\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Existence results \hfil EJDE--2001/24} {EJDE--2001/24\hfil Eduardo Hern\'{a}ndez M. \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2001}(2001), No. 24, pp. 1--14. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Existence results for a class of semi-linear evolution equations % \thanks{ {\em Mathematics Subject Classifications:} 35A05, 34G20, 34A09. \hfil\break\indent {\em Key words:} Banach spaces, semigroup of linear operators, abstract differential equations, \hfil\break\indent fractional powers of closed operators, regular solutions. \hfil\break\indent \copyright 2001 Southwest Texas State University. \hfil\break\indent Submitted December 14, 2000. Published April 10, 2001. \hfil\break\indent Partially supported by grant 13394-5 from Fapesp Brazil } } \date{} % \author{ Eduardo Hern\'{a}ndez M. } \maketitle \begin{abstract} We prove the existence of regular solutions for the quasi-linear evolution $$\frac{d}{dt}(x(t)+g(t,x(t))=Ax(t)+f(t,x(t)),$$ where $A$ is the infinitesimal generator of an analytic semigroup of bounded linear operators defined on a Banach space and the functions $f, g$ are continuous. \end{abstract} \newtheorem{Lemma}{Lemma} \newtheorem{Theorem}{Theorem} \newtheorem{Corollary}{Corollary} \newtheorem{Definition}{Definition} \renewcommand{\theequation}{\thesection.\arabic{equation}} \catcode@=11 \@addtoreset{equation}{section} \catcode@=12 \section{Introduction}\label{sec1} The class of equations considered in this paper have the form $$\label{ne} \begin{array}{c} \frac{d}{dt} (x(t)+g(t,x(t))=Ax(t)+f(t,x(t)), \quad t>0, \\ x(0)=x_0\,. \end{array}$$ We consider this system as a Cauchy problem on a Banach space $X$, where $A$ is the infinitesimal generator of an analytic semigroup of bounded linear operators $(T(t))_{t\geq 0}$; $f,g:[0,T]\times \Omega \to X$ are appropriate continuous functions and $\Omega$ is an open subset of $X$. The case $g\equiv 0$ has an extensive literature. The books of Pazy \cite{PA}, Krein \cite{KE}, Goldstein \cite{GO} and the references contained therein, give a good account of important results. Throughout this paper $X$ will be a Banach space equipped with the norm $\|\cdot\|$ and the operator $A:D(A)\subset X\to X$ will be the infinitesimal generator of an analytic semigroup of bounded linear operators $(T(t))_{t\geq 0}$ on $X$. For the theory of strongly continuous semigroups, refer to \cite{PA} and \cite{GO}. We mention here only some notation and properties essential to our purpose. In particular, it is well known that there exist $\tilde{M}\geq 1$ and a real number $w$ such that $$\| T(t) \|\leq \tilde{M}e^{wt}, \quad t\geq 0\,.