\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 13, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/13\hfil Existence and uniqueness of solutions] {Existence and uniqueness of solutions for quasi-linear differential equations with deviating arguments} \author[R. Haloi, D. Bahuguna, D. N. Pandey \hfil EJDE-2012/13\hfilneg] {Rajib Haloi, Dhirendra Bahuguna, Dwijendra N. Pandey} \address{Rajib Haloi \newline Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin 208016, India\newline Tel. +91-512-2597053, Fax +91-512-2597500} \email{rajib.haloi@gmail.com} \address{Dhirendra Bahuguna \newline Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin 208016, India} \email{dhiren@iitk.ac.in} \address{Dwijendra N. Pandey \newline Department of Mathematics, Indian Institute of Technology Roorkee, Pin 247667, India} \email{dwij.iitk@gmail.com} \thanks{Submitted August 15, 2011. Published January 17, 2012.} \subjclass[2000]{34G20, 34K30, 35K90, 47N20} \keywords{Analytic semigroup; parabolic equation; deviated argument;\hfill\break\indent Banach fixed point theorem} \begin{abstract} We prove the existence and uniqueness of a local solution to a quasi-linear differential equation of parabolic type with deviated argument in an arbitrary Banach space. The results are obtained by applying the Sobolevski\u{i}-Tanabe theory of parabolic equations, fractional powers of operators, and the Banach fixed point theorem. We include an example that illustrates the theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} Differential equations with a deviating argument are differential equations in which the unknown function and its derivative appear under different values of the argument. Differential equations with a deviating argument have many applications in science and technology. These includes the theory of automatic control, the theory of self-oscillating systems, the problems of long-term planning in economics, the study of problems related with combustion in rocket motion, a series of biological problems, and many other areas of science and technology, the number which is steadily expanding, for more details we refer to \cite{norkin,gal,gal2008,Grimm,TJ,kwaspisz} and references cited therein. We shall study the existence and uniqueness of a local solution for the following differential equation in a Banach space $(X,\|\cdot\|)$, \begin{equation} \begin{gathered} \frac{du}{dt}+A(t,u(t))u(t) = f(t,u(t),u(h(t,u(t)))),\quad t>0; \\ u(0)= u_0. \end{gathered} \label{mainp01} \end{equation} Here, we assume that $-A(t,x)$, for each $t\geq 0$ and $x\in X$, generates an analytic semigroup of bounded linear operators on $X$. The nonlinear $X$-valued functions $f$ and $h$ satisfy suitable growth conditions in their arguments stated in Section 2. The existence and uniqueness of solutions for a quasi-linear differential equation in Banach spaces have been studied by many authors (see