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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 183, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/183\hfil Adjoint systems and Green functionals]
{Adjoint systems and Green functionals for second-order
linear integro-differential equations with nonlocal conditions}

\author[A. Sirma \hfil EJDE-2015/183\hfilneg]
{Ali Sirma}

\address{Ali Sirma \newline
Department of Mathematics,
Faculty of Arts and Sciences, Yuzuncu Yil University,
 65000 Van, Turkey. \newline
Department of Mathematics Engineering,
Faculty of Arts and Sciences,
Istanbul Technical University, 34469,
Istanbul, Turkey}
\email{alisirma01@gmail.com}

\thanks{Submitted February 12, 2015. Published July 2, 2015.}
\subjclass[2010]{35A24, 65N80, 34B27}
\keywords{Adjoint system; Green's functional; $p$-integrability,
\hfill\break\indent nonlocal boundary conditions}

\begin{abstract}
 In this work, we generalize so called Green's functional concept in
 literature to second-order linear integro-differential equation with
 nonlocal conditions. According to this technique, a linear completely
 nonhomogeneous nonlocal problem for a second-order integro-differential
 equation is reduced to one and one integral equation  to identify
 the Green's solution. The coefficients of the equation are assumed to be
 generally nonsmooth functions satisfying some general properties such as
 $p$-integrability and boundedness. We obtain new adjoint system and Green's
 functional for second-order linear integro-differential equation with nonlocal
 conditions. An application illustrate the adjoint system and the  Green's
 functional. Another application shows when the Green's functional
 does not exist.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks



\section{Introduction}

Let $\mathbb{R}$ be the set of real numbers. Let $G=(x_0,x_1)$ be an open 
 bounded interval in $\mathbb{R}$. Let $L^p(G)$ with $1\le p < \infty$ be the 
space of $p-$integrable functions on $G$ and let 
$W^{2,p}(G)$ with $1\le p< \infty$ be the space of all classes of functions
 $u\in L^p(G)$ of $x$ having derivatives $d^k/dx^k \in L^p(G)$, where $k=1,2$. 
The norm on the space $W^{2,p}(G)$ is defined as
$$
\| u \|_{W^{2,p}(G)}=\sum_{k=0}^{k=2}{\| \frac{d^ku}{dx^k}  \|_{L^p(G)}}. 
$$
We consider the second-order integro-differential equation
\begin{equation}
\begin{split}
(V_2)(x)&\equiv u''(x)+A_1(x)u'(x)+A_0(x)u(x)\\
&\quad +\int_{x_0}^{x_1}{[B_1(x,\xi)u'(\xi)+B_0(x,\xi)u(\xi)]d\xi}
=z_2(x), \quad x\in G
\end{split} \label{e1}
\end{equation}
subject to the nonlocal boundary conditions
\begin{equation}
\begin{gathered}
V_1u \equiv a_1u(x_0)+b_1u'(x_0)+\int_{x_o}^{x_1}{g_1(\xi)u''(\xi)d\xi}=z_1,
\\
V_0u\equiv a_0u(x_0)+b_0u'(x_0)+\int_{x_o}^{x_1}{g_0(\xi)u''(\xi)d\xi}=z_0.
\end{gathered}\label{e2}
\end{equation}
We investigate for a solution to the problem in the space $W_p=W^{2,p}(G)$. 
Furthermore, we assume that the following conditions are satisfied: 
$A_i\in L^p(G)$, $B_i \in L^1(G\times G)$ and $g_i\in L^p(G)$ for $i=0,1$ 
are given functions with $B_i^0 \in L^p(G)$, where 
$B_i^0(x)=\int_{x_0}^{x_1} | B_i(x, \xi)| d\xi$; $a_i, b_i$
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.

\begin{remark} \rm
In \cite{a1}, second-order linear integro-differential equation \eqref{e1} 
is studied with the generally nonlocal multipoint conditions
$$ 
V_i\equiv \sum _{k=0}^{n}[a_{i,k}u(\beta_k)+b_{i,k}u'(\beta_k)]=z_i, \quad  i=0,1  
$$
where $a_{i,k}$ and $b_{i,k}$ are given numbers; $\beta_k \in \bar{G}$ are 
given points with $x_0=\beta_0<\dots<\beta_n=x_1$ and $z_0$ and $z_1$ are given
real numbers.