$$ In what follows we assume that $\| T(t)\|$ is uniformly bounded by $\tilde{M}$ and that $0 \in \rho(A)$. In this case it is possible to define the fractional power $(-A)^\alpha$, for $0<\alpha<1$, as a closed linear operator with domain $D((-A)^\alpha)$. Furthermore, the subspace $D((-A)^\alpha)$ is dense in $X$ and the expression $$\| x \|_\alpha = \| (-A)^\alpha x \|$$ defines a norm on $D((-A)^\alpha)$. Hereafter we represent by $X_\alpha$ the space $D((-A)^\alpha)$ endowed with the norm $\| \cdot \|_\alpha$. The following properties are well known (see \cite{PA}). \begin{Lemma}\label{an} Under the above conditions we have \begin{enumerate} \item If $0<\alpha\leq 1$, then $X_\alpha$ is a Banach space. \item If $0< \beta \leq \alpha$, then $X_\alpha \to X_{\beta}$ is continuous and compact when the resolvent operator of $A$ is compact. \item For every constant $a>0$, there exists $C_{a}>0$ such that $$\| (-A)^\alpha T(t)\| \leq \frac{C_{a}}{t^\alpha}, \quad 00 there exists a positive constant C'_{a} such that$$ \| (T(t)-I)(-A)^{-\alpha} \| \leq C'_{a}t^\alpha, \quad 00$such that $$\| f(t,x)-f(s,y)\|\leq N ( | t-s| +\| x-y\| )$$ for every$0\leq s,t\leq T$and$x,y\in \Omega$. \end{itemize} Then there exists a unique mild solution$x(\cdot,x_0)$of the abstract Cauchy problem (\ref{ne}) defined on$[0,r]$for some$00$and$0<\gamma_1,\gamma_2<1$such that for every$(t,x_1),(s,x_2)\in [0,T]\times\Omega_\alpha $we have $$\displaylines{ \| (-A)^{\beta}g(t,x_1)-(-A)^{\beta}g(s,x_2)\| \leq L\{| t-s |^{\gamma_1} + \| x_1-x_2\|_\alpha \}, \cr \| f(t,x_1)-f(s,x_2)\| \leq N\{| t-s |^{\gamma_2} + \| x_1-x_2\|_\alpha \}, \cr L\| (-A)^{\alpha-\beta}\| < 1\,. }$$ \begin{Theorem}\label{nclass} Let$x_0\in \Omega_\alpha $and assume that$f$and$g$satisfy the above assumptions, that$g$is$D(A)$-valued continuous and that$1-\beta<\min\{\beta-\alpha,\gamma_1, \gamma_2\}$. Then there exists a unique S-classical solution$x(\cdot,x_0)\in C([0,r]:X)$for some$00$such that $$V=\{(t,x)\in [0,r_1]\times X_\alpha :\,\, \| (-A)^\alpha x-(-A)^\alpha x_0\| <\delta \}\subset [0,T)\times\Omega_\alpha .$$ Assuming that the functions$f$and$(-A)^{\beta}g$are bounded on$V$by$C_1>0$, we choose$0From the choice of $r$ we conclude that $$\|\Psi(y)-(-A)^\alpha x_0\|_{r} \leq \delta$$ so that $\Psi(y) \in S$. On the other hand for $x(\cdot),\, y(\cdot) \in S$ and $t \in [0,r]$, \begin{eqnarray*} \lefteqn{\| \Psi(y)(t)- \Psi(x)(t)\|}\\ &\leq& \| (-A)^\alpha g(t,(-A)^{-\alpha}y(t))-(-A)^\alpha g(t,(-A)^{-\alpha}x(t))\|\\ &+& \int_0^{t}\frac{C_{1-\beta+\alpha}}{(t-s)^{1-\beta+\alpha}}\| (-A)^{\beta}g(s,(-A)^{-\alpha}y(s))-(-A)^{\beta}g(s,(-A)^{-\alpha}x(s))\| ds \\ &+& \int_0^{t}\frac{C_\alpha }{(t-s)^\alpha}\| f(s,(-A)^{-\alpha}y(s))-f(s,(-A)^{-\alpha}x(s))\| ds \\ &\leq& \| (-A)^{\alpha-\beta}\| L \| y(t)- x(t) \| + \int_0^{t}\{\frac{LC_{1-\beta+\alpha}}{(t-s)^{1-\beta+\alpha}}+ \frac{NC_\alpha }{(t-s)^\alpha}\}\| y-x\|_{r}ds, \end{eqnarray*} thus $$\| \Psi(y)-\Psi(x) \|_{r}\leq \left( L\| (-A)^{\alpha-\beta}\| +LC_{1-\beta+\alpha}\frac{r^{\beta-\alpha}}{\beta-\alpha}+ NC_\alpha \frac{r^{1-\alpha}}{1-\alpha}\right) \| y-x\|_{r}.