}[\cite{Akh07}] \label{thm6.2} 
If there exists a Green's functional, then it is unique. Additionally,  
a Green's functional exists if and only if $Vu=0$ has only the trivial solution.
\end{theorem}

From Theorems \ref{thm5.1}, \ref{thm6.1}, \ref{thm6.2} can be easily shown.

\noindent\textbf{Remark.}
If $Vu=0$ has a nontrivial solution, then a Green's functional corresponding 
to $Vu=z$ does not exist due to Theorem \ref{thm6.1}. In this case,
 $Vu=z$ usually has no solution unless $z$ is of a specific type. 
Therefore, a representation of the existing solution of $Vu=z$ are constructed
by a concept of the generalized Green's functional \cite{AkhOruc02,Akh07}.

It must be noted that the proposed Green's functional approach can 
also be employed some classes of nonlinear equations involving linear 
nonlocal conditions to transform into the corresponding integral equations 
and then solve them. The corresponding integral equations will naturally
 become of nonlinear type. These nonlinear integral equations can be solved 
approximately even if they can not be solved exactly.

\section{Some applications}

In this section, some applications to such problems involving nonlocal boundary 
conditions are implemented in order to emphasize the preferability of the
 presented approach.

\begin{example} \label{examp7.1} \rm
First, we seek for the Green's solution to the following problem, which has been 
considered in~\cite{RoSt09}:
\begin{gather} \label{e7.1}
u''(x)=-f(x),\quad x\in G=(0,1),\\
u(0)=\gamma_0 u'(\xi_0),\quad u(1)=\gamma_1u'(\xi_1), \label{e7.2}
\end{gather}
where $f(x)\in L_p(G)$, $\xi_0,\xi_1\in \overline{G}$ and
 $\gamma_0,\gamma_1\in\mathbb{R}$. We can rewrite this problem as
\begin{gather*}
(V_2u)(x) \equiv u''(x)=-f(x)=z_2(x),\quad x\in G=(0,1),\\
V_1u \equiv u(1)-\gamma_1u'(\xi_1)=0=z_1,\\
V_0u \equiv u(0)-\gamma_0u'(\xi_0)=0=z_0.
\end{gather*}
Thus, we have
\begin{gather*}
a_1=1,\quad b_1=1-\gamma_1,\quad g_1(\xi)=1-\xi-\gamma_1H(\xi_1-\xi),\\
a_0=1,\quad b_0=-\gamma_0,\quad g_0(\xi)=-\gamma_0H(\xi_0-\xi),\hspace{0.8cm}
\end{gather*}
and $A_i(x)=z_i=0$ for $i=0,1$, where $H(\xi_1-\xi)$ and $H(\xi_0-\xi)$ are 
Heaviside functions on $\mathbb{R}$.

Consequently, the special adjoint system \eqref{e6.3} corresponding to this
 problem can be constructed in the  form
\begin{gather} 
f_2(\xi)+f_1\{1-\xi-\gamma_1H(\xi_1-\xi)\}-f_0\gamma_0H(\xi_0-\xi)
 = (x-\xi)H(x-\xi),  \label{e7.3} \\
f_1(1-\gamma_1)-f_0\gamma_0 = x, \label{e7.4} \\
f_1+f_0 = 1, \label{e7.5}
\end{gather}
where $\xi\in (0,1)$. We firstly determine $f_1$ and $f_0$ with using 
only \eqref{e7.4} and \eqref{e7.5} under the condition 
$\Delta_1=1-\gamma_1+\gamma_0\neq 0$ in order to solve 
\eqref{e7.3}-\eqref{e7.5}. Thus, we have
\[
f_1=\frac{x+\gamma_0}{\Delta_1},\quad f_0=\frac{1-\gamma_1-x}{\Delta_1}.
\]
After substituting $f_1$ and $f_0$ into equation \eqref{e7.3}, $f_2(\xi)$ becomes
\begin{align*}
f_2(\xi)
&= (x-\xi)H(x-\xi)+\frac{(1-\gamma_1-x)}{\Delta_1} \gamma_0 H(\xi_0-\xi)\\
&\quad -\frac{(x+\gamma_0)}{\Delta_1}\{1-\xi-\gamma_1 H(\xi_1-\xi)\}.
\end{align*}
Thus, the Green's functional $f(x)=(f_2(\xi,x),f_1(x),f_0(x))$ for the 
problem has been determined. The first component $f_2(\xi,x)=f_2(\xi)$ 
is the Green's function for the problem. After substituting $\xi=s$ 
for notational compatibility, $f_2(\xi,x)$ is equal to the Green's 
function constructed in~\cite{RoSt09} for the problem. 
By \eqref{e6.5} in Theorem \ref{thm6.1}, the representation of the existing solution 
for the problem can be given as
\begin{align*}
u(x)&= \int_0^{1}-[(x-\xi)H(x-\xi)+\frac{(1-\gamma_1-x)}{\Delta_1} 
 \gamma_0 H(\xi_0-\xi)\\
& \quad -\frac{(x+\gamma_0)}{\Delta_1}\{1-\xi-\gamma_1 H(\xi_1-\xi)\}]f(\xi)d\xi.
\end{align*}
\end{example}