In the nonlocal boundary conditions \eqref{e2} if we take
\begin{gather*}
a_i=\sum_{k=0}^n{a_{i,k}},\quad  b_i
=\sum_{k=1}^{n}{a_{i,k}(\beta_k-x_0)}+\sum_{k=0}^n{b_{i,k}}, \\
g_i(\xi)=\sum_{k=1}^{n}{a_{i,k}(\beta_k-\xi)H(\beta_k-\xi)}
 +\sum_{k=0}^n{b_{i,k}H(\beta_k-\xi)}
\end{gather*}
 where $H(x)$ is the heaviside function on $\mathbb{R}$, then  \eqref{e1}-\eqref{e2}
is reduced to the problem studied in \cite{a1}. Therefore \eqref{e1}-\eqref{e2}
is a generalization of the problem studied in \cite{a1}.
\end{remark}

\begin{remark} \rm
In \eqref{e1} if we take $B_1=B_2\equiv 0$, then  \eqref{e1}-\eqref{e2} 
is reduced to the problem studied in \cite{o1}.
\end{remark}

\begin{remark} \rm
In \cite{o4}, the ordinary differential equation
\begin{equation}
u''(x)+A_0(x)u(x)+A_2(x)u(x_0)=z_2(x), \quad x\in G \label{ali}
\end{equation}
is studied with the nonlocal boundary conditions \eqref{e2}. 
In \eqref{e1} if we take $A_1\equiv 0$, $B_1(x,\xi)=\frac{A_2(x)(\xi-x_1)}{(x_0-x_1)}$ 
and $B_0(x,\xi)=A_2(x)$, then  \eqref{e1}-\eqref{e2} is reduced to the problem 
studied in \cite{o4}.
\end{remark}

So the second-order linear integro-differential equation \eqref{e1} 
with nonlocal conditions \eqref{e2} is a generalization of the problems 
studied in \cite{a1,o1,o4}. For more information about adjoint system and 
Green's functional method we refer to the references in this article and 
the references therein.

