\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 88, pp. 1--8.\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-2005/88\hfil Functional differential equations] {Functional differential equations with non-local boundary conditions} \author[A. Guezane-Lakoud\hfil EJDE-2005/88\hfilneg] {Assia Guezane-Lakoud} \address{Assia Guezane-Lakoud\hfill\break Department of Mathematics, Faculty of Sciences, University Badji Mokhtar, B. P. 12, 23000, Annaba, Algeria} \email{a\_guezane@yahoo.fr} \date{} \thanks{Submitted May 6, 2005. Published August 15, 2005.} \subjclass[2000]{35F15, 35B45, 35R05, 34G10} \keywords{Boundary-value problems; functional differential equations} \begin{abstract} In this work, we study an abstract boundary-value problem generated by an evolution equation and a non-local boundary condition. We prove the existence and uniqueness of the strong generalized solution and its continuity to respect to the parameters. The proofs are obtained via a priori estimates in non classical functional spaces and on the density of the range of the operator generated by the considered problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} The aim of this paper is to study a class of first order equations whose operator coefficients have variable domains and the boundary conditions are non-local. Here boundary-value problems (BVP) are called non-local if certain relations between traces of a solution are set at the boundary of the domain. Many authors have studied evolution equations with Cauchy conditions; see for example \cite{k1,k2,k3,l1,l2}. In most of these papers the operator coefficients are assumed to be infinitesimal generators of analytic semi groups and have constant domains. For similar problems, various important results were proved under different assumptions in \cite{g1,g2} for hyperbolic problems and \cite{l3} for homogeneous Cauchy boundary conditions. Actually, it is difficult to construct a strict solution of the posed BVP, for this reason we prove the existence and uniqueness of the strong generalized solution, then we establish its continuity to respect to the parameters. Summary of this article is as follows: In section 1, we give the statement of the problem, the basic assumptions then we define the functional spaces in where the posed problem will be solved, then its abstract formulation. In section 2, we prove the uniqueness and continuous dependence to respect to the data of the strong generalized solution when it exists. Section 3 is devoted to prove the existence Theorem. The continuity of the strong generalized solution to respect to the parameter is proved in section 4. Finally, we give an application of the results obtained in this work for a mixed problem. \section{Statement of the problem, main assumptions, and functional spaces} In the interval $I=] 0,T[$, with $00$ independent of $t$ and $u$ such that $$\label{e3} \| u\| _{\mu }^{2}\leq C\| L_{\mu }u\| _{E}^{2},\quad \forall u\in D(L_{\mu }).