\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{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