\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 92, pp. 1--30.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/92\hfil Time-dependent domains] {Time-dependent domains for nonlinear evolution operators and partial differential equations} \author[C.-Y. Lin\hfil EJDE-2011/92\hfilneg] {Chin-Yuan Lin} \dedicatory{Dedicated to Professor Jerome A. Goldstein on his 70th birthday} \address{Chin-Yuan Lin \newline Department of Mathematics \\ National Central University \\ Chung-Li 320, Taiwan} \email{cylin@math.ncu.edu.tw} \thanks{Submitted June 15, 2011. Published July 18, 2011.} \subjclass[2000]{47B44, 47H20, 35J25, 35K20} \keywords{Dissipative operators; evolution equations; \hfill\break\indent parabolic and elliptic equations} \begin{abstract} This article concerns the nonlinear evolution equation \begin{gather*} \frac{du(t)}{dt} \in A(t)u(t), \quad 0 \leq s < t < T, \\ u(s) = u_0 \end{gather*} in a real Banach space $X$, where the nonlinear, time-dependent, and multi-valued operator $ A(t) : D(A(t)) \subset X \to X$ has a time-dependent domain $D(A(t))$. It will be shown that, under certain assumptions on $ A(t) $, the equation has a strong solution. Illustrations are given of solving quasi-linear partial differential equations of parabolic type with time-dependent boundary conditions. Those partial differential equations are studied to a large extent. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} \label{S:A} Let $ (X, \|\cdot\|) $ be a real Banach space with the norm $ \|\cdot\|$, and let $ T > 0$, $\omega $ be two real constants. Consider the nonlinear evolution equation \begin{equation} \label{E:A} \begin{gathered} \frac{du(t)}{dt} \in A(t)u(t), \quad 0 \le s < t < T, \\ u(s) = u_0, \end{gathered} \end{equation} where \[ A(t) : D(A(t)) \subset X \to X \] is a nonlinear, time-dependent, and multi-valued operator. To solve \eqref{E:A}, Crandall and Pazy \cite{Cran} made the following hypotheses of (H1)--(H3) and the $ t $-dependence hypothesis of either (H4) or (H5), for each $ 0 \le t \le T $. \begin{itemize} \item[(H1)] $ A(t) $ is dissipative in the sense that \[ \|u - v\| \leq \|(u - v) - \lambda (g - h)\| \] for all $ u, v \in D(A(t)) $, $ g \in (A(t) - \omega)u, h \in (A(t) - \omega)v $, and for all $ \lambda > 0 $. Equivalently, \[ \Re(\eta(g - h)) \leq 0 \] for some $ \eta \in G(u - v) \equiv \{\xi \in X^{*} : \|u - v\|^2 = \xi(u - v) = \|\xi\|_{X^{*}}^2 \} $, the duality map of $ (u - v) $ \cite{Mi}. Here $ (X^{*}, \|.\|_{X^{*}}) $ is the dual space of $X$ and $ \Re(z) $ is the real part of a complex number $ z $. \item[(H2)] The range of $ (I - \lambda A(t)) $ contains the closure $ \overline{D(A(t))} $ of $ D(A(t)) $ for small $ 0 < \lambda < \lambda_0 $ with $ \lambda_0\omega < 1 $. \item[(H3)] $ \overline{D(A(t))} = \overline{D} $ is independent of $ t$. \item[(H4)] There are a continuous function $ f : [0, T] \to X $ and a monotone increasing function $ L : [0, \infty) \to [0, \infty) $, such that \[ \|J_{\lambda}(t)x - J_{\lambda}(\tau)x\| \leq \lambda \|f(t) - f(\tau)\| L(\|x\|) \] for $ 0 < \lambda < \lambda_0, 0 \leq t, \tau \leq T, $ and $ x \in \overline{D} $. Here $ J_{\lambda}(t)x \equiv (I - \lambda A(t))^{-1} $ exists for $ x \in \overline{D} $ by (H1) and (H2). \item[(H5)] There is a continuous function $ f : [0, T] \to X $, which is of bounded variation on $ [0, T] $, and there is a monotone increasing function $ L : [0, \infty) \to [0, \infty) $, such that \[ \|J_{\lambda}(t)x - J_{\lambda}(\tau)x\| \leq \lambda \|f(t) - f(\tau)\| L(\|x\|) (1 + |A(\tau)x|) \] for $ 0 < \lambda < \lambda_0, 0 \leq t, \tau \leq T, $ and $ x \in \overline{D} $. Here \[ |A(\tau)x| \equiv \lim_{\lambda \to 0}\|\frac{(J_{\lambda}(\tau) - I)x}{\lambda}\| \] by (H1) and (H2), which can equal $ \infty $ \cite{Crand,Cran}. \end{itemize} By defining the generalized domain $ \hat{D}(A(t)) \equiv \{x \in \overline{D(A(t))} : |A(t)x| < \infty \} $ \cite{Crand,Wes}, they \cite{Cran} proved, among other things, that the limit \begin{equation} \label{E:DvpdesA} U(t, s)x \equiv \lim_{n \to \infty} \prod_{i=1}^{n} J_{\frac{t - s}{n}}(s + i \frac{t - s}{n})x \end{equation} exists for $ x \in \overline{D} $ and that $ U(t, s)u_0 $ is a unique solution, in a generalized sense, to the equation \eqref{E:A} for $ u_0 \in \overline{D} $. Because of the restriction in (H3) that $\overline{D(A(t))} = \overline{D} $ is independent of $ t $, the boundary condition in the example in \cite{Cran} does not depend on time. In this paper, in order to enlarge the scope of applications, we will consider a different set of hypotheses, the dissipativity condition (H1), the range condition (H2'), and the time-regulating condition (HA) below. Here a similar set of hypotheses was considered in \cite{Lin2} but the results were not satisfactory. \begin{itemize} \item[(H2')] The range of $ (I - \lambda A(t)) $, denoted by $ E $, is independent of $ t $ and contains $ \overline{D(A(t))} $ for all $ t \in [0, T] $ and for small $ 0 < \lambda < \lambda_0 $ with $ \lambda_0 \omega < 1 $. \item[(HA)] There is a continuous function $ f : [0, T] \to \mathbb{R} $, of bounded variation, and there is a nonnegative function $ L $ on $ [ 0, \infty) $ with $ L(s) $ bounded for bounded $ s $, such that, for each $ 0 < \lambda < \lambda_0 $, we have \[ \{J_{\lambda}(t)x - J_{\lambda}(\tau)y : 0 \leq t, \tau \leq T, x, y \in E \} = S_1(\lambda) \cup S_2(\lambda). \] Here $ S_1(\lambda) $ denotes the set \begin{align*} &\big\{ J_{\lambda}(t)x - J_{\lambda}(\tau)y : 0 \leq t, \tau \leq T, x, y \in E, \\ &\quad \|J_{\lambda}(t)x -J_{\lambda}(\tau)y\| \leq L(\|J_{\lambda}(\tau)y\|)|t - \tau| \big\}, \end{align*} and $ S_2(\lambda) $ denotes the set \begin{align*} &\big\{J_{\lambda}(t)x - J_{\lambda}(\tau)y : 0 \leq t, \tau \leq T, x, y \in E, \|J_{\lambda}(t)x - J_{\lambda}(\tau)y\|\\ & \leq (1 - \lambda \omega)^{-1}[ \|x - y\| + \lambda |f(t) - f(\tau)| L(\|J_{\lambda}(\tau)y\|)( 1 + \frac{\|(J_{\lambda}(\tau) - I)y\|}{\lambda})] \big\}. \end{align*} \end{itemize} We will show that the limit in \eqref{E:DvpdesA} for $ x \in \overline{\hat{D}(A(s))} = \overline{D(A(s))} $ exists, and that this limit for $ x = u_0 \in \hat{D}(A(s)) $ is a strong solution to the equation \eqref{E:A}, if $ A(t) $ satisfies additionally an embedding property in \cite{Li} of embeddedly quasi-demi-closedness. We then apply the abstract theory to quasi-linear, parabolic partial differential equations with boundary conditions depending on time $ t $. We finally show that, in those applications, each quantity \[ J_{\frac{t - s}{n}}(s + i\frac{t - s}{n})h = [I - \frac{t - s}{n} A(s + i \frac{t - s}{n})]^{-1}h, \quad i = 1, 2, \ldots, n \] is the limit of a sequence where each term in the sequence is an explicit function $ F(\phi) $ of the solution $ \phi = \pounds_0^{-1}(h, \varphi) $ to the elliptic equation with $ \varphi \equiv 0 $: \begin{equation} \label{E:TimeC} \begin{gathered} -\Delta v(y) = h, \quad y \in \Omega, \\ \frac{\partial v}{\partial \nu} + v = \varphi, \quad y \in \partial \Omega. \end{gathered} \end{equation} Here for the dimension of the space variable $ y $ equal to 2 or 3, the $ \phi = \pounds_0^{-1}(h, 0) $ and the solution $ \pounds_0^{-1}(h, \varphi) $ to \eqref{E:TimeC} can be computed numerically and efficiently by the boundary element methods \cite{Gau,Sch}. See Sections \ref{S:D} and \ref{S:E} for more details of these, including how $ F(\phi) $ depends on $ \phi $, and for other aspects of the treated partial differential equations. There are many related works, to cite a few, we mention \cite{Ba,Br,Bre,Bro,Cr,Cra,Cran,En,Eng,Gol,Lie,Hi,Kat,Kato,Li,Lin0,Lin1,Lin2,Lin4,Mi,Oh,Pa,Paz,Ro,Roy,We}, especially the \cite{Lin4} for the recent development on nonlinear evolution equations where the hypothesis (H2) is relaxed. The rest of this article will be organized as follows. Section \ref{S:VaryA} obtains some preliminary estimates, and Section \ref{S:B} deals with the main results, where the nonlinear operator $ A(t) $ is equipped with time-dependent domain $ D(A(t)) $. The Appendix in Section \ref{S:VaryB} examines the difference equations theory in our papers \cite{Lin0,Lin1,Lin4}, whose results, together with those in Section \ref{S:VaryA}, will be used to prove the main results in Section \ref{S:B}. Section \ref{S:D} studies applications to linear or nonlinear partial differential equations of parabolic type, in which each corresponding elliptic solution $ J_{\frac{t - s}{n}}(s + i \frac{t - s}{n})h $ will be derived theoretically. Finally, Section \ref{S:E} follows Section \ref{S:D} but derives each elliptic solution $ J_{\frac{t - s}{n}}(s + i \frac{t - s}{n})h $ as the limit of a sequence where each term in the sequence is an explicit function of the solution $ \phi $ to the elliptic equation \eqref{E:TimeC} with $ \varphi \equiv 0 $. In either Section \ref{S:D} or Section \ref{S:E}, other aspects of the treated partial differential equations are considered. \section{Some preliminary estimates} \label{S:VaryA} Within this section and the Sections \ref{S:B} and \ref{S:VaryB}, we can assume, without loss of generality, that $ \omega \ge 0 $ where $ \omega $ is the $ \omega $ in the hypothesis (H1). This is because the case $ \omega < 0 $ is the same as the case $ \omega = 0 $. This will be readily seen from the corresponding proofs. To prove the main results Theorems \ref{T:XA} and \ref{T:XB} in Section \ref{S:B}, we need to make two preparations. One preparation is this section, and the other is the Appendix in Section \ref{S:VaryB}. \begin{proposition} \label{P:A} Let $ A(t) $ satisfy the dissipativity condition (H1), the range condition (H2') , and the time-regulating condition (HA), and let $ u_0 $ be in $ D(A(s)) \subset E $ % ($ \subset where $ 0 \le s \le T $. Let $ 0 < \epsilon < \lambda_0 $ be so chosen that $ 0 < \epsilon \omega < 1 $, and let $ 0 \le t_i = s + i \epsilon \le T $ where $ i \in \mathbb{N} $. Then \begin{equation} \label{E:VaryA} \|u_i - u_0\| \le \eta^{i}L(\|u_0\|)(i\epsilon) + [\eta^{i - 1}b_1 + \eta^{i - 2}b_2 + \dots + \eta b_{i - 1} + b_i] \end{equation} and \begin{equation} \label{E:VaryB} \begin{split} \|\frac{u_i - u_{i-1}}{\epsilon}\| &\leq [(c_ic_{i - 1} \dots c_2)L(\|u_0\|) \quad \text{or $ (c_ic_{i - 1} \dots c_3) L(\|u_1\|)$ or $ \dots $} \\ &\quad \text{or $ c_iL(\|u_{i - 2}\|) $ or $ L(\|u_{i - 1}\|) $}] + [(c_ic_{i -1}\dots c_1)a_0 \\ &\quad + (c_ic_{i - 1} \dots c_2)d_1 + (c_ic_{i - 1}\dots c_3)d_2 + \dots + c_id_{i - 1} + d_i]. \end{split} \end{equation} Here $u_i = \prod_{j = 1}^{i}J_{\epsilon}(t_{j})u_0$ exists uniquely by the hypotheses (H1) and (H2'); \\ $\eta = (1 - \epsilon \omega)^{-1} > 1$; \\ $b_i = \eta \epsilon \|v_0\| + \eta \epsilon |f(t_i) - f(s)| L(\|u_0\|)(1 + \|v_0\|)$, where $ v_0 $ is any element in $ A(s)u_0 $; \\ $c_i= \eta [1 + L(\|u_{i - 1}\|)|f(t_i) - f(t_{i-1})|]$; \\ $d_i = \eta L(\|u_{i - 1}\|)|f(t_i) - f(t_{i-1})|$; \\ the right sides of \eqref{E:VaryB} are interpreted as $[L(\|u_0\|)] + [c_1a_0 + d_1]$ for $ i = 1 $; \\ $[c_2L(\|u_0\|)$ or $L(\|u_1\|)] + [c_2c_1a_0 + c_2d_1 + d_2] $ for $ i = 2 $; \dots, and so on; and \[ a_0 = \|\frac{u_0 - u_{-1}}{\epsilon}\|, \] where $ u_{-1} $ is defined by $u_0 - \epsilon v_0 = u_{-1}$, with $ v_0 $ any element in $ A(s)u_0 $. \end{proposition} \begin{proof} We will use the method of mathematical induction. Two cases will be considered, and for each case, we divide the proof into two steps. \subsection*{Case 1} Here \eqref{E:VaryA} is considered \textbf{Step 1.} Claim that \eqref{E:VaryA} is true for $ i = 1 $. This will follow from the arguments below. If $ (u_1 - u_0)\in S_1(\epsilon) $ (defined in Section \ref{S:A}), then \[ \|u_1 - u_0\| = \|J_{\epsilon}(t_1)u_0 - J_{\epsilon}(s)(I - \epsilon A(s))u_0\| \leq L(\|u_0\|)|t_1 - s| \leq L(\|u_0\|)\epsilon, \] which is less than or equal to the right-hand side of \eqref{E:VaryA} with $ i = 1 $. On the other and, if $ (u_1 - u_0) \in S_2(\epsilon) $ (defined in Section \ref{S:A}), then \[ \|u_1 - u_0\| \le \eta \|u_0 - u_0\| +\eta \epsilon \|v_0\| +\eta \epsilon |f(t_1) - f(s)|L(\|u_0\|)(1 + \|v_0\|), \] which is less than or equal to the right-hand side of \eqref{E:VaryA} with $ i = 1 $. Here $ v_0 $ is any element in $ A(s)u_0 $. \textbf{Step 2.} By assuming that \eqref{E:VaryA} is true for $ i = i - 1 $, we shall show that it is also true for $ i = i $. If $ (u_i - u_0) \in S_1(\epsilon) $, then \[ \|u_i - u_0\| = \|J_{\epsilon}(t_i)u_{i - 1} - J_{\epsilon}(s)(I - \epsilon A(s))u_0\| \le L(\|u_0\|)|t_i - s| = L(\|u_0\|)(i\epsilon), \] which is less than or equal to the right side of \eqref{E:VaryA} with $ i = i $ because of $ \eta^{i} > 1 $. On the other hand, if $ (u_i - u_0) \in S_2(\epsilon) $, then \[ \|u_i - u_0\| \le \eta \|u_{i - 1} - u_0\| + b_i \] where $\eta = (1 - \epsilon \omega)^{-1}$ and \[ b_i = \eta \epsilon \|v_0\| + \eta \epsilon|f(t_i) - f(s)| L(\|u_0\|)(1 + \|v_0\|). \] This recursive inequality, combined with the induction assumption, readily gives \begin{align*} \|u_i - u_0\| &\le \eta \{\eta^{i - 1}L(\|u_0\|)(i - 1)\epsilon + [\eta^{i - 2}b_1 + \eta^{i - 3}b_2 + \dots + \eta b_{i - 2} + b_{i - 1}]\} + b_i \\ &= \eta^{i}L(\|u_0\|)(i - 1)\epsilon + [\eta^{i - 1}b_1 + \eta^{i - 2}b_2 + \dots + \eta b_{i - 1} + b_i], \end{align*} which is less than or equal to the right-hand side of \eqref{E:VaryA} with $ i = i $ because of $ (i - 1)\epsilon \le i\epsilon $. \subsection*{Case 2} Here \eqref{E:VaryB} is considered. \textbf{Step 1.} Claim that \eqref{E:VaryB} is true for $i = 1 $. This follows from the Step 1 in Case 1, because there it was shown that \[ \|u_1 - u_0\| \le L(\|u_0\|)\epsilon \quad \text{or $ b_1 $}, \] which, when divided by $ \epsilon $, is less than or equal to the right side of \eqref{E:VaryB} with $ i = 1 $. Here $ a_0 = \|v_0\| $, in which $ a_0 = (u_0 - u_{-1})/\epsilon $ and $ u_{-1} \equiv u_0 - \epsilon v_0 $. \textbf{Step 2.} By assuming that \eqref{E:VaryB} is true for $ i = i - 1 $, we will show that it is also true for $ i = i $. If $ (u_i - u_{i - 1})\in S_1(\epsilon)$, then \[ \|u_i - u_{i - 1}\| \le L(\|u_{i - 1}\|) |t_i - t_{i - 1}| = L(\|u_{i - 1}\|) \epsilon. \] This, when divided by $ \epsilon $, has its right side less than or equal to one on the right-hand sides of \eqref{E:VaryB} with $ i = i $. If $ (u_i - u_{i-1}) \in S_2(\epsilon) $, then \begin{align*} \|u_i - u_{i-1}\| &\leq (1 - \epsilon \omega)^{-1}[\|u_{i-1} - u_{i-2}\| \\ &\quad + \epsilon |f(t_i) - f(t_{i-1})| L(\|u_{i-1}\|)(1 + \frac{\|u_{i-1} - u_{i-2}\|}{\epsilon})]. \end{align*} By letting \begin{gather*} a_i = \frac{\|u_i - u_{i-1}\|} {\epsilon}, \\ c_i = (1 - \epsilon \omega)^{-1}[1 + L(\|u_{i - 1}\|)|f(t_i) - f(t_{i-1})|], \quad \text{and} \\ d_i = L(\|u_{i - 1}\|)(1 - \epsilon \omega)^{-1}|f(t_i) - f(t_{i-1})|, \end{gather*} it follows that $a_i \leq c_ia_{i - 1} + d_i$. Here notice that \[ u_0 - \epsilon v_0 = u_{-1}; \quad a_0 = \|\frac{u_0 - u_{-1}}{\epsilon}\| = \|v_0\|. \] The above inequality, combined with the induction assumption, readily gives \begin{align*} a_i &\leq c_i\big\{[(c_{i - 1}c_{i - 2} \dots c_2)L(\|u_0\|) \quad \text{or $ (c_{i - 1}c_{i - 2} \dots c_3) L(\|u_1\|)$ or $ \dots $} \\ &\quad \text{or $ c_{i - 1}L(\|u_{i - 3}\|) $ or $ L(\|u_{i - 2}\|) $}] + [(c_{i - 1}c_{i -2}\dots c_1)a_0 \\ &\quad + (c_{i - 1}c_{i - 2} \dots c_2)d_1 + (c_{i - 1}c_{i - 2}\dots c_3)d_2 + \dots \\ &\quad + c_{i - 1}d_{i - 2} + d_{i - 1}]\big\} + d_i \\ &\leq [(c_ic_{i - 1} \dots c_2)L(\|u_0\|) \quad \text{or $ (c_ic_{i - 1} \dots c_3) L(\|u_1\|) $ or $ \dots $} \\ &\quad \text{or $ c_iL(\|u_{i - 2}\|) $}] + [(c_ic_{i -1}\dots c_1)a_0 \\ &\quad + (c_ic_{i - 1} \dots c_2)d_1 + (c_ic_{i - 1}\dots c_3)d_2 + \dots + c_id_{i - 1} + d_i], \end{align*} each of which is less than or equal to one on the right sides of \eqref{E:VaryB} with $ i = i $. The induction proof is now complete. \end{proof} \begin{proposition} \label{P:VaryA} Under the assumptions of Proposition \ref{P:A}, the following are true if $ u_0 $ is in $ \hat{D}(A(s)) = \{y \in \overline{D(A(s))}:|A(s)y| < \infty\} $: \begin{gather*} \|u_i - u_0\| \le K_1(1 - \epsilon \omega)^{-i}(2i + 1)\epsilon \le K_1e^{(T - s)\omega}(3)(T - s); \\ \|\frac{u_i - u_{i - 1}}{\epsilon}\| \le K_3; \end{gather*} where the constants $ K_1 $ and $ K_3 $ depend on the quantities: \begin{gather*} K_1 = K_1(L(\|u_0\|), (T - s), \omega, |A(s)u_0|, K_{B}); \\ K_2 = K_2(K_1, (T - s), \omega, \|u_0\|); \\ K_3 = K_3(L(K_2), (T - s), \omega, \|u_0\|, |A(s)u_0|, K_{B}); \\ K_{B} \text{ is the total variation of $ f $ on $ [0, T] $}. \end{gather*} \end{proposition} \begin{proof} We divide the proof into two cases. \subsection*{Case 1} Here $ u_0 \in D(A(s)) $. It follows immediately from Proposition \ref{P:A} that \begin{gather*} \|u_i - u_0\| \le N_1(1 - \epsilon \omega)^{-i}(2i + 1)\epsilon \le N_1e^{(T - s)\omega}(3)(T - s); \\ \|\frac{u_i - u_{i - 1}}{\epsilon}\| \le N_3; \end{gather*} where the constants $ N_1 $ and $ N_3 $ depend on the quantities: \begin{gather*} N_1 = N_1(L(\|u_0\|), (T - s), \omega, \|v_0\|, K_{B}); \\ N_2 = N_2(N_1, (T - s), \omega, \|u_0\|); \\ N_3 = N_3(L(N_2), (T - s), \omega, \|u_0\|, \|v_0\|, K_{B}); \\ K_{B} \text{ is the total variation of $ f $ on $ [0, T] $}. \end{gather*} We used here the estimate in \cite[Page 65]{Cran} \[ c_i\dots c_1 \leq e^{i\epsilon \omega} e^{e_i + \dots + e_1}, \] where $ e_i = L(\|u_{i - 1}\|)|f(t_i) - f(t_{i-1})| $. \subsection*{Case 2} Here $ u_0 \in \hat{D}(A(s)) $. This involves two steps. \textbf{Step 1.} Let $ u_0^{\mu} = (I - \mu A(s)) ^{-1}u_0 $ where $ \mu > 0 $, and let \[ u_i = \prod_{j = 1}^{i}J_{\epsilon}(t_{j})u_0; \quad u_i^{\mu} = \prod_{j = 1}^{i}J_{\epsilon}(t_{j})u_0^{\mu}. \] As in \cite[Lemma 3.2, Page 9]{Paz}, we have, by letting $ \mu \to 0 $, \[ u_0^{\mu} \to u_0; \] here notice that $ D(A(s)) $ is dense in $ \hat{D}(A(s)) $. Also it is readily seen that \[ u_i^{\mu} = \prod_{k=1}^{i}(I - \epsilon A(t_{k})) ^{-1}u_0^{\mu} \to u_i = \prod_{k=1}^{i}(I - \epsilon A(t_{k}))^{-1} u_0 \] as $ \mu \to 0 $, since $ (A(t) - \omega) $ is dissipative for each $ 0 \leq t \leq T $. \textbf{Step 2.} Since $ u_0^{\mu} \in D(A(s)) $, Case 1 gives \begin{equation} \label{E:TimeE} \begin{gathered} \|u_i^{\mu} - u_0^{\mu}\| \le N_1(L(\|u_0^{\mu}\|), (T - s), \omega, \|v_0^{\mu}\|, K_{B}) (1 - \epsilon \omega)^{-i}(2i + 1)\epsilon \\ \frac{\|u_i^{\mu} - u_{i-1}^{\mu}\|}{\epsilon} \leq N_3(L(N_2), (T - s), \omega, \|u_0^{\mu}\|, \|v_0^{\mu}\|, K_{B}), \end{gathered} \end{equation} where \[ N_2 = N_2(N_1, (T - s), \omega, \|u_0^{\mu}\|), \] and $ v_0^{\mu} $ is any element in $ A(s)(I - \mu A(s))^{-1}u_0 $. We can take \[ v_0^{\mu} = w_0^{\mu} \equiv \frac{(J_{\mu}(s) - I)u_0}{\mu}, \] since $ w_0^{\mu} \in A(s)(I - \mu A(s))^{-1}u_0 $. On account of $ u_0 \in \hat{D}(A(s)) $, we have \[ \lim_{\mu \to 0}\|\frac{(J_{\mu}(s) - I)u_0}{\mu}\| = |A(s)u_0| < \infty. \] Thus, by letting $ \mu \to 0 $ in \eqref{E:TimeE} and using Step 1, the results in the Proposition \ref{P:VaryA} follow. The proof is complete. \end{proof} \section{Main results} \label{S:B} Using the estimates in Section \ref{S:VaryA}, together with the difference equations theory, the following result will be shown in in Section \ref{S:VaryB}. \begin{proposition} \label{P:VaryB} Under the assumptions of Proposition \ref{P:XA}, the following inequality is true \[ a_{m, n} \le \begin{cases} L(K_2)|n\mu - m\lambda|, \quad &\text{if $ S_2(\mu) = \emptyset$}; \\ c_{m, n} + s_{m, n} + d_{m, n} + f_{m, n} + g_{m, n}, \quad &\text{if $ S_1(\mu) = \emptyset $}; \end{cases} \] where $ a_{m, n}, c_{m, n}, s_{m, n}, f_{m, n}, g_{m, n} $ and $ L(K_2) $ are defined in Proposition \ref{P:XA}. \end{proposition} In view of this and Proposition \ref{P:A}, we are led to the following claim. \begin{proposition} \label{P:XA} Let $ x \in \hat{D}(A(s)) $ where $ 0 \le s \le T $, and let $ \lambda , \mu > 0 $, $ n, m \in \mathbb{N}, $ be such that $ 0 \le (s +m \lambda), (s + n \mu) \le T $, and such that $ \lambda_0 > \lambda \geq \mu > 0 $ for which $ \mu \omega, \lambda \omega < 1 $. If $ A(t) $ satisfies the dissipativity condition (H1), the range condition (H2'), and the time-regulating condition (HA), then the inequality is true: \begin{equation} \label{E:VaryC} a_{m, n} \le c_{m, n} + s_{m, n} + d_{m, n} + e_{m, n} + f_{m, n} + g_{m, n}. \end{equation} Here \begin{gather*} a_{m, n} \equiv \|\prod_{i = 1}^{n} J_{\mu}(s + i \mu)x - \prod_{i = 1}^{m}J_{\lambda}(s + i \lambda)x\|; \\ \gamma \equiv (1 - \mu \omega)^{-1} > 1; \quad \alpha \equiv \frac{\mu}{\lambda}; \quad \beta \equiv 1 - \alpha; \\ c_{m, n} = 2K_1\gamma^{n}[(n\mu - m \lambda) + \sqrt{(n \mu - m \lambda)^2 + (n \mu)(\lambda - \mu)}]; \\ s_{m, n} = 2K_1\gamma^{n}(1 - \lambda \omega)^{-m} \sqrt{(n \mu - m \lambda)^2 + (n \mu)(\lambda - \mu)}; \\ d_{m, n} = [K_4\rho(\delta)\gamma^{n}(m \lambda)] + \{K_4\frac{\rho(T)}{\delta^2}\gamma^{n}[(m \lambda)(n \mu - m \lambda)^2 + (\lambda - \mu)\frac{m(m + 1)}{2}\lambda^2]\}; \\ e_{m, n} = L(K_2)\gamma^{n}\sqrt{(n \mu - m \lambda)^2 + (n \mu)(\lambda - \mu)}; \\ f_{m, n} = K_1[\gamma^{n}\mu + \gamma^{n}(1 - \lambda \omega)^{-m}\lambda]; \\ g_{m, n} = K_4\rho(|\lambda - \mu|)\gamma^{n}(m\lambda); \\ K_4 = \gamma L(K_2)(1 + K_3); \quad \delta > 0 \quad \text{is arbitrary}; \\ \rho(r) \equiv \sup\{|f(t) - f(\tau)| : 0 \leq t, \tau \leq T, |t - \tau| \leq r \} \end{gather*} where $\rho(r)$ is the modulus of continuity of $ f $ on $ [0, T] $; and $ K_1, K_2 $, and $ K_3 $ are defined in Proposition \ref{P:VaryA}. \end{proposition} \begin{proof} We will use the method of mathematical induction and divide the proof into two steps. Step 2 will involve six cases. \textbf{Step 1.} \eqref{E:VaryC} is clearly true by Proposition \ref{P:VaryA}, if $ (m, n)= (0, n) $ or $ (m, n) = (m, 0) $. \textbf{Step 2.} By assuming that \eqref{E:VaryC} is true for $ (m, n) = (m - 1, n - 1) $ or $ (m, n) = (m, n - 1) $, we will show that it is also true for $ (m, n) = (m, n) $. This is done by the arguments below. Using the nonlinear resolvent identity in \cite{Cr}, we have \begin{align*} a_{m, n} &= \|J_{u}(s + n \mu)\prod_{i = 1} ^{n - 1}J_{\mu}(s + i \mu)x\\ &\quad - J_{\mu}(s + m \lambda) [\alpha \prod_{i = 1}^{m - 1}J_{\lambda}( s + i \lambda)x + \beta \prod_{i = 1} ^{m}J_{\lambda}(s + i \lambda)x)]\|. \end{align*} Here $ \alpha = \mu/\lambda$ and $ \beta = (\lambda - \mu)/\lambda$. Under the time-regulating condition (HA), it follows that, if the element inside the norm of the right side of the above equality is in $ S_1(\mu) $, then, by Proposition \ref{P:VaryA} with $ \epsilon = \mu $, \begin{equation} \label{E:VaryMain2} a_{m, n} \le L(\|\prod_{i = 1}^{n}J_{\mu} (s + i \mu)x\|)|m \lambda - n \mu| \le L(K_2)|m \lambda - n \mu|, \end{equation} which is less than or equal to the right-hand side of \eqref{E:VaryC} with $ (m, n) = (m, n) $, where $ \gamma ^{n} > 1 $. If that element instead lies in $ S_2(\mu) $, then, by Proposition \ref{P:VaryA} with $ \epsilon = \mu $, \begin{equation} \label{E:VaryMain} \begin{split} a_{m, n} &\le \gamma (\alpha a_{m - 1, n - 1} +\beta a_{m, n - 1}) + \gamma \mu |f(s + m \lambda) - f(s + n \mu)|\\ &\quad\times L(\|\prod_{i = 1}^{n}J_{\mu}(s + i \mu)x\|)[1 + \|\frac{\prod_{i = 1}^{n}J_{\mu} (s + i \mu)x - \prod_{i = 1}^{n - 1} J_{\mu}(s + i \mu)x}{\mu}\|] \\ &\le [\gamma \alpha a_{m - 1, n - 1} + \gamma \beta a_{m, n - 1}] + K_4\mu \rho(|n \mu - m \lambda|), \end{split} \end{equation} where $ K_4 = \gamma L(K_2)(1 + K_3) $ and $ \rho(r) $ is the modulus of continuity of $ f $ on $ [0, T] $. From this, it follows that proving the relations is sufficient under the induction assumption: \begin{gather} \gamma \alpha p_{m - 1, n - 1} + \gamma \beta p_{m, n - 1} \le p_{m, n}; \label{E:VaryD} \\ \gamma \alpha q_{m - 1, n - 1} + \gamma \beta q_{m, n - 1} + K_4\mu \rho(|n \mu - m \lambda|) \le q_{m, n}; \label{E:VaryE} \end{gather} where $ q_{m, n} = d_{m, n} $, and $ p_{m, n} = c_{m, n} $ or $ s_{m, n} $ or $ e_{m, n} $ or $ f_{m, n} $ or $ g_{m, n} $. Now we consider five cases. \textbf{Case 1.} Here $ p_{m, n} = c_{m, n} $. Under this case, \eqref{E:VaryD} is true because of the calculations, where \[ b_{m, n} = \sqrt{(n\mu - m\lambda)^2 + (n\mu)(\lambda - \mu)} \] was defined and the Schwartz inequality was used: \begin{gather*} \alpha[(n - 1)\mu - (m - 1)\lambda] + \beta[(n - 1)\mu - m\lambda] = (n \mu - m \lambda); \\ \begin{aligned} \alpha b_{m - 1, n - 1} + \beta b_{m, n - 1} &= \sqrt{\alpha}\sqrt{\alpha}b_{m - 1, n - 1} + \sqrt{\beta}\sqrt{\beta} b_{m, n - 1} \\ &\quad \le (\alpha + \beta)^{1/2}(\alpha b_{m - 1, n - 1}^2 + \beta b_{m, n - 1}^2)^{1/2} \\ &\quad \le \{(\alpha + \beta)(n\mu - m\lambda)^2 + 2(n\mu - m \lambda)[\alpha(\lambda - \mu) - \beta \mu] \\ &\quad + [\alpha(\lambda - \mu)^2 + \beta \mu^2] + (n - 1)\mu(\lambda - \mu)\}^{1/2} \\ &= b_{m, n}. \end{aligned} \end{gather*} Here \[ \alpha + \beta = 1; \quad \alpha(\lambda - \mu) - \beta \mu = 0; \quad \alpha(\lambda - \mu)^2 + \beta \mu^2 = \mu(\lambda - \mu). \] \textbf{Case 2.} Here $ p_{m, n} = s_{m, n} $. Under this case, \eqref{E:VaryD} is true, as is with the Case 1, by noting that \[ (1 - \lambda \omega)^{-(m - 1)} \le (1 - \lambda \omega)^{-m}. \] \textbf{Case 3.} Here $ q_{m, n} = d_{m, n} $. Under this case, \eqref{E:VaryE} is true because of the calculations: \begin{align*} &\gamma \alpha d_{m - 1, n - 1} + \gamma \beta d_{m, n - 1} + K_4 \mu \rho(|n \mu - m \lambda|) \\ &\le \{\gamma \alpha[K_4\rho(\delta)\gamma^{n - 1}(m - 1)\lambda] + \gamma \beta[K_4 \rho(\delta)\gamma^{n - 1}(m \lambda)]\} \\ & \quad + \gamma \alpha \{K_4\frac{\rho(T)}{\delta^2} \gamma^{n - 1}[(m - 1)\lambda\left((n - 1)\mu - (m - 1)\lambda\right)^2 + (\lambda - \mu)\frac{(m - 1)m}{2}\lambda^2]\} \\ &\quad + \gamma \beta\{K_4\frac{\rho(T)}{\delta^2} \gamma^{n - 1}[(m \lambda)\left((n - 1)\mu - m\lambda\right)^2 + (\lambda - \mu)\frac{m(m + 1)}{2}\lambda^2]\} \\ &\quad + K_4\mu \rho(|n \mu - m \lambda|) \\ &= K_4\rho(\delta)\gamma^{n}[(\alpha + \beta)(m\lambda) - \alpha \lambda] \\ &\quad + K_4\frac{\rho(T)}{\delta^2} \gamma^{n}\{\alpha[(n\mu - m\lambda)^2 + 2(n \mu - m \lambda)(\lambda - \mu) + (\lambda - \mu)^2](m \lambda - \lambda) \\ &\quad + [\alpha (\lambda - \mu)\frac{m(m + 1)}{2}\lambda^2 - \alpha(\lambda - \mu)m\lambda^2] \\ &\quad + \beta[(n \mu - m \lambda)^2 - 2(n \mu - m \lambda)\mu + \mu^2](m \lambda) \\ &\quad +[\beta (\lambda - \mu)\frac{m(m + 1)}{2}\lambda^2]\} + K_4\mu \rho(|n\mu - m\lambda|) \\ &\quad \le K_4\rho(\delta)\gamma^{n}[(m\lambda) - \mu] + K_4\mu \rho(|n \mu - m \lambda|) \\ &\quad + K_4\frac{\rho(T)}{\delta^2}\gamma^{n}[(m\lambda)(n \mu - m \lambda)^2 + (\lambda - \mu)\frac{m(m + 1)}{2}\lambda^2 - \mu(n \mu - m \lambda)^2] \\ & \equiv r_{m, n}, \end{align*} where the negative terms $ [2(n\mu - m\lambda)(\lambda - \mu) + (\lambda - \mu)^2](-\lambda)$ were dropped, \[ \alpha 2(n\mu - m\lambda)(\lambda - \mu) - \beta 2(n\mu - m\lambda)\mu = 0, \] and \[ [\alpha(\lambda - \mu)^2 + \beta \mu^2](m\lambda) = (m\lambda)\mu(\lambda - \mu), \] which cancelled \[ -\alpha(\lambda - \mu)m\lambda^2 = - (m\lambda)\mu(\lambda - \mu); \] it follows that $ r_{m, n} \le d_{m, n} $, since \begin{align*} &K_4\mu \rho(|n\mu - m\lambda|) \\ & \le \begin{cases} K_4\mu \rho(\delta) \le K_4\mu\rho(\delta)\gamma^{n}, \quad &\text{if $ |n\mu - m\lambda| \le \delta $}; \\[3pt] K_4\mu\rho(T)\frac{(n\mu - m\lambda)^2}{\delta^2} \le K_4\mu\rho(T)\gamma^{n}\frac{(n\mu - m\lambda)^2}{\delta^2}, \quad &\text{if $ |n\mu - m\lambda| > \delta $}. \end{cases} \end{align*} \textbf{Case 4.} Here $ p_{m, n} = e_{m, n} $. Under this case, \eqref{E:VaryD} is true, as is with the Case 1. \textbf{Case 5.} Here $ p_{m, n} = f_{m, n} $. Under this case, \eqref{E:VaryD} is true because of the calculations: \begin{align*} \gamma \alpha f_{m - 1, n - 1} + \gamma \beta f_{m, n - 1} &= \gamma \alpha K_1[\gamma^{n - 1}\mu + \gamma^{n - 1} (1 - \lambda \omega)^{-(m - 1)}\lambda] \\ & \quad + \gamma \beta K_1[\gamma^{n - 1} \mu + \gamma^{n - 1}(1 - \lambda \omega)^{-m}\lambda] \\ & \le K_1[(\alpha + \beta)\gamma^{n}\mu + (\alpha + \beta)\gamma^{n}(1 - \lambda \omega)^{-m}\lambda], \\ &= f_{m, n}. \end{align*} \textbf{Case 6.} Here $ p_{m, n} = g_{m, n} $. Under this case, \eqref{E:VaryD} is true because of the calculations: \begin{align*} \gamma \alpha g_{m - 1, n - 1} + \gamma \beta g_{m, n - 1} &\le K_4\gamma^{n}\rho(|\lambda - \mu|)\alpha(m - 1)\lambda + K_4\gamma^{n}\rho(|\lambda - \mu|)\beta(m \lambda) \\ & \le K_4\gamma^{n}\rho(|\lambda - \mu|)(\alpha + \beta)(m\lambda) \\ &= g_{m, n}. \end{align*} Now the proof is complete. \end{proof} Here is one of our two main results: \begin{theorem} \label{T:XA} If the nonlinear operator $ A(t) $ satisfies the dissipativity condition {\rm (H1)}, the range condition {\rm (H2')}, and the time-regulating condition {\rm (HA)} , then \[ U(s + t, s)u_0 \equiv \lim_{n \to \infty}\prod_{i = 1}^{n} J_{\frac{t}{n}}(s + i \frac{t}{n})u_0 \] exists for $ u_0 \in \overline{\hat{D}(A(s))} = \overline{D(A(s))} $ where $ s,t \ge 0 $ and $ 0 \le (s + t) \le T $, and is the so-called a limit solution to the equation \eqref{E:A} . Furthermore, this limit $ U(s + t, s)u_0 $ has the Lipschitz property \[ \|U( s + t, s)u_0 - U(s + \tau, s)u_0\| \le k |t - \tau| \] for $ 0 \le s + t, s + \tau \le T $ and for $ u_0 \in \hat{D}(A(s)) $. \end{theorem} \begin{proof} For $ x \in \hat{D}(A(s)) $, it follows from Proposition \ref{P:XA}, by setting $ \mu = \frac{t}{n}, \lambda = \frac{t}{m}, $ and $ \delta^2 = \sqrt{\lambda - \mu} $ that, as $ n, m \to \infty $, $ a_{m, n} $ converges to $ 0 $, uniformly for $ 0 \le (s + t) \le T $. Thus \[ \lim_{n \to \infty} \prod_{i = 1}^{n}J_{\frac{t}{n}}(s + i\frac{t}{n})x \] exists for $ x \in \hat{D}(A(s)) $. This limit also exits for $ x \in \overline{\hat{D}(A(s))} = \overline{D(A(s)))} $, on following the limiting arguments in Crandall-Pazy \cite{Cran}. On the other hand, setting $ \mu = \lambda = t/n$, $m = [\frac{t}{\mu}]$ and setting $ \delta^2 = \sqrt{\lambda - \mu} $, it follows that \begin{equation} \label{E:TimeF} \lim_{n \to \infty}\prod_{i = 1}^{n}J_{\frac{t}{n}}(s + i \frac{t}{n})u_0 = \lim_{\mu \to 0}\prod_{i = 1}^{[\frac{t}{\mu}]}J_{\mu}(s + i\mu)u_0. \end{equation} Now, to show the Lipschitz property, \eqref{E:TimeF} and Crandall-Pazy \cite[Page 71]{Cran} will be used. From Proposition \ref{P:VaryA}, it is derived that \begin{gather*} \begin{aligned} \|u_{n} - u_{m}\| &\le \|u_{n} - u_{n - 1}\| + \|u_{n - 1} - u_{n - 2}\| + \dots + \|u_{m + 1} - u_{m}\| \\ &\le K_3 \mu (n - m) \quad \text{for } x \in \hat{D}(A(s)), \end{aligned}\\ u_{n} = \prod_{i = 1}^{n}J_{\mu}(s + i \mu)x, \quad u_{m} = \prod_{i = 1}^{m}J_{\mu}(s + i \mu)x, \end{gather*} where $ n = [t/\mu]$, $m = [\tau/\mu]$, $t > \tau $ and $ 0 < \mu < \lambda_0 $. The proof is completed by making $ \mu \to 0 $ and using \eqref{E:TimeF}. \end{proof} Now discretize \eqref{E:A} as \begin{equation} \label{E:TimeB} \begin{gathered} u_i - \epsilon A(t_i)u_i \ni u_{i-1}, \\ u_i \in D(A(t_i)), \end{gathered} \end{equation} where $ n \in \mathbb{N} $ is large, and $ \epsilon $ is such that $ s \le t_i = s + i \epsilon \le T $ for each $ i = 1, 2, \ldots, n $. Here notice that, for $ u_0 \in E $, $ u_i $ exists uniquely by the hypotheses (H1) and (H2'). Let $ u_0 \in \hat{D}(A(s)) $, and construct the Rothe functions \cite{Ka,Ro}. Let \begin{gather*} \chi^{n}(s) = u_0, \quad C^{n}(s) = A(s), \\ \chi^{n}(t) = u_i, \quad C^{n}(t) = A(t_i) \quad \text{for } t \in (t_{i-1}, t_i], \end{gather*} and let \begin{gather*} u^{n}(s) = u_0, \\ u^{n}(t) = u_{i-1} + (u_i - u_{i-1}) \frac{t - t_{i-1}}{\epsilon} \quad \text{for } t \in (t_{i-1}, t_i] \subset [s, T]. \end{gather*} Since $ \|\frac{u_i - u_{i-1}}{\epsilon}\| \leq K_3 $ for $ u_0 \in \hat {D}(A(s)) $ by Proposition \ref{P:A}, it follows that, for $ u_0 \in \hat{D}(A(s)) $, \begin{equation} \label{E:B} \begin{gathered} \lim_{n \to \infty}\sup_{t \in [0, T]}\|u^{n}(t) - \chi^{n}(t)\| = 0, \\ \|u^{n}(t) - u^{n}(\tau)\| \leq K_3|t - \tau|, \end{gathered} \end{equation} where $ t, \tau \in (t_{i-1}, t_i] $, and that, for $ u_0 \in \hat{D}A(s)) $, \begin{equation} \label{E:C} \begin{gathered} \frac{du^{n}(t)}{dt} \in C^{n}(t)\chi^{n}(t), \\ u^{n}(s) = u_0, \end{gathered} \end{equation} where $ t \in (t_{i-1}, t_i] $. Here the last equation has values in $ B([s, T]; X) $, which is the real Banach space of all bounded functions from $ [s, T] $ to $X$. \begin{proposition} \label{P:B} If $ A(t) $ satisfies the assumptions in Theorem \ref{T:XA}, then \[ \lim_{n \to \infty}u^{n}(t) = \lim_{n \to \infty}\prod_{i = 1}^{n} J_{\frac{t - s}{n}}(s + i \frac{t}{n})u_0 \] uniformly for finite $ 0 \le (s +t) \le T $ and for $ u_0 \in \hat{D}(A(s)) $. \end{proposition} \begin{proof} The asserted uniform convergence will be proved by using the Ascoli-Arzela Theorem \cite{Roy}. Pointwise convergence will be proved first. For each $ t \in [s, T) $, we have $ t \in [t_i, t_{i+1}) $ for some $ i $, and so $ i = [\frac{t - s}{\epsilon}] $, the greatest integer that is less than or equal to $ \frac{t - s}{\epsilon} $. That $ u_i $ converges is because, for each above $ t $, \begin{equation} \label{E:TimeG} \begin{aligned} \lim_{\epsilon \to 0}u_i &= \lim_{\epsilon \to 0}\prod_{k=1}^{i}(I - \epsilon A(t_{k}))^{-1}u_0 \\ &= \lim_{n \to \infty} \prod_{k=1}^{n}[I - \frac{t - s}{n}A(s + k\frac{t - s}{n})]^{-1}u_0 \end{aligned} \end{equation} by \eqref{E:TimeF}, which has the right side convergent by Theorem \ref{T:XA}. Since \[ \|\frac{u_i - u_{i - 1}}{\epsilon}\| \le K_3 \] for $ u_0 \in \hat{D}(A(s)) $, we see from the definition of $ u^{n}(t) $ that \[ \lim_{n \to \infty}u^{n}(t) = \lim_{\epsilon \to 0}u_i = \lim_{n \to \infty}\prod_{i = 1}^{n} J_{\frac{t - s}{n}}(s + i\frac{t - s}{n})u_0 \] for each $ t $. On the other hand, due to \[ \|\frac{u_i - u_{i - 1}}{\epsilon}\| \le K_3 \] again, we see that $ u^{n}(t) $ is equi-continuous in $ C([s, T]; X) $, the real Banach space of all continuous functions from $ [s, T] $ to $X$. Thus it follows from the Ascoli-Arzela theorem \cite{Roy} that, for $ u_0 \in \hat{D}(A(s)) $, some subsequence of $ u^{n}(t) $ (and then itself) converges uniformly to some \[ u(t) = \lim_{n \to \infty}\prod_{i = 1}^{n} J_{\frac{t - s}{n}}(s + i \frac{t - s}{n})u_0 \in C([s, T]; X). \] This completes the proof. \end{proof} Now consider a strong solution. Let $ (Y, \|\cdot\|_{Y}) $ be a real Banach space, into which the real Banach space $ (X, \|\cdot\|) $ is continuously embedded. Assume additionally that $ A(t) $ satisfies the embedding property of embeddedly quasi-demi-closedness: \begin{itemize} \item[(HB)] If $ t_{n} \in [0, T] \to t $, if $ x_{n} \in D(A(t_{n})) \to x $, and if $ \|y_{n}\| \leq k $ for some $ y_{n} \in A(t_{n})x_{n} $, then $ \eta(A(t)x) $ exists and \[ |\eta(y_{n_{l}}) - z| \to 0 \] for some subsequence $ y_{n_{l}} $ of $ y_{n} $, for some $ z \in \eta(A(t)x) $, and for each $ \eta \in Y^{*} \subset X^{*} $, the real dual space of $ Y $. \end{itemize} Here is the other main result. \begin{theorem} \label{T:XB} Let $ A(t) $ satisfy the dissipativity condition {\rm (H1)}, the range condition {\rm (H2')}, the time-regulating condition {\rm (HA)}, and the embedding property {\rm (HB)}. Then equation \eqref{E:A}, for $ u_0 \in \hat{D}(A(s)) $, has a strong solution \[ u(t) = \lim_{n \to \infty} \prod_{i = 1}^{n}J_{\frac{t - s}{n}}(s + i \frac{t}{n})u_0 \] in $ Y $, in the sense that \begin{gather*} \frac{d}{dt}u(t) \in A(t)u(t) \quad \text{in $ Y $ for almost every $ t \in (0, T) $}; \\ u(s) = u_0. \end{gather*} The solution is unique if $ Y \equiv X $. Furthermore, \[ \|u(t) - u(\tau)\|_{X} \le K_3|t - \tau| \] for $ 0 \le s \le t$, $\tau \le T $, a result from Theorem \ref{T:XA}. \end{theorem} The results in the above theorem follow from Theorem \ref{T:XA} and the proof in \cite[page 364]{Li}, \cite[pages 262-263]{Lin2}. \begin{remark} \label{T:XC} \rm The results in Sections \ref{S:VaryA} and \ref{S:B} are still true if the range condition (H2') is replaced by the weaker condition (H2'') below, provided that the initial conditions $ u_0 \in \hat{D}(A(s)) (\supset D(A(s)) ) $ and $ u_0 \in \overline{\hat{D}(A(s))} = \overline{D(A(s))} (\supset D(A(s)) )$ are changed to the condition $ u_0 \in D(A(s)) $. This is readily seen from the corresponding proofs. Here \begin{itemize} \item[(H2'')] The range of $ (I - \lambda A(t)) $, denoted by $ E $, is independent $ t $ and contains $ D(A(t)) $ for all $ t \in [0, T] $ and for small $ 0 < \lambda < \lambda_0 $ with $ \lambda_0\omega < 1 $. \end{itemize} \end{remark} \section{Applications to partial differential equations (I)} \label{S:D} Within this section, $ K $ will denote a constant that can vary with different occasions. Now we make the following assumptions: \begin{itemize} \item[(A1)] $ \Omega $ is a bounded smooth domain in $ \mathbb{R}^{n}, n \geq 2, $ and $ \partial \Omega $ is the boundary of $ \Omega $. \item[(A2)] $ \nu(x) $ is the unit outer normal to $ x \in \partial \Omega $, and $ \mu $ is a real number such that $ 0 < \mu < 1 $. \item[(A3)] $ \alpha(x, t, p) \in C^2(\overline \Omega \times \mathbb{R}^{n}) $ is true for each $ t \in [0, T] $, and is continuous in all its arguments. Furthermore, $ \alpha(x, t, p) \geq \delta_0 > 0 $ is true for all $ x, z $, and all $ t \in [0, T] $, and for some constant $ \delta_0 > 0 $. \item[(A4)] $ g(x, t, z, p) \in C^2(\overline \Omega \times \mathbb{R} \times \mathbb{R}^{n}) $ is true for each $ t \in [0, T] $, is continuous in all its arguments, and is monotone non-increasing in $ z $ for each $ t , x $, and $ p $. \item[(A5)] $ \frac{g(x,t, z, p)}{\alpha(x, t, p)} $ is of at most linear growth in $ p $, that is , \[ | \frac{g(x, t, z, p)}{\alpha(x, t, p)} | \leq M(x, t, z)(1 + |p|) \] for some continuous function $ M $ and for all $ t \in [0, T] $ when $ |p| $ is large enough. \item[(A6)] $ \beta(x, t, z) \in C^{3}(\Omega \times \mathbb{R}) $ is true for each $ t \in [0, T] $, is continuous in all its arguments, and is strictly monotone increasing in $ z $ so that $ \beta_{z} \geq \delta_0 > 0 $ for the constant $ \delta_0 > 0 $ in (A3). \item[(A7)] \begin{gather*} |\alpha(x, t, p) - \alpha(x, \tau, p)| \leq |\zeta(t) - \zeta(\tau)| N_1(x, |p|), \\ |g(x, t, z, p) - g(x, \tau, z, p)| \leq |\zeta(t) - \zeta(\tau)|N_2(x, |z|, |p|), \\ |\beta(x, t, z) - \beta(x, \tau, z)| \leq |t - \tau|N_3(x, |z|) \end{gather*} are true for some continuous positive functions $ N_1, N_2, N_3 $ and for some continuous function $ \zeta $ of bounded variation. \end{itemize} Define the $t$-dependent nonlinear operator $ A(t) : D(A(t)) \subset C(\overline \Omega) \to C(\overline \Omega) $ by \begin{gather*} D(A(t)) = \{ u \in C^{2 + \mu}(\overline \Omega) : \frac{\partial u}{\partial \nu} + \beta(x, t, u) = 0 \quad \text{on $ \partial \Omega $} \} \quad \text{and} \\ A(t)u = \alpha(x, t, Du) \Delta u + g(x, t, u, Du) \quad \text{for $ u \in D(A(t)) $}. \end{gather*} \begin{example} \label{T:XD} \rm Consider the equation \begin{equation} \label{E:XA} \begin{gathered} \frac{\partial}{\partial t}u(x, t) = \alpha(x, t, Du)\Delta u + g(x, t, u, Du), \quad (x, t) \in \Omega \times (0, T), \\ \frac{\partial}{\partial \nu}u + \beta(x, t, u) = 0 , \quad x \in \partial \Omega, \\ u(x, 0) = u_0, \end{gathered} \end{equation} for $ u_0 \in D(A(0)) $. The above equation has a strong solution \[ u(t) = \lim_{n \to \infty}\prod_{i = 1}^{n} J_{\frac{t}{n}}(i \frac{t}{n})u_0 \] in $ L^2(\Omega) $ with \[ \frac{\partial}{\partial \nu}u(t) + \beta(x, t, u(t)) = 0, \quad x \in \partial \Omega, \] and the solution $ u(t) $ satisfies the property \[ \sup_{t \in [0, T]}\|u(t)\|_{C^{1 + \mu} (\overline{\Omega})} \le K \] for some constant $ K $. \end{example} \begin{proof} It was shown in \cite[Pages 264-268]{Lin2} that $ A(t) $ satisfies the dissipativity condition (H1), the range condition (H2'') with $ E = C^{\mu}(\overline{\Omega}) $ for any $ 0 < \mu < 1 $, and satisfies the time-regulating condition (HA) and the embedding property (HB). Here the third line on \cite[Page 268]{Lin2}: \[ \times [\|N_2(z, \|v\|_{\infty}, \|Dv\|_{\infty})\|_{\infty} + \frac{\|N_1(z, \|Dv\|_{\infty})\|_{\infty}}{\delta_1} \|A(\tau)v\|_{\infty})] \] should have $ \|A(\tau)v\|_{\infty} $ replaced by \[ [\|A(\tau)v\|_{\infty} + \|g(z, \tau, v, Dv)\|_{\infty}]. \] Hence Remark \ref{T:XC} and Theorems \ref{T:XA} and \ref{T:XB} are applicable. It remains to prove that $ u(t) $ satisfies the mentioned property and the middle equation in \eqref{E:XA} in $ C(\overline{\Omega}) $. This basically follows from \cite[pages 264-268]{Lin2}. To this end, the $ u_i $ in \eqref{E:TimeB} will be used. Since $ A(t) $ satisfies (H1), (H2''), and (HA), it follows from Proposition \ref{P:VaryA} and Remark \ref{T:XC} that \[ \|\frac{u_i - u_{i - 1}}{\epsilon}\| = \|A(t_i)u_i\|_{\infty} \le K_3 \quad \text{and} \quad \|u_i\|_{\infty} \leq K_2. \] Thus, from linear $ L^{p} $ elliptic theory \cite{Tr,Gi2}, it follows that $\|u_i\|_{W^{2, p}} \le K$ for some constant $ K $, whence \begin{equation} \label{E:TimeLA} \|u_i\|_{C^{1 + \eta}} \le K \end{equation} for any $ 0 < \eta < 1 $ by the Sobolev embedding theorem \cite{Gi2}. This, together with the interpolation inequality \cite{Gi2} and the Ascoli-Arzela theorem \cite{Gi2, Roy}, implies that a convergent subsequence of $ u_i $ converges in $ C^{1 + \mu}(\overline{\Omega}) $ for any $ 0 < \lambda < \eta < 1 $. Therefore, on account of \eqref{E:TimeG} and Proposition \ref{P:B}, \[ \sup_{t \in [0, T]}\|u(t)\|_{C^{1 + \mu}} \le K \] results for $ u_0 \in D(A(0)) $, and $ u(t) $ satisfies the middle equation in \eqref{E:XA} in $ C(\overline{\Omega}) $. The proof is complete \end{proof} Consider the linear equation \begin{equation} \label{E:ZA} \begin{gathered} \frac{\partial u(x,t)}{\partial t} = \sum_{i, j = 1}^{n}a_{ij}(x, t)D_{ij}u(x,t) + \sum_{i = 1}^{n}b_i(x, t)D_iu(x, t) + c(x, t)u(x, t) \\ \quad \text{for $ (x, t) \in \Omega \times (0, T) $}, \\ \frac{\partial}{\partial \nu}u + \beta(x, t) u = 0, \quad x \in \partial \Omega, \\ u(x, 0) = u_0, \end{gathered} \end{equation} in which the following are assumed. Let $ a_{ij}(x, t) = a_{ji}(x, t) $, and let \[ \lambda_{\rm min} |\xi|^2 \leq \sum_{i, j}^{n}a_{ij}(x, t)\xi_i\xi_{j} \le \lambda_{\rm max} |\xi|^2 \] for some positive constants $ \lambda_{\rm min}$, $\lambda_{\rm max} $, for all $ \xi \in \mathbb{R}^{n} $, and for all $ x, t $. Let \[ a_{ij}(x, t),\; b_i(x, t),\; c(x, t) \in C^{\mu}(\overline{\Omega}) \] uniformly for all $ t $, be continuous in all their arguments, and be of bounded variation in $ t $ uniformly for $ x $. Let $c(x, t) \le 0$ for all $ x, t $, \[ \beta(x, t) \in C^{1 + \mu}(\overline{\Omega}), \quad 0 < \mu < 1 \] for all $ t $, and $ \beta(x, t) \ge \delta > 0 $ for some constant $ \delta > 0 $. Finally, let $ \beta(x, t) $ and $ c(x, t) $ be continuous in all its arguments, and let $ \beta(x, t) $ be Lipschitz continuous in $ t $ uniformly for $ x $. \begin{example} \label{T:ZA} \rm If $ \sum_{i, j}a_{ij}(x, t)D_{ij}u(x, t) = a_0(x, t)\Delta u(x, t) $ for some $ a_0(x, t) $, then the equation \eqref{E:ZA}, for $ u_0 \in D(A(0)) $, has a strong solution \[ u(t) = \lim_{n \to \infty}\prod_{i = 1}^{n} J_{\frac{t}{n}}(i \frac{t}{n})u_0 \] in $ L^2(\Omega) $ with \[ \frac{\partial}{\partial \nu}u(t) + \beta(x, t)u(t) = 0, \quad x \in \partial \Omega, \] and $ u(t) $ satisfies the property \[ \sup_{t \in [0, T]}\|u(t)\|_{C^{1 + \mu}(\overline{\Omega})} \le K. \] \end{example} \begin{proof} Linear elliptic equation theory \cite[Pages 128-130]{Gi2} shows that the corresponding operator $ A(t) $ satisfies the range condition (H2'') with $ E = C^{\mu}(\overline{\Omega}) $. The arguments in \cite[Pages 267-268]{Lin2} shows that $ A(t) $ satisfies the dissipativity condition (H1), the time-regulating condition (HA), and the embedding property (HB). The proof is complete, after applying Remark \ref{T:XC}, Theorems \ref{T:XA} and \ref{T:XB}, and the proof for Theorem \ref{T:XD}. \end{proof} \begin{example} \label{T:ZB} \rm Suppose that \[ a_{ij}(x),\; b_i(x),\; c(x) \in C^{1 + \mu}(\overline{\Omega}),\; \beta(x) \in C^{2 + \mu}(\overline{\Omega}) \] are independent of $ t $, where $ 0 < \mu < 1 $. Then equation \eqref{E:ZA} has a unique classical solution \[ u(t) = \lim_{n \to \infty}\prod_{i = 1}^{n} J_{\frac{t}{n}}(i \frac{t}{n})u_0 = \lim_{n \to \infty}(I - \frac{t}{n}A)^{-n}u_0 \] for $u_0 \in D(A) $ with $ Au_0 \in D(A) $, and the solution has the properties that $ \frac{du(t)}{dt} $ is Lipschitz continuous in $ t $, and that \[ \|\frac{du}{dt}\|_{C^{1 + \mu}(\overline{\Omega})} \le K. \] Furthermore, $ \frac{d}{dt}u $ is differentiable in $ t $ and $ \frac{d^2}{dt^2}u(t) $ is Lipschitz continuous in $ t $, if $ u_0 $ is in $ D(A^{3}) $ such that $ A^{3}u_0 \in D(A) $. More regularity of $ \frac{du}{dt} $ in $ t $ can be obtained iteratively. \end{example} \begin{remark} \label{rmk2} \rm In order for $ u_0 $ to be in $ D(A^2) $, more smoothness assumptions should be imposed on the coefficient functions $ a_{ij}(x), b_i(x), c(x) $ and $ \beta(x) $. \end{remark} \begin{proof} Here observe that the operator $ A $ is not closed, and so \cite[Theorem 1 Page 363]{Li} does not apply directly. The $ u_i $ in \eqref{E:TimeB} will be used, and $ u_0 \in D(A) $ with $ Au_0 \in D(A) $ will be assumed for a moment. It follows that \[ Au_i = \frac{u_i - u_{i - 1}}{\epsilon} = (I - \epsilon A)^{-i}(Au_0), \] and hence, by \eqref{E:TimeLA} which is for the proof of Theorem \ref{T:XD}, \[ \|Au_i\|_{C^{1 + \eta}(\overline{\Omega})} = \|(I - \epsilon A)^{-i}(Au_0)\|_{C^{1 + \eta}(\overline{\Omega})} \le K \] for $ Au_0 \in D(A) $ and for any $ 0 < \eta < 1 $. This implies \[ \|u_i\|_{C^{3 + \eta}(\overline{\Omega})} \le K \] by the Schauder global estimate with more smoothness in the linear elliptic theory \cite{Gi2}. Consequently, on using the interpolation inequality \cite{Gi2} and the Ascoli-Arzela theorem \cite{Gi2,Roy}, we have \[ Au_i \to Au(t) = U(t)(Au_0) \] through some subsequence with respect to the topology in $ C^{1 + \lambda}(\overline{\Omega}) $ for any $ 0 < \lambda < \eta < 1 $. Here \[ U(t)u_0 \equiv \lim_{n \to \infty} (I - \frac{t}{n}A)^{-n}u_0. \] The rest follows from \cite[Page 363]{Li}, where the Lipschitz property in Theorem \ref{T:XA} and Remark \ref{T:XC} will be used. \end{proof} Now consider the linear equation with the space dimension $ 1 $: \begin{equation} \label{E:MA} \begin{gathered} \frac{\partial u}{\partial t} = a(x, t)u_{xx} + b(x, t)u_{x} + c(x, t)u, \quad (x, t) \in (0, 1) \times (0, T), \\ u'(j, t) = (-1)^{j}\beta_{j}(j, t)u(j, t), \quad j = 0, 1, \\ u(x, 0) = u_0(x). \end{gathered} \end{equation} Here we assume that $ a, b, c $ are jointly continuous in $ x \in [0, 1]$, $t \in [0, T] $, and are of bounded variation in $ t $ uniformly for all $ x $, that $ c(x, t) \le 0 $ and $ a(x, t) \ge \delta_0 $ for some constant $ \delta_0 > 0 $, and finally that $ \beta_{j} \ge \delta_0 > 0, j = 0, 1 $ are jointly continuous in $ x, t $, and are Lipschitz continuous in $ t $, uniformly over $ x $. Let $ A(t) $: $ D(A(t)) \subset C[0, 1] \to C[0, 1] $ be the operator defined by \begin{gather*} A(t)u \equiv a(x, t)u'' + b(x, t)u' + c(c, t)u \quad \text{for $ u \in D(A(t)) $ where} \\ D(A(t)) \equiv \{v\in C^2[0, 1]: v'(j) = (-1)^{j} \beta_{j}(j, t)v(j), j = 0, 1 \}. \end{gather*} Following \cite{Li} and the proof for the previous case of higher space dimensions, and applying linear ordinary differential equation theory \cite{Co, LinA} and Theorem \ref{T:XB}, the next example is readily proven. Here the range condition (H2') is satisfied with $ E = C[0, 1] \supset \overline{D(A(t))} $ for all $ t $. \begin{example} \label{T:MA} \rm Equation \eqref{E:MA} has a strong solution \[ u(t) = \lim_{n \to \infty} (I - \frac{t}{n}A)^{-n}u_0 \] in $ L^2(0, 1) $ for $ u_0 \in \hat{D}(A(0)) $, and $ u(t) $ satisfies the middle equation in \eqref{E:MA} and the Lipschitz property \[ \|u(t) -u(\tau)\|_{\infty} \le k|t - \tau| \] for $ u_0 \in \hat{D}(A(0)) $ and for $ 0 \le t, \tau \le T $. \end{example} In the case that $ a, b, c, \beta_{j}$, for $j = 0, 1$, are independent of $ t $, the Theorem 1 in \cite[Page 363]{Li}, together with the Lipschitz property in the Theorem \ref{T:XA} in this paper, will readily deliver the following example. Here it is to be observed that the corresponding operator $ A $ is closed. \begin{example} \label{T:MB} \rm If the coefficient functions $ a, b, c, \beta_{j}, j = 0, 1 $ are independent of $ t $, then the equation \eqref{E:MA} has a unique classical solution \[ u(t) = \lim_{n \to \infty}(I - \frac{t}{n}A)^{-n}u_0 \] for $ u_0 \in D(A) $ with $ Au_0 \in \overline{D(A)} $. This $ u(t) $ has this property that the function $ \frac{du}{dt} $ is continuous in $ t $. Furthermore, $ u(t) $ is Lipschitz continuous in $ t $ for $ u_0 \in \hat{D}(A) $, and $ \frac{du}{dt} $ is Lipschitz continuous in $ t $ for $ u_0 \in D(A) $ with $ Au_0 \in \hat{D}(A) $, and is differentiable in $ t $ for $ u_0 \in D(A^2) $ with $ A^2u_0 \in \overline{D(A)} $. More regularity of $ \frac{du}{dt} $ can be obtained iteratively. \end{example} \begin{remark}\label{rmk3} \rm In order for $ u_0 $ to be in $ D(A^2) $, more smoothness assumptions should be imposed on the coefficient functions $ a(x), b(x), c(x) $, and $ \beta_{j}, j = 0, 1 $. \end{remark} \section{Applications to partial differential equations (II)} \label{S:E} In this section, it will be further shown that, for each concrete $ A(t) $ in Section \ref{S:D}, the corresponding quantity \[ J_{\frac{t}{n}}(i\frac{t}{n})h = [I - \frac{t}{n}A(i\frac{t}{n})]^{-1}h, \quad i = 1, 2, \ldots, n \] is the limit of a sequence where each term in the sequence is an explicit function of the solution $ \phi $ to the elliptic equation \eqref{E:TimeC} with $ \varphi \equiv 0 $. We start with the case of linear $ A(t) $ and consider the parabolic equation \eqref{E:ZA}. \begin{proposition} \label{T:NA} For $ h \in C^{\mu}(\overline{\Omega}) $, the solution $ u $ to the equation \begin{equation} \label{E:TimeXA} [I - \epsilon A(t)]u = h \end{equation} where $ 0 \le t \le T $ and $ \epsilon > 0 $, is the limit of a sequence where each term in the sequence is an explicit function of the solution $ \phi $ to the elliptic equation \eqref{E:TimeC} with $ \varphi \equiv 0 $. Here $ A(t) $ is the linear operator corresponding to the parabolic equation \eqref{E:ZA}. \end{proposition} \begin{proof} The linear operator $ A(t): D(A(t)) \subset C(\overline{\Omega}) \to C(\overline{\Omega}) $ is defined by \begin{gather*} A(t)u \equiv \sum_{i, j}a_{ij}(x, t)D_{ij} u + \sum_ib_i(x, t)D_iu + c(x, t)u \\ \text{for $ u \in D(A(t)) \equiv \{u \in C^{2 + \mu}(\overline{\Omega}): \frac{\partial u}{\partial \nu} + \beta(x, t)u = 0 \quad $ on $ \partial \Omega \} $}. \end{gather*} Solvability of \eqref{E:TimeXA} follows from \cite[Pages 128-130]{Gi2}, where the method of continuity \cite[Page 75]{Gi2} is used. By writing out fully how the method of continuity is used, it will be seen that the solution $ u $ is the limit of a sequence where each term in the sequence is an explicit function of the solution $ \phi $ to the elliptic equation \eqref{E:TimeC} with $ \varphi \equiv 0 $. To this end, set \begin{gather*} U_1 = C^{2 + \mu}(\overline{\Omega}), \quad U_2 = C^{\mu}(\overline{\Omega}) \times C^{1 + \mu}(\partial \Omega), \\ L_{\tau}u = \tau [u - \epsilon A(t)u] + (1 - \tau)(- \Delta u) \quad \text{in} \quad \Omega, \\ N_{\tau}u = \tau[\frac{\partial u}{\partial \nu} + \beta(x, t)u] + (1 - \tau) (\frac{\partial u}{\partial \nu} + u) \quad \text{on} \quad \partial \Omega, \end{gather*} where $ 0 \le \tau \le 1 $. Define the linear operator $ \pounds_{\tau}: U_1 \to U_2 $ by \[ \pounds_{\tau}u = (L_{\tau}u, N_{\tau}u) \] for $ u \in U_1 $, and assume that $ \pounds_{s} $ is onto for some $ s \in [0, 1] $. It follows from \cite[Pages 128-130]{Gi2} that \begin{align} \label{E:PA} \|u\|_{U_1} \le C \|\pounds_{\tau}u\|_{U_2}, \end{align} where the constant $ C $ is independent of $ \tau $. This implies that $ \pounds_{s} $ is one to one, and so $ \pounds_{s}^{-1} $ exists. By making use of $ \pounds_{s}^{-1} $, the equation, for $ w_0 \in U_2 $ given, \[ \pounds_{\tau}u = w_0 %(h, 0) \] is equivalent to the equation \[ u = \pounds_{s}^{-1}w_0 + (\tau - s)\pounds_{s}^{-1}( \pounds_0 - \pounds_1)u, \] from which a linear map $S: U_1 \to U_1$, \[ Su = S_{s}u \equiv \pounds_{s}^{-1}w_0 + (\tau - s)\pounds_{s}^{-1}( \pounds_0 - \pounds_1)u \] is defined. The unique fixed point $ u $ of $ S = S_{s}$ will be related to the solution of \eqref{E:TimeXA}. By choosing $ \tau \in [0, 1] $ such that \begin{align} \label{E:PB} |s - \tau| < \delta \equiv [C( \|\pounds_0\|_{U_1 \to U_2} + \|\pounds_1\|_{U_1 \to U_2})]^{-1}, \end{align} it follows that $ S = S_{s} $ is a strict contraction map. Therefore $ S $ has a unique fixed point $ w $, and the $ w $ can be represented by \[ \lim_{n \to \infty}S^{n}0 = \lim_{n \to \infty}(S_{s})^{n}0 \] because of $ 0 \in U_1 $. Thus $ \pounds_{\tau} $ is onto for $ |\tau - s| < \delta $. It follows that, by dividing $ [0, 1] $ into subintervals of length less than $ \delta $ and repeating the above arguments in a finite number of times, $ \pounds_{\tau} $ becomes onto for all $ \tau \in [0, 1] $, provided that it is onto for some $ \tau \in [0, 1] $. Since $ \pounds_0 $ is onto by the potential theory \cite[Page 130]{Gi2}, we have that $ \pounds_1 $ is also onto. Therefore, for $ w_0 = (h, 0) $, the equation \[ \pounds_1u = w_0 \] has a unique solution $ u $, and the $ u $ is the seeked solution to \eqref{E:TimeXA}. Here it is to be observed that $ \phi \equiv \pounds_0^{-1}(h, 0) $ is the unique solution $ \pounds_0^{-1}(h, \varphi) $ to the elliptic equation \eqref{E:TimeC} with $ \varphi \equiv 0 $: \begin{gather*} -\Delta v = h, \quad x \in \Omega, \\ \frac{\partial v}{\partial \nu} + v(x) = 0 \quad \text{on} \quad \partial \Omega, \end{gather*} and that \begin{gather*} S0 = S_00 = \pounds_0^{-1}(h, 0), \\ S^20 = (S_0)^20 = \pounds_0^{-1}(h, 0) + \pounds_0^{-1}[|\tau -0| (\pounds_0 - \pounds_1)\pounds_0^{-1}(h, 0)], \\ \dots. \end{gather*} The proof is complete. \end{proof} \begin{remark} \label{rmk4}\rm $\bullet$ The solution $ u $ is eventually represented by %this integral formula \[ u(x) = \pounds_0^{-1}H((h, 0)), %\int_{\Omega}G(x, y)h(y)(y) \, dy \] where $ H((h, 0)) $ is a convergent series in which each term is basically obtained by, repeatedly, applying the linear operator $ (\pounds_0 - \pounds_1)\pounds_0^{-1} $ to $ (h, 0) $ for a certain number of times. $\bullet$ The quantity $ \pounds_0^{-1}(h, \varphi) $, for each $ (h, \varphi) \in U_2 $ given, can be computed numerically and efficiently by the boundary element methods \cite{Gau,Sch}, if the dimension of the space variable $ x $ equals $ 2 $ or $ 3 $. $\bullet$ The constant $ C $ above in \eqref{E:PA} and \eqref{E:PB} depends on $ n, \mu, \lambda_{\rm min}, \Omega $, and on the coefficient functions $ a_{ij}(x, t), b_i(x, t), c(x, t), \beta(x, t) $, and is not known explicitly \cite{Gi2}. % \cite[First Edition, Page 134]{Gi2}), Therefore, the corresponding $ \delta $ cannot be determined in advance, and so, when dealing with the elliptic equation \eqref{E:TimeXA} in Proposition \ref{T:NA} numerically, it is more possible, by choosing $ \tau \in [0, 1] $ such that $ |s - \tau| $ is smaller, that the sequence $ S^{n}0 $ will converge, for which $|s - \tau| < \delta$ occurs. \end{remark} Next, we extend the above techniques to the case of nonlinear $ A(t) $, and consider the nonlinear parabolic equation \eqref{E:XA}; more work is required in this case. \begin{proposition} \label{T:OA} For $ h \in C^{\mu}(\overline{\Omega}) $, the solution $ u $ to the equation \eqref{E:TimeXA} \[ [I - \epsilon A(t)]u = h \] where $ 0 \le t \le T $ and $ \epsilon > 0 $, is the limit of a sequence where each term in the sequence is an explicit function of the solution $ \phi $ to the elliptic equation \eqref{E:TimeC} with $ \varphi \equiv 0 $. Here $ A(t) $ is the nonlinear operator corresponding to the parabolic equation \eqref{E:XA}, and $ \beta(x, t, 0) \equiv 0 $ is assumed additionally. \end{proposition} \begin{proof} The nonlinear operator $ A(t): D(A(t)) \subset C(\overline{\Omega}) \to C(\overline{\Omega}) $ is defined by \begin{gather*} D(A(t)) = \{u \in C^{2 + \mu}(\overline{\Omega}): \frac{\partial u}{\partial \nu} + \beta(x, t, u) = 0 \quad \text{on} \quad \partial \Omega\}, \\ A(t)u = \alpha(x, t, Du)\Delta u + g(x, t, u, Du), \quad u \in D(A(t)). \end{gather*} Equation \eqref{E:TimeXA} with the nonlinear $ A(t) $ has been solved in \cite{Lin2}, but here the proof will be based on the contraction mapping theorem as in the proof of Proposition \ref{T:NA}. To this end, set \begin{gather*} U_1 = C^{2 + \mu}(\overline{\Omega}), \\ U_2 = C^{\mu}(\overline{\Omega}) \times C^{1 + \mu}(\partial \Omega), \\ L_{\tau}u = \tau[u - \epsilon A(t)u] + (1 - \tau)(u - \Delta u), \quad x \in \Omega, \\ N_{\tau}u = \tau[\frac{\partial u}{\partial \nu} + \beta(x, t, u)] + (1 - \tau)( \frac{\partial u}{\partial \nu} + u) \quad \text{on } \partial \Omega, \end{gather*} where $ 0 \le \tau \le 1 $. Define the nonlinear operator $ \pounds_{\tau}: U_1 \to U_2 $ by \[ \pounds_{\tau}u = (L_{\tau}u, N_{\tau}u) \] for $ u \in U_1 $, and assume that $ \pounds_{s} $ is onto for some $ s \in [0, 1]$. As in proving that $ A(t) $ satisfies the dissipativity (H1) where the maximum principle was used, $ \pounds_{s} $ is one to one, and so $ \pounds_{s}^{-1} $ exists. By making use of $ \pounds_{s}^{-1} $, the equation, for $ w_0 \in U_2 $ given, $\pounds_{\tau}u = w_0$ is equivalent to the equation \[ u = \pounds_{s}^{-1}[w_0 + (\tau - s)(\pounds_0 - \pounds_1)u], \] from which a nonlinear map \begin{gather*} S: U_1 \to U_1, \\ Su = S_{s}u \equiv \pounds_{s}^{-1}[w_0 + (\tau - s)(\pounds_0 - \pounds_1)u] \quad \text{for $ u \in U_1 $} \end{gather*} is defined. The unique fixed point of $ S = S_{s} $ will be related to the solution of \eqref{E:TimeXA} with nonlinear $ A(t) $. By restricting $ S = S_{s} $ to the closed ball of the Banach space $ U_1 $, \[ B_{s, r, w_0} \equiv \{u \in U_1: \|u - \pounds_{s}^{-1}w_0\|_{C^{2 + \mu}} \le r > 0\}, \] and choosing small enough $ |\tau - s| $, we will show that $ S = S_{s} $ leaves $ B_{s, r, w_0} $ invariant. This will be done by the following steps 1 to 4. \textbf{Step 1.} It follows as in \cite[Pages 265-266]{Lin2} that, for $ \pounds_{\tau} v = (f, \chi) $, \begin{equation} \label{E:Time1A} \begin{gathered} \|v\|_{\infty} \le k_{\{\|f\|_{\infty}, \|\chi\|_{C(\partial \Omega)}\}}, \\ \|Dv\|_{C^{\mu}} \le k_{\{\|v\|_{\infty}\}}\|Dv\|_{\infty} + k_{\{\|v\|_{\infty}, \|f\|_{\infty}, \|\chi\|_{C(\partial \Omega)}\}}, \\ \|v\|_{C^{1 + \mu}} \le k_{\{\|\chi\|_{C(\partial \Omega)}, \|f\|_{\infty}\}}, \\ \|v\|_{C^{2 + \mu}} \le K \|\pounds_{\tau}v\|_{U_2} = K \|\pounds_{\tau}v\|_{C^{\mu}(\overline{\Omega}) \times C^{1 + \mu}(\partial \Omega)} \end{gathered} \end{equation} where $ k_{\{\|f\|_{\infty}\}} $ is a constant depending on $ \|f\|_{\infty} $, and similar meaning is defined for other constants $ k $'s; further, $ K $ is independent of $ \tau $, but depends on $ n, \delta_0, \mu, \Omega $, and on the coefficient functions $ \alpha(x, t, Dv), g(x, t, v, Dv), \beta(x, t, v) $, which have incorporated the dependence of $ v, Dv $ into $ \|\pounds_{\tau}v\|_{U_2} $. \textbf{Step 2.} It is readily seen that, for $ v \in C^{2 + \mu}(\overline{\Omega}) $ with $ \|v\|_{C^{2 + \mu}} \le R > 0 $, we have \begin{equation} \label{E:Time2A} \|\pounds_{\tau}v\|_{U_2} \le k_{\{R\}}\|v\|_{C^{2 + \mu}}, \end{equation} where $ k_{\{R\}} $ is independent of $ \tau $. \textbf{Step 3.