e.g. \cite{Amann,And81,Kato75,Kato93,Kob89,Murphy,Pazy,Sane79,Sane89,SOB}). Using fixed point argument, Pazy \cite{Pazy} obtained the mild and classical solution to the following homogeneous quasi-linear differential equation in a Banach space $(X, \|\cdot\|)$, \begin{gather*} \frac{du}{dt}+A(t,u(t))u(t)=0, \quad 00$ (independent of $t$ and $\lambda$) such that $$ \|R(\lambda ;A(t))\|\leq \frac{C}{|\lambda|+1},\quad \operatorname{Re} \lambda \leq 0, \; t \in [0,T]. $$ \item[(A3)] There are constants $C>0$ and $\rho \in(0,1]$, such that $$ \|[A(t)-A(\tau)]A^{-1}(s)\|\leq C |t-\tau|^\rho, $$ for $t,s,\tau \in [0,T]$. Here, $C$ and $\rho$ are independent of $t,\tau$ and $s$. \end{itemize} It is well known that assumption (A2) implies that for each $s\in [0,T]$, $-A(s)$ generates a strongly continuous analytic semigroup $\{ e^{-tA(s)}: t\geq 0 \}$ in $\mathcal{L}(X)$, where $\mathcal{L}(X)$ denotes the Banach algebra of all bounded linear operators on $X$. Then there exist positive constants $C$ and $d$ such that \begin{gather} \|e^{-tA(s)}\| \leq C e^{-dt},\quad t\geq 0;\label{fr}\\ \|A(s)e^{-tA(s)}\| \leq C e^{-dt}/t,\quad t > 0,\label{der} \end{gather} for all $s\in [0,T]$. It is to be noted that the assumption (A3) implies that there exists a constant $C>0$ such that \begin{equation}\label{A1} \|A(t)A^{-1}(s)\|\leq C, \end{equation} for all $ 0\leq s, t\leq T $. Hence, for each $t$, the functional $y\to \|A(t)y\|$ defines an equivalent norm on $D(A)=D(A(0))$ and the mapping $t\to A(t)$ from $[0,T]$ into $\mathcal{L}(X_1,X)$ is uniformly H\"{o}lder continuous. Consider the homogeneous Cauchy problem \begin{equation}\label{hom} \frac{du}{dt}+A(t)u=0; \quad u(t_0)=u_0. \end{equation} Then the solution to this problem is given by the following Theorem. \begin{theorem}[\cite{AF,SOB}] \label{pq1} Let the Assumptions {\rm (A1)--(A3)} hold. Then there exists a unique fundamental solution $\{U(t,s): 0\leq s \leq t \leq T\}$ to \eqref{hom} that possesses the following properties: \begin{itemize} \item[(i)] $U(t,s)\in \mathcal{L}(X)$ and $U(t,s)$ is strongly continuous in $t,s$ for all $0\leq s \leq t \leq T$. \item[(ii)] $U(t,s)x\in D(A) $ for each $x\in X$, for all $0\leq s \leq t \leq T$. \item[(iii)] $U(t,r)U(r,s)=U(t,s)$ for all $0\leq s \leq r \leq t \leq T$. \item[(iv)] the derivative $\partial U(t,s)/ \partial t$ exists in the strong operator topology and belongs to $\mathcal{L}(X)$ for all $0\leq s < t \leq T$, and strongly continuous in $s$ and $t$, where $ s < t \leq T$. \item[(v)] $\frac{\partial U(t,s)}{\partial t}+A(t)U(t,s)=0$ and $U(s,s)=I$ for all $0\leq s < t \leq T$. \end{itemize} \end{theorem} For $t_0 \geq 0$, let $C^\beta([t_0,T];X)$ denote the space of all $X$-valued functions $h(t)$, that are uniformly H\"{o}lder continuous on $[0,T]$ with exponent $\beta$, where $ 0<\beta \leq 1$. Define $$ [h]_{\beta}=\sup _{t_0\leq t, s \leq T} \|h(t)-h(s)\|/|t-s|^\beta. $$ Then $C^\beta([t_0,T];X)$ is a Banach space with respect to the norm $$ \|h\|_{C^\beta([t_0,T];X)}=\sup _{t_0 \leq t \leq T}\|h(t)\| + [h]_\beta. $$ Consider the inhomogeneous Cauchy problem \begin{equation}\label{aux} \frac{du}{dt}+A(t)u=f(t); \quad u(t_0)=u_0. \end{equation} \begin{theorem}[\cite{AF,SOB}] \label{pq2} Let the assumptions {\rm (A1)-(A3)} hold. If $f\in C^\beta([t_0,T];X)$, then there exists a unique solution of \eqref{aux}. Furthermore, the solution can be written as $$ u(t)= U(t,t_0)u_0+\int_{t_{0}}^t U(t,s)f(s)ds, \quad t_0\leq t \leq T, $$ and $u:[t_0,T]\to X$ is continuously differentiable on $(t_0,T]$. \end{theorem} We shall use the following assumption to establish the existence and uniqueness of a local solution to \eqref{mainp01}. \begin{itemize} \item[(F1)] The operator $A_0=A(0,u_0)$ is closed operator with domain $ D_0$ ($D_0$ denote domain of $A_0 $) dense in $X$ and there exists a constant $C>0$ independent of $\lambda$ such that \begin{equation} \|(\lambda I-A_0)^{-1}\|\leq \frac{C}{1+|\lambda|}; \quad \text{for all $\lambda$ with Re } \lambda \leq 0. \end{equation} \end{itemize} Assumption (F1) allows us to define negative fractional powers of the operator $A_0$. For $\alpha>0$, define negative fractional powers $A_0^{-\alpha}$ by the formula $$ A_0^{-\alpha}= \frac{1}{\Gamma (\alpha)} \int_0^{\infty} e^{-t A_0} t ^{\alpha -1}d t . $$ Then $A_0^{-\alpha}$ is one-to-one and bounded linear operator on $X$. Thus, there exists an inverse of the operator $A_0^{-\alpha}$. We define the positive fractional powers of $A_0$ by $A_0^\alpha\equiv[A_0^{-\alpha}]^{-1}$. Then $A_0^\alpha$ is closed linear operator with dense domain $D(A_0^\alpha)$ in $X$ and $D(A_0^\alpha)\subset D(A_0^\beta)$ if $ \alpha > \beta$. For $0<\alpha \leq 1$, let $X_\alpha= D(A_0^\alpha)$ and equip this space with the graph norm $$ \|x\|_\alpha=\|A_0^\alpha x\|. $$ Then $X_\alpha$ is a Banach space with respect to this norm. If $0<\alpha \leq 1$, the embedding $X_1 \hookrightarrow X_\alpha \hookrightarrow X$ are dense and continuous. We define, for each $\alpha>0,~ X_{-\alpha}=(X_\alpha)^*$, the dual space of $X_\alpha$, endowed with the natural norm $$ \|x\|_{-\alpha}=\|A_0^{-\alpha}x\|. $$ Let $R,R'>0$ and $B_\alpha=\{x\in X_\alpha: \|x\|_\alpha 0$ and $0< \theta_1\leq1$, such that the nonlinear map $f:[0,T]\times B_\alpha \times B_{\alpha-1}\to X$ satisfies the condition \begin{equation} \|f(t,x,x')-f(s,y,y')\| \le L_f (|t-s|^{\theta_1}+\|x-y\|_\alpha+ \|x'-y'\|_{\alpha-1}),\label{f20q} \end{equation} (b) There exist constants $L_h=L_h(t, R)> 0$ and $0< \theta_2\leq 1$, such that $h(\cdot ,0)=0$, $h:B_\alpha \times [0,T] \to [0,T]$ satisfies the following condition, \begin{equation} |h(x,t)-h(y,s)| \le L_h (\|x-y\|_\alpha + |t-s|^{\theta_2}),\label{h20q} \end{equation} for all $x,y\in B_\alpha$ and for all $s,t \in [0,T] $. \item[(F4)] Let $u_0\in X_\beta$ for some $\beta>\alpha$ and \begin{equation} \| u_0\|_\alpha 0$. \end{lemma} We have the following corollary. \begin{corollary} For $y\in X_1$, define $$ P(y;h)=U(t,0)y+ \int_0^t U(t,s)h(s)ds,~0\leq t\leq T. $$ Then $P$ is a bounded linear mapping from $X_1\times C^\beta([t_0,T];X)$ into $C([t_0,T];X_1)$. \end{corollary} \section{Existence of a Solution} In this section, we will establish the existence and uniqueness of a local solution to \eqref{mainp01}. Let $I=[0,\delta]$ for some positive number $\delta$ to be specified later. Let $C_\alpha$, $0\leq \alpha \leq 1$, denote the space of all $ X_{\alpha}$-valued continuous functions on $I$, endowed with the sup-norm, $ \sup_{t\in I}\|\psi(t)\|_{\alpha}$, $ \psi \in C(I;X_\alpha)$. Let $$ Y_\alpha=C_{L_\alpha}(I;X_\alpha) =\{\psi \in C_\alpha: \|\psi(t)-\psi(s)\|_{\alpha-1}\le L_\alpha |t-s|,\;\text{for all}\; t,s\in I\}, $$ where $L_\alpha$ is a positive constant to be specified later. It is clear that $Y_\alpha$ is a Banach space with the sup-norm of $C_\alpha$. \begin{definition}\rm Given $u_0\in X_\alpha $, by a solution of problem \eqref{mainp01}, we mean a function $u:I\to X$ that satisfies the followings: \begin{itemize} \item[(i)] $u(.)\in C_{L_\alpha}(I;X_\alpha) \cap C^1((0,\delta);X) \cap C(I;X)$; \item [(ii)] $u(t)\in X_\alpha$, for all $t\in (0,\delta)$; \item[(iii)] $ \frac{du}{dt}+A(t,u(t))u(t)=f(t,u(t),u([h(u(t),t)]))$, for all $t\in (0,\delta)$; \item[(iv)] $u(0)=u_0$. \end{itemize} \end{definition} Let $K>0$ and $0<\eta <\beta-\alpha $ be fixed constants. Let $$ \mathcal{S}_{\alpha}=\{y\in C_\alpha \cap Y_\alpha: y(0)=u_0, \|y(t)-y(s)\|_\alpha \leq K |t-s|^\eta\}. $$ Then $\mathcal{S}_\alpha$ is a non-empty closed and bounded subset of $C_\alpha$. Now we prove the following theorem concerning the existence and uniqueness of a local solution to \eqref{mainp01}. The proof is based on ideas from Gal \cite{gal} and Sobolevski\u i \cite{SOB} \begin{theorem} \label{ms201} Let $u_0\in X_\beta$, where $0<\alpha<\beta \leq 1$. Let the assumptions {\rm (F1)--(F4)} hold. Then there exists a positive number $\delta=\delta( \alpha,u_0), 0<\delta \leq T$ and a unique solution $u(t)$ to \eqref{mainp01} in $[0,\delta]$ such that $u\in \mathcal{S}_{\alpha}\cap C^1((0,\delta); X)$. \end{theorem} \begin{proof} Let $v\in \mathcal{S}_{\alpha} $. Then from the assumption (F4), it follows that if $\delta>0$ is sufficiently small, then \begin{equation} \|v(t)\|_\alpha 0$ is a constant independent of $\delta$ and of the particular $v\in \mathcal{S}_\alpha$. It is also to be noted that \begin{equation} A_v(0)=A_0. \end{equation} If $\delta>0$ is sufficiently small, then from assumption (F1) and inequality \eqref{H1}, we have \begin{equation} \|(\lambda I-A_v(t))^{-1}\|\leq \frac{C}{1+|\lambda|}; \quad \text{for $\lambda$ with Re } \lambda \leq 0,\; t\in I. \end{equation} Also from assumption (F2), it follows that \begin{equation}\label{H12} \|[A_v(t)-A_v(s)]A^{-1}_v(\tau)\|\leq C |t-s|^\mu, \quad \text{if } t,\tau, s \in I. \end{equation} Thus the operator $A_v(t)$ satisfies