\section{Adjoint space of the solution space}

Problem \eqref{e1}-\eqref{e2} is a linear nonhomogeneous problem which can 
be considered as an operator equation
\begin{equation}
Vu=z  \label{e13}
\end{equation}
with the linear operator $V=(V_2,V_1,V_0)$ and $z=(z_2(x),z_1,z_0)$.
In order that the linear operator $V$ defined from the normed space $W_p$ 
into the Banach space $E_p\equiv L^p(G)\times\mathbb{R}^2$ have an adjoint operator, 
first of all the linear operator $V$ should be a bounded operator. Since
\begin{align*}
&\| V_2u \|_{L^p(G)}\\
&=\Big(\int_{x_0}^{x_1}| V_2u(x)|^pdx\Big)^{1/p} \\
&=\Big(\int_{x_0}^{x_1}\Bigl| u''(x)+A_1(x)u'(x)+A_0(x)u(x) 
\\
&\quad +\int_{x_0}^{x_1}{[B_1(x,\xi)u'(\xi)+B_0(x,\xi)u(\xi)]d\xi} 
\Bigr|^pdx\Big)^{1/p}\\
&\le \Big(\int_{x_0}^{x_1}\Bigl[| u''(x)| +| A_1(x)u'(x)|
 +| A_0(x)u(x)|   \\
&\quad +\int_{x_0}^{x_1}{[| B_1(x,\xi)u'(\xi)| 
 +| B_0(x,\xi)u(\xi)| ]d\xi}\Bigr] ^pdx\Big)^{1/p} \\
&\le \| u\|_{W_p}\Big(\int_{x_0}^{x_1}\Bigl[1 +| A_1(x)|
 +| A_0(x)| +\int_{x_0}^{x_1}{[| B_1(x,\xi)| +| B_0(x,\xi)| ]d\xi}\Bigr] ^pdx
 \Big)^{1/p} \\
&\le \| u\|_{W_p}\Big( \Big(\int_{x_0}^{x_1}| A_1(x)|^pdx \Big)^{1/p} 
+\Big(\int_{x_0}^{x_1}| A_0(x)|^pdx \Big)^{1/p} \\
&\quad +\Big(\int_{x_0}^{x_1}\Big[\int_{x_0}^{x_1}| B_1(x,\xi)|
  d\xi\Big]^pdx\Big)^{1/p}
 +\Big(\int_{x_0}^{x_1}\Big[\int_{x_0}^{x_1}| B_0(x,\xi)| d\xi\Big]^pdx
 \Big)^{1/p}\Big) \\  
&\le \| u\|_{W_p}\Big( \Big(\int_{x_0}^{x_1}| A_1(x)|^pdx \Big)^{1/p} 
+\Big(\int_{x_0}^{x_1}| A_0(x)|^pdx \Big)^{1/p}\\
&\quad +\Big(\int_{x_0}^{x_1}[B_1^0(x)]^pdx\Big)^{1/p}
 +\Big(\int_{x_0}^{x_1}[B_0^0(x)]^pdx\Big)^{1/p}\Big)
\end{align*}
and $B_i^0\in L^p(G)$, $A_i\in L^p(G)$, for $i=0,1$ then $V_2$ is bounded 
in $L^p(G)$. And, since
$$
\| Vu\|_{E_p}=\| V_2u \|_{L^p(G)}+| V_1u| +| V_0u|,
$$
then $V$ is bounded from $W_p$ into the Banach space 
$E_p\equiv L^p(G)\times\mathbb{R}^2$ consisting of elements 
$z=(z_2(x),z_1,z_0)$ with norm
$$
\| z\|_{E_p}=\| z_2 \|_{L^p(G)}+| z_1| +| z_0|,  \quad  1\le p<\infty. 
$$
Problem \eqref{e1}-\eqref{e2} is studied by means of a new concept of the 
adjoint problem. This concept is introduced in  \cite{o1,o4} using 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}
u(x)=u(\alpha)+u'(\alpha)(x-\alpha)+\int_{\alpha}^x{(x-\xi)u''(\xi)d\xi} \label{e3}
\end{equation}
where $\alpha$ is a given point in $\bar{G}$ 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 point $\gamma$ of $\bar{G}$. In addition, the values $u(\alpha)$, 
$u'(\alpha)$ and the second 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 $\nu \in L_p(G)$, there exists one and only one $u\in W_p$ 
such that $u(\alpha)=\nu_0$, $u'(\alpha)=\nu_1$ and $u''(\alpha)=\nu_2(x)$. 
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{a1}] \label{thm2.1}
If $1\le p <\infty$, then any linear bounded functional $F\in W_p^*$ 
can be expressed as
\begin{equation}
F(x)=\int_{x_0}^{x_1}{u''(x)\varphi_2 (x)dx}+u'(x_0)\varphi_1+u(x_0)
\varphi_0 \label{e4}
\end{equation}
with a unique element $\varphi= (\varphi_2(x),\varphi_1, \varphi_0)\in E_q$ 
where $\frac{1}{p}+\frac{1}{q}=1$.
\end{theorem}

\begin{proof}
To give the proof, a bounded linear bijective operator 
$$
Nu=(u''(x),u'(x_0),u(x_0))
$$ 
is constructed from the space $W_p$ into the space $E_p$.
 Since the adjoint operator $N^*$ is also a bounded linear bijective operator 
from the space $E^*_p$ to the space $W^*_p$ then using the fact that
 $E^*_p=E_q$ for $\frac{1}{p}+\frac{1}{q}=1$, the conclusion follows. 
For the detail of the proof, see \cite{a1}.
\end{proof}

