\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 78, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2013/78\hfil Delay differential equations] {Delay differential equations with homogeneous integral conditions} \author[A. Raheem, D. Bahuguna \hfil EJDE-2013/78\hfilneg] {Abdur Raheem, Dhirendra Bahuguna} % in alphabetical order \address{Abdur Raheem \newline Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur -208016, India} \email{araheem@iitk.ac.in} \address{Dhirendra Bahuguna \newline Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur -208016, India} \email{dhiren@iitk.ac.in} \thanks{Submitted April 6, 2012. Published March 22, 2013.} \subjclass[2000]{35D35, 34K05, 34K07, 34K10, 34K30} \keywords{Method of semidiscretization; delay differential equation; \hfill\break\indent strong solution} \begin{abstract} In this article we prove the existence and uniqueness of a strong solution of a delay differential equation with homogenous integral conditions using the method of semidiscretization in time. As an application, we include an example that illustrates the main result. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} This article concerns the delay differential equation having homogeneous integral conditions, \begin{gather} \frac{\partial u}{\partial t}- \frac{\partial^2 u}{\partial x^2}-\lambda \frac{\partial^3 u}{\partial x^2 \partial t}= F(x,t,u_t) \quad \text{on } (0,1)\times (0,T],\label{eq1}\\ u(x,t)=\phi (x,t) \quad \text{on } (0,1)\times [-T,0],\label{eq2} \end{gather} with integral conditions \begin{gather} \int_{0}^{1}u(x,t)\,dx=0, \quad t \in [0,T], \label{eq3} \\ \int_{0}^{1}xu(x,t)\,dx=0,\quad t \in [0,T], \label{eq4} \end{gather} where $00$, the nonlinear operator $A$ is single-valued and $m$-accretive defined from the domain $D(A)\subset X$ into $X$, the nonlinear map $f$ is defined from $[0,T]\times X \times \mathcal{C}_0:=C([-\tau,0];X)$ into $X$, the map $h$ is defined from $\mathcal{C}_0$ and $\mathcal{C}_0$. Bahuguna, Abbas, and Dabas \cite{b1} applied the method of semidiscretization to a semilinear functional partial differential equation with an integral condition. Our problem is motivated by the work of Lakoud and Belakroum \cite{la3} and Dubey \cite{sd1}. Lakoud and Balakroum \cite{la3}\ established the existence and uniqueness of a weak solution for the integro-differential evolution with a memory term, $\frac{\partial v}{\partial t}- \frac{\partial^2 v}{\partial x^2}-\lambda \frac{\partial^3 v}{\partial x^2 \partial t}=g(x,t)+\int_{0}^{t}a(t-s)k(s,v(x,s))\,ds \quad \text{on }(0,1)\times (0,T],$ subject to the initial conditions $v(x,0)=V_0(x),$ and integral conditions \begin{gather*} \int_{0}^{1}v(x,t)\,dx=E(t), \\ \int_{0}^{1}xv(x,t)\,dx=G(t), \end{gather*} where $f,V_0,G,E$ are given functions and $T, \lambda$ are positive constants. The plan of the paper is as follows. In section 2, we state all the assumptions and preliminaries. In section 3, we state the main result. In section 4, we state and prove all the lemmas that are required to prove the main result and at the end of this section, we prove the