conditions (A1), (A2) and (A3). Hence, there exists a fundamental solution $U_v(t,s)$ corresponding to $A_v(t)$ and satisfies all estimates derived in Theorem \ref{pq1} uniformly with respect to $v\in \mathcal{S}_\alpha$. Put $f_v(t)=f(t,v(t),v([h(v(t),t)]))$. Then the assumption (F3) implies that $f_v$ is H\"{o}lder continuous on $I$ of exponent $\gamma =\min \{\theta_1,\theta_2, \eta\}$. Now consider the equation \begin{equation} \begin{gathered} \frac{dw}{dt}+A(t,v(t))w(t) = f_v(t), \quad t\in I; \\ w(0) = u_0. \end{gathered}\label{Pv01} \end{equation} By Theorem \ref{pq2}, there exists a unique solution $w_v$ to \eqref{Pv01} that is given by \begin{equation} w_v(t)= U_v(t,0)u_0+\int_{0}^t U_v(t,s)f_v(s)ds, \quad t \in I. \end{equation} For each $v\in \mathcal{S}_\alpha $, define a map $F$ by \begin{equation} \label{approxi12301} Fv(t) = U_v(t,0)u_0+\int_{0}^tU_v(t,s)f_v(s)ds, \quad \text{for each } t\in I. \end{equation} By Lemma \ref{lem1}, the map $F$ is well defined. We will claim that $F$ maps from $\mathcal{S}_{\alpha}$ into itself, for sufficiently small $\delta>0$. Indeed, if $t_1,t_2\in I$ with $t_2>t_1$, then we have \begin{equation} \begin{aligned} \|Fv(t_2)-Fv(t_1)\|_{\alpha-1} &\leq \|[U_v(t_2,0)-U_v(t_1,0)]u_0\|_{\alpha-1} \\ &\quad + \|\int_{0}^{t_2}U_v(t_2,s)f_v(s)ds - \int_{0}^{t_1}U_v(t_1,s)f_v(s)ds\| _{\alpha-1}. \end{aligned}\label{new201} \end{equation} We will use the bounded inclusion $X\subset X_{\alpha-1}$ to estimate each of the term on the right-hand side of \eqref{new201}. The first term on the right-hand side of \eqref{new201} is estimated as follows \cite[see Lemma II. 14.1]{AF}, \begin{equation} \|(U_v(t_2,0)-U_v(t_1,0))u_0\|_{\alpha-1} \leq C_1 \|u_0\|_\alpha (t_2-t_1), \label{new1201} \end{equation} where $C_1$ is some positive constant. We have the following estimate for the second term on the right hand side of \eqref{new201} \cite[Lemma II. 14.4]{AF}, \begin{equation} \begin{aligned} &\|\int_{0}^{t_2}U_v(t_2,s)f_v(s)ds- \int_{0}^{t_1}U_v(t_1,s)f_v(s)ds\|_{\alpha-1}\\ & \leq C_2 N_1 (t_2-t_1)(|\log(t_2-t_1)|+1), \end{aligned}\label{new2201} \end{equation} where $N_1= \sup_{s\in [0,T]} \|f_v(s)\|$ and $C_2$ is some positive constant. Using estimates \eqref{new1201} and \eqref{new2201}, from \eqref{new201}, we obtain \begin{equation} \|Fv(t_2)-Fv(t_1)\|_{\alpha-1} \leq L_{\alpha}|t_2-t_1|,\label{new821} \end{equation} where $L_{\alpha}= \max \{C_1 (t_2-t_1)^{\alpha-1}\|u_0\|_\alpha, C_2 N_1(|\log(t_2-t_1)|+1)\} $ that depends on $C_1,C_2,N_1,\delta$. Our next aim is to show that $\|Fv(t+h)-Fv(t)\|_\alpha \leq K h^\eta$, for some constant $K>0$ and $0<\eta<1 $. If $0\leq \alpha <\beta \leq 1$ and $0 \leq t \leq t+h \leq \delta $, then \begin{align*} \|Fv(t+h)-Fv(t)\|_\alpha &\leq \|[U_v(t+h,0)-U_v(t,0)]u_0\|_\alpha \\ &\quad + \|\int_{0}^{t+h} U_v(t+h,s)f_v(s)ds-\int_{0}^{t} U_v(t,s)f_v(s)ds\|_\alpha. \end{align*} Using \cite[Lemmas II.14.1, II.14.4]{AF}, we obtain the