} It will be shown that, if \[ \|u\|_{C^{2 + \mu}} \le R, \quad \|v\|_{C^{2 + \mu}} \le R > 0, \] then \begin{equation} \label{E:Time3A} \|\pounds_{\tau}u - \pounds_{\tau}v\|_{U_2} \le k_{\{R\}}\|u -v\|_{C^{2 + \mu}}. \end{equation} It will be also shown that, if \[ \pounds_{\tau}u = (f, \chi_1), \quad \pounds_{\tau}v = (w, \chi_2), \] then \begin{equation} \label{E:Time3B} \begin{aligned} \|u - v\|_{C^{2 + \mu}} &\le k_{\{\|\pounds_{\tau}u\|_{U_2}, \|\pounds_{\tau}v\|_{U_2}\}} [\|f - w\|_{C^{\mu}} + \|\chi_1 - \chi_2\|_{C^{1 + \mu}}] \\ &= k_{\{\|\pounds_{\tau}u\|_{U_2}, \|\pounds_{\tau}v\|_{U_2}\}}\|\pounds_{\tau}u - \pounds_{\tau}v\|_{U_2}. \end{aligned} \end{equation} Here $ K_{\{R\}} $ and $ K_{\{\|\pounds_{\tau}u\|_{U_2}, \|\pounds_{\tau}v\|_{U_2}\}} $ are independent of $ \tau $. Using the mean value theorem, we have that \begin{gather*} \begin{aligned} f - w &= L_{\tau}u - L_{\tau}v \\ &= (u - v) - (1 - \tau)\Delta (u - v) - \tau \epsilon [\alpha \Delta(u - v) \\ &\quad + \alpha_{p}(x,t,p_1)(Du - Dv)\Delta v + g_{p}(x, t, u, p_2)(Du - Dv) \\ &\quad + g_{z}(x, t, z_1, Dv)(u - v)], \quad x \in \Omega, \end{aligned} \\ \frac{\partial (u - v)}{\partial \nu} + [\beta(x, t, u) - \beta(x, t, v)] = \chi_1 - \chi_2 \quad \text{on } \partial \Omega , \end{gather*} were $ p_1, p_2 $ are some functions between $ Du $ and $ Dv $, and $ z_1 $ is some function between $ u $ and $ v $. It follows as in \eqref{E:Time2A} that \[ \|\pounds_{\tau}u - \pounds_{\tau}v\|_{U_2} \le k_{\{R\}}\|u - v\|_{C^{2 + \mu}}, \] which is the desired estimate. On the other hand, the maximum principle yields \[ \|u - v\|_{\infty} \le k_{\{\|f - w\|_{\infty}, \|\chi_1 - \chi_2\|_{\infty}\}} \] and \eqref{E:Time1A} yields \[ \|u\|_{C^{2 + \mu}} \le K \|\pounds_{\tau}u\|_{U_2}, \quad \|v||_{C^{2 + \mu}} \le K \|\pounds_{\tau}v\|_{U_2}. \] Thus, it follows from the Schauder global estimate \cite{Gi2} that \[ \|u - v\|_{C^{2 + \mu}} \le k_{\{\|\pounds_{\tau}u\|_{U_2}, \|\pounds_{\tau}\|_{U_2}\}} \|\pounds_{\tau}u - \pounds_{\tau}v\|_{U_2}, \] which is the other desired estimate. \textbf{Step 4.} Consequently, for $ u \in B_{s, r, w_0} $, we have that, by \eqref{E:Time1A}, \begin{equation} \label{E:Time4A} \|u\|_{C^{2 + \mu}} \le r + \|\pounds_{s}^{-1}w_0\|_{C^{2 + \mu}} \le r + K\|w_0\|_{U_2} \equiv R_{\{r, \|w_0\|_{U_2}\}}, \end{equation} and that \begin{align*} & \|Su - \pounds_{s}^{-1}w_0\|_{C^{2 + \mu}} \\ &\le k_{\{\|w_0\|_{U_2}, \|w_0 + (\tau - s) (\pounds_0 - \pounds_1)u\|_{U_2}\}} \|(\tau - s)(\pounds_0 - \pounds_1)u\|_{U_2} \quad \text{by \eqref{E:Time3B}} \\ &\le |\tau - s|k_{\{\|w_0\|_{U_2}, R_{\{r, \|w_0\|_{U_2}\}}\}} \quad \text{by \eqref{E:Time2A} and \eqref{E:Time4A}}. \end{align*} Here the constant $ k_{\{\|w_0\|_{U_2}, R_{\{r, \|w_0\|_{U_2}\}}\}} $ when $ w_0 $ given and $ r $ chosen, is independent of $ \tau $ and $ s $. Hence, by choosing some sufficiently small $ \delta_1 > 0 $, there results \[ S = S_{s}: B_{s, r, w_0} \subset U_1 \to B_{s, r, w_0} \subset U_1 \] for $ |\tau -s| < \delta_1 $; that is, $ B_{s, r, w_0} $ is left invariant by $ S = S_{s}$. Next, it will be shown that, for small $ |\tau - s| $, $ S = S_{s} $ is a strict contraction on $ B_{s, r, w_0} $, from which $ S = S_{s} $ has a unique fixed point. Because, for $ u, v \in B_{s, r, w_0} $, \[ \|u\|_{C^{2 + \mu}} \le R_{\{r, \|w_0\|_{U_2}\}}, \quad \|v\|_{C^{2 + \mu}} \le R_{\{r, \|w_0\|_{U_2}\}} \quad \text{by \eqref{E:Time4A}}, \] it follows that, by \eqref{E:Time2A}, \begin{equation} \label{E:Time5A} \begin{gathered} \|w_0 + (\tau - s)(\pounds_0 - \pounds_1)u\|_{U_2} \le k_{\{\|w_0\|_{U_2}, R_{\{r, \|w_0\|_{U_2}\}}\}}, \\ \|w_0 + (\tau - s)(\pounds_0 - \pounds_1)v\|_{U_2} \le k_{\{\|w_0\|_{U_2}, R_{\{r, \|w_0\|_{U_2}\}}\}}, \end{gathered} \end{equation} and that, by \eqref{E:Time3A}, \begin{equation} \label{E:Time5B} \|(\tau - s)[(\pounds_0 - \pounds_1)u - (\pounds_0 - \pounds_1)v]\|_{U_2} \le |\tau - s|k_{\{R_{\{r, \|w_0\|_{U_2}\}}\}}\|u - v\|_{C^{2 + \mu}}. \end{equation} Therefore, on account of \eqref{E:Time3B}, \eqref{E:Time5A}, and \eqref{E:Time5B}, we obtain \[ \|Su - Sv\|_{C^{2 + \mu}} \\ \le |\tau - s|k_{\{R_{\{r, \|w_0\|_{U_2}\}}, \|w_0\|_{U_2}\}} k_{\{ R_{\{r, \|w_0\|_{U_2}\}}\}} \|u - v\|_{C^{2 + \mu}}. \] Here the constant $ k_{\{R_{\{r, \|w_0\|_{U_2}\}}, \|w_0\|_{U_2}\}} k_{\{R_{\{r, \|w_0\|_{U_2}\}}\}} $ when $ w_0 $ given and $ r $ chosen, is independent of $ \tau $ and $ s $. Hence, by choosing some sufficiently small $ \delta_2 > 0 $, it follows that \[ S = S_{s}: B_{s, r, w_0} \to B_{s, r, w_0} \] ia a strict contraction for \[ |\tau - s| < \delta_2 \le \delta_1. \] Furthermore, the unique fixed point $ w $ of $ S = S_{s} $ can be represented by \[ \lim_{n \to \infty}S^{n}0 = \lim_{n \to \infty}(S_{s})^{n}0 \] if $ \beta(x, t, 0) \equiv 0 $ and if $ r= r_{\{K\|w_0\|_{U_2}\}} $ is chosen such that \begin{equation} \label{E:TimeQA} r = r_{\{K\|w_0\|_{U_2}\}} \ge K\|w_0\|_{U_2} \ge \|\pounds_{s}^{-1}w_0\|_{C^{2 + \mu}} \end{equation} (by \eqref{E:Time4A}); this is because $ 0 $ belongs to $ B_{s, r, w_0} $ in this case. Thus $ \pounds_{\tau} $ is onto for $ |\tau - s|< \delta_2$. It follows that, by dividing $ [0, 1] $ into subintervals of length less than $ \delta_2 $ and repeating the above arguments in a finite number of times, $ \pounds_{\tau} $ becomes onto for all $ \tau \in [0, 1] $, provided that it is onto for some $ \tau \in [0, 1]$. Since $ \pounds_0 $ is onto by linear elliptic theory \cite{Gi2}, we have that $ \pounds_1 $ is also onto. Therefore, the equation, for $ w_0 = (h, 0) $, \[ \pounds_1u = w_0 \] has a unique solution $ u $, and the $ u $ is the sought solution to \eqref{E:TimeXA}. Here it is to be observed that $ \psi \equiv \pounds_0^{-1}(h, 0) $ is the unique solution to the elliptic equation \begin{gather*} v -\Delta v = h, \quad x \in \Omega, \\ \frac{\partial v}{\partial \nu} + v(x) = 0 \quad \text{on} \quad \partial \Omega, \end{gather*} and that, by Proposition \ref{T:NA}, $ \psi $ is the limit of a sequence where each term in the sequence is an explicit function of the solution $ \phi $ to the elliptic equation \eqref{E:TimeC} with $ \varphi \equiv 0 $. It is also to be observed that \begin{gather*} S0 = S_00 = \pounds_0^{-1}(h, 0), \\ S^20 = (S_0)^20 = \pounds_0^{-1}[(h, 0) + |\tau -0| (\pounds_0 - \pounds_1)\pounds_0^{-1}(h, 0)], \\ \dots, \end{gather*} where $ (\pounds_0 - \pounds_1)\pounds_0^{-1} $ is a nonlinear operator. The proof is complete. \end{proof} \begin{remark} \label{rmk5} \rm The constants $ k_{\{R_{\{r, \|w_0\|_{U_2}\}}\}} $ and $ k_{\{R_{\{r, \|w_0\|_{U_2}\}}, \|w_0\|_{U_2}\}} k_{\{ R_{\{r, \|w_0\|_{U_2}\}}\}} $, when $ w_0 $ is given and when $ r $ is chosen and conditioned by \eqref{E:TimeQA}, is not known explicitly, and so the corresponding $ \delta_2 $ cannot be determined in advance. Hence, when dealing with the elliptic equation \eqref{E:TimeXA} in Proposition \ref{T:OA} numerically, it is more possible, by choosing $ \tau \in [0, 1] $ such that $ |\tau - s| $ is smaller, that the sequence $ S^{n}0 $ will converge, for which $|\tau - s| < \delta_2 \le \delta_1$ occurs. \end{remark} Finally, what will be considered is the linear equation \eqref{E:MA} of space dimension $1$. \begin{proposition} \label{T:PA} For $ h \in C[0, 1] $, the solution $ u $ to the equation \eqref{E:TimeXA} \[ [I - \epsilon A(t)]u = h \] where $ 0 \le t \le T $ and $ \epsilon > 0 $, is the limit of a sequence where each term in the sequence is an explicit function of the solution $ \phi $ to the ordinary differential equation \begin{equation} \label{E:TimeYA} \begin{gathered} v - v'' = h \quad x \in (0, 1), \\ v'(j) = (-1)^{j}v(j), \quad j = 0, 1. \end{gathered} \end{equation} Here $ A(t) $ is the linear operator corresponding to the parabolic equation \eqref{E:MA}. \end{proposition} \begin{proof} The linear operator $ A(t): D(A(t)) \subset C[0, 1] \to C[0, 1] $ is defined by \begin{gather*} A(t)u \equiv a(x, t)u'' + b(x, t)u' + c(x, t)u \quad \text{for $ u \in D(A(t)) $ where} \\ D(A(t)) \equiv \{v\in C^2[0, 1]: v'(j) = (-1)^{j} \beta_{j}(j, t)v(j), \quad j = 0, 1 \}. \end{gather*} The contraction mapping theorem in the proof of Proposition \ref{T:NA} will be used in order to solve the equation \eqref{E:TimeXA}. To this end, set, for $ 0 \le \tau \le 1 $, \begin{gather*} U_1 = C^2[0, 1], \quad U_2 = C[0, 1] \times {\mathbb R}^2, \\ L_{\tau}u = \tau[u - \epsilon A(t)u] + (1 - \tau)(u - u''), \\ \begin{aligned} N_{\tau}u &= \Big(\tau [u'(0) - \beta_0(0, t)u(0)] + (1 - \tau)[u'(0) - u(0)], \\ &\quad \tau [u'(1) + \beta_1(1, t)u(1)] + (1 - \tau)[u'(1) + u(1)]\Big). \end{aligned} \end{gather*} Define the linear operator $\pounds_{\tau} : U_1 \to U_2 $ by \[ \pounds_{\tau}u = (L_{\tau}u, N_{\tau}u) \] for $ u \in U_1 $, and assume that $ \pounds_{s} $ is onto for some $ s \in [0, 1] $. The following will be readily derived. $\bullet$ For $ u \in C^2[0, 1] $, we have \begin{equation} \label{E:TimeWB} \|\pounds_{\tau}u\|_{U_2} = \|\pounds_{\tau}u\|_{C[0, 1] \times {\mathbb R}^2} \le k_{\{a, b, c, \beta_0, \beta_1\}}\|u\|_{C^2}, \end{equation} where $ k_{\{a, b, c, \beta_0, \beta_1\}} $ is independent of $ \tau $, and can be computed, depending on the given $ a(x, t), b(x, t), c(x, t), \beta_0(0, t) $, and $ \beta_1(1, t) $. $\bullet$ For $ \pounds_{\tau}u = (h, (r, s)) $, the maximum principle shows \[ \|u\|_{\infty} \le \|h\|_{\infty} + |\frac{r}{\beta_0(0, t)}| + |\frac{s}{\beta_1(1, t)}|. \] This, together with the known interpolation inequality \cite[Page 65]{Gol} or \cite[Pages 7-8]{Mi} \[ \|u'\|_{\infty} \le \frac{2}{\lambda} \|u\|_{\infty} + \frac{\lambda}{2}\|u''\|_{\infty} \] for any $ \lambda > 0 $, applied to $ \pounds_{\tau}u = (h, (r, s)) $, it follows that, by choosing small enough $ \lambda = \lambda_1 $, \begin{equation} \label{E:TimeWA} \|u\|_{C^2} \le k_{\{\lambda_1, a, b, c, \beta_0, \beta_1\}}(\|h\|_{\infty} + |r| + |s|) = k_{\{\lambda_1, a, b, c, \beta_0, \beta_1\}}\|\pounds_{\tau} u\|_{U_2}, \end{equation} where $ k_{\{\lambda_1, a, b, c, \beta_0, \beta_1\}} $ is independent of $ \tau $ and can be computed