main result. In the last section, we give an application of the main result. Throughout the paper we denote a generic constant by $C$. This constant may have different values in the same discussion. \section{Preliminaries} We will use the following assumptions: \begin{itemize} \item[(H1)] The nonlinear map $F:(0,T] \times \mathcal{C}_0 \to B(0,1)$ satisfies a local Lipschitz condition $$\|F(t_1,\psi_1)-F(t_2,\psi_2)\|_B \leq L_F(r)[|t_1-t_2|+\|\psi_1-\psi_2\|_0],$$ for all $t_1,t_2 \in (0,T]$ and $\psi_1,\psi_2 \in \mathcal{C}_0$ with $\|\psi_i-\phi(0)\|_0 \leq r$, $i=1,2$ and $L_F(r)$ is a nondecreasing function of $r>0$. \item[(H2)] The history function $\phi :[-T,0] \to L^2(0,1)$ is uniformly Lipschitz continuous with Lipschitz constant $K>0$; i.e., $\|\phi(t)-\phi(s)\| \leq K |t-s|$. \item[(H3)] $\int_{0}^{1} \phi(x,0) \,dx=0$, $\int_{0}^{1} x \phi(x,0)\,dx=0$. \end{itemize} \begin{lemma} \label{lm1} If $-A$ is the infinitesimal generator of a $C_0$-semigroup of contractions in a Banach space $X$ then $A$ is m-accretive; i.e., $$(Au,J(u))\geq 0, \quad \text{for }u\in D(A),$$ where $J$ is the duality mapping and $R(I+\lambda A)=X$ for $\lambda >0$, $I$ is the identity operator on $X$ and $R(\cdot)$ is the range of an operator. \end{lemma} The proof of the above lemma follows from Lumer Phillips theorem \cite[Thm. 1.4.3]{pz}. \section{Main result} \begin{theorem} \label{t1} Suppose that assumptions {\rm (H1)--(H3)} are satisfied. Then problem \eqref{eq1}-\eqref{eq4} has a unique strong solution on the interval $[-T,T]$. \end{theorem} \section{Discretization and a priori estimates} To apply the method of semidiscretization we divide the interval $[0,T]$ into the subintervals of length $h_n=\frac{T}{n}$. We set $u_0^n=\phi(0)$ for all $n\in \mathbb{N}$ and define $\{u_j^n\}$ successively as the unique solution of the problem \begin{gather} \delta u^n_j-\frac{\partial^2 u^n_j}{\partial x^2}-\lambda \frac{\partial^2 \delta u^n_j}{\partial x^2} =F(t_j^n,\tilde{u}_{j-1}^n), \label{eq5}\\ \int_{0}^{1}u^n_j\,dx = 0, \label{eq6}\\ \int_{0}^{1}xu^n_j\,dx = 0, \label{eq7} \end{gather} where $\tilde{u}^n_0=\phi(t)$ for $t\in [-T,0]$ and $2 \leq j \leq n$, $\tilde{u}^n_{j-1}(t) = \begin{cases} \phi(t^n_{j-1}+t), & \text{if } t\in [-T,-t^n_{j-1}] \\ u^n_{i-1}+(t^n_{j-1}+t-t^n_{i-1})\delta u^n_i, & \text{if } t\in[-t^n_{j-i},-t_{j-i-1}^n],\; 1\leq i \leq j-1, \end{cases}$ and $\delta u^n_j=\frac{u^n_j-u^n_{j-1}}{h_n}.$ Let $w_j^n=u^n_j+\lambda \delta u^n_j$. This implies that $\delta u^n_j=\frac{1}{h_n+\lambda}w^n_j-\frac{1}{h_n+\lambda}u^n_{j-1}$. So \eqref{eq5}-\eqref{eq7} will reduce to \begin{gather} -\frac{\partial^2w^n_j}{\partial x^2} +\frac{1}{h_n+\lambda}w_j^n = f^n_j, \label{eq8}\\ \int_{0}^{1}w^n_j\,dx=0 \label{eq9},\\ \int_{0}^{1}xw^n_j\,dx=0 \label{eq10}, \end{gather} where $$f^n_j=\frac{1}{h_n+\lambda}u^n_{j-1}+F(t^n_j,\tilde{u}^n_{j-1}).$$ Now we show the existence and uniqueness of functions $w^n_j$ satisfying \eqref{eq8}-\eqref{eq10}. For this consider $H=L^2(0,1)$, the Hilbert space of all real valued square integrable functions on the interval $(0,1)$. Let the linear operator $A$ be defined by $$D(A):=\{u \in H: u'' \in H, \int_{0}^{1}u(x)\,dx=\int_{0}^{1}xu(x)\,dx=0\},\quad Au=-u''.