\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small {\em
Electronic Journal of Differential Equations},
 Vol. 2005(2005), No. 28, 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  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/28\hfil A new Green function concept]
{A new Green function concept for fourth-order differential equations}

\author[K. Orucoglu \hfil EJDE-2005/28\hfilneg]
{Kamil Orucoglu}

\address{Istanbul Technical University,
Faculty of Science, Maslak 34469,
Istanbul, Turkey}
\email{koruc@itu.edu.tr}

\date{}
\thanks{Submitted November 19, 2004. Published March 6, 2005.}
\subjclass[2000]{34A30, 34B05, 34B10, 34B27, 45A05, 45E35, 45J05}
\keywords{Green function; linear operator; multipoint;
nonlocal problem; \hfill\break\indent
nonsmooth coefficient; differential equation}

\begin{abstract}
 A linear completely nonhomogeneous generally nonlocal multipoint
 problem is investigated for a fourth-order differential equation
 with generally nonsmooth coefficients satisfying some general
 conditions such as $p$-integrability and boundedness. A system of
 five integro-algebraic equations called an adjoint system is
 introduced for this problem. A concept of a Green functional is
 introduced as a special solution of the adjoint system. This new
 type of Green function concept, which is more natural than the
 classical Green-type function concept, and an integral form of the
 nonhomogeneous problems can be found more naturally. Some
 applications are given for elastic bending problems.
\end{abstract}

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

The Green functions of linear boundary-value problems for
ordinary differential equations with sufficiently smooth coefficients
have been investigated in detail in several studies
\cite{k1,n1,s1,s2,t1}. In
this work, a linear, generally nonlocal multipoint problem is
investigated for a differential equation of fourth-order. The
coefficients of the equation are assumed to be generally nonsmooth
functions satisfying some general conditions 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 on a space of distributions
\cite{h1,s1}. 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 application of the
classical methods for such a problem. As it follows from
\cite[p. 87]{k1}, 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 new approach is introduced for the investigation of
the considered problem and other similar problems. This approach
is based on \cite{a1,a2,a3} and on methods of functional analysis.
The main idea of this approach is related to the use of a new
concept of the adjoint problem named ``adjoint system''. Such
an adjoint system, in fact, includes five ``integro-algebraic''
equations with an unknown elements $(f_4(\zeta ), f_3, f_2, f_1,
f_0)$ in which $f_4(\zeta )$ is a function, and $f_j$, $j=0, 1, 2,
3$ are real numbers. One of these equations is an integral
equation with respect to $f_4(\zeta )$ and generally includes
$f_j$ as parameters. The other four  can be considered a
system of four algebraic equations with respect to $(f_0, f_1
f_2, f_3)$, and they may include some integral functionals defined
on $f_4(\zeta )$. The form of our adjoint system depends on the
operators of the equation and the conditions. The role of our
adjoint system is similar to that of the adjoint operator equation
in the general theory of the linear operator equations in
Banach spaces \cite{b1,k1,k2}. The integral
representation of the solution is obtained by a concept of the
``Green functional'' which is introduced as a special solution
$f(x)=(f_4(\zeta , x), f_3(x), f_2(x), f_1(x), f_0(x))$ of the
corresponding adjoint system having a special free term depending
on $x$ as a parameter. The superposition principle for the
equation is given by the first element $f_4(\zeta , x)$ of the
Green functional $f(x)$; the other four elements $f_j(x)$, $(j=0, 1,
2, 3)$  correspond to the unit effects of the conditions.
If the homogeneous problem has a nontrivial solution, then the
Green functional does not exist. The present approach for
the Green functionals is constructive.  In principle, this
approach is different from the classical methods for
constructing  Green type functions \cite{s2}.


\section{Statement of the problem}

Let $\mathbb{R}$ be the set of the real numbers.
Let $G=(x_0, x_1)$ be a bounded open interval in $\mathbb{R}$,
Let $L_p(G)$, with  $1\le 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 $W_p^{(4)}(G)$, $1\le p\le
\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 $k=1,\dots , 4$.
The norm in the space $W_p^{(4)}(G)$ is defined as
$$
\Vert u\Vert_{W_{p}^{(4)}(G)}=\sum_{k=0}^{4}\Vert {d^ku\over
dx^k}\Vert_{L_{p}(G)}\,.
$$
We consider the differential equation
\begin{equation}
(V_4u)(x)\equiv u^{(iv)}(x)+A_0(x)u (x)+A_1(x)u'
(x)+A_2(x)u''(x) +A_3(x)u'''(x)=z_4(x),
\label{e2.1}
\end{equation}
$x\in G$, subject to the following generally nonlocal multipoint-boundary
conditions
\begin{equation}
\begin{gathered}
V_0u\equiv u(x_0)=z_0; \\
V_1u\equiv u' (x_0)=z_1;\\
V_2u\equiv \alpha_1u(\beta
)+\alpha_2u''(x_1)+\alpha_3u' (x_1)=z_2;\\
V_3u\equiv u(x_1)=z_3. \end{gathered}
\label{e2.2}
\end{equation}
Problem \eqref{e2.1}-\eqref{e2.2} is considered in the space $W_p=W_p^{(4)}(G)$.
Furthermore, it is assumed that the following conditions are satisfied:
$A_j\in L_p(G)$ are given functions, where
$j=0, 1, 2, 3$; $\alpha_j$ are given numbers; $\beta \in \bar{G}$ is
given point with $x_0<\beta <x_1$; $z_4\in L_p(G)$ is given
function, and $z_j$ are given numbers.