\section{Adjoint operator and adjoint system of integro-algebraic equations}

In this section we consider an explicit form for the adjoint operator
 $V^*$ of $V$. To this end, we take any linear bounded functional 
$f=(f_2(x),f_1,f_0)\in E_q$. We can also assume that
\begin{equation}
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.\label{e9}
\end{equation}
By substituting expressions \eqref{e1}-\eqref{e2} and expression \eqref{e3}
 (for $\alpha=x_0$) of $u\in W_p$ into \eqref{e9}, we obtain the equation
\begin{align*}
f(Vu)&\equiv \int_{x_0}^{x_1} f_2(x)\Big\{u''(x)+A_1(x)
\Big[u'(x_0)+\int_{x_0}^x{u''(\xi)d\xi}\Big]\\   
&\quad +A_0(x)\Big[u(x_0)+u'(x_0)(x-x_0)
+\int_{x_0}^x{(x-\xi)u''(\xi)d\xi}\Big]\\ 
&\quad + \int_{x_0}^{x_1} B_1(x,s)\Big[u'(x_0)+\int_{x_0}^s{u''(\xi)d\xi}
 \Big]ds\\    
&\quad +\int_{x_0}^{x_1} B_0(x,s)\Big[u(x_0)+u'(x_0)(s-x_0)
 +\int_{x_0}^s{(s-\xi)u''(\xi)d\xi}\Big]ds 
 \Big\} dx \\
&\quad +f_1\Big\{a_1u(x_0)+b_1u'(x_0)+\int_{x_0}^{x_1}{g_1(\xi)u''(\xi)d\xi}
 \Big\} \\
&\quad  +f_0\Big\{a_0u(x_0)+b_0u'(x_0)+\int_{x_0}^{x_1} g_0(\xi)u''(\xi)d\xi
\Big\}.
\end{align*}  
After some calculations, we obtain
\begin{equation}
\begin{split}
f(Vu)
&\equiv \int_{x_0}^{x_1}{f_2(x)(V_2u)(x)dx}+f_1(V_1u)+f_0(V_0u)\\
&=\int_{x_0}^{x_1}(\omega_2f)(\xi)u''(\xi)d\xi+(\omega_1f)u'(x_0)+(\omega_0f)u(x_0)\\
&\equiv(\omega f)(u), \quad \text{for any $f\in E_q$ and any 
$u\in W_p$, $1\le p\le\infty$},
\end{split}\label{e11}
\end{equation}
where
\begin{equation}
\begin{gathered}
\begin{aligned}
(\omega_2f)(\xi)
&=f_2(\xi)+f_1g_1(\xi)+f_0g_0(\xi)+\int_{\xi}^{x_1}f_2(s)[A_0(s)(s-\xi)+A_1(s)
]ds\\
&\quad +\int_{x_0}^{x_1}f_2(x)\Bigl[\int_{\xi}^{x_1}B_1(x,s)ds+\int_{\xi}^{x_1}B_0(x,s)(s-\xi)ds\Bigr]dx,\\
\omega_1f&=b_1f_1+b_0f_0+\int_{x_0}^{x_1}f_2(x)[A_0(x)(x-x_0)+A_1(x)
]dx\\
&\quad +\int_{x_0}^{x_1}\int_{x_0}^{x_1}f_2(x)[B_0(x,s)(s-x_0)+B_1(x,s)]dsdx,
\end{aligned} \\
\omega_0f=a_1f_1+a_0f_0+\int_{x_0}^{x_1}f_2(x)A_0(x)dx
+\int_{x_0}^{x_1}\int_{x_0}^{x_1}f_2(x)B_0(x,s)ds dx.
\end{gathered} \label{e12}
\end{equation}

As shown in the beginning of the second section, the linear operator $V$ 
defined from the normed space $W_p$ into the Banach space $E_p$ is bounded, 
its adjoint should be also be linear and bounded. As in the section two, 
the boundedness of the linear operators $\omega_2$, $\omega_1$, $\omega_0$ 
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, can be shown. 
Therefore, the operator $\omega =(\omega_2, \omega_1,\omega_0):E_q\to E_q$
 represented by $\omega f=(\omega_2f,\omega_1f,\omega_0f)$ is linear and bounded.
 By \eqref{e11} and Theorem \ref{thm2.1}, the operator $\omega$ is an adjoint 
operator for the operator $V$ when $1\le p<\infty$, in other words, $V^*=\omega$.