$$ Then we know that $-A$ is the infinitesimal generator of a $C_0$-semigroup $S(t),t\geq 0$ of contractions in $H$. The existence of unique $w^n_j$ satisfying equations \eqref{eq8}-\eqref{eq10} is a consequence of Lemma \ref{lm1}. As $$u^n_j=\frac{h_n}{\lambda+h_n}w^n_j+\frac{\lambda}{\lambda+h_n}u^n_{j-1},$$ there exist unique $u^n_j \in D(A)$ satisfying \eqref{eq5}--\eqref{eq7}. Now we define $$U^n(t)= \begin{cases} \phi(t), & \text{if } t\in [-T,0] \\ u^n_{j-1}+(t-t^n_{j-1})\frac{u^n_j-u^n_{j-1}}{h_n}, &\text{if } t\in (t^n_{j-1},t^n_j]. \end{cases}\label{eq11}$$ \begin{lemma} \label{lem4.1} For $n \in \mathbb{N}$ and $j=1,2,\cdots n$, $$\|u^n_j-\phi(0)\| \leq C,$$ where $C$ is a generic constant independent of $n,j,h_n$. \end{lemma} \begin{proof} Now for any $\psi \in V$, from \eqref{eq5} we have $$(\delta u^n_j,\psi)_B-\Big(\frac{\partial^2 u^n_j}{\partial x^2},\psi \Big)_B-\lambda \Big(\frac{\partial^2 \delta u^n_j}{\partial x^2},\psi \Big)_B=(F^n_j,\psi)_B, \label{eq12}$$ where $F^n_j=F(t^n_j,\tilde{u}^n_{j-1})$. By the definition of the inner product $(,)_B$, we have \begin{eqnarray} \Big(\frac{\partial^2 u^n_j}{\partial x^2},\psi \Big)_B=-\int_{0}^{1}u^n_j \psi \,dx=-(u^n_j,\psi).\label{eq13} \end{eqnarray} So \eqref{eq12} reduces to $$(\delta u^n_j,\psi)_B+(u^n_j,\psi)+\lambda(\delta u^n_j,\psi)=(F^n_j,\psi)_B. \label{eq14}$$ Taking $j=1,\psi=u_1^n-u^n_0$ in \eqref{eq14}, \begin{align*} &(u^n_1-u^n_0,u_1^n-u^n_0)_B+h_n(u^n_1,u_1^n-u^n_0) +\lambda( u^n_1-u^n_0,u_1^n-u^n_0) \\ &=h_n(F^n_1,u_1^n-u^n_0)_B . \end{align*} Now using \eqref{eq13}, we obtain \begin{align*} &(u^n_1-u^n_0,u_1^n-u^n_0)_B+h_n(u^n_1-u^n_0,u_1^n-u^n_0) +\lambda( u^n_1-u^n_0,u_1^n-u^n_0)\\ &=h_n\left(F^n_1+\frac{d^2u^n_0}{dx^2},u_1^n-u^n_0\right)_B. \end{align*} Now, we obtain $\|u^n_1-u^n_0\|^2_B+ h_n \|u^n_1-u^n_0\|^2+ \lambda \|u^n_1-u^n_0\|^2 \leq h_n \big[\|F^n_1\|_B+ \|\frac{d^2u^n_0}{dx^2} \|_B\big]\|u^n_1-u^n_0\|_B.$ By ignoring first two terms on the left hand side, we obtain $\lambda \|u^n_1-u^n_0\|^2 \leq h_n \big[\|F^n_1\|_B+h_n \|\frac{d^2u^n_0}{dx^2} \|_B\big]\|u^n_1-u^n_0\|_B.$ As $\|u^n_1-u^n_0 \|_B \leq \frac{1}{\sqrt{2}}\|u^n_1-u^n_0\|$, we have $\|u^n_1-u^n_0\| \leq \frac{h_n}{\lambda \sqrt{2}}\big[\|F^n_1\|_B+ \|\frac{d^2u^n_0}{dx^2}\|_B\big].$ By using assumption (H1), and the inequality $h_n \leq T$, we obtain $\|u^n_1-u^n_0\| \leq \frac{T}{\lambda \sqrt{2}} \big[L_F(r)(T+r)+\|F(0,\phi(0))\|_B+ \|\frac{d^2u^n_0}{dx^2}\|_B\big] \leq C.$ By putting $\psi =u^n_j-u^n_0$ in \eqref{eq14}, we obtain \begin{align*} &(u^n_j-u^n_{j-1},u^n_j-u^n_0)_B+h_n(u^n_j,u^n_j-u^n_0) +\lambda(u^n_j-u^n_{j-1},u^n_j-u^n_0)\\ &=h_n(F^n_j,u^n_j-u^n_0)_B. \hspace{6.7cm} \end{align*} Using \eqref{eq13}, we obtain \begin{align*} &\big(u^n_j-u^n_0,u^n_j-u^n_0\big)_B+h_n\big(u^n_j-u^n_0,u^n_j-u^n_0\big) +\lambda\big(u^n_j-u^n_0,u^n_j-u^n_0\big) \\ &=h_n\big(F^n_j,u^n_j-u^n_0\big)_B +\big(u^n_{j-1}-u^n_0,u^n_j-u^n_0\big)_B+h_n \big(\frac{d^2u^n_0}{dx^2},u^n_j-u^n_0 \big)_B\\ &\quad +\lambda \big(u^n_{j-1}-u^n_0,u^n_j-u^n_0\big). \end{align*} By ignoring the first two terms on the left hand side, we obtain \begin{align*} \lambda\|u^n_j-u^n_0\|^2 & \leq h_n \|F^n_j\|_B \|u^n_j-u^n_0\|_B +\|u^n_{j-1}-u^n_0\|_B \|u^n_j-u^n_0\|_B \\ &\quad +h_n \| \frac{d^2u^n_0}{dx^2}\|_B\|u^n_j-u^n_0\|_B+\lambda \|u^n_{j-1}-u^n_0\| \|u^n_j-u^n_0\|. \end{align*} As $\|u^n_j-u^n_0\|_B \leq \frac{1}{\sqrt{2}} \|u^n_j-u^n_0\|$, we have $$\|u^n_j-u^n_0\| \leq \frac{1}{\lambda}\big[\frac{h_n}{\sqrt{2}}(\|F^n_j\|_B + \| \frac{d^2u^n_0}{dx^2}\|_B)+(\frac{1}{2}+\lambda)\|u^n_{j-1}-u^n_0\|\big]. \label{eq15}$$ By assumption (H1), we have \begin{aligned} \|F^n_j\|_B&= \|F(t^n_j,\tilde{u}^n_{j-1})\|_B \\ &= \|F(t^n_j,\tilde{u}^n_{j-1})-F(0,\phi(0))\|_B +\|F(0,\phi(0))\|_B \\ &\leq L_F(r)[|t^n_j|+\|\tilde{u}^n_{j-1}-\phi(0)\|_0] +\|F(0,\phi(0))\|_B\\ &\leq L_F(r)[T+r]+\|F(0,\phi(0))\|_B. \end{aligned} \label{eq16} Using \eqref{eq16} in \eqref{eq15}, we obtain \begin{gather*} \begin{aligned} \|u^n_j-u^n_0\| &\leq \frac{h_n}{\lambda \sqrt{ 2}}\big[L_F(r)(T+r)+\|F(0,\phi(0))\|_B +\| \frac{d^2u^n_0}{dx^2}\|_B \big]\\ &\quad +\Big(1+\frac{1}{2\lambda}\Big)\|u^n_{j-1}-u^n_0\| \end{aligned} \\ \|u^n_j-u^n_0\| \leq h_n K+\Big(1+\frac{1}{2\lambda}\Big)\|u^n_{j-1}-u^n_0\|, \end{gather*} where $K=\frac{1}{\lambda \sqrt{2}}\big[L_F(r)(T+r)+\|F(0,\phi(0))\|_B+\| \frac{d^2u^n_0}{dx^2}\|_B\big]$. Repeating the above procedure, we obtain $$\|u^n_j-u^n_0\| \leq C.$$ This completes the proof. \end{proof} \begin{lemma}\label{lm2} For $j=1,2, \cdots , n$, $$\|\frac{u^n_j-u^n_{j-1}}{h_n} \| \leq C.$$ \end{lemma} \begin{proof} By putting $j=1$, $\psi=u^n_j-u^n_0$ in \eqref{eq14} and using \eqref{eq13}, we obtain \begin{align*} &\Big(\frac{u^n_1-u^n_0}{h_n},u^n_1-u^n_0 \Big)_B + (u^n_1-u^n_0,u^n_1-u^n_0)+ \lambda \Big(\frac{u^n_1-u^n_0}{h_n},u^n_1-u^n_0 \Big) \\ &=(F_1^n,u^n_1-u^n_0)_B+ \Big(\frac{d^2u^n_0}{dx^2},u^n_1-u^n_0\Big)_B. \end{align*} By ignoring the first two terms on the left hand sides, we obtain \begin{eqnarray} \frac{\lambda}{h_n}\|u^n_1-u^n_0\|^2 \leq \Big(F^n_1+\frac{d^2u^n_0}{dx^2},u^n_1-u^n_0 \Big)_B. \end{eqnarray} As $\|u^n_j-u^n_0\|_B \leq \frac{1}{\sqrt{2}} \|u^n_j-u^n_0\|$, we have $\|\frac{u^n_1-u^n_0}{h_n}\| \leq \frac{1}{\lambda\sqrt{2}} \big[\|F^n_1 \|_B + \| \frac {d^2u^n_0}{dx^2} \|_B\big].$ Using assumption (H1), we obtain $\|\frac{u^n_1-u^n_0}{h_n} \| \leq \frac{1}{\lambda \sqrt{2}}\big[L_F(r)(T+r)+\|F(0,\phi(0))\|_B + \| \frac {d^2u^n_0}{dx^2} \|_B \big] \leq C.$ Subtracting \eqref{eq14} written for $j$, from the same identity for $j-1$ and then putting $\psi=u^n_j-u^n_{j-1}$, we obtain \begin{align*} &(\delta u^n_j,u^n_j-u^n_{j-1})_B + (u^n_j-u^n_{j-1},u^n_j-u^n_{j-1}) + \lambda(\delta u^n_j,u^n_j-u^n_{j-1}) \\ &=(F^n_j-F^n_{j-1},u^n_j-u^n_{j-1})_B+ (\delta u^n_{j-1},u^n_j-u^n_{j-1})_B +\lambda \left(\delta u^n_{j-1},u^n_j-u^n_{j-1} \right ). \end{align*} Ignoring the first two terms on the left hand side, \begin{align*} \frac{\lambda}{h_n}\| u^n_j-u^n_{j-1}\| ^2 & \leq \|F^n_j-F^n_{j-1}\|_B \|u^n_j-u^n_{j-1}\|_B+\|\frac{u^n_{j-1}-u^n_{j-2}}{h_n} \|_B \| u^n_j-u^n_{j-1}\| _B \\ &\quad +\lambda\| \frac{u^n_{j-1}-u^n_{j-2}}{h_n} \|\| u^n_j-u^n_{j-1}\|. \end{align*} As $\|\psi\|_B \leq \|\psi\|/\sqrt{2}$, we have \begin{eqnarray} \|\frac{u^n_j-u^n_{j-1}}{h_n} \| \leq \frac{1}{\lambda \sqrt{2}}\|F^n_j-F^n_{j-1}\|_B +(1+\frac{1}{2\lambda})\|\frac{u^n_{j-1}-u^n_{j-2}}{h_n} \|. \label{eq17} \end{eqnarray} Now using assumption (H1), \begin{align*} \|F^n_j-F^n_{j-1}\|_B &= \|F(t^n_j,\tilde{u}^n_{j-1})-F(t^n_{j-1},\tilde{u}^n_{j-2})\|_B \\ &\leq L_F(r)[|t^n_j-t^n_{j-1}|+\|\tilde{u}^n_{j-1}-\tilde{u}^n_{j-2}\|_0] \\ &\leq L_F(r) [T+2r]. \end{align*} Using the above inequality in \eqref{eq17}, we obtain \begin{eqnarray} \|\frac{u^n_j-u^n_{j-1}}{h_n} \| \leq \frac{1}{\lambda \sqrt{2}}L_F(r) [T+2r] +(1+\frac{1}{2\lambda})\|\frac{u^n_{j-1}-u^n_{j-2}}{h_n} \|. \end{eqnarray} Repeating the above procedure, we finally obtain \begin{eqnarray} \|\frac{u^n_j-u^n_{j-1}}{h_n} \| \leq C. \end{eqnarray} This completes the proof. \end{proof} Now we introduce a sequence of step functions $\{X^n(t)\}$ defined by $$X^n(t)= \begin{cases} \phi(0), & \text{if } t=0 \\ u^n_j, & \text{if } t\in (t^n_{j-i},t_j^n]. \end{cases}\label{eq18}$$ \begin{remark} \label{rk1} \rm From Lemma \ref{lm2} it follows that the functions $U^n$ are uniformly Lipschitz continuous on $[-T,T]$ and $U^n(t)-X^n(t)\to 0$, as $n \to \infty$ on $[0,T]$. \end{remark} Let $F^n(t)=F(t^n_j,\tilde{u}^n_{j-1})$. By assumption (H1) and remark \ref{rk1}, we see that $F^n(t) \to F(t,u_t)$. Using \eqref{eq11} and \eqref{eq18}, in \eqref{eq5}, we obtain \begin{eqnarray} \frac{d^-}{dt}U^n(t)-\frac{\partial^2}{\partial x^2}X^n(t)-\lambda \frac{\partial^3}{\partial x^2 \partial t}X^n(t)=F^n(t). \label{eq19} \end{eqnarray} Integrating with respect to $t$, we obtain $$-\int_{0}^{t}\big[\frac{\partial^2}{\partial x^2}X^n(s)+\lambda \frac{\partial^3}{\partial x^2 \partial s}X^n(s)\big]\,ds=\phi(0)-U^n(t)+\int_{0}^{t}F^n(s)\,ds.\label{eq20}$$ \begin{lemma} \label{lem4.4} There exists $u \in C([-T,T];B(0,1))$ such that $U^n(t) \to u(t)$ uniformly on $[-T,T]$. Moreover $u(t)$ is Lipschitz continuous on $[-T,T]$. \end{lemma} \begin{proof} From \eqref{eq19}, we have \begin{align*} &\Big(\frac{d^-}{dt}U^n(t)-\frac{d^-}{dt}U^k(t),U^n(t)-U^k(t) \Big)_B+(X^n(t)-X^k(t),U^n(t)-U^k(t)) \hspace{.2cm}\hspace{.5cm}\\ &+\lambda \Big(\frac{\partial}{\partial t}X^n(t)-\frac{\partial}{\partial t}X^k(t),U^n(t)-U^k(t)\Big)\\ &=(F^n(t)-F^k(t),U^n(t)-U^k(t))_B. \end{align*} Now, \begin{align*} &\frac{1}{2}\frac{d^-}{dt}\|U^n(t)-U^k(t)\|^2_B+\|X^n(t)-X^k(t)\|^2 +\frac {\lambda}{2} \frac{\partial}{\partial t}\|X^n(t)-X^k(t)\|^2\\ &=(X^n(t)-X^k(t),X^n(t)-X^k(t)-U^n(t)+U^k(t))\\ &\quad + \lambda \Big(\frac{\partial}{\partial t}(X^n(t)-X^k(t)), X^n(t)-X^k(t)-U^n(t)+U^k(t)\Big)\\ &\quad +(F^n(t)-F^k(t),U^n(t)-U^k(t))_B. \end{align*} By ignoring the last two terms on the left hand side, we obtain $\frac{1}{2}\frac{d^-}{dt}\|U^n(t)-U^k(t)\|^2_B \leq \delta_{nk}(t) + \|F^n(t)-F^k(t)\|_B \|U^n(t)-U^k(t)\|_B.