Problem \eqref{e2.1}-\eqref{e2.2} is a linear completely nonhomogeneous problem
which can be considered an operator equation:
\begin{equation}
Vu=z, \label{e2.3}
\end{equation}
with the linear operator $V=(V_4, V_3, V_2, V_1, V_0)$ and $z=(z_4(x), z_3,
z_2, z_1, z_0)$.

The conditions given above show that $V$ is bounded from $W_p$
to the Banach space
$E_p=L_p(G)\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}$
consisting of element $z=(z_4(x), z_3, z_2, z_1, z_0)$ with
$$
\Vert z\Vert_{E_p}=\Vert z_4\Vert_{L_p(G)}+\vert z_3\vert +\vert z_2\vert
+\vert z_1\vert +\vert z_0\vert , \quad 1\le p\le \infty .
$$
If, for a given $z\in E_p$,  problem \eqref{e2.1}-\eqref{e2.2} has a unique
solution $u\in W_p$ with $\Vert u\Vert_{W_p}\le c_0\Vert z\Vert_{E_p}$, 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 $V$ is a (linear) homeomorphism  between $W_p$ and $E_p$.

\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 following \cite{a2,a3} by the
adjoint operator $V^\star $ of $V$. Furthermore, some isomorphic
decompositions of the space $W_p$ of the solutions and its adjoint space
$W_p^\star $ will be employed.

It is well known that any function $u\in W_p$
can be represented as
\begin{equation}
\begin{aligned}
u(x)=&u(\alpha )+u' (\alpha )(x-\alpha )+ u''(\alpha
){(x-\alpha )^2\over 2}\\
&+u'''(\alpha ){(x-\alpha )^3\over 6}
+\int_\alpha ^x {(x-\zeta )^3\over 6}u^{(iv)}(\zeta )d\zeta ,
\end{aligned}\label{e3.1}
\end{equation}
where $\alpha \in \bar{G}$ is a given point. Furthermore, the trace or
the value operators  $D_0u=u(\gamma )$, $D_1u=u' (\gamma )$,
$D_2u=u''(\gamma )$ and $D_3u=u'''(\gamma )$
are bounded and surjective from $W_p$ onto $\mathbb{R}$ for a given
$\gamma\in \bar{G}$. In addition, the values $u(\alpha )$, $u' (\alpha )$,
$u''(\alpha )$, $u'''(\alpha )$ and  $u^{(iv)}(x)$ are unrelated elements
of the function $u\in W_p$ in the following sense:
For arbitrary numbers $\nu_j$ and
an arbitrary function $\nu_4\in L_p(G)$,
there exists one and only one
$u\in W_p$ such that $u(\alpha )=\nu_0$, $u' (\alpha )=\nu_1$,
$u''(\alpha )=\nu_2$, $u'''(\alpha )=\nu_3$, and
$u^{(iv)}(x)=\nu_4(x)$.
These assertions show that there exists a
linear homeomorphism  between $W_p$ and $E_p$. That is, the space $W_p$ has the
isomorphic decomposition $W_p=L_p(G)\times \mathbb{R}\times
\mathbb{R}\times \mathbb{R}\times \mathbb{R}$.

\begin{theorem} \label{thm3.1}
If $1\le p<\infty $, then any linear bounded functional
$F\in W_p^\star $ can be represented as
\begin{equation}
F(u)=\int_{x_{0}}^{x_{1}}u^{(iv)}(x)\varphi_4(x)dx+u'''
(x_0)\varphi_3 +u''(x_0)\varphi_2+u'
(x_0)\varphi_1+u(x_0)\varphi_0
\label{e3.2}
\end{equation}
with a unique element $\varphi =(\varphi_4(x), \varphi_3,  \varphi_2,  \varphi_1,
\varphi_0)\in E_q$,
where $p+q=pq$. Any linear bounded functional $F\in W_\infty^\star $
can be represented as
\begin{equation}
F(u)=\int_{x_{0}}^{x_{1}}u^{(iv)}(x)d\varphi_4+u'''
(x_0)\varphi_3 +u''(x_0)\varphi_2+u'
(x_0)\varphi_1+u(x_0)\varphi_0
\label{e3.3}
\end{equation}
with a unique element $\varphi =(\varphi_4(e), \varphi_3,
\varphi_2, \varphi_1, \varphi_0)\in {\hat E_1} =(BA(\Sigma ,\mu
))\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}
\times\mathbb{R}$, where $\mu $ is the Lebesque measure on
$\mathbb{R}$, $\Sigma $ is $\sigma $-algebra of the $\mu
$-measurable subsets $e\subset G$ and $BA(\Sigma , \mu )$ is the
space of  bounded additive functions $\varphi_4(e)$ defined on
$\Sigma $ with $\varphi_4(e)=0$ when $\mu (e)=0$ \cite[p.
192]{k2}. The inverse is also valid, that is, if $\varphi \in
E_q$, then \eqref{e3.2} is bounded on $W_p$, $1\le p<\infty $, and
if $\varphi \in {\hat E_1}$, then \eqref{e3.3} is bounded on
$W_\infty $.
\end{theorem}