$$ The last estimate and the choice of $r$ imply that $\Psi$ is a contraction mapping on $S$. Let $y(\cdot)$ be the unique fixed point of the operator $\Psi$ in $S$. We affirm that $y(\cdot)$ is locally H\"{o}lder continuous. In fact, let $\vartheta$ be a real number with $0<\vartheta < \min\{1-\alpha,\beta-\alpha\}$ and $\vartheta+\beta>1$, and let $\tilde{C}>0$ be the constant guaranteed in Lemma \ref{an}, such that for all $0\leq s\leq t \leq T$ and $00$ sufficiently small, \begin{eqnarray*} \lefteqn{\| y(t+h)-y(t)\|}\\ &\leq& \| (T(h)-I)(-A)^\alpha T(t)(x_0-g(0,x_0))\| \\ &&+\| (-A)^{\alpha-\beta}\| L \{ h^{\gamma_1}+\|y(t+h)-y(t)\| \}\\ && + \int_0^{t}\|(T(h)-I)(-A)^{1-\beta+\alpha}T(t-s)(-A)^{\beta} g(s,(-A)^{-\alpha}y(s))\| ds\\ && + \int_{t}^{t+h}\| (-A)^{1-\beta+\alpha}T(t+h-s)(-A)^{\beta}g(s,(-A)^{-\alpha}y(s))\| ds\\ && + \int_0^{t}\| (T(h)-I)(-A)^\alpha T(t-s)f(s,(-A)^{-\alpha}y(s))\| ds \\ && + \int_{t}^{t+h}\| (-A)^\alpha T(t+h-s)f(s,(-A)^{-\alpha}y(s))\| ds \nonumber \\ &\leq& \frac{\tilde{C}}{t^{(\alpha +\vartheta)}}\| x_0-g(0,x_0)\| h^{\vartheta} + L\| (-A)^{\alpha-\beta}\| \{ h^{\gamma_1}+\| y(t+h)-y(t)\| \} \\ && + \int_0^{t}\frac{\tilde{C}h^{\vartheta}C_1}{(t-s)^{1-\beta+\alpha+\vartheta}}ds + \int_{t}^{t+h}\frac{C_{1-\beta+\alpha}C_1}{(t+h-s)^{1-\beta+\alpha}}ds \nonumber\\ && +\int_0^{t}\frac{\tilde{C}h^{\vartheta}C_1}{(t-s)^{\alpha+\vartheta}}ds + \int_{t}^{t+h}\frac{C_\alpha C_1}{(t+h-s)^\alpha}ds \\ &\leq& \frac{C(x_0)h^{\vartheta}}{t^{\vartheta+\alpha}}+ C_2h^{\gamma_1}+L\| (-A)^{\alpha-\beta}\| \| y(t+h)-y(t)\|+C_{3}h^{\vartheta}\\ &&+C_{4}h^{\beta-\alpha}+C_{5}h^{1-\alpha} \end{eqnarray*} where the constants $C_{i}$ are independent of $t$. If $\bar{\rho}=\min\{ \vartheta,\gamma_1\}$, the last inequality can be rewritten in the form $$\| y(t+h)-y(t)\| \leq \frac{C(\alpha,\beta,\vartheta,t,x_0)}{1-\mu}h^{\bar{\rho}}$$ since $\mu=L\| (-A)^{\alpha-\beta}\|<1$. Therefore the function $y(\cdot)$ is locally $\bar{\rho}$-H\"{o}lder continuous on $(0,r)$, moreover, we can to assume that $\bar{\rho}+\beta>1$. Now it is easy to show that $s\to (-A)^{\beta}g(s,(-A)^{-\alpha}y(s))$ and $s\to f(s,(-A)^{-\alpha}y(s))$ are $\rho$-H\"{o}lder continuous on $(0,r)$, where $\rho=\min\{\bar{\rho},\gamma_2\}$ and $\rho+\beta>1$. From this remark, in \cite[Theorem 2.4.1]{GO} and Lemma \ref{regc} below, we infer that the function \begin{eqnarray} x(t)&=& T(t)(x_0+g(0,x_0)) - g(t,(-A)^{-\alpha}y(t))\nonumber \\ && +\int_0^{t}(-A)^{1-\beta}T(t-s)(-A)^{\beta}g(s,(-A)^{-\alpha}y(s))ds \label{reggol} \\ && + \int_0^{t}T(t-s)f(s,(-A)^{-\alpha}y(s))ds \nonumber \end{eqnarray} is $X_\alpha$-valued, that the integral terms in (\ref{reggol}) are functions in $C^{1}([0,r]:X)$ and that $x(t)\in D(A)$ for all $t\in (0,r)$. Operating on $x(\cdot)$ with $(-A)^\alpha$, we conclude that $(-A)^{-\alpha}y=x$ and hence that $x(t)+g(t,x(t))$ is a $C^{1}$ function on $(0,b)$. The proof is completed.