$$ \end{theorem} \begin{proof} To obtain the a priori estimate \eqref{e3}, we introduce the family of abstract smoothing operators $A_{\varepsilon }^{-1}(t)=(I+\varepsilon A(t))^{-1}$, $\varepsilon >0$, with range $D(A(t))$, they have the following properties: \begin{itemize} \item[(P1)] The operators $A_{\varepsilon }^{-1}(t)$ are strongly differentiable for almost all $t$ and the derivatives $\frac{dA_{\varepsilon }^{-1}(t)}{dt}\in L_{\infty }(I,\mathcal{L}(H))$ and when $\varepsilon \rightarrow 0$, \begin{equation*} |u-A_{\varepsilon }^{-1}(t)u|=|\varepsilon A(t)A_{\varepsilon }^{-1}(t)u|\rightarrow 0,\quad \forall u\in H \end{equation*} \item[(P2)] We approximate the unbounded operators $A(t)$ using bounded operators $A(t)A_{\varepsilon }^{-1}(t)$ which are strongly differentiable for almost all $t\in I$ and \begin{equation*} \frac{d(A(t)A_{\varepsilon }^{-1}(t))}{dt}=\frac{-1}{\varepsilon } \frac{dA_{\varepsilon }^{-1}(t)}{dt}. \end{equation*} \end{itemize} Note that $(A_{\varepsilon }^{-1}(t)) ^{\ast }$ has the properties (P1)-(P2). Now, we multiply equation \eqref{e1} by $B(t)=e^{c(s-t)}( A_{\varepsilon}^{-1}(t)) ^{\ast } A_{\varepsilon }^{-1}(t)u$. Then we integrate the double real part of the result equation over the interval $I_{s}=] 0,s[\subset I$, to obtain \label{e4} \begin{aligned} &2\mathop{\rm Re}\int_{0}^{s}e^{c(s-t)}( \mathcal{L}u,( A_{\varepsilon }^{-1}(t)) ^{\ast }A_{\varepsilon }^{-1}(t)u) dt\\ &=2\mathop{\rm Re}\int_{0}^{s}e^{c(s-t)}( u_{t},( A_{\varepsilon }^{-1}(t)) ^{\ast }A_{\varepsilon }^{-1}(t)u) dt \\ &\quad +2\mathop{\rm Re}\int_{0}^{s}e^{c(s-t)}( A(t)u,( A_{\varepsilon }^{-1}(t)) ^{\ast }A_{\varepsilon }^{-1}(t)u) dt, \end{aligned} which is equivalent to \label{e5} \begin{aligned} | A_{\varepsilon }^{-1}(t)u| _{t=s}^{2} &=e^{cs}| A_{\varepsilon }^{-1}(t)u| _{t=0}^{2}+2\mathop{\rm Re} \int_{0}^{s}e^{c(s-t)}(A_{\varepsilon }^{-1}(t)\mathcal{L}u,A_{\varepsilon }^{-1}(t)u)dt \\ &\quad -c\mathop{\rm Re}\int_{0}^{s}e^{c(s-t)}| A_{\varepsilon }^{-1}(t)u| ^{2}dt\\ &\quad +\mathop{\rm Re}\int_{0}^{s}e^{c(s-t)}( u, \frac{d}{dt}(A_{\varepsilon }^{-1\ast }(t)A_{\varepsilon }^{-1}(t))u)dt \\ &\quad -2\mathop{\rm Re}\int_{0}^{s}e^{c(s-t)}(Au,( A_{\varepsilon }^{-1\ast }(t)) A_{\varepsilon }^{-1}(t)u)dt\,. \end{aligned} Using the Cauchy inequality and passing to the limit as $\varepsilon \to 0$, and using (A2) and the properties of the smoothing operators, we arrive at \label{e6} \begin{aligned} &| u| _{t=s}^{2}+\int_{0}^{s}e^{c(s-t)}|u| _{t}^{2}dt\\ &\leq e^{cs}| u|_{t=0}^{2}+\int_{0}^{s}e^{c(s-t)}| \mathcal{L}u| ^{2}dt+(1-c)\int_{0}^{s}e^{c(s-t)}| u| ^{2}dt, \end{aligned} To eliminate the last term on the right hand side of \eqref{e6}, we choose $% c=1$, consequently $$\label{e7} | u| _{t=s}^{2}+\int_{0}^{s}e^{(s-t)}| u| _{t}^{2}dt\leq e^{s}| u| _{t=0}^{2}+\int_{0}^{s}e^{(s-t)}| \mathcal{L}u| ^{2}dt,$$ which implies $$\label{e8} | u| _{t=s}^{2}+\int_{0}^{s}| u| _{t}^{2}dt\leq e^{T}| u| _{t=0}^{2}+e^{T}\int_{0}^{T}| \mathcal{L}u| ^{2}dt.$$ Repeating steps already employed, but on the interval $] s,T[$, we obtain $$e^{s-T}| u| _{t=T}^{2}+\int_{s}^{T}e^{(s-t)}| u| _{t}^{2}dt\leq | u| _{t=s}^{2}+\int_{s}^{T}e^{(s-t)}| \mathcal{L}u| ^{2}dt.$$ Since the exponential function is increasing, $$\label{e10} | u| _{t=T}^{2}+\int_{s}^{T}| u|_{t}^{2}dt\leq e^{T}| u| _{t=s}^{2}+e^{T}\int_{0}^{T}| \mathcal{L}u| ^{2}dt.$$ Multiplying \eqref{e8} and \eqref{e10} by $e^{2T}$ and then adding up, we have \label{e11} \begin{aligned} &(e^{2T}-e^{T})| u|_{t=s}^{2}+e^{2T}\int_{0}^{s}| u| _{t}^{2}dt+\int_{s}^{T}| u| _{t}^{2}dt\\ &\leq e^{3T}| u| _{t=0}^{2}-| u| _{t=T}^{2}+(e^{3T}+e^{T})\| \mathcal{L}u\| ^{2}. \end{aligned} \begin{lemma}[\cite{c1}] \label{lem2} Let $g$ be a function from $\overline{I}$ into $H$ and let $h$ be an element of $H$ such that $$h=g(0)-\mu g(T).$$ If the parameter $\mu$ satisfies \$| \mu |^{2}