$ where \begin{align*} \delta_{nk} (t) &=\|X^n(t)-X^k(t)\|[\|X^n(t)-U^n(t)\|+\|X^k(t)-U^k(t)\|] \\ &\quad +\frac{\lambda}{2} \|\frac{\partial}{\partial t}(X^n(t)-X^k(t)) \| \big[\|X^n(t)-U^n(t)\|+\|X^k(t)-U^k(t)\|\big]. \end{align*} By Remark~\ref{rk1}, it is clear that $\delta_{nk} (t) \to 0$ as $n,k \to \infty$ uniformly on the interval $[0,T]$. Now by assumption (H1), we have \begin{eqnarray} \|F^n(t)-F^k(t)\|_B &= \|F(t^n_j,\tilde{u}^n_{j-1})-F(t^k_l,\tilde{u}^k_{l-1})\|_B \\ &\leq \delta'_{nk}(t) + L_F(r) \|U^n(t)-U^k(t)\|_B, \end{eqnarray} where $$\delta'_{nk}(t) =L_F(r)[|t^n_j-t^k_l|+\|U^n(t)-\tilde{u}^n_{j-1}\|_0 +\|U^k(t)-\tilde{u}^k_{l-1}\|_0].$$ Clearly $\delta'_{nk}(t) \to 0$ as $n,k \to \infty$ uniformly on $[0,T]$. This implies that for a.e. $t \in [0,T]$, \begin{eqnarray} \frac{1}{2}\frac{d^-}{dt}\|U^n(t)-U^k(t)\|^2_B \leq \delta'_{nk} (t) + L_F(r)\|U^n(t)-U^k(t)\|^2_B, \end{eqnarray} Integrating the above inequality over $(0,t)$ with $0 \leq t\leq T$, we obtain $\|U^n(t)-U^k(t)\|^2_B \leq 2\delta'_{nk}T + 2L_F(r)\int_{0}^{t}\|U^n(s)-U^k(s)\|^2_B\,ds.$ Applying Gronwall's inequality, we obtain that $U^n \to u$ in $C([-\tau,T],B(0,1))$. As each $U^n$ is uniformly Lipschitz continuous, and by assumption (H2), $u$ is Lipschitz continuous. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{t1}] Taking limits as $n \to \infty$ in \eqref{eq20}, we obtain $-\int_{0}^{t}\big[\frac{\partial^2}{\partial x^2}u(t) +\lambda \frac{\partial^3}{\partial x^2 \partial s}u(t)\big]\,ds =\phi(0)-u(t)+\int_{0}^{t}F(s,u_t)\,ds.$ This implies that $\frac{\partial u(t)}{\partial t}- \frac{\partial^2 u(t)}{\partial x^2}-\lambda \frac{\partial^3 u(t)}{\partial x^2 \partial t}= F(t,u_t),\quad\text{a.e. }t\in [0,T].$ Clearly $u(t)$ is differentiable with $u(t) \in V$ a.e. on $[0,T]$ and $u(t)=\phi(t)$, $t \in [-T,0]$. This implies that $u(t)(x)=u(x,t)$ is a strong solution of \eqref{eq1}--\eqref{eq4}. Now we show the uniqueness of the strong solution. To do this, suppose that $u_1,u_2$ are two strong solutions of \eqref{eq1}-\eqref{eq4}. Let $u=u_1-u_2$, then for $\psi \in V$, we have $\Big(\frac{\partial u}{\partial t}, \psi \Big)_B+ (u, \psi) + \lambda \Big(\frac{\partial u}{\partial t}, \psi \Big) =\Big( F(t,(u_1)_t)-F(t,(u_2)_t), \psi \Big)_B.$ Putting $\psi=u$ and ignoring last two terms in the left hand side, we obtain $\Big(\frac{\partial u}{\partial t}, u \Big)_B \leq \Big(F(t,(u_1)_t)-F(t,(u_2)_t), u \Big)_B.$ By using assumption (H1), we obtain \begin{align*} \frac{1}{2} \frac{\partial}{\partial t} \|u(t)\|^2_B &\leq L_F(r) \|(u_1)_t-(u_2)_t\|_B \|u(t)\|_B \\ &\leq L_F(r) \sup_{-T \leq t+\theta \leq t} \|u_t(\theta)\|_B\sup_{-T \leq \theta \leq t} \|u(\theta)\|_B. \end{align*} Integrating between $0$ and $t$, we obtain \begin{gather*} \sup_{-T \leq\theta \leq t}\|u(\theta)\|^2_B \leq 2 L_F(r) \int_{0}^{t} \|u\|^2_s \,ds \\ \|u\|^2_t \leq 2 L_F(r) \int_{0}^{t} \|u\|^2_s \,ds. \end{gather*} Applying Gronwall's inequality, we obtain $u=0$ on $[-T,T]$. Hence we obtain a unique strong solution of problem \eqref{eq1}-\eqref{eq4} on the interval $[-T,T]$. \end{proof} \section{Application} Consider the partial differential equation $$\begin{gathered} \frac{\partial v}{\partial t}- \frac{\partial^2 v}{\partial x^2}-\lambda \frac{\partial^3 v}{\partial x^2 \partial t}= g(x,t)+\int_{0}^{t}a(t-s)k(s,v(x,s))\,ds \quad \text{on } (0,1)\times (0,T], \\ v(x,t)= \phi (x,t) \quad \text{on} \quad (0,1)\times [-T,0], \end{gathered} \label{eq21}$$ with the integral conditions \begin{gather} \int_{0}^{1}v(x,t)\,dx=0,\label{eq22} \\ \int_{0}^{1}xv(x,t)\,dx=0.