\hfill$\diamondsuit$\medskip The proof of the next Lemma is analogous to the proof in \cite[Theorem 2.4.1]{GO}. However there are some differences that require special attention and we include the principal ideas of this proof for completeness. \begin{Lemma}\label{regc} Let $0<\beta<1$ and $g\in C([0,T]:X_{1-\beta})$. Assume that $g:[0,T]\to X$ is $\theta$-H\"older continuous in $(0,T)$ with $\beta+\theta>1$. If $y:[0,T]\to X$ is defined by $$y(t)=\int_0^{t}(-A)^{1-\beta}T(t-s)g(s)ds,$$ then $y(t)\in D(A)$ for every $t\in [0,T)$ and $\dot{y} \in C([0,T):X)$. \end{Lemma} \paragraph{Proof.} For $t\in [0,T)$ we rewrite $y(t)$ in the form $$\label{leii} \int_0^{t}(-A)^{1-\beta}T(t-s)(g(s)-g(t))ds + \int_0^{t}(-A)^{1-\beta}T(t-s)g(t)ds=v(t)+w(t).$$ Clearly, $Aw(t)=T(t)(-A)^{1-\beta}g(t)-(-A)^{1-\beta}g(t) \in C([0,T]:X)$. For $\epsilon >0$, sufficiently small we define the function $$v_{\epsilon}(t):= \left\{ \begin{array}{ll} \int_0^{t-\epsilon}(-A)^{1-\beta}T(t-s)(g(s)-g(t))ds\,, & \hbox{for } t\in[\epsilon,T),\\[3pt] 0 & \hbox{for } t \in [0,\epsilon).\end{array} \right.$$ It is clear that $v_{\epsilon}(t)\in D(A)$. Moreover for $0<\delta_1<\delta_2$ \begin{eqnarray*} \| Av_{\delta_2}(t)-Av_{\delta_1}(t)\| &\leq& \int_{t-\delta_2}^{t-\delta_1} \| (-A)^{2-\beta}T(t-s)(g(s)-g(t))\| ds \\ &\leq& C_{2-\beta}(\delta_2^{\beta+\theta -1}-\delta_1^{\beta+\theta-1}). \end{eqnarray*} The last inequality proves that $Av_{\delta}$ is convergent, $\beta+\theta>1$, and therefore \begin{eqnarray} A(v(t))=\int_0^{t}A^{2-\beta}T(t-s)( g(s)-g(t) ) ds \end{eqnarray} since $A$ is a closed operator. From the previous remark it follows that $y(t)\in D(A)$ for $t\in [0,T]$. The continuity of $\partial_{t}y$ follows as in \cite[Theorem 2.4.1]{GO}. \hfill$\diamondsuit$\medskip In the rest of this paper for a function $j:[0,b]\times X\to$ $X$ and $h\in {{\bf I\kern -1.6pt{\bf R}}}$ we denote by $\partial_{h}j$ to the function $$\partial_{h}j(t)=\frac{j(t+h)-j(t)}{h}.$$ Moreover, if $j$ is differentiable we will employ the decomposition: $$\label{descomi} j(t+s,y)-j(t,y)=D_1j(t,y)s+W_1(j,t,t+s,y)$$ and $$\label{descomii} j(t,y+y_1)-j(t,y)=D_2j(t,y)\cdot y_1+W_2(j,t,y,y+y_1)$$ where \begin{eqnarray*} \frac{W_1(j,t,t+s,y)}{| s |}\to 0&& \mbox{as}\quad s\to 0\\ \frac{W_2(j,t,y,y+y_1)}{\| y_1\|}\to 0 &&\mbox{as}\quad y_1\to 0\,. \end{eqnarray*} To prove the next theorem, we need a preliminary result which is interesting in its own right. \begin{Lemma}\label{lipsolutions} Under the assumptions in Theorem \ref{lip}, if $x_0\in D(A)$ and $g(0,x_0)\in D(A)$, then $x(\cdot)=x(\cdot,x_0)$ is Lipschitz on closed intervals. \end{Lemma} \paragraph{Proof.