\documentclass[twoside]{article} \pagestyle{myheadings} \markboth{\hfil High-order mixed-type differential equations\hfil EJDE--2000/60} {EJDE--2000/60\hfil M. Denche \& A. L. Marhoune \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~60, pp.~1--10. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % High-order mixed-type differential equations with weighted integral boundary conditions % \thanks{ {\em Mathematics Subject Classifications:} 35B45, 35G10, 35M10. \hfil\break\indent {\em Key words:} Integral boundary condition, energy inequalities, equation of mixed type. \hfil\break\indent \copyright 2000 Southwest Texas State University. \hfil\break\indent Submitted March 27, 2000. Published September 21, 2000.} } \date{} % \author{ M. Denche \& A. L. Marhoune } \maketitle \begin{abstract} In this paper, we prove the existence and uniqueness of strong solutions for high-order mixed-type problems with weighted integral boundary conditions. The proof uses energy inequalities and the density of the range of the operator generated. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Let $\alpha$ be a positive integer and $Q$ be the set $(0,1)\times ( 0,T)$. We consider the equation $$\mathcal{L}u:=\frac{\partial ^{2}u}{\partial t^{2}}+(-1)^{\alpha }a(t)\frac{% \partial ^{2\alpha +1}u}{\partial x^{2\alpha }\partial t}=f(x,t), \label{1}$$ where the function $a(t)$ and its derivative are bounded on the interval $[ 0,T]$: \begin{eqnarray} 0