\begin{proof} The operator $N$ given by $Nu=(u^{(iv)}(x),
u'''(x_0), u''(x_0), u' (x_0), u(x_0))$
is bounded from $W_p$ onto $E_p$, and has a bounded inverse
$N^{-1}$ defined as
%\begin{equation}
\begin{align*}
u(x)=&(N^{-1}g)(x)\\
\equiv &\int_{x_0}^{x}{(x-\zeta )^3\over 6}g_4(\zeta )d\zeta
+g_3{(x-x_0)^3\over 6} +g_2{(x-x_0)^2\over 2}
+g_1(x-x_0)+g_0,\\
 &g=(g_4(x), g_3, g_2, g_1, g_0)\in E_p.
\end{align*} %\label{e3.4} \end{equation}
Clearly, the kernel of $N$ is trivial and the image of $N$ is equal
to $E_p$. Therefore, there exists a bounded adjoint
operator $N^\star :E^\star_p\to  W_p^\star $ with $\ker  N^\star
=\{ 0\} $ and $\mathop{\rm Im} N^\star =W_p^\star $. That is,
for a given $F\in W_p^\star $ there exists a unique $\psi \in E_p^\star $
such that
\begin{equation}
F=N^\star \psi \quad \hbox{or} \quad F(u)=\psi (Nu),\quad u\in W_p.
\label{e3.5}
\end{equation}
If $1\le p<\infty $, then $E_p^\star =E_q$ (in the sense of a isomorphism
(see \cite[p. 191]{k2}). Therefore, the functional $\psi $ can be represented as
\begin{equation}
\psi (g)=\int_{x_{0}}^{x_{1}}\varphi_4(x)g_4(x)dx
+\varphi_3g_1+\varphi_2g_2+\varphi_1g_1+\varphi_0g_0,  \quad g\in E_p,
\label{e3.6}
\end{equation}
with a unique element $\varphi =(\varphi_4(x), \varphi_3, \varphi_2, \varphi_1,
\varphi_0 )\in E_q$.
Part 2 of \eqref{e3.5} and \eqref{e3.6} show that any $F\in W_p^\star $ is uniquely
represented as \eqref{e3.2}. Clearly, for a given $\varphi \in E_q$, the
functional $F$ given by \eqref{e3.2} is bounded on $W_p$.
Thus, \eqref{e3.2} is a general form
of the functionals $F\in W_p^\star $. The case $p=\infty $ can be proven in a
similar way.
\end{proof}

Theorem \ref{thm3.1} shows that $W_p^\star =E_q$ for all $1\le p<\infty $, and
$W_\infty ^\star =E_\infty ^\star ={\hat E_1}$. Furthermore, we can also
consider the space $E_1$ as a subspace of the space ${\hat E_1}$.

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

The question of finding an explicit form of the adjoint operator
$V^\star $ is considered in this section.
For this reason, any element $f=(f_4(x),f_3, f_2, f_1, f_0)\in E_q$ is
considered as a linear bounded functional on $E_p$. Furthermore, it is
also assumed that
\begin{equation}
f(Vu)=\int_{x_{0}}^{x_{1}}f_4(x)(V_4u)(x)dx+f_3(V_3u)+f_2(V_2u)
+f_1(V_1u)+f_0(V_0u),
\label{e4.1}
\end{equation}
$u\in W_p$. By substituting the expressions \eqref{e2.1} and \eqref{e2.2} of
$V_4$ and $V_i$, $i=0, 1, 2, 3$,
and also the expression \eqref{e3.1} (with $\alpha =x_0$) of $u\in W_p$
into \eqref{e4.1}, we
obtain