} Initially we prove that $x(\cdot)$ is $\beta$-H\"{o}lder continuous on a closed interval $[0,b]$. Using the continuity of $(-A)^{\beta}g$ and $f$ we can to assert that $(-A)^{\beta}g(s,x(s))$ and $f(s,x(s))$ are bounded by $C_1>0$ on $[0,b]$. Employing that $x_0\in D(A)$ and that $g(0,x_0)\in D(A)$; for $t\in [0,b)$ and $h>0$ we have \begin{eqnarray*} \lefteqn{\|x(t+h)-x(t)\|}\\ &\leq& C_2h + \| g(t+h,x(t+h))-g(t,x(t))\| \\ &&+\int_0^{t}\frac{C_{1-\beta}}{(t-s)^{{1-\beta}}}\| (-A)^{\beta}g(s+h,x(s+h))-(-A)^{\beta}g(s,x(s))\| ds \\ &&+\int_0^{h}\| (-A)^{1-\beta}T(t+h-s)(-A)^{\beta}g(s,x(s))\| ds \\ && +\tilde{M}\int_0^{t}\| f(s+h,x(s+h))-f(s,x(s))\| ds +\tilde{M}\int_0^{h}\| f(s,x(s))\| ds \end{eqnarray*} thus \begin{eqnarray*} \| x(t+h)-x(t) \| &\leq& C_{3}h^{\beta} +\| (-A)^{-\beta}\| L\| x(t+h)-x(t)\| \nonumber \\ && +\int_0^{t}\{\frac{C_{1-\beta}L}{(t-s)^{1-\beta}}+N\tilde{M}\}\| x(s+h)-x(s)\| ds. \nonumber \end{eqnarray*} Since $\| (-A)^{-\beta}\| L<1$, the Gronwall-Bellman inequality \cite[Lemma 5.6.7]{PA} implies that $x(\cdot)$ is $\beta$-H\"{o}lder continuous. Reiterating the previous estimates and using that $x(\cdot)$ is $\beta$-H\"{o}lder; if $t\in [0,T)$ and $h>0$ we get \begin{eqnarray*} \lefteqn{\|x(t+h)-x(t) \|}\\ && \leq C_{4}h +\| (-A)^{-\beta}\| L\| x(t+h)-x(t)\| \\ &&+\int_0^{t}\{\frac{C_{1-\beta}L}{(t-s)^{1-\beta}}+N\tilde{M}\}\| x(s+h)-x(s)\| ds\\ &&+\int_0^{h}\frac{C_{1-\beta}L}{(t+h-s)^{1-\beta}}\| (-A)^{\beta}g(s,x(s))-(-A)^{\beta}g(0,x_0)\| ds \\ &&+\int_0^{h}\| T(t+h-s)(-A)g(0,x_0)\| ds \end{eqnarray*} then \begin{eqnarray*} \| x(t+h)-x(t) \| &\leq& C_{5} h^{2\beta} +\| (-A)^{-\beta}\| L\| x(t+h)-x(t)\| \nonumber \\ && +\int_0^{t}\{\frac{C_{1-\beta}L}{(t-s)^{1-\beta}}+N\tilde{M}\}\| x(s+h)-x(s)\| ds.\nonumber \end{eqnarray*} The assumption $\| (-A)^{-\beta}\| L<1$ and Gronwall Bellman inequality, implies that $x(\cdot)$ is $2\beta$-H\"{o}lder continuous. Clearly the previous routine permit to infer that $x(\cdot)$ is Lipschitz continuous, this completes the proof. \hfill$\diamondsuit$\medskip In the next theorem we establish the existence of classical solutions using some usual regularity assumptions on the functions $f$ and $(-A)^{\beta}g$. \begin{Theorem}\label{class} Assume that $(-A)^{1-\beta}g(\cdot)$ and $f(\cdot)$ are continuously differentiable functions on $[0,T]\times \Omega$.