Following the articles \cite{a1,o1,o4}, equation \eqref{e13} can be transformed  
into the equivalent equation
\begin{equation}
VSh=z, \label{e14}
\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, \eqref{e11} can be written as
\begin{equation}
\begin{split}
f(VSh)&\equiv \int_{x_0}^{x_1}{f_2(x)(V_2Sh)(x)dx}+f_1(V_1Sh)+f_0(V_0Sh)\\
&=\int_{x_0}^{x_1}(\omega_2f)(\xi)h_2(\xi)d\xi+(\omega_1f)h_1+(\omega_0f)h_0\\
&\equiv (\omega f)(h)\quad \text{for any } f\in E_q, \quad \text{for any } u\in W_p, \quad 1\le p\le\infty.
\end{split}\label{e15}
\end{equation}
Therefore the operator $VS$ is the adjoint of the operator $\omega$. Consequently, the equation
\begin{equation}
\omega f=\varphi \label{e16}
\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{e13} and \eqref{e14} 
for all $1\le p \le\infty$. Equation \eqref{e16} can be written in explicit
 form as the system of equations
\begin{equation}
\begin{gathered}
(\omega_2f)(\xi)=\varphi_2(\xi), \quad \xi\in G,\\
\omega_1f=\varphi_1, \\
\omega_0f=\varphi_0.
\end{gathered}\label{e17}
\end{equation}

\section{Solvability conditions for the completely nonhomogeneous problem}

Using the argument in the articles \cite{a1,a3}, we consider the operator 
$Q=\omega -I_q$, where $I_q$ is the identity operator on $E_q$. 
This operator can also be defined as $Q=(Q_2,Q_1,Q_0)$ with
\begin{equation}
\begin{gathered}
(Q_2f)(\xi)=(\omega_2f)(\xi)-f_2(\xi), \quad \xi\in G;\\
Q_1f=\omega_1f-f_1,\\
Q_0f=\omega_0f-f_0.
\end{gathered}\label{e18}
\end{equation}
The expressions in \eqref{e12} and the conditions imposed on $A_i$ and $b_i$ 
show that $Q_2$ is a compact operator from $E_q$ into $L_q(G)$ and also $Q_1$ 
and $Q_0$ are compact operators from $E_q$ into $\mathbb{R}$, where $1<p<\infty$. 
Therefore, $Q:E_q \to E_q$ is a compact operator and therefore has a compact 
adjoint operator $Q^*:E_p\to E_p$. Since $\omega =Q+I_q$ and $VS=Q^*+I_p$, 
where $I_p=I_q^*$, we have that \eqref{e14} and \eqref{e16} are canonical 
Fredholm type equations. Consequently, we have the following result.

\begin{theorem}[\cite{a1}] \label{thm4.1}
 Assume that $1<p<\infty $. Then the homogenous equation $Vu=0$ has either 
only the trivial solution or a finite number of linearly independent 
solutions in $W_p$:

(1) If $Vu=0$ has only the trivial solution in $W_p$ then also $\omega f=0$ 
has only the trivial solution in $E_q$. Then the operators $V:W_p \to E_p$ and 
$\omega: E_q\to E_q$ become linear homeomorphisms.

(2) If $Vu=0$ has $m$ linear independent solutions $u_1,u_2,\dots,u_m$ in $W_p$,
 then the equation $\omega f=0$ also has $m$ linear independent solutions
\begin{equation*}
f^{(1)}=(f_2^{(1)}(x),f_1^{(1)},f_0^{(1)}),\dots,f^{(m)}
=(f_2^{(m)}(x),f_1^{(m)},f_0^{(m)})
\end{equation*}
in $E_q$. In this case, \eqref{e13} and \eqref{e16} 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{gather}
\int_{x_0}^{x_1}{f_2^{(i)}(\xi)z_2(\xi)d_\xi}+f_1^{(i)}z_1+f_0^{(i)}z_0=0, 
\quad i=1,2,\dots,m, \\
\int_{x_0}^{x_1}{\varphi_2(\xi)u_1''(\xi)d\xi}
+\varphi_1u_i'(x_0)+\varphi_0u_i(x_0)=0,\quad i=1,2,\dots ,m,
\end{gather}
are satisfied.
\end{theorem}