following two estimates \begin{gather}\label{A101} \|[U_v(t+h,0)-U_v(t,0)]u_0\|_\alpha \leq C(\alpha,u_0)h^{\beta-\alpha}; \\ \label{B101} \|\int_0^{t+h} U_v(t+h,s)f_v(s)ds-\int_{0}^{t} U_v(t,s)f_v(s)ds\|_\alpha \leq C(\alpha)N_1 h^{1-\alpha} (1+|\log h|). \end{gather} From \eqref{A101} and \eqref{B101}, it is clear that $$ \|Fv(t+h)-Fv(t)\|_\alpha \leq h^\eta[C(\alpha,u_0) \delta^{\beta-\alpha-\eta}+C(\alpha)N_1 \delta^\nu h^{1-\alpha -\eta -\nu}(|\log h|+1)] $$ for any $\nu >0, \nu<1-\alpha-\eta$. Hence, for sufficiently small $\delta>0$ , we have $$ \|Fv(t+h)-Fv(t)\|_\alpha \leq K h^\eta, $$ for some constant $K>0$. Thus, we have shown that $F$ maps $\mathcal{S}_\alpha$ into itself. Finally, we will show that $F$ is a contraction map. For $v_1,v_2 \in S_\alpha$, put $z_1(t)=w_{v_1}(t)$ and $z_2(t)=w_{v_2}(t)$. Thus, for $j=1,2$, we have \begin{equation} \begin{gathered} \frac{dz_j}{dt}+A_{v_j}(t)z_j(t) =f_{v_j}(t), \quad t\in I; \\ z_j(0) = u_0. \end{gathered}\label{Pv01c} \end{equation} It follows from \eqref{Pv01c} that \begin{equation} \frac{d}{dt}(z_1-z_2)+A_{v_1}(t)(z_1-z_2) =[A_{v_2}(t)-A_{v_1}(t)]z_2+[f_{v_1}(t)-f_{v_2}(t)]. \end{equation} Using \cite[Lemmas II.14.3, II.14.5]{AF}, we obtain that $A_0(t)z_2(t)$ is uniformly H\"{o}lder continuous for $\tau \leq t \leq \delta $, $\tau >0$. Also from Lemma \ref{lem1}, $A_0 \int^t_0 U_{v_2}(t,s) f_{v_2}(s)ds$ is a bounded function, and hence we have the bound \begin{equation}\label{bo} \|A_0z_2(t)\|\leq C t^{\beta-1}. \end{equation} Further, in view of \eqref{A1} and \eqref{H12}, the operator $[A_{v_2}(t)-A_{v_1}(t)]A^{-1}_0$ is uniformly H\"{o}lder continuous for $\tau \leq t \leq \delta $, $\tau >0$. Hence, $[A_{v_2}(t)-A_{v_1}(t)]z_2(t)$ is uniformly H\"{o}lder continuous for $\tau \leq t \leq \delta $, $\tau >0$. Applying Theorem \ref{pq1}, we get that for any $\tau \leq t \leq \delta $, $\tau >0$, \begin{equation} \label{co} \begin{aligned} z_1(t)-z_2(t) &= U_{v_1}(t,\tau)[z_1(\tau)-z_2(\tau)] \\ &\quad+ \int_\tau ^t U_{v_1}(t,s)\{[A_{v_2}(s)-A_{v_1}(s)]z_2(t) +[f_{v_1}(s)-f_{v_2}(s)]\}ds. \end{aligned} \end{equation} The bound in \eqref{bo} allows us to take $\tau \to 0$ in \eqref{co}, and passing to the limit, we obtain \begin{align*} z_1(t)-z_2(t) = \int_0 ^t U_{v_1}(t,s)\{[A_{v_2}(s)-A_{v_1}(s)]z_2(t) +[f_{v_1}(s)-f_{v_2}(s)]\}ds. \end{align*} Now using \eqref{Ah}, \eqref{f20q}, \eqref{h20q} and \cite[inequality (1.65), page 23]{SOB}, we obtain \begin{equation} \begin{aligned} \|Fv_1(t)-Fv_2(t)\|_\alpha & \leq C_3 C(R)\int _0^t (t-s)^{-\alpha}(\|v_1(s)-v_2(s) \|_\alpha s^{\beta-1} ds\\ &\quad +C_4L_f \int _0^t (t-s)^{-\alpha}\{\|v_1(s)-v_2(s)\|_\alpha \\ &\quad +\|v_1([h(v_1(s),s)])-v_2([h(v_2(s),s)])\|_{\alpha-1}\}ds\\ &\leq C_3 C(R)\int _0^t (t-s)^{-\alpha}(\|v_1(s)-v_2(s) \|_\alpha s^{\beta-1} ds\\ &\quad + \frac{C_4}{1-\alpha }L_f (2+L_\alpha L_h)\delta^{1-\alpha}\sup_{t\in I} \|v_1(t)-v_2(t)\|_\alpha\\ &\leq \widetilde{K} \delta^{\beta-\alpha} \sup_{t\in