\begin{example} \label{examp7.2} \rm
Next, we construct the Green's solution to the following problem, which has 
been considered in \cite{RoSt09}:
\begin{gather} \label{e7.6}
u''(x)=-f(x),\quad x\in G=(0,1), \\
u(0)=\gamma _0\int_0^{1}(1+t)u(t)dt,\quad u(1)=\gamma _1\int_0^{1}u(t)dt, \label{e7.7}
\end{gather}
where $f(x)\in L_p(G)$ and $\gamma_0,\gamma_1\in\mathbb{R}$. 
We can rewrite this problem as
\begin{gather*}
(V_2u)(x) \equiv u''(x)=-f(x)=z_2(x),\quad x\in G=(0,1),\\
V_1u \equiv u(1)-\gamma _1\int_0^{1}u(t)dt=0=z_1,\\
V_0u \equiv u(0)-\gamma _0\int_0^{1}(1+t)u(t)dt=0=z_0.
\end{gather*}
Then, we have
\begin{gather*}
a_1=1-\gamma_1,\quad b_1=1-\frac{\gamma_1}{2},\quad 
 g_1(\xi)=1-\xi-\gamma_1(\frac{1}{2}-\xi+\frac{\xi^{2}}{2}),\\
a_0=1-\frac{3}{2}\gamma_0,\quad b_0=-\frac{5}{6}\gamma_0,\quad 
g_0(\xi)=-\gamma_0(\frac{\xi^{3}}{6}+\frac{\xi^{2}}{2}-\frac{3}{2}\xi+\frac{5}{6}),
\end{gather*}
and $A_i(x)=z_i=0$ for $i=0,1$.