\section{Green's functional}

Consider the equation
\begin{equation}
(\omega f)(u)=u(x),\quad \forall u\in W_p,\label{e19}
\end{equation}
given in the form of a functional identity, where 
$f=(f_2(\xi),f_1,f_0)\in E_q$ is an unknown triple and $x\in \bar{G}$ is a parameter.

\begin{definition}[\cite{a1}] \label{def5.1} \rm
Suppose that $f(x)=(f_2(\xi,x),f_1(x),f_0(x))\in E_q$ is a triple with a 
parameter $x\in \bar{G}$. If for a given $x\in \bar{G}$, $f=f(x)$ is a solution 
of functional equation \eqref{e19} then $f(x)$ is called a Green's functional
of $V$ or a Green's functional of  \eqref{e13}.
\end{definition}

Due to the operator $I_{W_p,C}$ of the imbedding of $W_p$ into the space 
$C(\bar{G})$ of continuous functions on $\bar{G}$ is bounded, the linear 
functional $\eta (x)$ defined by $\eta(x)(u)=u(x)$ is bounded on $W_p$ 
for a given $x\in\bar{G}$. On the other hand, $(\omega f)(u)=(V^*f)(u)$. 
Thus, \eqref{e19} can also be written as, \cite{a2,a3},
$$ 
(V^*f)=\eta(x).
 $$
In other words, \eqref{e19} can be considered as a special case of the adjoint 
equation $V^*f=\psi$ for some $\psi=\eta(x)$.

By substituting $\alpha=x_0$ into \eqref{e3} and using \eqref{e11}, we can 
write \eqref{e19} as
\begin{equation}
\begin{split}
&\int_{x_0}^{x_1}(\omega_2f)(\xi)u''(\xi)d\xi
 +(\omega_1f)u'(x_0)+(\omega_0f)u(x_0)\\
&=\int_{x_0}^{x}(x-\xi)u''(\xi)d\xi+u'(x_0)(x-x_0)+u(x_0),
\end{split}\label{e20}
\end{equation}
for any $f\in E_q$ and any $u\in W_p$.
The elements $u''\in L_p(G)$, $u'(x_0)\in \mathbb{R}$ and
 $u(x_0)\in \mathbb{R}$ of the function $u\in W_p$ are unrelated. 
Then, we can construct the system
\begin{equation}
\begin{gathered}
(\omega_2f)(\xi)=(x-\xi)H(x-\xi), \quad \xi\in G, \\
(\omega_1 f)=(x-x_0),\\
(w_0f)=1,
\end{gathered}\label{e21}
\end{equation}
where $H(x-\xi)$ is the Heaviside function on $\mathbb{R}$.

Equation \eqref{e19} is equivalent to the system \eqref{e21} which is a 
special case for the adjoint system \eqref{e17} 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 the system \eqref{e21} for an arbitrary $x\in \bar{G}$.
For a solution $u\in W_p$ of \eqref{e13}, we can rewrite \eqref{e11} as
\begin{equation}
\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}\label{e22}
\end{equation}
The right side of \eqref{e22} is equal to $u(x)$. Therefore, we can state 
the following theorem.

\begin{theorem}[\cite{a1}] \label{thm5.2}
 If \eqref{e13} has at least one Green's functional $f(x)$, then any
solution $u\in W_p$ of \eqref{e13} can be represented by
\begin{equation}
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}
In particular the homogenous equation $Vu=0$ has only the trivial solution.
\end{theorem}

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

\begin{theorem}[\cite{a1}] \label{thm5.3}
 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}

\section{Applications}

In this section we present some applications of the theory investigated
 above.