\begin{align*}
f(Vu)=&\int_{x_{0}}^{x_{1}}f_4(x)\{u^{(iv)}(x)+A_0(x)\{ u(x_0)+u'
(x_0)(x-x_0)\\
&+u''(x_0){(x-x_0)^2\over 2}
+u'''(x_0){(x-x_0)^3\over 6}+\int_{x_0}^x{(x-\zeta )^3\over
6}u^{(iv)}(\zeta )d\zeta \} \\
&+A_1(x)\{ u' (x_0)+u''(x_0)(x-x_0)
+u'''(x_0){(x-x_0)^2\over 2}\\
&+\int_{x_{0}}^{x}{(x-\zeta )^2\over 2} )u^{(iv)}(\zeta )d\zeta \}
+A_2(x)\{ u''(x_0) +u'''(x_0)(x-x_0)\\
&+\int_{x_{0}}^{x}(x-\zeta )u^{(iv)}(\zeta )d\zeta \}
+A_3(x)\{ u'''(x_0)+\int_{x_{0}}^{x}u^{(iv)}(\zeta )d\zeta
\} \}dx  \\
&+f_3\{ u(x_0)+u' (x_0)(x_1-x_0) +u''(x_0){(x_1-x_0)^2\over
2}+u'''(x_0){(x_1-x_0)^3\over 6}\\ &+
\int_{x_{0}}^{x_{1}}{(x_1-\zeta )^3\over 6}u^{(iv)}(\zeta )d\zeta \}
+f_2\{ \alpha_1 [u(x_0)+u' (x_0)(\beta -x_0)\\
&+u''(x_0){(\beta -x_0)^2\over 2}+u'''(x_0){(\beta -x_0)^3\over 6}
+\int_{x_{0}}^{\beta} {(\beta -\zeta )^3\over 6}u^{(iv)}(\zeta )d\zeta ]\\
&+\alpha_2[ u''(x_0)+u'''(x_0)(x_1-x_0)
+\int_{x_{0}}^{x_{1}}( x _ 1 -\zeta )u^{(iv)}(\zeta )d\zeta ]\\
&+\alpha_3 [u' (x_0)+u''(x_0)(x_1-x_0)+
u'''(x_0){(x_1-x_0)^2\over 2}\\
&+\int_{x_{0}}^{x_{1}}{(x_1 -\zeta )^2\over 2}u^{(iv)}(\zeta )d\zeta
]+f_1u' (x_0)+f_0u(x_0).
\end{align*}
After some calculations, the following identity is obtained
\begin{equation}
\begin{aligned}
f(Vu)&\equiv \int_{x_{0}}^{x_{1}}f_4(x)(V_4u)(x)dx+\sum_{i=0}^3f_i(V_iu)
=\int_{x_{0}}^{x_{1}}(\omega_4f)(\zeta )u^{(iv)}(\zeta )d\zeta \\
&\quad+(\omega_3f)u'''(x_0)+(\omega_2f)u^{\prime\prime
}(x_0)+(\omega_1f)u' (x_0)+(\omega_0f)u(x_0)\\
&\equiv (\omega
f)(u),\quad \forall f\in E_q, \quad \forall u\in W_p,  \ 1\le p\le \infty ,
\end{aligned} \label{e4.2}
\end{equation}
where
\begin{align*}
(\omega_4f)(\zeta )
=&f_4(\zeta )+\int_{\zeta }^{x_{1}}f_4(s)[A_0(s){(s-\zeta
 )^3\over 6}+A_1(s){(s-\zeta )^2\over 2}\\
&+A_2(s)(s-\zeta ) +A_3(s)]ds +f_3{(x_1-\zeta )^3\over 6}\\
&+f_2[\alpha_1{(\beta -\zeta )^3\over
6}H(\beta -\zeta )+\alpha_2(x_1-\zeta )+\alpha_3{(x_1-\zeta )^2\over 2}];
\end{align*}
\begin{align*}
\omega_3f=&\int_{x_{0}}^{x_{1}}f_4(s)[A_0(s){(s-x_0)^3\over
 6}+A_1(s){(s-x_0)^2\over 2}\\
&+A_2(s)(s-x_0)+A_3(s)]ds +f_3{(x_1-x_0)^3\over 6}\\
&+f_2[\alpha_1{(\beta -x_0)^3\over
6}+\alpha_2(x_1-x_0)+\alpha_3{(x_1-x_0)^2\over 2}];
\end{align*}
\begin{equation} \label{e4.3}
\begin{aligned}
\omega_2f=&\int_{x_{0}}^{x_{1}}f_4(s)[A_0(s){(s-x_0)^2\over
2}+A_1(s)(s-x_0)+A_2(s)]ds \\
&+f_3{(x_1-x_0)^2\over 2}+f_2[\alpha_1{(\beta -x_0)^2\over
2}+\alpha_2+\alpha_3(x_1-x_0)];
\end{aligned}
\end{equation}
\begin{align*}
\omega_1f=&\int_{x_{0}}^{x_{1}}f_4(s)[A_0(s)(s-x_0)
+A_1(s)]ds +f_3(x_1-x_0)\\
&+f_2[\alpha_1(\beta -x_0) +\alpha_3]+f_1;
\end{align*}
\[
\omega_0f=\int_{x_0}^{x_1}f_4(s)A_0(s)ds +f_3 +f_2\alpha_1+f_0
\]
and $H(x)$ is the Heaviside function on $\mathbb{R}$.

The operators $\omega_4$, $\omega_3$, $\omega_2$, $\omega_1$, and $\omega_0$
are linear and bounded from
the space $E_q$ consisting  of the element $(f_4(x), f_3, f_2, f_1, f_0)$
into the spaces $L_q(G)$, $\mathbb{R}$, $\mathbb{R}$, $\mathbb{R}$ and $\mathbb{R}$,
 respectively. Therefore, the operator
 $\omega =(\omega_4, \omega_3, \omega_2, \omega_1, \omega_0)$  given by
 $\omega f=(\omega_4f, \omega_3f, \omega_2f, \omega_1, \omega_0)$
becomes linear and bounded from $E_q$ into itself. The identity
\eqref{e4.2} and Theorem \ref{thm3.1} shows that when $1\le
p<\infty $, the operator $\omega $ is an adjoint operator for the
operator $V$, that is, $V^\star =\omega $. When $p=\infty $ the
operator $\omega $ is bounded from $E_1$ into $E_1$; in this case,
$\omega $ becomes the restriction of the adjoint operator
$V^\star:E_\infty ^\star \to  W_\infty ^\star $ of $V$ onto
$E_1\subset E_\infty ^\star $.