\, If $x_0,\, g(0,x_0) \in D(A)$ and $\| D_2 g(0,x_0) \|_{{\mathcal{L}}(X)}<1$ then $\dot{x}(\cdot, x_0) \in C([0,b]:X)$ for some $0< b< T$. \end{Theorem} \paragraph{Proof:} Let $x(\cdot)=x(\cdot,x_0)$ and $z(\cdot)$ be a solution of the integral equation \begin{eqnarray} z(t)&=&T(t)(Ax_0+ A g(0,x_0) + f(0,x_0))+h(t) - D_2g(t,x(t))(z(t)) \nonumber\\ &&+\int_0^{t}(-A)^{1-\beta}T(t-s)D_2(-A)^{\beta}g(s,x(s))( z(s))ds \label{reg2}\\ &&+\int_0^{t}T(t-s)D_2f(s,x(s))(z(s))ds \nonumber \end{eqnarray} where \begin{eqnarray}\label{inicond} z(0)=Ax_0+A g(0,x_0)+f(0,x_0) -D_1g(0,x_0)-D_2g(0,x_0)(z(0))\nonumber \end{eqnarray} and \begin{eqnarray*} h(t)&=& - D_1g(t,x(t))+\int_0^{t}(-A)^{1-\beta}T(t-s)D_1(-A) ^{\beta}g(s,x(s))ds \\ &&+\int_0^{t}T(t-s)D_1f(s,x(s))ds\,. \end{eqnarray*} The existence and uniqueness of local solution for (\ref{reg2}), is consequence of the contraction mapping principle and the condition $\| D_2g(0,x_0)\|_{{\mathcal{L}}(X)}<1$; we omit details. In what follows we assume that $x(\cdot)$ and $z(\cdot)$ are defined on $[0,2b]$ where $0<2b0$ sufficiently small, we have \begin{eqnarray*} \lefteqn{ \| \xi(t,h) \| }\\ &=&\|\frac{x(t+h)-x(t)}{h}-z(t)\| \\ &\leq& \| T(t)(\frac{T(h)-I}{h}x_0-A(x_0))\| \nonumber\\ &&+ \| \frac{1}{h}\int_0^{h}T(t+h-s)f(s,x(s))ds-T(t)f(0,x_0) \| \nonumber\\ &&+\| T(t)(\frac{T(h)-I}{h})g(0,x_0)+\frac{1}{h}\int_0^{h} (-A)T(t+h-s)g(s,x(s))ds \|\nonumber \\ && + \| D_1g(t,x(t+h))-D_1g(t,x(t))\| + \| D_2g(t,x(t))(\xi(t,h))\| \nonumber\\ && +\| \frac{W_1(g,t,t+h,x(t+h))}{h}\| + \|\frac{W_2(g,t,x(t),x(t+h))}{h}\| \nonumber\\ &&+ \int_0^{t}\frac{C_{1-\beta}}{(t-s)^{1-\beta}}\| D_1(-A)^{\beta}g(s,x(s+h))- D_1(-A)^{\beta}g(s,x(s))\| ds\nonumber \\ && +\int_0^{t}\frac{C_{1-\beta}}{(t-s)^{1-\beta}}\| D_2(-A)^{\beta}g(s,x(s))(\xi(s,h))\| ds \nonumber \\ && +\int_0^{t}\frac{C_{1-\beta}}{(t-s)^{1-\beta}}\| \frac{W_1((-A)^{\beta}g,s,s+h,x(s+h))}{h}\| ds \nonumber\\ && + \int_0^{t}\frac{C_{1-\beta}}{(t-s)^{1-\beta}}\| \frac{W_2((-A)^{\beta}g,s,x(s),x(s+h))}{h}\| ds \nonumber \\ &&+ \int_0^{t}\tilde{M}\| D_1f(s,x(s+h))-D_1f(s,x(s))\| ds \nonumber \\ &&+\int_0^{t}\tilde{M}\| D_2f(s,x(s))\|\|\xi(s,h)\|ds +\int_0^{t}\tilde{M}\|\frac{W_1(f,s,s+h,x(s+h))}{h}\|ds \nonumber \\ && + \tilde{M}\int_0^{t}\| \frac{W_2(f,s,x(s),x(s+h))}{h}\| ds.\nonumber \end{eqnarray*} On the other hand, from lemma \ref{lip} we know that $x(\cdot)$ is Lipschitz continuous; therefore, $$\frac{W_2((-A)^{\beta}g,s,x(s),x(s+h))}{\| x(s+h)-x(s)\|}\cdot \frac{\| x(s+h)-x(s)\|}{h}\to 0 \quad\mbox{as}\quad h \to 0$$ and $$\frac{W_2(f,s,x(s),x(s+h))}{\| x(s+h)-x(s)\|}\cdot \frac{\| x(s+h)-x(s)\|}{h}\to 0 \quad\mbox{as} \quad h \to 0$$ uniformly for $s\in [0,b]$. This enables us to rewrite the last inequality in the form \begin{eqnarray*} \lefteqn{ \| \xi(t,h) \| }\\ &=&\| \frac{x(t+h)-x(t)}{h}-z(t)\| \\ &\leq& \rho (t,h)+\frac{1}{h}\int_0^{h} \frac{C_{1-\beta}}{(t+h-s)^{1-\beta}}\|(-A)^{\beta}g(0,x_0)-(-A)^{\beta}g(s,x(s))\|ds \\ \, &&+\| D_2 g(t,x(t))( \xi(t,h))\| +\int_0^{t}\frac{C_{1-\beta}}{(t-s)^{1-\beta}}\| D_2(-A)^{\beta}g(s,x(s))\| \|\xi(s,h)\|ds \\ &&+\tilde{M}\int_0^{t}\| D_2f(s,x(s)\| \|\xi(s,h)\|ds \nonumber \end{eqnarray*} where $\rho(t,h)\to 0$ as $h \to 0$, uniformly for $t\in [0,b].