\begin{example} \label{examp6.1} \rm
First let us consider the  problem
\begin{gather}
u''(x)+xu(\frac{1}{2})=g(x), \quad x\in G=(0,1) \label{e23}\\
u(0)=\frac{1}{4}u'(\frac{1}{3}), \quad u'(0)=\frac{1}{5}u(\frac{1}{6})\label{e24}
\end{gather}
where $g\in L_p(G)$. Using the identities
\begin{gather*}
u(\alpha)=\int_{0}^{1}\frac{1}{\alpha}H(\alpha-\xi)\xi u'(\xi)d\xi
+\int_{0}^{1}\frac{1}{\alpha}H(\alpha-\xi)u(\xi)d\xi, \quad \alpha\in G=(0,1),\\
u(c)=u(0)+cu'(0)+\int_{0}^{1}(c-\xi)H(c-\xi)u''(\xi)d\xi,\quad c\in G=(0,1),\\
u'(c)=u'(0)+\int_{0}^{1}H(c-\xi)u''(\xi)d\xi,\quad c\in G=(0,1),
\end{gather*}
for $x\in G=(0,1)$. We can rewrite this problem as
\begin{gather*}
(V_2u)(x)=u''(x)+\int_{0}^{1}[2x\xi u'(\xi)+2xu(\xi)]H(\frac{1}{2}-\xi)d\xi
=g(x)=z_2(x), \\
(V_1u)=u(0)-\frac{1}{4}u'(0)-\int_{0}^{1}\frac{1}{4}H(\frac{1}{3}-\xi)
 u''(\xi)d\xi=0=z_1, \\
(V_0u)=-\frac{1}{5}u(0)+\frac{29}{30}u'(0)+\int_{0}^{1}
(\frac{1}{6}-\xi)H(\frac{1}{6}-\xi)u''d\xi
=0=z_0.
\end{gather*}
Therefore, we have 
\begin{gather*}
A_1(x)=A_0(x)=0, \quad B_1(x,\xi)=2x\xi H(\frac{1}{2}-\xi),\\
B_0(x,\xi)=2x H(\frac{1}{2}-\xi), \quad
a_1=1,\quad b_1=-\frac{1}{4},\\
g_1(\xi)=-\frac{1}{4}H(\frac{1}{3}-\xi), \\
a_0=-\frac{1}{5},\quad b_0=\frac{29}{30},\quad 
g_0(\xi)=(\frac{1}{6}-\xi)H(\frac{1}{3}-\xi), \\
z_2(x)=g(x), \quad z_1=z_0=0.
\end{gather*}
 Thus, the adjoint system corresponding to the problem \eqref{e23}-\eqref{e24} is
\begin{equation}
\begin{gathered}
\begin{aligned}
(\omega_2f)(\xi)
&=f_2(\xi)-f_1\frac{1}{4}H(\frac{1}{3}-\xi)+f_0(\frac{1}{6}-\xi)H(\frac{1}{3}-\xi) \\
&\quad +\int_{0}^{1}f_2(x)\Bigl[\int_{\xi}^{1}2xsH(\frac{1}{2}-s)ds
+\int_{\xi}^{1}2x H(\frac{1}{2}-s)(s-\xi)ds\Bigr]dx\\
&=\varphi_2(\xi),
\end{aligned}\\
\omega_1f=-\frac{1}{4}f_1+\frac{29}{30}f_0
+\int_{0}^{1}\int_{0}^{1}f_2(x)4xsH(\frac{1}{2}-s)dsdx=\varphi_1(\xi),\\
\omega_0f=f_1-\frac{1}{5}f_0+\int_{0}^{1}\int_{0}^{1}f_2(x)2x H(\frac{1}{2}-s)dsdx=\varphi_0(\xi),
\end{gathered}\label{e25}
\end{equation}
where $f=(f_2(x),f_1,f_0)\in E_q$ is unknown function and 
$\varphi=(\varphi_2(x),\varphi_1,\varphi_0)\in E_q$ is a given function.
In \eqref{e25}, if we take $\varphi_2(x)=(x-\xi)H(x-\xi)$, $\varphi_1=x$ and 
$\varphi_0=1$ then we can obtain the special adjoint system corresponding 
to the problem \eqref{e23}-\eqref{e24} as
\begin{gather}
\begin{aligned}
&f_2(\xi)-\frac{1}{4}H(\frac{1}{3}-\xi)f_1+(\frac{1}{6}-\xi)H(\frac{1}{3}-\xi)f_0\\
&+\int_{0}^{1}\int_{\xi}^{1}f_2(x)[4xs
-2x\xi]H(\frac{1}{2}-s)dsdx=(x-\xi)H(x-\xi),
\end{aligned} \label{e28} \\
-\frac{1}{4}f_1+\frac{29}{30}f_0
+\int_{0}^{1}\int_{0}^{1}f_2(x)4xsH(\frac{1}{2}-s)dsdx=x, \label{e29}
\\
f_1-\frac{1}{5}f_0+\int_{0}^{1}\int_{0}^{1}f_2(x)2x H(\frac{1}{2}-s)dsdx=1,
\label{e30}
\end{gather}
where $\xi \in (0,1)$. To solve the system of equations \eqref{e28}, \eqref{e29}), 
\eqref{e30}, first we solve the equations \eqref{e29} and \eqref{e30}
 to determine $f_0$ and $f_1$ with respect to $f_2$, then we find that
\begin{gather*}
f_0=\frac{3}{11}(4x+1)-\frac{9}{11}K(x), \\
f_1=\frac{1}{55}(12x+58)-\frac{64}{55}K(x),
\end{gather*}
where $K(\alpha)=\int_{0}^{1}xf_2(x,\alpha)dx$. 
After substituting $f_1$ and $f_0$ into the equation \eqref{e28}, $f_2(\xi)$
can be found as
\begin{equation}
\begin{split}
f_2(\xi)&=\Big(-\frac{14}{55}+\frac{19}{110}-\frac{17}{110}K(x)+\frac{3}{11}(4x+1)
-\frac{9\xi}{11}K(x)\Big) H(\frac{1}{3}-\xi)\\
&\quad -\int_{0}^{1}\int_{\xi}^{1}f_2(x)[4xs
-2x\xi]H(\frac{1}{2}-s)dsdx+(x-\xi)H(x-\xi),
\end{split}
\end{equation}
Thus, the Green's functional $f(x)=(f_2(\xi,x),f_1(x),f_0(x))$ 
for the problem has been determined. Therefore, by Theorem \ref{thm5.2}, 
a solution $u\in W_p$ of the problem \eqref{e23}-\eqref{e24}
 can be represented as
$$
u(x)=\int_{x_0}^{x_1}f_2(\xi,x)g(\xi)d\xi.
$$
\end{example}

