\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