$ Since $x(\cdot )$ is Lipschitz and $\| D_2g(\cdot,x(\cdot )) \|_{b} <\eta$, follow that \begin {eqnarray} \| \xi(t,h) \| & \leq & \frac{1}{1-\eta}\rho (t,h) +\frac{C_{1-\beta}LCh^{\beta}}{\beta} \nonumber\\ && +\frac{1}{1-\eta}\int_0^{t}\frac{C_{1-\beta}}{(t-s)^{1-\beta}}\| D_2(-A)^{\beta}g(s,x(s))\| \|\xi(s,h)\| ds \nonumber \\ &&+ \frac{1}{1-\eta}\tilde{M}\int_0^{t}\| D_2f(s,x(s)\| \| \xi(s,h)\| ds. \nonumber \end{eqnarray} Finally, the Gronwall's inequality \cite[Lemma 5.6.7]{PA} shows that $\xi(t,h)\to 0$ as $h\to 0$. Therefore, $\dot{x}(\cdot ,x_0)=z(\cdot)$. This completes the proof. \hfill$\diamondsuit$ \begin{Corollary}\label{classical} If $g$ is a $D(A)$-valued continuous function then there exits a unique classical solutions of (\ref{ne}) defined on $[0,b]$ for some $0< b< T$. \end{Corollary} \paragraph{Proof:} From Theorem \ref{class} we know that $x(\cdot )=x(\cdot ,x_0)\in C^{1}([0,b]:X)$ for some $00,&\\ \label{Sobolevequa'} &u(0)=u_0, \quad u_0\in D(B)\,,& \end{eqnarray} where$A, B$are closed linear operators on a Banach space$X$. The literature includes different and complete results concerning to existence, uniqueness and qualitative properties of mild, strong and classical solutions for (\ref{Sobolevequa})-(\ref{Sobolevequa'}) (see \cite{Brill, ligh, Showalter2, Showalter1}). Some usual assumptions on the operators$A,B$(see for example \cite{Brill, ligh}) are \begin{itemize} \item$A,\,B$are closed linear operators. \item$D(B)\subset D(A)$and$B$has a continuous inverse. \end{itemize} >From these assumptions and the Closed Graph Theorem it follows that$AB^{-1}$is a bounded linear operator on$X$. In this case the approach is to consider the related integral equation $$\label{absobolev0} x(t)=T(t)Bx_0+ \int_0^{t}T(t-s)f(s,B^{-1}x(s))ds,$$ where$T(t)$with$t\geq 0$is the semigroup generated by$AB^{-1}$. We shall consider the abstract Cauchy problem \begin{eqnarray}\label{absobolev21} &\frac{d}{dt}(u(t)+Bu(t))=Au(t)+f(t,u(t)), \quad t>0,&\\ \label{absobolev22} & u(0)= u_0, \quad u_0\in D(B),& \end{eqnarray} where$A,B$are closed linear operators on a Banach space$X$and \begin{itemize} \item$D(A)\subset D(B)$and$B$has a continuous inverse \item$AB^{-1}$is the infinitesimal generator of an analytic semigroup of bounded linear operators on$X$. \end{itemize} Under these conditions, we consider the associated system \begin{eqnarray}\label{Sobolevequa2} &\frac{d}{dt}(u+ B^{-1}u(t))=AB^{-1}u(t)+f(t,B^{-1}u(t)), \quad t>0,&\\ \label{Sobolevequa2'} &u(0)=B^{-1}u_0, \quad u_0\in D(B).