\label{eq23} \end{gather} In the above problem, we identify the unknown function $v:(0,T] \to B(0,1)$, by $v(t)(x)=v(x,t)$, $g: (0,T] \to B(0,1)$ by $g(t)(x)=g(x,t)$, $k:(0,T]\times \mathbb{R} \to B(0,1)$ by $k(t,v(x,t))=k(t,v(t))(x)$ and the history function $\phi:[-T,0] \to B(0,1)$ by $\phi(t)(x)=\phi(x,t)$. Also we take $$V=\big\{\phi \in L^2(0,1):\int_{0}^{1}\phi(x)\,dx=\int_{0}^{1}x\phi (x)\,dx=0 \big\}.$$ Putting $t=s-\eta$ in the integral term, problem \eqref{eq21}-\eqref{eq23} reduces to $$\begin{gathered} \frac{\partial v}{\partial t}- \frac{\partial^2 v}{\partial x^2}-\lambda \frac{\partial^3 v}{\partial x^2 \partial t}= g(t)+\int_{-t}^{0}a(-\eta)k(t+\eta,v(t+\eta))d\eta \quad \text{on } (0,T], \\ v(t)= \phi (t), \quad t \in [-T,0]. \end{gathered}\label{eq24}$$ Now we consider the following assumptions: \begin{itemize} \item[(i)] There exists $k_1>0$, such that for all $t,s \in (0,T]$ $$\|g(t)-g(s)\|_B \leq k_1 |t-s|.$$ \item[(ii)] There exists $k_2>0$, such that for all $t,s \in (0,T]$ and $\psi_1,\psi_2 \in \mathcal{C}_0$, $$\|k(t,\psi_1(t))-k(s,\psi_2(s))\|_B \leq k_2[|t-s|+\|\psi_1-\psi_2\|_0].$$ \item[(iii)] Also there exist $M>0$, such that $$\|a(t)\|_B \leq M, \quad t \in [-T,0].$$ \end{itemize} Now we define $G :(0,T]\times \mathcal{C}_0 \to B(0,1)$ by $$G(t,\psi)=g(t)+\int_{-t}^{0}a(-\eta)k(t+\eta,\psi)d\eta.$$ Thus \eqref{eq24} reduces to $$\begin{gathered} \frac{\partial v}{\partial t}- \frac{\partial^2 v}{\partial x^2}-\lambda \frac{\partial^3 v}{\partial x^2 \partial t} = G(t,v_t) \quad \text{on } (0,T], \\ v(t)= \phi (t), \quad t \in [-T,0]. \end{gathered}\label{eq25}$$ Now we show that $G$ satisfies assumption (H1). For this take $t,s \in (0,T]$ and $\psi_1, \psi_2 \in \mathcal{C}_0$ \begin{align*} &\|G(t,\psi_1)-G(s,\psi_2)\|_B\\ &\leq \|g(t)-g(s)\|_B + \big\|\int_{-t}^{0}a(-\eta)k(t+\eta,\psi_1(\eta))d\eta -\int_{-s}^{0}a(-\eta)k(s+\eta,\psi_2(\eta))d\eta \big \|_B. \end{align*} Using the given conditions on $g,a$ and $k$, we obtain \begin{align*} \|G(t,\psi_1)-G(s,\psi_2)\|_B &\leq k_1 |t-s|+Mk_2\int_{-t}^{0} \{|t-s|+\|\psi_1-\psi_2\|_0\}d\eta \\ &\quad +M\int_{-s}^{-t}\|k(s+\eta,\psi_2(\eta))\|_Bd\eta. \end{align*} After some simplifications, we obtain \begin{align*} \|G(t,\psi_1)-G(s,\psi_2)\|_B &\leq (k_1+Mk_2T+MK)|t-s|+Mk_2T \|\psi_1-\psi_2\|_0 \\ &\leq L [|t-s|+\|\psi_1-\psi_2\|_0], \end{align*} where $$L = \max\{(k_1+Mk_2T+MK),Mk_2T\},\quad \|k(s+\eta,\psi_2(\eta))\|_B \leq K.