Equation \eqref{e2.3} can be reduced to the following equivalent equation:
\begin{equation}
VSg=z \label{e4.4}
\end{equation}
with an unknown $g=(g_4, g_3, g_2, g_1, g_0)\in E_p$ by the
transformation $u=Sg$, where $S=N^{-1}$. If $u=Sg$, then
$u^{(iv)}(x)=g_4(x)$, $u'''(x_0)=g_3$, $u''(x_0)=g_2$, $u'
(x_0)=g_1$ and $u(x_0)=g_0$. Therefore, \eqref{e4.2} can be
rewritten as
\begin{equation}
\begin{aligned}
f(VSg)\equiv &\int_{x_{0}}^{x_{1}}f_4(x)(V_4Sg)(x)dx+\sum_{i=0}^3f_i(V_iSg)
=\int_{x_{0}}^{x_{1}}(\omega_4f)(\zeta )g_4(\zeta )d\zeta \\
&+(\omega_3f)g_3+(\omega_2f)g_2+(\omega_1f)g_1+(\omega_0f)g_0\equiv (\omega
f)(g),\\
&\forall f\in E_q, \quad \forall g\in E_p,  \ 1\le p\le \infty .
\end{aligned}
\label{e4.5}
\end{equation}
This shows that $V^\star =(VS)^\star =\omega $ if $1\le p<\infty $, and
$\omega^\star =VS$ if $1<p\le \infty $. That is, at least one of the operators
$VS$ and $\omega $ becomes an adjoint operator for the other one of them.
Therefore, the  equation
\begin{equation}
\omega f=\varphi \label{e4.6}
\end{equation}
with an unknown function $f=(f_4(x), f_3, f_2, f_1, f_0)\in E_q$
and a given function $\varphi =(\varphi_4(x), \varphi_3,
\varphi_2, \varphi_1, \varphi_0)$ in $E_q$ can be considered as an
adjoint equation of \eqref{e4.4} (or of \eqref{e2.3}) for all
$1\le p\le \infty $. Equation \eqref{e4.6} can be written in
explicit form as the  system of equations
\begin{equation}
\begin{gathered}
(\omega_2f)(\zeta )=\varphi_2(\zeta ), \quad \zeta \in X;\\
\omega_3f=\varphi_3, \\
\omega_2f=\varphi_2, \\
\omega_1f=\varphi_1, \\
\omega_0f=\varphi_0.\end{gathered} \label{e4.7}
\end{equation}
The expressions \eqref{e4.3} show that the first equation in
\eqref{e4.7} is an integral equation with respect to $f_4(\zeta )$
and it includes $f_3$ and $f_2$ as parameters; furthermore,
equations 2 and 3 in \eqref{e4.7} and equations 4 and 5 in
\eqref{e4.7}
become a system of four algebraic equations
with respect to $(f_3, f_2, f_1, f_0)$ and these equations include
some integral functionals defined on $f_4(\zeta )$. That is,
\eqref{e4.7} is a system of five integro-algebraic equations. This
system is introduced by the identity \eqref{e4.3} which, in fact,
is an integration by parts formula in a nonclassical form. The
traditional type adjoint problem is defined by the classical
Green's formula of the integration by parts \cite{s2}, and,
therefore, has a sense only for some restricted classes of the
problems.

\section{Solvability conditions of completely nonhomogeneous problems}

The operator is taken as $Q=\omega -I_q$, 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 $Q=( Q_4, Q_3, Q_2, Q_1, Q_0)$ with
\begin{equation}
\begin{gathered}
(Q_4f)(\zeta )=(\omega_4f)(\zeta )-f_4(\zeta ), \quad \zeta \in G;\\
Q_if=\omega_if-f_i, \quad i=0,\dots  3.
\end{gathered}
\label{e5.1}
\end{equation}
The expressions \eqref{e4.3} and the conditions imposed on $A_i$
show that $Q_4$ is a compact operator from $E_q$ into $L_q(G)$,
and also $Q_i$ are compact operators from $E_q$ into $\mathbb{R}$,
where $1<p< \infty $. That is, $Q:E_q\to  E_q$ is a compact
operator, and, therefore, has a compact adjoint operator $Q^\star
:E_p\to  E_p$. Since $\omega =Q+I_q$ and $VS=Q^\star +I_p$, where
$I_p=I_q^\star $, we have that the equations \eqref{e4.4} and
\eqref{e4.6} are canonical Fredholm type equations; furthermore,
$S$ becomes a right regularizer of \eqref{e2.3} (see \cite[p.
52]{k1}). Consequently, the following theorem is proven.

\begin{theorem} \label{thm5.1}
 Assume that $1< p<\infty $. Then $Vu=0$ has
either only the trivial solution or a finite number linearly independent
solutions in $W_p$:
\\
(i) If $Vu=0$ has only the trivial solution in $W_p$, then $\omega f=0$
also 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.
\\
(ii) If $Vu=0$ has $m$ linearly independent solutions $u_1,\dots , u_m$ in
$W_p$, then $\omega f=0$ has also $m$ linearly independent solutions
\[
f^{(1)}=(f_4^{(1)}(x), f_3^{(1)}, f_2^{(1)}  f_1^{(1)}, f_0^{(1)}), \dots
, f^{(m)}=(f_4^{(m)}(x), f_3^{(1)}, f_2^{(1)},
f_1^{(m)}, f_0^{(m)})
\]
in $E_q$. In this case, the equations \eqref{e2.3} and
\eqref{e4.6} have the 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}
\int_{x_{0}}^{x_{1}}f_4^{(i)}(\zeta )z_4(\zeta )d\zeta
+f_3^{(i)}z_3+f_2^{(i)}z_2 +f_1^{(i)}z_1+f_0^{(i)}z_0=0, \quad i=1, \dots , m,
\label{e5.2}
\end{equation}
and
\begin{equation}
\int_{x_{0}}^{x_{1}}\varphi_4(\zeta )u_i^{(iv)}(\zeta )d\zeta
+\varphi_3u_i'''
(x_0)+\varphi_2u_i''(x_0) +\varphi_1u_i'
(x_0)+\varphi_0u_i(x_0)=0,
 \label{e5.3}
\end{equation}
$i=1, \dots , m$,
are satisfied, respectively.
\end{theorem}

\section{Green Functional}

The following equation given in the form of the functional identity is
considered
\begin{equation}
(\omega f)(u)=u(x), \quad \forall u\in W_p \label{e6.1}
\end{equation}
in which $f=(f_4(\zeta ),f_3, f_2, f_1, f_0)\in E_q$ is an unknown element and
$x\in \bar{G}$ is a parameter. \smallskip