I} \|v_1(t)-v_2(t)\|_\alpha, \end{aligned} \label{con1} \end{equation} where $\widetilde{K}=\max \{\frac{C_3 C(R)}{1-\alpha }, \frac{C_4}{1-\alpha }L_f (2+L_\alpha L_h)\}$. Choose $\delta >0$ such that $$ \widetilde{K} \delta^{\beta-\alpha} <\frac{1}{2}. $$ Then, from \eqref{con1}, it is clear that $F$ is a contraction map. Since $\mathcal{S}_\alpha$ is a complete metric space, by the Banach fixed-point theorem, there exists $u\in \mathcal{S}_\alpha$ such that $Fu=u$. It follows from Sobolevski\u{i} \cite[Theorem 5]{SOB} that $ u \in C^1((0,\delta);X)$. Thus $u$ is a solution to \eqref{mainp01} on $[0,\delta ]$. \end{proof} \section{Example} Consider the quasi-linear parabolic differential equation with a deviated argument \begin{equation} \begin{gathered} \frac{\partial u}{\partial t} + a(x,t,u,\frac{\partial u}{\partial x}) \frac{\partial^2 u}{\partial x^2}= \widetilde{H}(x,u(t,x))+\widetilde{G}(t,x,u(t,x)); \\ u(t,0)= u(t,1), \quad t>0; \\ u(0,x)= u_{0}(x), \quad x\in (0,1), \end{gathered} \label{ex201} \end{equation} where $a(\cdot, \cdot,\dots ,\cdot)$ is a continuously differentiable real valued function in all variables. Here, $\widetilde{H}(x,u(t,x))= \int_{0}^{x}K(x,y) u(\widetilde{g}(t)|u(t,y)|,y) dy$ for all $(t,x)\in (0,\infty)\times (0,1)$. Assume that $\widetilde{g}:\mathbb{R_{+}}\to \mathbb{R_{+}}$ is locally H\"{o}lder continuous in $t$ with $\widetilde{g}(0)=0$ and $K\in C^{1}([0,1]\times [0,1];\mathbb{R})$. The function $\widetilde{G}: \mathbb{R_{+}} \times [0,1]\times \mathbb{R} \to \mathbb{R}$ is measurable in $x$, locally H\"{o}lder continuous in $t$, locally Lipschitz continuous in $u$, uniformly in $x$. Here, the parabolically means that for any real vector $\xi \neq 0$ and for arbitrary values of $u, \frac{\partial u}{\partial x}$, it holds $$ -a(x,t,u, \frac{\partial u}{\partial x})\xi^2>0. $$ Let $ A(t,u)u(t)=a(x,t,u,\frac{\partial u}{\partial x})\frac{\partial^2 u }{\partial x^2}$. If $u_0\in C^1(0,1)$, then $$ A_0 u \equiv a\big(x,0,u_0,\frac{\partial u_0}{\partial x} \big)\frac{\partial^2 u}{\partial x^2} $$ is strongly elliptic operator with continuous coefficient. Let $X=L^{2}((0,1);\mathbb{R})$. Then $X_1=D(A_0)=H^{2}(0,1)\cap H^{1}_{0}(0,1)$, $X_{1/2}=D((A_0)^{1/2})=H^{1}_{0}(0,1)$ and $X_{-1/2}=H^{-1}(0,1)$. It is well known that the assumption (F1) is satisfied. The assumption on $a$ implies that $A(t,x)$ satisfies \eqref{Ah}. For $x\in (0,1)$, we define $f: \mathbb{R_{+}}\times H^{1}_0(0,1)\times H^{-1}(0,1)\to L^{2}(0,1) $ by $$ f(t,\phi,\psi)=\widetilde{H}(x,\psi)+\widetilde{G}(t,x,\phi), $$ where $\widetilde{H}(x,\psi(x,t)) = \int _{0}^{x}K(x,y)\psi(y,t)dy$. We also assume that $\widetilde{G}:\mathbb{R}_+ \times[0,1]\times H^{-1}(0,1)\to L^{2}(0,1) $ satisfies $$ \|\widetilde{G}(t,x,u)-\widetilde{G}(s,x,v)\|_{L^2(0,1)} \leq C (|t-s|^{\theta_1}+\|u-v\|_{H^{-1}(0,1)}), $$ for some constant $C>0$. 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