Consequently, the special adjoint system \eqref{e6.3} corresponding to this problem 
is of the form
\begin{gather} \label{e7.8}
\begin{aligned}
&f_2(\xi)+f_1\{1-\xi-\gamma_1(\frac{1}{2}-\xi+\frac{\xi^{2}}{2})\}\\
&-f_0\gamma_0(\frac{\xi^{3}}{6}+\frac{\xi^{2}}{2}-\frac{3}{2}\xi+\frac{5}{6})
= (x-\xi)H(x-\xi),
\end{aligned}\\
(1-\frac{\gamma_1}{2})f_1-\frac{5}{6}\gamma_0f_0 = x, \label{e7.9}\\
(1-\gamma_1)f_1+(1-\frac{3}{2}\gamma_0)f_0 = 1, \label{e7.10}
\end{gather}
where $\xi\in (0,1)$ and, $H(x-\xi)$ is Heaviside function on $\mathbb{R}$.
 We firstly determine $f_1$ and $f_0$ with using only \eqref{e7.9} and \eqref{e7.10}
 under the condition 
$\Delta_2=(1-\frac{\gamma_1}{2})(1-\frac{3}{2}\gamma_0)
+\frac{5}{6}\gamma_0(1-\gamma_1)\neq 0$ in order to solve \eqref{e7.8}-\eqref{e7.10}. 
Hence, we have
\[
f_1=\frac{(1-\frac{3}{2}\gamma_0)x+\frac{5}{6}\gamma_0}{\Delta_2},\quad 
f_0=\frac{1-\frac{\gamma_1}{2}-x(1-\gamma_1)}{\Delta_2}.
\]
After substituting $f_1$ and $f_0$ into equation \eqref{e7.8}, $f_2(\xi)$ becomes
\begin{align*}
f_2(\xi)
&= (x-\xi)H(x-\xi)+\frac{[1-\frac{\gamma_1}{2}-x(1-\gamma_1)]}{\Delta_2}
 \gamma_0(\frac{\xi^{3}}{6}+\frac{\xi^{2}}{2}-\frac{3}{2}\xi+\frac{5}{6})\\
&\quad -\frac{[(1-\frac{3}{2}\gamma_0)x+\frac{5}{6}\gamma_0]}{\Delta_2}
 \{1-\xi-\gamma_1(\frac{1}{2}-\xi+\frac{\xi^{2}}{2})\}.
\end{align*}
Thus, the Green's functional $f(x)=(f_2(\xi,x),f_1(x),f_0(x))$ 
for the problem has been determined. The first component $f_2(\xi,x)=f_2(\xi)$ 
is the Green's function for the problem. After substituting $\xi=s$ 
for notational compatibility, $f_2(\xi,x)$ is equal to the Green's 
function constructed in~\cite{RoSt09} for the problem. 
By \eqref{e6.5} in Theorem \ref{thm6.1}, the representation of the existing solution 
for the problem can be given as
\begin{align*}
u(x)
&= \int_0^{1}-[(x-\xi)H(x-\xi)+\frac{[1-\frac{\gamma_1}{2}-x(1-\gamma_1)]}{\Delta_2}
\gamma_0(\frac{\xi^{3}}{6}+\frac{\xi^{2}}{2}-\frac{3}{2}\xi+\frac{5}{6})\\
& \quad -\frac{[(1-\frac{3}{2}\gamma_0)x+\frac{5}{6}\gamma_0]}{\Delta_2}
 \{1-\xi-\gamma_1(\frac{1}{2}-\xi+\frac{\xi^{2}}{2})\}]f(\xi)d\xi.
\end{align*}
\end{example}

\begin{example} \label{examp7.3} \rm
Finally, we consider the following problem to reduce an integral equation 
by using the Green's functional concept:
\begin{gather}
u''(x)+\lambda u(x)=f(x),\quad x\in G=(0,1), \label{e7.11} \\
u(0)=\alpha_{00}u(\beta_{00})+\alpha_{01}u'(\beta_{01})+\gamma _0\int_0^{1}u(t)dt,
 \label{e7.12} \\
u(1)=\alpha_{10}u(\beta_{10})+\alpha_{11}u'(\beta_{11})+\gamma _1\int_0^{1}u(t)dt,
\label{e7.13}
\end{gather}
where $f(x)\in L_p(G)$, $\alpha_{00},\alpha_{01},\alpha_{10},\alpha_{11},
\gamma_0,\gamma_1\in\mathbb{R}$, 
$\beta_{00},\beta_{01},\beta_{10},\beta_{11}\in \overline{G}$ and, 
$\lambda$ is a constant. We can rewrite this problem as
\begin{gather*}
(V_2u)(x) \equiv u''(x)+\lambda u(x)=f(x)=z_2(x),\quad x\in G=(0,1),\\
V_1u \equiv u(1)-\alpha_{10}u(\beta_{10})-\alpha_{11}u'(\beta_{11})
 -\gamma _1\int_0^{1}u(t)dt=0=z_1,\\
V_0u \equiv u(0)-\alpha_{00}u(\beta_{00})-\alpha_{01}u'(\beta_{01})
 -\gamma _0\int_0^{1}u(t)dt=0=z_0.
\end{gather*}
Then, we have
\begin{gather*}
a_1=1-\alpha_{10}-\gamma_1,\quad 
 b_1=1-\alpha_{10}\beta_{10}-\alpha_{11}-\frac{\gamma_1}{2}, \\
a_0=1-\alpha_{00}-\gamma_0,\quad 
 b_0=-\beta_{00}-\alpha_{01}-\frac{\gamma_0}{2}, \\
g_1(\xi)=1-\xi-\alpha_{10}(\beta_{10}-\xi)H(\beta_{10}-\xi)
 -\alpha_{11}H(\beta_{11}-\xi)-\gamma_1\frac{(1-\xi)^{2}}{2},\\
g_0(\xi)=-\alpha_{00}(\beta_{00}-\xi)H(\beta_{00}-\xi)-\alpha_{01}H(\beta_{01}-\xi)
 -\gamma_0\frac{(1-\xi)^{2}}{2},
\end{gather*}
and $A_0(x)=\lambda, A_1(x)=z_1=z_0=0$ and, $H(\cdot)$ is Heaviside
 function on $\mathbb{R}$.

