\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 $$(V_4u)(x)\equiv u^{(iv)}(x)+A_0(x)u (x)+A_1(x)u' (x)+A_2(x)u''(x) +A_3(x)u'''(x)=z_4(x), \label{e2.1}$$ $x\in G$, subject to the following generally nonlocal multipoint-boundary conditions $$\begin{gathered} V_0u\equiv u(x_0)=z_0; \\ V_1u\equiv u' (x_0)=z_1;\\ V_2u\equiv \alpha_1u(\beta )+\alpha_2u''(x_1)+\alpha_3u' (x_1)=z_2;\\ V_3u\equiv u(x_1)=z_3. \end{gathered} \label{e2.2}$$ 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