\begin{example} \label{exam6.2} \rm
Now, let us consider the problem
\begin{gather}
u''(x)+u(x)-\frac{1}{2}u(0)=g(x), \quad x\in G=(0,\pi) \label{e26}\\
u(\pi)=0\quad u'(0)=0\label{e27}
\end{gather}
where $g\in L_p(G)$. Using the identities given in Example \ref{examp6.1}, 
for $x\in G=(0,\pi)$ the problem \eqref{e26}-\eqref{e27} can be written as
\begin{gather}
u''(x)+u(x)-\frac{1}{2}\int_{0}^{1}[\xi u'(\xi)+u(\xi)]d\xi=g(x)=z_2(x), \label{e31}
\\
u(0)+\pi u'(0)+\int_{0}^{\pi}(\pi-\xi)u''(\xi)d\xi=0=z_1,\quad
u'(0)=0=z_0. \label{e32}
\end{gather}
As is given in Theorem \ref{thm5.3}, in order for problem \eqref{e26}-\eqref{e27} 
or \eqref{e31}-\eqref{e32} to have a Green's functional, the corresponding 
homogenous problem should have only the trivial solution. But, the corresponding 
homogenous problem
\begin{gather}
u''(x)+u(x)-\frac{1}{2}\int_{0}^{1}[\xi u'(\xi)+u(\xi)]d\xi=0, 
\quad x\in G=(0,\pi), \\
u(0)+\pi u'(0)+\int_{0}^{\pi}(\pi-\xi)u''(\xi)d\xi=0,\quad
u'(0)=0,
\end{gather}
has a solution $u(x)=5\cos{x}+5$, other than the trivial solution. 
So  problem \eqref{e26}-\eqref{e27} or  problem \eqref{e31}-\eqref{e32} 
does not have any Green's functionals in accordance with Definition 
\ref{def5.1}.
\end{example}

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\end{document}