\noindent\textbf{Definition} % 6.1.
Assume that $f(x)=(f_4(\zeta , x), f_3(x), f_2(x), f_1(x), f_0(x))\in E_q$ is an
element with the parameter $x\in \bar{G}$. If $f=f(x)$ is the solution
of \eqref{e6.1} for a given $x\in \bar{G}$, then $f(x)$ is called as a
Green functional of $V$ (or of \eqref{e2.3}). \smallskip

The operator $I_{W_p,  C}$ of the imbedding of $W_p$ into the space
$C(\bar{G})$
of the continuous functions on $\bar{G}$ is bounded. Then, the linear
functional $\theta (x)$ given by  $\theta (x)(u)=u(x)$ is bounded on $W_p$ for
a given $x\in \bar{G}$. This and $(\omega f)(u)=(V^\star f)(u)$ show that the
equation \eqref{e6.1} can also be written as (see \cite{a3,a4})
\[
V^\star f=\theta (x).
\]
That is, the equation \eqref{e6.1} can be considered as a special case of the
 adjoint equation $V^\star f=\psi $ when $\psi =\theta (x)$.

Now, by employing \eqref{e3.1} with $x=x_0$ and \eqref{e4.3}, the equation
\eqref{e6.1} is written as
\begin{equation}
\begin{aligned}
&\int_{x_{0}}^{x_{1}}(\omega_4f)(\zeta )u^{(iv)}(\zeta )d\zeta +
(\omega_3f)u''' (x_0)+(\omega_2f)u''(x_0)\\
&+(\omega_1f)u' (x_0) +(\omega_0f)u(x_0)\\
&=\int_{x_{0}} ^{x} {(x-\zeta )^3\over 6}u^{(iv)}(\zeta )d\zeta +
u'''(x_0 ){(x-x_0)^3\over 6}\\
&\quad +u''(x_0) {(x-x_0 )^2\over 2}+
u' (x_0)(x-x_0)+u(x_0), \quad \forall u\in W_p.
\end{aligned} \label{e6.2}
\end{equation}
The elements $u^{(iv)}(\zeta )\in L_p(G)$, $u''' (x_0),
u''(x_0), u'(x_0)\in \mathbb{R}$ and
$u(x_0)\in \mathbb{R}$ of the functions $u\in W_p$ are unrelated. Then,
\begin{equation}
\begin{gathered}
(\omega_4f)(\zeta )={(x-\zeta )^3\over 6}H(x-\zeta ), \quad \zeta \in G;\\
\omega_3f={(x-x_0)^3\over 6}; \\
\omega_2f={(x-x_0)^2\over 2}; \\
\omega_1f=(x-x_0); \\
\omega_0f=1.\end{gathered} \label{e6.3}
\end{equation}
This shows that the equation \eqref{e6.1} is equivalent to the
system \eqref{e6.3} which is a special case of the adjoint system
\eqref{e4.7} when
\begin{gather*}
\varphi_4(\zeta )={(x-\zeta )^3\over 6}H(x-\zeta ), \quad
 \varphi_3={(x-x_0)^3\over 6},\\
 \varphi_2={(x-x_0)^2\over 2},\quad
 \varphi_1=(x-x_0), \quad \varphi_0=1\,.
\end{gather*}
Therefore, $f(x)$ is the Green functional if and only if it is a
solution of the integro-algebraic equations \eqref{e6.3} for an
arbitrary $x\in \bar{G}$. For a solution $u$ of \eqref{e2.3} and a
Green functional $f(x)$, the identity \eqref{e4.2} can be written
as
\begin{equation}
\begin{aligned}
&\int_{x_{0}}^{x_{1}}f_4(\zeta , x)z_4(\zeta )d\zeta +f_3(x)z_3+f_2(x)z_2+
f_1(x)z_1+f_0(x)z_0\\
&=\int_{x_{0}}^{x_{1}}{(x-\zeta )^3\over 6}H(x-\zeta )u^{(iv)}(\zeta
)d\zeta +u'''(x_0){(x-x_0)^3\over 6}\\
&\quad +u''(x_0){(x-x_0)^2\over 2}+u'(x_0)(x-x_0)+u(x_0).
\end{aligned} \label{e6.4}
\end{equation}
The right-hand side of \eqref{e6.4} is equal to $u(x)$. Therefore, the following
theorem holds.

\begin{theorem} \label{thm6.1}
If \eqref{e2.3} has at least one Green functional $f(x)$, then an
arbitrary solution $u\in W_p$ of \eqref{e2.3} can be represented as
\begin{equation}
u(x)=\int_{x_{0}}^{x_{1}}f_4(\zeta , x)z_4(\zeta )d\zeta +f_3(x)z_3+f_2z_2+
f_1(x)z_1+f_0(x)z_0.
\label{e6.5}
\end{equation}
Furthermore, $Vu=0$ has only one trivial solution.
\end{theorem}

If at least one of the operators $V:W_p\to  E_p$ or $\omega
:E_q\to  E_q$ is a homeomorphism, then the other one is also
a homeomorphism;
furthermore, there exists a unique Green functional, where $1\le p\le \infty $.
The Green functional exists and is unique.
The necessary and sufficient conditions for the existence of a
Green functional are given by the following theorem for the case $1<p<\infty $.

