\documentclass[reqno]{amsart} \usepackage{amssymb} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 136, pp. 1--13.\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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/136\hfil Approximations of solutions] {Approximations of solutions to retarded integrodifferential equations} \author[D. Bahuguna, M. Muslim\hfil EJDE-2004/136\hfilneg] {Dhirendra Bahuguna, Malik Muslim} % in alphabetical order \address{Dhirendra Bahuguna \hfill\break Department of Mathematics \\ Indian Institute of Technology\\ Kanpur - 208 016, India} \email{d\_bahuguna@yahoo.com} \address{Malik Muslim \hfill\break Department of Mathematics \\ Indian Institute of Technology\\ Kanpur - 208 016, India} \email{malik\_iitk@yahoo.com} \date{} \thanks{Submitted October 21, 2002. Published November 23, 2004.} \subjclass[2000]{34K30, 34G20, 47H06} \keywords{Retarded differential equation; analytic semigroup; \hfill\break\indent approximate solution; convergence of solutions; Faedo-Galerkin approximation} \begin{abstract} In this paper we consider a retarded integrodifferential equation and prove existence, uniqueness and convergence of approximate solutions. We also give some examples to illustrate the applications of the abstract results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Consider the semilinear retarded differential equation with a nonlocal history condition in a separable Hilbert space $H$: \begin{equation} \begin{gathered} u'(t)+Au(t)=Bu(t)+Cu(t-\tau) +\int_{-\tau}^{0}a(\theta)Lu(t+\theta)d\theta,\quad 0 < t \le T<\infty, \tau > 0, \\ u(t)=h(t), \quad t \in [-\tau,0]. \end{gathered}\label{ex1} \end{equation} Here $u$ is a function defined from $[-\tau,\infty)$ to the space $H$, $h:[-\tau,0]\to H$ is a given function and the kernel $a\in L^{p}_{\rm loc}(-\tau,0)$. For each $t\geq 0$, $u_{t}:[-\tau,0]\to H$ is defined by $u_t(\theta)=u(t+\theta),$ $\theta \in [-\tau,0]$ and the operators $A:D(A)\subseteq H\to H$, is a linear operator. The operators $B:D(B)\subseteq H\to H$, \ $C:D(C)\subseteq H\to H$, and $L:D(L)\subseteq H\to H$ are non-linear continuous operators. For $t \in [0,T]$, we shall use the notation $\mathcal{C}_t:=C([-\tau,t];H)$ for the Banach space of all continuous functions from $[-\tau,t]$ into $H$ endowed with the supremum norm $$ \|\psi\|_t :=\sup_{-\tau \le \eta \le t}\|\psi(\eta)\|. $$ The existence, uniqueness and regularity of solutions of (\ref{ex1}) under different conditions have been considered by Blasio \cite{GE} and Jeong \cite{JES} and some of the papers cited therein. For the initial work on the existence, uniqueness and stability of various types of solutions of differential and functional differential equations, we refer to Bahuguna \cite{db1,db2}, Balachandran and Chandrasekaran \cite{bc}, Lin and Liu \cite{ll}. The related results for the approximation of solutions may be found in Bahuguna, Srivastava and Singh \cite{bss} and Bahuguna and Shukla \cite{bs}. Initial study concerning existence, uniqueness and finite-time blow-up of solutions for the following equation, \begin{equation} \label{eq2} \begin{gathered} u' (t) + Au(t) = g(u(t)), \quad t \ge 0, \\ u(0) = \phi, \end{gathered} \end{equation} have been considered by Segal \cite{is}, Murakami \cite{hm}, Heinz and von Wahl \cite{hw}. Bazley \cite{nb1,nb2} has considered the following semilinear wave equation \begin{equation} \label{eq3} \begin{gathered} u'' (t) + Au(t) = g(u(t)), \quad t \ge 0, \\ u(0) = \phi, \quad u' (0) = \psi, \end{gathered} \end{equation} and has established the uniform convergence of approximations of solutions to~(\ref{eq3}) using the existence results of Heinz and von Wahl \cite{hw}. Goethel \cite{rg} has proved the convergence of approximations of solutions to~(\ref{eq2}) but assumed $g$ to be defined on the whole of $H$. Based on the ideas of Bazley \cite{nb1,nb2}, Miletta \cite{pdm} has proved the convergence of approximations to solutions of~(\ref{eq2}). In the present work, we use the ideas of Miletta \cite{pdm} and Bahuguna \cite{bs, bss} to establish the convergence of finite dimensional approximations of the solutions to~(\ref{ex1}). \section{Preliminaries and Assumptions} Existence of a solution to (\ref{ex1}) is closely associated with the existence of a function $u \in \mathcal{C}_{\tilde{T}}$, $0< \tilde{T}\le T$ satisfying $$ u(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[Bu(s)\\ +Cu(s-\tau)+\int^{0}_{-\tau}a(\theta)Lu(s+\theta)d\theta]ds, & t \in [0,\tilde{T}] \end{cases} $$ and such a function $u$ is called a {\em mild solution} of (\ref{ex1}) on $[-\tau,\tilde{T}]$. A function $u\in \mathcal{ C}_{\tilde{T}}$ is called a {\em classical solution} of (\ref{ex1}) on $[-\tau,\tilde{T}]$ if $u \in C^1((0,\tilde{T}];H)$ and $u$ satisfies (\ref{ex1}) on $[-\tau,\tilde{T}]$. We assume in~(\ref{ex1}), that the linear operator $A$ satisfies the following. \vskip .4cm \begin{itemize} \item[(H1)] $A$ is a closed, positive definite, self-adjoint linear operator from the domain $D(A) \subset H$ into $H$ such that $D(A)$ is dense in $H,$ $A$ has the pure point spectrum $$0< \lambda_0 \le \lambda_1 \le \lambda_2 \le \dots$$ and a corresponding complete orthonormal system of eigenfunctions $\{ \phi_i \}$, i.e., $$ A\phi_i=\lambda_i \phi_i \quad \mbox{and} \quad (\phi_i,\phi_j) = \delta_{ij}, $$ where $\delta_{ij} =1$ if $i=j$ and zero otherwise. \end{itemize} If (H1) is satisfied then $-A$ is the infinitesimal generator of an analytic semigroup $\{e^{-tA}:t\geq 0\}$ in $H$ (cf. Pazy \cite[pp. 60-69]{ap}). It follows that the fractional powers $A^\alpha$ of $A$ for $0 \le \alpha \le 1$ are well defined from $D(A^\alpha) \subseteq H$ into $H$ (cf. Pazy \cite[pp. 69-75]{ap}). Hence for convenience, we suppose that $$ \|e^{-tA}\| \le M \quad \mbox{for all} \quad t \ge 0 $$ and $0 \in \rho(-A)$, where $\rho(-A)$ is the resolvent set of $-A$. $D(A^\alpha)$ is a Banach space endowed with the norm $\|x\|_\alpha = \|A^\alpha x\|$. %\label{dan} For $t \in [0,T]$, we denote by $\mathcal{C}_t^\alpha:=C([-\tau,t];D(A^\alpha))$ endowed with the norm $$ \|\psi\|_{t,\alpha}:=\sup_{-\tau \le \nu \le t}\|\psi(\nu)\|_\alpha. $$ Further, we assume the following. \begin{itemize} \item[(H2)] $h \in \mathcal{C}_0^\alpha$ and $h$ is locally h\"older continuous on $[-\tau,0]$. \item[(H3)] We shall assume that the map $B:D(A^\alpha)\to H$ satisfies the following Lipschitz condition on balls in $D(A^\alpha)$: for each $\eta>0$ and some $0<\alpha<1$ there exists a constant $K_1(\eta)$ such that\\ (i) $\|B(\psi)\|\leq K_1(\eta)$ for $\psi\in D(A^\alpha)$ with $\|A^\alpha\psi\|\leq \eta$, \\ (ii) $\|B(\psi_1)-B(\psi_2)\|\leq K_1(\eta)\|A^\alpha(\psi_1-\psi_2)\|$ for $\psi_1,\psi_2\in D(A^\alpha)$ with $\|A^\alpha\psi_i\|\leq \eta$ for $i=1,2$. \item[(H4)] The map $C:D(A^\alpha)\to H$ satisfies the following Lipschitz condition on balls in $D(A^\alpha)$: For each $\eta>0$ and some $0<\alpha<1$ there exists a constant $K_2(\eta)$ such that\\ (iii) $\|C(\psi)\|\leq K_2(\eta)$ for $\psi\in D(A^\alpha)$ with $\|A^\alpha\psi\|\leq \eta$, \\ (iv) $\|C(\psi_1)-C(\psi_2)\|\leq K_2(\eta)\|A^\alpha(\psi_1-\psi_2)\|$ for $\psi_1,\psi_2\in D(A^\alpha)$ with $\|A^\alpha\psi_i\|\leq \eta$ for $i=1,2$. \item[(H5)] The map $L:D(A^\alpha)\to H$ satisfies the following Lipschitz condition on balls in $D(A^\alpha)$: For each $\eta>0$ and some $0<\alpha<1$ there exists a constant $K_3(\eta)$ such that\\ (v) $\|L(\psi)\|\leq K_3(\eta)$ for $\psi\in D(A^\alpha)$ with $\|A^\alpha\psi\|\leq \eta$, \\ (vi) $\|L(\psi_1)-L(\psi_2)\|\leq K_3(\eta)\|A^\alpha(\psi_1-\psi_2)\|$ for $\psi_1,\psi_2\in D(A^\alpha)$ with $\|A^\alpha\psi_i\|\leq \eta$ for $i=1, 2$. \item[(H6)] $a\in L^{p}_{\rm loc}(-\tau,0)$ for some $1< p <\infty$ and $a_T=\int^0_{-\tau}|a(\theta)|\,d\theta$. \end{itemize} \section{Approximate Solutions and Convergence} Let $H_n$ denote the finite dimensional subspace of $H$ spanned by $\{ \phi_0,\phi_1,\cdots,\phi_n \}$ and let $P^n:H \longrightarrow H_n$ be the corresponding projection operator for $n=0,1,2,\cdots$. Let $0< {T_0} \le T$ be such that \begin{equation} \sup_{0\leq t\leq T_0}\|(e^{-tA}-I)A^\alpha h(0)\| \le {R \over 2},\label{ex7} \end{equation} where $R > 0$ be a fixed quantity. Let us define $$ \bar{h}(t)= \begin{cases} h(t), & \mbox{if } t \in [-\tau,0], \\ h(0), & \mbox{if } t \in [0,T]. \end{cases} $$ We set \begin{equation} T_0 < \min\big[\{{R \over 2}(1- \alpha) ({K({\eta}_0) C_\alpha})^{-1}\}^{1 \over {1-\alpha}},\; \{{1 \over 2}(1- \alpha) ({K(\eta_0)C_\alpha})^{-1}\}^{1\over{1-\alpha}}\big],\label{mild1} \end{equation} where \begin{eqnarray} K(\eta_0)=[K_1(\eta_0)+K_2(\eta_0)+K_3(\eta_0)a_{T}]\label{ex3} \end{eqnarray} and $C_\alpha$ is a positive constant such that $\|A^\alpha e^{-tA}\| \le C_\alpha t^{-\alpha}$ for $t>0$. We define $B_n: H\longrightarrow H$ such that $$B_nx = BP^nx, \; x\in H. $$ Similarly $C_n$ and $L_n $ are given by $$ C_nx = CP^nx, \; x\in H, \quad L_nx = LP^nx, \; x\in H. $$ Let $A^\alpha:\mathcal{C}_{t}^\alpha \to \mathcal{ C}_{t}$ be given by $(A^\alpha \psi)(s)=A^\alpha (\psi(s))$, $s \in [-\tau,t]$, $t \in [0,T]$. We define a map $F_n$ on $B_R(\mathcal{ C}_{T_0}^\alpha, \bar{h})$ as follows $$ (F_nu)(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[B_nu(s)\\ +C_nu(s-\tau)+\int^{0}_{-\tau}a(\theta)L_nu(s+\theta)d\theta]ds, & t \in [0,{T_0}], \end{cases} $$ for $u \in B_R(\mathcal{C}_{T_0}^\alpha,\bar{h}).