& \end{eqnarray} If in addition$B^{-1}$is$D(AB^{-1})$-valued continuous and$f\$ is continuously differentiable, the existence of classical solutions for (\ref{Sobolevequa2})-(\ref{Sobolevequa2'}) and consequently for (\ref{absobolev21})-(\ref{absobolev22}), follows from Corollary \ref{classical}. \begin{thebibliography}{00} {\frenchspacing \bibitem{Brill} Brill, Heinz., A semilinear Sobolev evolution equation in a Banach space. {\sl J. Differential Equations} 24 (1977), no. 3, 412--425. \bibitem{GO} Goldstein, Jerome A., Semigroups of linear operators and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985. \bibitem{HK} Hale, Jack K.; Kato, Junji., Phase space for retarded equations with infinite delay. {\sl Funkcial. Ekvac.} 21 (1978), no. 1, 11--41. \bibitem{HH1} Hern\'{a}ndez, Eduardo; Henr\'{\i}quez, Hern\'{a}n R., Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay. {\sl J. Math. Anal. Appl.} 221 (1998), no. 2, 499--522. \bibitem{HH2} Hern\'{a}ndez, Eduardo; Henr\'{\i}quez, Hern\'{a}n R., Existence results for partial neutral functional-differential equations with unbounded delay. {\sl J. Math. Anal. Appl.} 221 (1998), no. 2, 452--475. \bibitem{HH3} Hern\'{a}ndez, Eduardo., Regular Solutions for Partial Neutral Functional Differential Equations with Unbounded Delay. Preprint. \bibitem{HMN} Hino, Yoshiyuki; Murakami, Satoru; Naito, Toshiki., Functional-differential equations with infinite delay. Lecture Notes in Mathematics, 1473. Springer-Verlag, Berlin, 1991. \bibitem{KE} Krein, S. G., Linear differential equations in Banach space. Translated from the Russian by J. M. Danskin. {\sl Translations of Mathematical Monographs, Vol. 29. American Mathematical Society, Providence, R.I.}, 1971. \bibitem{Lagnese} Lagnese, John E., General boundary value problems for differential equations of Sobolev type. {\sl SIAM J. Math. Anal.} 3 (1972), 105--119. \bibitem{ligh} Lightbourne, James H., III; Rankin, Samuel M., III., A partial functional-differential equation of Sobolev type. {\sl J. Math. Anal. Appl.} 93 (1983), no. 2, 328--337. \bibitem{NACH} Nachbin, L., Introduction to functional Analysis: Banach Space and Differential Calculus. Marcel Dekker, New York, 1981. \bibitem{PA} Pazy, A., Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York-Berlin, 1983. \bibitem{Showalter2} Showalter, R. E., A nonlinear parabolic-Sobolev equation. {\sl J. Math. Anal. Appl.} 50 (1975), 183--190. \bibitem{Showalter1} Showalter, R. E., Degenerate parabolic initial-boundary value problems. {\sl J. Differential Equations} 31 (1979), no. 3, 296--312. }\end{thebibliography} \noindent{\sc Eduardo Hern\'{a}ndez M.} \\ Departamento de Matem\'atica \\ Instituto de Ci\^encias Matem\'aticas de S\~ao Carlos \\ Universidade de S\~ao Paulo \\ Caixa Postal 668 \\ 13560-970 S\~ao Carlos, SP. Brazil \\ e-mail: lalohm@icmc.sc.usp.br \end{document}