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

\begin{proof}
If there exists a Green functional, then $Vu=0$ has the unique
solution $u=0$ (Theorem \ref{thm6.1}). In this case $\omega :E_q\to  E_q$ becomes
a homeomorphism (Theorem \ref{thm5.1}). Therefore, the Green functional, as a solution
of \eqref{e6.3}, is unique. The second part of the theorem follows from
Theorem \ref{thm5.1}.
\end{proof}

\noindent\textbf{Remark.}% 6.1.
 Assume that $Vu=0$ has a nontrivial solution. Then
\eqref{e2.3} does not have a Green functional (Theorem \ref{thm6.1}).
In this case, $Vu=z$ usually has no solution unless the right-hand
side $z$ is a particular type.
For example, $Vu=z$ has no solution if
\begin{equation}
\int_{x_{0}}^{x_{1}}f_4(z)z_4(x)dx+f_3z_3+f_2z_2+f_1z_1+f_0z_0=0
\label{e6.6}
\end{equation}
is not true at least for one solution $f=(f_4(\zeta ), f_3, f_2, f_1, f_0)$ of
the homogenous adjoint equation $\omega f=0$. In this case, the
representation of the existing solution of $Vu=z$ is obtained by a concept of
the generalized Green functional \cite{a3}.

\section{Comparison with the classical Green type function}

Consider the following problem which is a special case of
\eqref{e2.3}:
\begin{equation}
\begin{gathered}
(V_4u)(x)\equiv u^{(iv)}(x)+A(x)u=z_4(x),\quad x\in G;\\
V_0u\equiv u(x_0)=z_0,\\
V_1u\equiv u'(x_0)=z_1,\\
V_1u\equiv u''(x_0)=z_2,\\
V_0u\equiv u(x_1)=z_3.
\end{gathered} \label{e7.1}
\end{equation}
In this case, system \eqref{e6.3} can be written as
\begin{align*}
(\omega_4f)(\zeta )&\equiv f_4(\zeta )+\int_{\zeta }^{x_{1}}f_4(s)A(s){(s-\zeta
)^3\over 6}ds+f_3{(x_1-\zeta )^3\over 6}+f_2(x_1-\zeta )\\
&={(x-\zeta )^3\over 6}H(x-\zeta ), \quad \zeta \in G;
\end{align*}
$$
\omega_3f\equiv \int_{x_{0}}^{x_{1}}f_4(s)A(s){(s-x_0)^3\over 6}ds
+f_3{(x_1-x_0)^3\over 6}+f_2(x_1-x_0)
={(x-x_0)^3\over 6};
$$
\begin{equation}
\omega_2f\equiv \int_{x_{0}}^{x_{1}}f_4(s)A(s){(s-x_0)^2\over 2}ds
+f_3{(x_1-x_0)^2\over 2}+f_2
={(x-x_0)^2\over 2}; \label{e7.2}
\end{equation}
\begin{gather*}
\omega_1f\equiv \int_{x_{0}}^{x_{1}}f_4(s)A(s)(s-x_0)ds+f_3(x_1-x_0)+f_1
=(x-x_0); \\
\omega_0f\equiv \int_{x_{0}}^{x_{1}}f_4(s)A(s)ds+f_3(x_1-x_0)+f_0=1.\\
\end{gather*}
 From parts 2 and 3 of \eqref{e7.2} and parts 4 and 5 of \eqref{e7.2}
 it is obtained that
\begin{equation}
\begin{aligned}
&f_3={3\Delta (x-x_0)^2-(x-x_0)^3\over 2\Delta^3}+
\int_{x_{0}}^{x_{1}}f_4(s)A(s)\{ {(s-x_0)^3)-3\Delta
(s-x_0)^2\over 2\Delta^3}\} ds;\\ &f_2={(x-x_0)^3-\Delta
(x-x_0)^2\over 4\Delta}+ \int_{x_{0}}^{x_{1}}f_4(s)A(s)\{ {\Delta
(s-x_0)^2-(s-x_0)^3\over
4\Delta}\} ds;\\
&f_1=(x-x_0)-f_3\Delta -
\int_{x_{0}}^{x_{1}}f_4(s)A(s)(s-x_0)ds;\\
&f_0=1-f_3-
\int_{x_{0}}^{x_{1}}f_4(s)A(s)ds,\quad
\Delta =x_1-x_0.\end{aligned}
\label{e7.3}
\end{equation}
Substituting parts 1 and 2 of \eqref{e7.3} into part 1 of \eqref{e7.2},
\begin{equation}
\begin{aligned}
&f_4(\zeta )+\int_{\zeta }^{x_{1}}f_4(s)A(s){(s-\zeta
)^3\over 6}ds\\ &+{(x_1-\zeta )^3\over 6}
\int_{x_{0}}^{x_{1}}f_4(s)A(s)\{{(s-x_0)^3-3\Delta (s-x_0)^2\over
2\Delta^3}\}ds\\
&+(x_1-\zeta )\int_{x_{0}}^{x_{1}}f_4(s)A(s)\{{\Delta (s-x_0)^2-(s-x_0)^3
\over 4\Delta}\}ds\\
&={(x-\zeta )\over 6}H(x-\zeta )-{(x_1-\zeta )^3\over 6}
({3\Delta (x-x_0)^2-(x-x_0)^3\over 2\Delta^3})\\
&\quad -(x_1-\zeta )({(x-x_0)^3-\Delta (x-x_0)^2\over 4\Delta})
,\quad  \zeta \in G.\end{aligned}\label{e7.4}
\end{equation}
That is, the first element $f_4(\zeta , x)$ of the Green functional
\[
f(x)=(f_4(\zeta , x), f_3(x), f_2(x) f_1(x), f_0(x))
\]
of  problem \eqref{e7.1} becomes the
solution of the independent integral equation \eqref{e7.4}; the latter four
elements
$f_3(x)$, $f_2(x)$, $f_1(x)$ and $f_0(x)$ of $f(x)$ can be obtained by \eqref{e7.3}.
The equation \eqref{e7.4}
has a unique solution $f_4(\zeta , x)\in L_q(G)$ (for given $x\in \bar{G}$) if
and only if $Vu=0$ has only the trivial solution (Theorem \ref{thm6.2}). If $Vu=0$ has a
nontrivial solution, then the Green functional does not exist.