$ \begin{theorem} \label{ms} Suppose that the conditions (H1)-(H6) are satisfied and $h(t)\in D(A)$ for all $t\in [-\tau,0]$. Then there exists a unique $u_n\in B_R(\mathcal{C}_{T_0}^\alpha,\bar{h})$ such that $F_nu_n=u_n$ for each $n=0, 1, 2, \cdots $, i.e., $u_n$ satisfies the approximate integral equation \begin{equation} \label{eqn000} %\label{ex9} u_n(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[B_nu_n(s) \\ +C_nu_n(s-\tau)+\int^{0}_{-\tau}a(\theta)L_nu_n(s+\theta)d\theta]ds, & t \in [0,{T_0}]. \end{cases} \end{equation} \end{theorem} \begin{proof} First we show that $F_n:B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})\to B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})$. For this first we need to show that the map $t\mapsto (F_nu)(t)$ is continuous from $[-\tau, T_0]$ into $D(A^\alpha)$ with respect to $\|\cdot \|_{\alpha}$ norm. Thus for any $u \in B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})$, and $t_1, t_2 \in [-\tau,0]$, we have \begin{equation} (F_nu)(t_1)-(F_nu)(t_2)=h(t_1)-h(t_2)\label{ex8}. \end{equation} Now for $t_1\;t_2\in (0,T_0]$ with $t_1< t_2$ we have \begin{equation} \label{ex4} \begin{aligned} & \|(F_nu)(t_2)-(F_nu)(t_1)\|_{\alpha}\\ &\leq \|(e^{-{t_2}A}-e^{-{t_1}A})h(0)\|_{\alpha} + \int_0^{t_1}\|(e^{-(t_2-s)A}-e^{-(t_1-s)A})A^{\alpha}\|\\ &\quad \times\big [ \|B_nu(s)\| + \|C_nu(s-\tau)\|+\int^{0}_{-\tau}|a(\theta)|\|L_nu(s+\theta)\|d\theta \big]ds \\ &\quad + \int_{t_1}^{t_2}\|(e^{-(t_2-s)A})A^{\alpha}\| \big[ \|B_nu(s)\| + \|C_nu(s-\tau)\| \\ &\quad +\int^{0}_{-\tau}|a(\theta)|\|L_nu(s+\theta)\|d\theta \big]ds. \end{aligned} \end{equation} Since part (d) of Theorem 6.13 in Pazy \cite[p. 74]{ap} states that for $0<\beta\leq 1$ and $x\in D(A^{\beta})$, $$ \|(e^{-tA}-I)x\|\leq C_{\beta}t^{\beta}\|A^{\beta}x\|. $$ Hence if $0<\beta<1$ is such that $0 < \alpha+\beta < 1$ then $A^{\alpha}y\in D(A^\beta)$. Therefore for $t,s\in (0,T_0]$, we have \begin{equation} \|(e^{-tA}-I)A^{\alpha}e^{-sA}x\| \leq C_{\beta}t^{\beta}\|A^{\alpha+\beta}e^{-sA}x\| \leq C_{\beta}C_{\alpha+\beta}t^{\beta}s^{-(\alpha+\beta)}\|x\|.\label{app12} \end{equation} We use the inequality (\ref{app12}) to obtain \begin{equation} \begin{aligned} &\int_0^{t_1}\|(e^{-(t_2-s)A}-e^{-(t_1-s)A})A^{\alpha}\|[\|B_nu(s)\| +\|C_nu(s-\tau)\| \\ &+\int^{0}_{-\tau}|a(\theta)|\|L_nu(s+\theta)\|d\theta ]ds \\ &\leq \int_0^{t_1}\|(e^{-(t_2-t_1)A}-I)e^{-(t_1-s)A}A^{\alpha}\|[ \|B_nu(s)\|+\|C_nu(s-\tau)\| \\ & \quad +\int^{0}_{-\tau}|a(\theta)|\|L_nu(s+\theta)\|d\theta ]ds \\ &\leq C_{\alpha,\beta}(t_2-t_1)^{\beta}, \end{aligned}\label{ex6} \end{equation} where $$ C_{\alpha,\beta}=C_{\beta}C_{\alpha +\beta}K(\eta_0){T_0^{1-({\alpha+\beta})}\over[1-({\alpha+\beta})]}, $$ $K(\eta_0)$ is given by (\ref{ex3}) and $\eta_0 = R+\|h\|_{0,\alpha}$. We calculate the second part of the integral (\ref{ex4}) as follows. We have \begin{equation} \begin{aligned} &\int_{t_1}^{t_2}\|e^{-(t_2-s)A}A^{\alpha}\| [\|B_nu(s)\|+\|C_nu(s-\tau)\| +\int^{0}_{-\tau}|a(\theta)|\|L_nu(s+\theta)\|d\theta]ds \\ &\leq C_{\alpha}K(\eta_0){(t_2-t_1)^{1-\alpha}\over (1-\alpha)}. \end{aligned} \label{ex5} \end{equation} Hence from (\ref{ex8}), (\ref{ex6}) and (\ref{ex5}) the map $t\mapsto (F_nu)(t)$ is continuous from $[-\tau,T_0]$ into $D(A^\alpha)$ with respect to $\|\cdot\|_\alpha$ norm. Now, for $t\in [-\tau,0]$, $(F_nu)(t)-\bar{h}(t)=0$.