$$ As $G$ satisfies a Lipschitz like condition, we apply the result of Theorem \ref{t1} to ensure the existence and uniqueness of a strong solution of \eqref{eq21}-\eqref{eq23} . \subsection*{Acknowledgements} The authors want to thank the anonymous referees for their valuable suggestions. The first author acknowledges the sponsorship from CSIR, India, under research grant 09/092 (0652) /2008-EMR-1. The second author acknowledges the financial help from the Department of Science and Technology, New Delhi, under research project SR/S4/MS:796/12. \begin{thebibliography}{99} \bibitem{b1} D. Bahuguna, S. Abbas, J. Dabas; \emph{Partial functional differential equation with an integral condition and applications to population dynamics,} Nonlinear Analysis 69, (2008), 2623--2635. \bibitem{b2} D. Bahuguna, V. Raghavendra; \emph{Rothe's method to parabolic integrodifferential equation via abstract integrodifferential equations,} Applicable Analysis, vol. 33, no., 3--4, 153--167, 1989. \bibitem{b3} D. Bahuguna, A. K. Pani, V. Raghavendra, \emph{Rothe's method to semilinear hyperbolic integrodifferential equations,} Journal of Applied Mathematics and Stochastic Analysis, vol. 3, no. 4, pp. 245--252, 1990. \bibitem{ab2} A. Bouziani, R. Mechri; \emph{ The Rothe's Method to a parabolic Integrodifferential Equation with a Nonclassical Boundary conditions,} Int. J. Stoch. Anal. 2010, Art. ID 519684, 1--16. \bibitem{sd1} Shruti A. Dubey; \emph{The method of lines applied to nonlnear nonlocal functional differential equations,} J. Math. Anal. Appl. 376(2011) 275--281. \bibitem{jk1} J. Kacur; \emph{Method of Rothe in Evolution Equations,} Teubner-Text zur Mathematik, Vol. 80, 1985. \bibitem{jk2} N. Kikuchi, J. Kacur; \emph{Convergence of Rothe's method in H\"lder spaces,} Appl. Math. 48 (2003), no. 5, 353–-365. \bibitem{la1} A. G. Lakoud, A. Chaoni; \emph{Rothe's Method Applied to semilinear Hyperbolic Integrodifferetial equation with integral condition,} Int. J. open Problem Compt. Math. Vol. 4, No. 1, March 2011. \bibitem{la2} A. G. Lakoud, M. S. Jasmati, A. Chaoni; \emph{Rothe's method for an integrodifferential equation with integral conditions,} Nonlinear Analysis 72 (2010) 1552--1530. \bibitem{la3} A. G. Lakoud, D. Belakroum; \emph{Time-descretization schema for an integrodifferential Sobolev type equation with integral conditions,} Appied Math. Comp. Vol. 218, 9 (2012), 4695--4702. \bibitem{ab1} N. Merazga, A. Bouziani; \emph{Rothe time-discretization method for a nonlocal problem arising in thermoelasticity,} Journal of Applied Mathematics and Stochastic Analysis 2005:1,(2005) 13--28. \bibitem{pz} A. Pazy; \emph{Semigroup of linear operators and application to partial differential equations,} Springer-Verlag, New York, 1983. \bibitem{kr1} K. Rektorys; \emph{The Method of Discretization in time and Partial Differential Equations,} D. Reidel Publishing Company, 1982. \bibitem{kr2} K. Rektorys; \emph{On application of direct variational metheods to the solution of parabolic boundary value problems of arbitrary order in the space variables,} Czech. Math. J., 21(96), 1971, 318--339. \bibitem{kr3} K. Rektorys; \emph{Numerical and theoretical treating of evolution problems by the method of discretization in time,} Equadiff 6 (Brno, 1985), 71--84, Lecture Notes in Math., 1192, Springer, Berlin, 1986. \end{thebibliography} \end{document}