Consequently, the special adjoint system \eqref{e6.3} corresponding to this
 problem is of the form
\begin{gather} \label{e7.14}
\begin{aligned}
&f_2(\xi)+\int_{\xi}^{1}f_2(s)\lambda(s-\xi)ds\\
&+ f_1\{1-\xi-\alpha_{10}(\beta_{10}-\xi)H(\beta_{10}-\xi)
 -\alpha_{11}H(\beta_{11}-\xi)-\gamma_1\frac{(1-\xi)^{2}}{2}\}\\
&+f_0\{-\alpha_{00}(\beta_{00}-\xi)H(\beta_{00}-\xi)
 -\alpha_{01}H(\beta_{01}-\xi)-\gamma_0\frac{(1-\xi)^{2}}{2}\}\\
&= (x-\xi)H(x-\xi),
\end{aligned}\\
 \label{e7.15}
(1-\alpha_{10}\beta_{10}-\alpha_{11}-\frac{\gamma_1}{2})f_1+(-\beta_{00}
-\alpha_{01}-\frac{\gamma_0}{2})f_0+\int_0^{1}f_2(s)\lambda sds=x,\\
(1-\alpha_{10}-\gamma_1)f_1+(1-\alpha_{00}-\gamma_0)f_0
+\int_0^{1}f_2(s)\lambda ds=1, \label{e7.16}
\end{gather}
where $\xi\in (0,1)$. We denote $\int_0^{1}f_2(s)\lambda sds$ and 
$\int_0^{1}f_2(s)\lambda ds$ by $E$ and $F$ respectively, and then 
determine $f_1$ and $f_0$ with using only \eqref{e7.15} and \eqref{e7.16}
 under the condition 
$\Delta_{3}=b_1a_0-b_0a_1=(1-\alpha_{10}\beta_{10}-\alpha_{11}
-\frac{\gamma_1}{2})(1-\alpha_{00}-\gamma_0)-(-\beta_{00}-\alpha_{01}
-\frac{\gamma_0}{2})(1-\alpha_{10}-\gamma_1)\neq 0$ 
in order to solve \eqref{e7.14}-\eqref{e7.16}. As a result, we have
\begin{equation} \label{e7.17}
f_1=\frac{a_0(x-E)-b_0(1-F)}{\Delta_{3}},\quad
f_0=\frac{b_1(1-F)-a_1(x-E)}{\Delta_{3}}.
\end{equation}
After substituting $f_1$ and $f_0$ into \eqref{e7.14}, we have
\begin{equation} \label{e7.18}
\begin{aligned}
&f_2(\xi)+\int_{\xi}^{1}f_2(s)\lambda(s-\xi)ds
+\{\frac{a_0(x-E)-b_0(1-F)}{\Delta_{3}}\}\\
&\times\{1-\xi-\alpha_{10}(\beta_{10}-\xi)H(\beta_{10}-\xi)
 -\alpha_{11}H(\beta_{11}-\xi)-\gamma_1\frac{(1-\xi)^{2}}{2}\}\\
&+\{\frac{b_1(1-F)-a_1(x-E)}{\Delta_{3}}\}\\
&\times \{-\alpha_{00}(\beta_{00}-\xi)H(\beta_{00}-\xi)
 -\alpha_{01}H(\beta_{01}-\xi)-\gamma_0\frac{(1-\xi)^{2}}{2}\}\\
&= (x-\xi)H(x-\xi).
\end{aligned}
\end{equation}
As can be seen from the denotations $E$ and $F$, \eqref{e7.18}
is an integral equation for $f_2(\xi)$. After $f_2(\xi)$ is determined
 by solving this integral equation, and then $f_1(x)$ and $f_0(x)$
by \eqref{e7.17}; the Green's functional $f(x)=(f_2(\xi,x),f_1(x),f_0(x))$
for the problem will have been constructed.
The first component $f_2(\xi,x)=f_2(\xi)$ will become the Green's
function for the problem. Consequently, it must be noticed that the
 Green's function is constructed under the condition $\Delta_{3}\neq 0$
in addition to the solvability conditions for the problem.
\end{example}

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\end{document}