\\ For $t \in (0,T_0]$, we have \begin{align*} &\|(F_nu)(t)- \bar{h}(t)\|_{\alpha}\\ &\leq\|(e^{-tA}-I)A^{\alpha}h(0)\| \\ &\quad +\int_0^t\|e^{-(t-s)A}A^{\alpha}\|\big[\|B_nu(s)\|+\|C_nu(s-\tau)\| +\int^{0}_{-\tau}|a(\theta)|\|L_nu(s+\theta)\|d\theta\big]ds \\ &\leq {R\over 2}+C_{\alpha}K(\eta_0){T_0^{1-\alpha}\over 1-\alpha}. \end{align*} Hence $\|F_nu- \bar{h}\|_{T_0,\alpha}\leq R$. Thus $F_n:B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})\to B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})$. Now, for any $u, v\in B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})$ and $t\in [-\tau,0]$ we have $F_nu(t) - F_nv(t)= 0$. For $t\in (0,T_0]$ and $u, v\in B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})$ we have \begin{align*} &\|F_nu(t) - F_nv(t)\|_{\alpha}\\ & \leq \int^t_0\|e^{-(t-s)A}A^{\alpha}\|\Big[\|B_nu(s)-B_nv(s)\| +\|C_nu(s-\tau)-C_nv(s-\tau)\| \\ &\quad +\int^0_{-\tau}|a(\theta)|\|L_nu(s+\theta)-L_nv(s+\theta)\|d\theta\Big]ds \\ &\leq \int^t_0C_{\alpha}(t-s)^{-\alpha}[K_1({\eta_0})\|u(s)-v(s)\|_{\alpha} +K_2({\eta_0})\|u(s-\tau)-v(s-\tau)\|_{\alpha} \\ &\quad +\int^0_{-\tau}|a(\theta)|K_3(\eta_0)\|u(s+\theta)-v(s+\theta)\|_{\alpha} d\theta ] ds \\ &\leq\int_0^tC_{\alpha}(t-s)^{-\alpha}K(\eta_0)\|u-v\|_{T_0,\alpha}ds \\ &\leq{1\over 2}\|u-v\|_{T_0,\alpha}. \end{align*} Taking the supremum on $t$ over $[-\tau,T_0]$ we get $$ \|F_nu-F_nv\|_{T_0,\alpha}\leq {1\over 2}\|u-v\|_{T_0,\alpha}. $$ Hence there exists a unique $u_n\in B_R(\mathcal{C}_{T_0}^\alpha, \bar{h})$ such that $F_nu_n=u_n$, which satisfies the approximate integral equation \eqref{eqn000}. This completes the proof of Theorem~\ref{ms}. \end{proof} \begin{corollary}\label{cr1} If all the hypotheses of the Theorem \ref{ms} are satisfied then $u_n(t)\in D(A^{\beta})$ for all $t\in [-\tau,T_0]$ where $0\le \beta<1$. \end{corollary} \begin{proof} From Theorem \ref{ms} there exists a unique $u_n\in B_R(\mathcal{C}_{T_0}^\alpha,\bar{h})$ satisfying \eqref{eqn000}. From \cite[Theorem 1.2.4]{ap} we have that $e^{-tA}x\in D(A)$ for $x\in D(A)$. Also from Part (a) of \cite[Theorem 2.6.13]{ap} we have $e^{-tA}:H\mapsto D(A^\beta)$ for $t>0$ and $0\le \beta <1$. H\"older continuity of $u_n$ follows from the similar arguments as used in (\ref{ex6}) and (\ref{ex5}). From \cite[Theorem 4.3.2]{ap}, for $0From Corollaries \ref{cr1} and \ref{cr3}, inequality (\ref{16}) becomes \begin{equation} \|u_n(t) - u_m(t)\|_{\alpha} \\ \leq C_{1}.t_{0}'+ {C_{2}\over \lambda_{m}^{\beta-\alpha}} +C_{\alpha}K(\eta_0)\int^t_{t_0'}(t-s)^{-\alpha}\|u_n-u_m\|_{s,\alpha}ds, \label{17} \end{equation} where $C_1=2C_{\alpha}(t_0-t_0')^{-\alpha}C K(\eta_0) $ and $C_2= {2K(\eta_0)C_{\alpha}T^{1-\alpha}\over (1-\alpha)}$. Now we replace $t$ by $t+\theta$ in inequality (\ref{17}) where $\theta\in [t_0'-t,0]$, we get \begin{equation} \begin{aligned} &\|u_n(t+\theta) - u_m(t+\theta)\|_{\alpha} \\ &\leq C_{1}.t_{0}'+ {C_{2}\over \lambda_{m}^{\beta-\alpha}} +C_{\alpha}K(\eta_0)\int^{t+\theta}_{t_0'}(t+\theta-s)^{-\alpha} \|u_n-u_m\|_{s,\alpha}ds. \end{aligned}\label{18} \end{equation} We put $s-\theta=\gamma$ in (\ref{18}) to get \begin{align*} &\|u_n(t+\theta) - u_m(t+\theta)\|_{\alpha} \\ &\leq C_{1}.t_{0}'+ {C_{2}\over \lambda_{m}^{\beta-\alpha}} +C_{\alpha}K(\eta_0)\int^{t}_{t_0'-\theta}(t-\gamma)^{-\alpha} \|u_n-u_m\|_{\gamma,\alpha}ds\\ &\leq C_{1}.t_{0}'+ {C_{2}\over \lambda_{m}^{\beta-\alpha}} +C_{\alpha}K(\eta_0)\int^{t}_{t_0'}(t-\gamma)^{-\alpha}\|u_n-u_m\|_{\gamma,\alpha}ds. %\label{19} \end{align*} Now \begin{equation} \begin{aligned} &\sup_{{t_0'-t}\le \theta\le 0}\|u_n(t+\theta) - u_m(t+\theta)\|_{\alpha} \\ & \leq C_{1}.t_{0}'+ {C_{2}\over \lambda_{m}^{\beta-\alpha}} +C_{\alpha}K(\eta_0)\int^{t}_{t_0'}(t-\gamma)^{-\alpha}\|u_n-u_m\|_{\gamma,\alpha}ds. \end{aligned}\label{20} \end{equation} We have \begin{align*} &\sup_{{-\tau-t}\le \theta\le 0}\|u_n(t+\theta) - u_m(t+\theta)\|_{\alpha} \\ &\le \sup_{0\le \theta+t\le t_0'}\|u_n(t+\theta) -u_m(t+\theta)\|_{\alpha} +\sup_{{t_0'-t}\le \theta\le 0}\|u_n(t+\theta) - u_m(t+\theta)\|_{\alpha}. \end{align*} Using inequalities (\ref{20}) and (\ref{16}) in the above inequality, we get \begin{align*} &\sup_{{-\tau}\le t+\theta\le t}\|u_n(t+\theta) - u_m(t+\theta)\|_{\alpha} \\ &\le(C_{1}+C_3)t_{0}'+ {(C_{2}+C_4) \over \lambda_{m}^{\beta-\alpha}} +C_{\alpha}K(\eta_0)\int^{t}_{t_0'}(t-\gamma)^{-\alpha}\|u_n-u_m\|_{\gamma,\alpha}ds, %\label{22} \end{align*} where $C_3$ and $C_4$ are constants. An application of Gronwall's inequality to the above inequality gives the required result. This completes the proof of the Theorem \ref{ms3}. \end{proof} With the help of Theorems \ref{ms} and \ref{ms3}, we may state the following existence, uniqueness and convergence result. \begin{theorem} \label{thmmain} Suppose that the conditions (H1)-(H6) are satisfied and $h(t)\in D(A)$ for all $t\in [-\tau,0]$ hold. Then there exist a function $u_n \in C([-\tau,T_0];H)$ and $u \in C([-\tau,T_0];H)$ satisfying \begin{equation} \label{approxsol} % \label{ex9} u_n(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[B_nu_n(s) \\ +C_nu_n(s-\tau)+\int^{0}_{-\tau}a(\theta)L_nu_n(s+\theta)d\theta]ds, & t \in [0,{T_0}] \end{cases} \end{equation} and \begin{equation} \label{sol} u(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[Bu(s)\\ +Cu(s-\tau)+\int^{0}_{-\tau}a(\theta)Lu(s+\theta)d\theta]ds, & t \in [0,\tilde{T}] \end{cases} \end{equation} such that $u_n \to u$ in $C([-\tau,T_0];H)$ as $n \to \infty$, where $B_n$, $C_n$ and $L_n$ are as defined earlier. \end{theorem} \section{Faedo-Galerkin Approximations} We know from the previous sections that for any $-\tau\leq T_0\leq T$, we have a unique $u\in C_{T_0}^{\alpha}$ satisfying the integral equation \begin{equation} \label{so23} u(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[Bu(s)\\ +Cu(s-\tau)+\int^{0}_{-\tau}a(\theta)Lu(s+\theta)d\theta]ds, & t \in [0,\tilde{T}]. \end{cases} \end{equation} Also, there is a unique solution $u\in C_{T_0}^{\alpha}$ of the approximate integral equation \begin{equation}\label{approxso2} %\label{ex9} u_n(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[B_nu_n(s) \\ +C_nu_n(s-\tau)+\int^{0}_{-\tau}a(\theta)L_nu_n(s+\theta)d\theta]ds, & t \in [0,{T_0}]. \end{cases} \end{equation} Faedo-Galerkin approximation ${\bar{u}}_n=P^n{u_n}$ is given by \begin{equation}\label{approxso3} %\label{ex119} {\bar{u}}_n(t) =\begin{cases} P^nh(t), & t \in [-\tau,0], \\[3pt] e^{-tA}P^nh(0)+\int_0^te^{-(t-s)A}P^n[B_nu_n(s) \\ +C_nu_n(s-\tau)+\int^{0}_{-\tau}a(\theta)L_nu_n(s+\theta)d\theta]ds, & t \in [0,{T_0}], \end{cases} \end{equation} where $B_n$, $C_n$ and $L_n$ are as defined earlier. If the solution $u(t)$ to (\ref{so23}) exists on $-\tau\le t\le T_0$ then it has the representation \begin{eqnarray} u(t)=\sum_{i=0}^{\infty}{\alpha}_{i}(t)\phi_i, \end{eqnarray} where ${\alpha}_{i}(t)=(u(t),\phi_i)$ for $i=0,1,2,3,\cdots$ and \begin{eqnarray} {\bar{u}}_n(t)=\sum_{i=0}^{n}{\alpha}_{i}^{n}(t)\phi_i, \end{eqnarray} where ${\alpha}_{i}^{n}(t)=({{\bar{u}_n}}(t),\phi_i)$ for $i=0,1,2,3,\cdots$. As a consequence of Theorem~\ref{ms} and Theorem~\ref{ms3}, we have the following result. \begin{theorem} \label{thmmain2} Suppose that the conditions (H1)-(H6) are satisfied and $h(t)\in D(A)$ for all $t\in [-\tau,0]$. Then there exist unique functions ${\bar{u}}_n \in C([-\tau,T_0];H_n)$ and $u \in C([-\tau,T_0];H)$ satisfying \[ %\label{ex9} {\bar{u}}_n(t) =\begin{cases} P^nh(t), & t \in [-\tau,0], \\[3pt] e^{-tA}P^nh(0)+\int_0^te^{-(t-s)A}P^n[B_nu_n(s) \\ +C_nu_n(s-\tau)+\int^{0}_{-\tau}a(\theta)L_nu_n(s+\theta)d\theta]ds, & t \in [0,{T_0}] \end{cases} \] and \[ u(t) =\begin{cases} h(t), & t \in [-\tau,0], \\[3pt] e^{-tA}h(0)+\int_0^te^{-(t-s)A}[Bu(s)\\ +Cu(s-\tau)+\int^{0}_{-\tau}a(\theta)Lu(s+\theta)d\theta]ds, & t \in [0,\tilde{T}], \end{cases} \] such that ${\bar{u}}_n \to u$ in $C([-\tau,T_0];H)$ as $n \to \infty$, where $B_n, C_n$ and $L_n$ are as defined earlier. \end{theorem} \begin{theorem} Let (H1)-(H6) hold. If $h(t)\in D(A)$ for all $t\in [-\tau,0]$ then for any $-\tau \le t \le T_0\le T$, $$ \lim_{n\to \infty}\sup_{-\tau\le t\le T_0} \big[\sum_{i=0}^{n}{\lambda}_i^{2\alpha}\{{\alpha}_i(t)-{\alpha}_i^n(t)\}^2\big]=0. $$ \end{theorem} \begin{proof} Let $\alpha_i^n(t)=0$ for $i >n$. We have \[ A^{\alpha}[u(t)-{\bar{u}}_n(t)] =A^{\alpha}\big[\sum_{i=0}^{\infty}\{{\alpha}_i(t)-{\alpha}_i^n(t)\}\phi_i\big] \\ =\sum_{i=0}^{\infty}\lambda_i^{\alpha}\{{\alpha}_i(t)-{\alpha}_i^n(t)\}\phi_i. \] Thus we have \begin{eqnarray} \|A^{\alpha}[u(t)-{\bar{u}}_n(t)\|^2\geq \sum_{i=0}^{n}{\lambda}_i^{2\alpha}|{\alpha}_i(t)-{\alpha}_i^n(t)|^2. \end{eqnarray} Hence as a consequence of Theorem \ref{thmmain} we have the required result. \end{proof} \section{Example} Consider the following partial differential equation with delay, \begin{equation} \begin{gathered} w_{t}(t,x)=w_{xx}(t,x)+b(w(t,x))w_{x}(t,x)+c(w(t-\tau,x))w_{x}(t-\tau,x)\\ +\int_{-\tau}^0a(s)l(w(t+s,x))w_{x}(t+s,x)ds, \quad t \ge 0, \; x \in (0,1),\\ w(t,x)=\tilde{h}(t,x), \quad t \in [-\tau,0], \; x \in (0,1),\\ w(t,0)=w(t,1)=0, \quad t \ge 0, \end{gathered} \label{ex786786} \end{equation} where the kernel $a\in L^{p}_{\rm loc}(-\tau,0)$, $b$, $c$, $l$ are smooth functions from $\mathbb{R}$ into $\mathbb{R}$, $\tilde{h}$ is a given continuous function and $\tau>0$ is a given number. We define an operator $A$, as follows, \begin{equation} Au=-u''\quad \mbox{with}\quad u\in D( A) = H_{0}^{1}( 0,1)\cap H^{2}( 0,1).\label{4.1} \end{equation} Here clearly the operator $A$ satisfies the hypothesis (H1) and is the infinitesimal generator of an analytic semigroup $\{e^{-tA}:t\ge 0\} $. For $ 0\le \alpha < 1$, and $t \in [0,T]$, we denote $C_t^{\alpha}:=C([-\tau,t]; D( A^{\alpha}))$, which is the Banach space endowed with the sup norm $$ \|\psi\|_{t,{\alpha}} :=\sup_{-\tau \le \eta \le t}\|\psi(\eta)\|_{\alpha}. $$ We observe some properties of the operators $A$ and $A^{\alpha}$ defined by (\ref{4.1}) (cf. \cite{bs} for more details). For $\phi\in D( A) $ and $\lambda \in \mathbb{R}$, with $A\phi=-\phi''=\lambda u$, we have $\langle A\phi,\phi\rangle =\langle \lambda \phi,\phi\rangle$; that is, \[ \left\langle -\phi'',\phi\right\rangle =| u'| _{L^{2}}^{2}=\lambda | \phi| _{L^{2}}^{2}, \] so $\lambda >0$. A solution $\phi$ of $A\phi=\lambda \phi$ is of the form \[ \phi( x) =C\cos ( \sqrt{\lambda }x) +D\sin ( \sqrt{\lambda }x) \] and the conditions $\phi( 0) =\phi( 1)=0 $ imply that $C=0$ and $\lambda=\lambda_n=n^2\pi^2$, $n\in \mathbb{N}$. Thus, for each $n \in \mathbb{N}$, the corresponding solution is \[ \phi_n( x) =D\sin ( \sqrt{\lambda _n}x). \] We have $\langle \phi_n,\phi_m\rangle =0$, for $n\neq m$ and $\langle \phi_n,\phi_n\rangle =1$ and hence $D=\sqrt{2}$. For $u\in D( A) $, there exists a sequence of real numbers $\{ \alpha_{n}\} $ such that $$ u( x) =\sum_{n\in \mathbb{N}} \alpha_{n}\phi_{n}( x) , \quad \sum_{n\in \mathbb{N}}(\alpha _{n}) ^{2}<+\infty \quad \mbox{and} \quad \sum_{n\in \mathbb{N}}( \lambda_{n}) ^{2}( \alpha _{n}) ^{2} <+\infty. $$ We have \[ A^{1/2}u( x) =\sum_{n\in {\mathbb{N}}}\sqrt{\lambda _n}\;\alpha _n\;\phi_n( x) \] with $u\in D( A^{1/2})=H_0^1(0,1) $; that is, $\sum_{n\in {\mathbb{N} }}\lambda _{n}( \alpha _{n}) ^{2}<+\infty $. Then equation(\ref{ex786786}) can be reformulated as the following abstract equation in a separable Hilbert space $H=L^2(0,1)$: \begin{gather*} u'(t)+Au(t)=Bu(t)+Cu(t-\tau)+\int_{-\tau}^{0}a(\theta)Lu(t+\theta)d\theta, \; 0 < t \le T<\infty, \tau > 0,\\ u(t)=h(t), \quad t \in [-\tau,0], %\label{ex1111} \end{gather*} where $u(t)=w(t, .)$ that is $u(t)(x)=w(t,x)$, $u_t(\theta)(x)=w(t+\theta,x)$, $t \in [0,T]$, $\theta \in [-\tau,0]$, $x \in (0,1)$, the operator $A$ is as define in equation (\ref{4.1}) and $h(\theta)(x)=\tilde{h}(\theta,x)$ for all $\theta \in [-\tau,0]$ and $x\in (0,1)$. The operators $B,C$ and $L$ are given by as follows: \\ $B:D(A^{1/2})\mapsto H$, where $Bu(t)(x)=b(w(t,x))w_{x}(t,x)$, \\ $C:D(A^{1/2})\mapsto H$, where $Cu(t-\tau)(x)=c(w(t-\tau,x))w_x(t-\tau,x)$, and \\ $L:D(A^{1/2})\mapsto H$, where $Lu(t+s)(x)=l(w(t+s,x))w_x(t+s,x)$, where $s\in [-\tau,0]$ and $x\in (0,1)$. Let $\alpha$ be such that $3/4 < \alpha <1$. For $u,v\in D(A^\alpha)$ with $\|A^\alpha u\|\le \eta$ and $\|A^{\alpha}v\|\le \eta$, we have \begin{align*} &|b(u(x))u_{x}(x)-b(v(x))v_{x}(x)| \\ &\le|b(u(x))-b(v(x))||u_{x}(x)|+|b(v(x))||u_{x}(x)-v_{x}(x)|\\ &\le L_b |u(x)-v(x)||u_x(x)|+b_1|u_{x}(x)-v_{x}(x)|, \end{align*} where $L_b$ is the Lipschitz constant for $b$ and $b_1=L_b{\eta \over \lambda_0^{1/2}}+|b(0)|$. For $u,v \in D(A^\alpha) \subset D(A^{1/2})$, we have \begin{eqnarray*} \|B(u)-B(v)\|^2\le \int_0^1|[L_b |u(x)-v(x)||u_x(x)|+b_1|u_{x}(x)-v_{x}(x)|]|^2dx. \end{eqnarray*} Thus, from \cite[Lemma 8.3.3]{ap}, we get \begin{align*} &\|B(u)-B(v)\|^2\\ &\le 2{L_b}^2\int_0^1 |u(x)-v(x)|^2|u_x(x)|^2dx +2{b_1}^2\int_0^1|u_{x}(x)-v_{x}(x)||^2dx\\ &\le 2{L_b}^2\|u-v\|_{\infty}^2\int_0^1|u_x(x)|^2dx +2{b_1}^2\int_0^1|u_{x}(x)-v_{x}(x)|^2dx\\ &\le 2{L_b}^2\|u-v\|_{\infty}^2\|A^{1/2}u\|^2+2{b_1}^2\|A^{1/2}(u-v)\|^2\\ &\le 2{L_b}^2c^2{\eta^2}\|A^{\alpha}(u-v)\|^2+2{b_1}^2\|A^{\alpha}(u-v)\|^2\\ &\le {M_b(\eta)}^2\|A^{\alpha}(u-v)\|^2, \end{align*} where $3/4<\alpha<1$, $\|A^{\alpha}u\|\le \eta$, $\|A^{\alpha}v\|\le \eta$, $M_b(\eta)=\sqrt{2}[L_bc \eta + b_1]$, $\|u\|_\infty=\sup_{0 \le x \le 1}|u(x)|$ and $\|u\|_\infty \le c \|A^\alpha u\|$ for any $u \in D(A^\alpha)$. Hence the operator $B$ restricted to $D(A^\alpha)$ satisfies the hypothesis (H3) for $K_1(\eta)=M_b(\eta)$. 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