In order to compare the Green functional with the classical type Green
function, equation \eqref{e7.4} is considered.
Assume that $A(x)$ is absolutely continuous
on $\bar{G}$. If a function $f_4(\zeta )=f_4(\zeta, x)\in L_q(G)$ is the
solution of \eqref{e7.4}, then $f_4(\zeta , x)$ is absolutely
continuous
on $\bar{G}$ with respect to $\zeta $ (for a given $x\in \bar{G}$). Therefore,
by differentiating \eqref{e7.4} with respect to $\zeta $, it is obtained
that
$f_4'''(\zeta )$ becomes absolutely continuous on $[x_0, x]$
and $[x, x_1]$ with respect to $\zeta $. Therefore,
\begin{equation}
(V_4^\star f_4)(\zeta )\equiv {d^4f_4(\zeta )\over d\zeta^4 }
+A(\zeta )f_4(\zeta )=0, \ \zeta \in (x_0, x)\cup (x, x_1).
\label{e7.5}
\end{equation}
The boundary conditions of \eqref{e7.5} can be obtained from \eqref{e7.4}
as
\begin{equation}
\begin{gathered}
f_4(x_0)=f_4(x_1)=0, \\
f_4(x+0)=f_4(x-0),\\
f_4'(x+0)=f_4'(x-0),\\
f_4''(x+0)=f_4''(x-0),\\
f_4'''(\zeta )\vert_{\zeta =x+0} =
f_4'''(\zeta )\vert_{\zeta =x-0}+1.
\end{gathered} \label{e7.6}
\end{equation}
That is, the solution of \eqref{e7.4} is equivalent to the solution of
problem \eqref{e7.5}-\eqref{e7.6}. In other words, $f_4(\zeta )$ is the
solution of problem \eqref{e7.5}-\eqref{e7.6}. Therefore, $f_4(\zeta , x)$
as a function of $\zeta $ is the
classical Green function for the corresponding traditional adjoint problem
given by $(V_4^\star f_4)(\zeta )=\psi_4 (\zeta )$, $\zeta \in G$, and
$f_4(x_0)=f_4'(x_0)=f_4''(x_1)=f_4(x_1)=0$, where
$\psi_4\in L_1(G)$ is a given function. It can
be easily proven that the function $f_4(\zeta , x)$ as a function of $x$ is the
classical Green function for the equation $\eqref{e7.1}_1$ with
$u(x_0)=u'(x_0)=u''(x_1)=u(x_1)=0$
(see  \cite[p.200]{s2}).

Let us considered some simple cases.
Let $A_j=0$, $j=0, 1, 2, 3$ and $x_0=0$, $x_1=l$ in the equations \eqref{e6.3}.
If it is taken $\beta =0$, $\alpha_2=1$ and $\vert \alpha_1\vert$ is sufficiently
small, then the system of equations \eqref{e6.3} has
unique solution. Some simple results are given below.

\noindent\textbf{(i)}
If $\alpha_1=0$, $\alpha_2=1$ and $\alpha_3=c=\mu /EI_x$
are taken in the equation $\eqref{e6.3}_1$, then the Green function of an
elastic beam having two ends which are fixed support and elastic
support is obtained as
\begin{equation}
\begin{aligned}
f_4(x, \zeta )&={(x-\zeta )^3\over 6}H(x-\zeta )-{(l-\zeta )^3\over
6}[{x^2\over l^2}-{2(1+cl)(x^3-x^2l)\over l^2(4l+cl^2)}]\\
&\quad
-(l-\zeta +{c(l-\zeta )^2\over 2})\{ {(x^3-4x^2l)\over (4l+cl^2)}\},
\end{aligned}\label{e7.7}
\end{equation}
where $\mu $, $I_x$, $E$ are elastic material constants \cite{i1}.

\noindent\textbf{(ii)} $\alpha_1=\alpha_2=0$ and $\alpha_3=1$
are taken in part 1 of \eqref{e6.3}, then the Green function
of an elastic beam having both ends fixed is obtained as
\begin{equation}
f_4(x, \zeta )={(x-\zeta )^3\over 6}H(x-\zeta )-{(l-\zeta )^3\over
6}{x^2\over l^2}(3-{2x\over l})- {x^2\over l^2}(x-l){(l-\zeta
)^2\over 2}. \label{e7.8}
\end{equation}
Note that  \eqref{e7.4} is a Fredholm's equation of the second
kind for a given $x\in \bar{G}$. Therefore, it can be solved approximately
by a known method \cite{b2,c1}. Thus, \eqref{e6.3} can also be used for the
approximate calculations of the Green functional and solution.
The present Green function concept can also be used to investigate some
classes of nonlinear equations associated with linear non-local conditions
\cite{g1,l1,m1}.
Thus, the nonlinear problem can be reduced to equivalent nonlinear integral
equations.

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 \end{document}