explicitly. \medskip On account of the estimate \eqref{E:TimeWA}, $ \pounds_{s} $ is one to one, and so $ \pounds_{s}^{-1} $ exists. Thus, making use of $ \pounds_{s}^{-1} $, the equation, for $ w_0 \in U_2 $ given, $\pounds_{\tau}u = w_0 $ is equivalent to the equation \[ u = \pounds_{s}^{-1}w_0 + (\tau - s)\pounds_{s}^{-1}( \pounds_0 - \pounds_1)u, \] from which a linear map \begin{gather*} S: U_1 = C^2[0, 1] \to U_1 = C^2[0, 1], \\ Su = S_{s}u \equiv \pounds_{s}^{-1}w_0 + (\tau - s)\pounds_{s}^{-1}(\pounds_0 - \pounds_1)u, \quad u \in U_1 \end{gather*} is defined. Because of \eqref{E:TimeWA} and \eqref{E:TimeWB}, it follows that this $ S $ is a strict contraction if \[ |\tau - s| < \delta = [k_{\{\lambda_1, a, b, c, \beta_0, \beta_1\}}2k_{\{a, b, c, \beta_0, \beta_1\}}]^{-1}. \] The rest of the proof will be the same as that for Proposition \ref{T:NA}, in which the equation, for $ w_0 = (h, (0, 0)) $, \[ \pounds_1u = w_0 \] has a unique solution $ u $, and the $ u $ is the sought solution. \end{proof} \begin{remark} \label{rmk6} \rm $\bullet$ The $ \delta = [k_{\{\lambda_1, a, b, c, \beta_0, \beta_1\}}2k_{\{a, b, c, \beta_0, \beta_1\}}]^{-1} $ in the above proof of Proposition \ref{T:PA} can be computed explicitly. $\bullet$ The quantity $ \pounds_0^{-1}(h, (0, 0)) $ is represented by the integral \[ \pounds_0^{-1}(h, (0, 0)) = \int_0^{1}g_0(x, y)h(y) \, dy, \] where $ g_0(x, y) $ is the Green function associated with the boundary value problem \begin{gather*} u - u'' = h \quad \text{in} \quad (0, 1), \\ u'(j) = (-1)^{j}u(j), \quad j = 0, 1. \end{gather*} This $ g_0(x, y) $ is known explicitly by a standard formula. $\bullet$ As before, we have \begin{gather*} S0 = S_00 = \pounds_0^{-1}(h, (0, 0)), \\ S^20 = S_0^20 = \pounds_0^{-1}(h, (0, 0)) + \pounds_0^{-1}[|\tau - 0|( \pounds_0 - \pounds_1)\pounds_0^{-1}(h, (0, 0))], \\ \dots. \end{gather*} \end{remark} \section{Appendix} \label{S:VaryB} In this section, the Proposition \ref{P:VaryB} in Section \ref{S:B} will be proved, using the theory of difference equations. We now introduce its basic theory \cite{Mic}. Let \[ \{b_{n}\} = \{b_{n}\}_{n \in \{0\}\cup {\mathbb N}} = \{ b_{n} \}_{n = 0}^{\infty} \] be a sequence of real numbers. For such a sequence $ \{b_{n}\} $, we further extend it by defining \[ b_{n} = 0 \quad \text{if $ n = -1, -2, \ldots. $}. \] The set of all such sequences $ \{b_{n}\} $'s will be denoted by $ S $. Thus, if $ \{a_{n}\} \in S $, then $0 = a_{-1} = a_{-2} = \dots$. Define a right shift operator $ E : S \to S $ by \[ E\{b_{n}\} = \{b_{n + 1}\} \quad \text{for $ \{b_{n}\} \in S $}. \] For $ c \in {\mathbb R} $ and $ c \ne 0 $, define the operator $ (E - c)^{*} : S \to S $ by \[ (E - c)^{*}\{b_{n}\} = \{c^{n}\sum_{i = 0}^{n - 1} \frac{b_i}{c^{i + 1}}\} \] for $ \{b_{n}\} \in S $. Here the first term on the right side of the equality, corresponding to $ n = 0 $, is zero. Define, for $ \{b_{n}\} \in S $, \begin{gather*} (E - c)^{i *}\{b_{n}\} = [(E - c)^{*}]^{i}\{b_{n}\}, \quad i = 1, 2, \ldots; \\ (E - c)^{0}\{b_{n}\} = \{b_{n}\}. \end{gather*} It follows that $ (E - c)^{*} $ acts approximately as the inverse of $ (E - c) $ in this sense \[ (E - c)^{*}(E - c)\{b_{n}\} = \{b_{n} - c^{n}b_0\}. \] Next we extend the above definitions to doubly indexed sequences. For a doubly indexed sequence $ \{\rho_{m, n}\} = \{\rho_{m, n}\}_{m, n = 0}^{\infty} $ of real numbers, let \[ E_1\{\rho_{m, n} \} = \{\rho_{m + 1, n} \}; \quad E_2\{\rho_{m, n} \} = \{\rho_{m, n + 1} \}. \] Thus, $ E_1 $ and $ E_2 $ are the right shift operators, which acts on the first index and the second index, respectively. It is easy to see that \[ E_1E_2 \{\rho_{m, n}\} = E_2E_1 \{\rho_{m, n}\}\,. \] Before we prove the Proposition \ref{P:VaryB}, we need the following four lemmas, which are proved in \cite{Lin1,Lin0,Lin1,Lin4}, respectively. \begin{lemma} \label{L:VaryA} If \eqref{E:VaryMain} is true, then \begin{equation} \label{E:VaryMain3} \begin{aligned} \{a_{m, n}\} &\leq (\alpha \gamma (E_2 - \beta \gamma)^{*})^{m} \{a_{0, n}\} + \sum_{i = 0}^{m - 1}(\gamma \alpha(E_2 - \gamma \beta)^{*})^{i} \{(\gamma \beta)^{n}a_{m - i, 0}\} \\ &\quad + \sum_{j = 1}^{m}(\gamma \alpha)^{j - 1}((E_2 - \gamma \beta)^{*})^{j} \{r_{m + 1 - j, n + 1}\}, \end{aligned} \end{equation} where $ r_{m, n} = K_4\mu \rho(|n\mu - m\lambda|) $. \end{lemma} \begin{lemma} \label{L:VaryB} The following equality holds: \[ ((E_2 - \beta \gamma)^{*})^{m}\{n \gamma^{n}\} = \{\frac{n \gamma^{n}}{\alpha^{m}} \frac{1}{\gamma^{m}} - \frac{m \gamma^{n}}{\alpha^{m + 1}}\frac{1}{\gamma^{m}} + \Big(\sum_{i = 0}^{m - 1} \binom{n}{i}\frac{\beta^{n - i}}{\alpha^{m + 1 - i}} (m - i)\frac{1}{\gamma^{m}}\Big) \gamma^{n}\}. \] Here $ \gamma, \alpha $ and $ \beta $ are defined in Proposition \ref{P:XA}. \end{lemma} \begin{lemma} \label{L:VaryC} The following equality holds: \[ ((E - \beta \gamma)^{*})^{j}\{\gamma^{n} \} = \big\{\Big(\frac{1}{\alpha^{j}} - \frac{1}{\alpha^{j}}\sum_{i = 0}^{j - 1}\binom{n}{i} \beta^{n - i}\alpha^{i}\Big)\gamma^{n - j}\big\} = \big\{\Big(\frac{1}{\alpha^{j}} \sum_{i = j}^{n}\beta^{n - i}\alpha^{i}\Big)\gamma^{n - j}\big\} \] for $ j \in \mathbb{N} $. Here $ \gamma, \alpha $ and $ \beta $ are defined in Proposition \ref{P:XA} \end{lemma} \begin{lemma} \label{L:VaryD} The following equality holds: \begin{align*} (E - \beta \gamma)^{m *}\{n^2\gamma^{n}\} &= \gamma^{n - m}\{\frac{n^2}{\alpha^{m}} - \frac{(2m)n}{\alpha^{m + 1}} + (\frac{m(m - 1)}{\alpha^{m + 2}} + \frac{m(1 + \beta)}{\alpha^{m + 2}}) \\ &\quad - \sum_{j = 0}^{m - 1}\big(\frac{(m - j)(m - j - 1)} {\alpha^{m - j + 2}} + \frac{(m - j)(1 + \beta)}{\alpha^{m - j + 2}}\big)\binom{n}{j} \beta^{n - j}\}. \end{align*} Here $ \gamma, \alpha $, and $ \beta $ are defined in Proposition \ref{P:XA}. \end{lemma} \begin{proof}[Proof of Proposition \ref{P:VaryB}] If $ S_2(\mu) = \emptyset $, then \eqref{E:VaryMain2} is true, and so \[ a_{m, n} \le L(K_2)|n\mu - m\lambda|. \] If $ S_1(\mu) = \emptyset $, then \eqref{E:VaryMain} is true, and so the inequality \eqref{E:VaryMain3} follows by Lemma \ref{L:VaryA}. Since, by Proposition \ref{P:VaryA}, \begin{gather*} a_{0, n} \le K_1\gamma^{n}(2n + 1)\mu; \\ a_{m - i, 0} \le K_1(1 - \lambda \omega)^{-m}[2(m - i) + 1]\lambda; \end{gather*} it follows from Lemma \ref{L:VaryC} and from the Proposition 3 and its proof of \cite[Pages 115-116]{Lin0} that the first two terms of the right side of the inequality \eqref{E:VaryMain3} is less than or equal to \[ c_{m, n} + s_{m, n} + f_{m, n}. \] We finally estimate the third term, denoted by $ \{t_{m, n}\} $, of the right-hand side of \eqref{E:VaryMain3}. Observe that, using the subadditivity of $ \rho $, we have \begin{align*} \{t_{m, n}\} &\leq \sum_{j = 1}^{m}(\gamma \alpha)^{j - 1} (E_2 - \gamma \beta)^{j *}K_4 \mu \{\rho(|\lambda - \mu|) + \rho(|n \mu -m \lambda + j \lambda|)\} \\ &\leq \sum_{j = 1}^{m}(\gamma \alpha)^{j - 1}(E_2 - \gamma \beta)^{ j *}K_4 \mu \{\gamma^{n}\rho(|\lambda - \mu|) + \gamma^{n}\rho(|n \mu - (m - j)\lambda|)\} \\ &\equiv \{u_{m, n}\} + \{v_{m, n}\}, \end{align*} where $ \gamma = ( 1 - \mu \omega)^{-1} > 1 $. It follows from Lemma \ref{L:VaryC} that \begin{align*} \{u_{m, n}\} &\le \{K_4\mu\gamma^{n}\rho(|\lambda - \mu|)\sum_{j = 1}^{m} \alpha^{j - 1}\frac{1}{\alpha^{j}}\sum_{i = 1}^{n}\binom{n}{i}\beta^{n - i} \alpha^{i}\} \\ &\le \{K_4\gamma^{n}\rho(|\lambda - \mu|)\mu \frac{1}{\alpha}m\} = \{K_4\rho(|\lambda - \mu|)\gamma^{n}(m\lambda)\}. \end{align*} To estimate $ \{v_{m, n}\} $, as in Crandall-Pazy \cite[page 68]{Cran}, let $ \delta > 0 $ be given and write \[ \{v_{m, n}\} = \{I^{(1)}_{m, n}\} + \{I^{(2)}_{m, n}\}, \] where $ \{I^{(1)}_{m, n}\} $ is the sum over indices with $ |n \mu - (m - j)\lambda| < \delta $, and $ \{I^{(2)}_{m, n}\} $ is the sum over indices with $ |n \mu - (m - j)\lambda| \geq \delta $. As a consequence of Lemma \ref{L:VaryC}, we have \begin{align*} \{I^{(1)}_{m, n}\} &\leq \{K_4 \mu \gamma^{n}\rho(\delta)\sum_{j = 1} ^{m}\alpha^{j - 1}\frac{1}{\alpha^{j}}\sum_{i = j}^{n}\binom{n}{i} \beta^{n - i}\alpha^{i}\} \\ &\leq \{K_4 \rho(\delta) \mu \gamma^{n}m \frac{1}{\alpha}\} = \{K_4 \rho(\delta)\gamma^{n}m \lambda \}. \end{align*} On the other hand, \begin{align*} \{I^{(2)}_{m, n}\} &\leq K_4\mu \rho(T)\sum_{j = 1}^{m}(\gamma \alpha)^{j - 1} (E_2 - \gamma \beta)^{j *}\{\gamma^{n}\} \\ &\leq K_4 \mu \rho(T)\sum_{j = 1}^{m}(\gamma \alpha)^{j - 1} (E_2 - \gamma \beta)^{j *} \{\gamma^{n}\frac{[n \mu - (m - j)\lambda]^2}{\delta^2}\}, \end{align*} which will be less than or equal to \[ \{K_4\frac{\rho(T)}{\delta^2}\gamma^{n}[(m\lambda) (n\mu -m\lambda)^2 + (\lambda - \mu)\frac{m(m + 1)}{2}\lambda^2]\} \] and so the proof is complete. This is because of the calculations, where Lemmas \ref{L:VaryB}, \ref{L:VaryC}, and \ref{L:VaryD} were used: \begin{gather*} [n\mu - (m - j)\lambda]^2 = n^2\mu^2 - 2(n\mu)(m - j)\lambda + (m - j)^2\lambda^2; \\ \begin{aligned} &\sum_{j = 1}^{m}(\gamma \alpha)^{j - 1} (E_2 - \gamma \beta)^{j *}\{\gamma^{n}n^2\}\mu^2 \\ & = \gamma^{n - 1}\sum_{j = 1}^{m}\alpha^{j - 1} \{\frac{n^2}{\alpha^{j}} - \frac{2jn}{\alpha^{j + 1}} +[\frac{j(j - 1)}{\alpha^{j + 2}} + \frac{j(1 + \beta)}{\alpha^{j + 2}}] \\ &\quad - \sum_{i = 0}^{j - 1}[\frac{(j - i)(j - i - 1)}{ \alpha^{j - i + 2}} + \frac{(j - i)(1 + \beta)}{\alpha^{j - i + 2}}] \binom{n}{i}\beta^{n - i}\}\mu^2 \\ & \le \gamma^{n}\sum_{j = 1}^{m}\{\frac{n^2}{\alpha} - \frac{2jn}{\alpha^2} + [\frac{j(j - 1)}{\alpha^{3}} + \frac{j(1 + \beta)}{\alpha^{3}}]\}\mu^2, \end{aligned} \end{gather*} where the negative terms associated with $ \sum_{i = 0}^{j - 1} $ were dropped; \begin{align*} &\sum_{j = 1}^{m}(\gamma \alpha)^{j - 1} (E_2 - \gamma \beta)^{j *}\{\gamma^{n}n\}[2\mu(m - j)\lambda](-1) \\ &= \sum_{j = 1}^{m}(\gamma \alpha)^{j - 1}\{\gamma^{n - j}[ \frac{n}{\alpha^{j}} - \frac{j}{\alpha^{j + 1}} \\ &\quad + \sum_{i = 0}^{j - 1}\binom{n}{i} \beta^{n - i}\alpha^{i - j - 1}(j - i)]\}[2\mu(m - j)\lambda](-1) \\ &\le \sum_{j = 1}^{m}\gamma^{n}\{\frac{n}{\alpha} - \frac{j}{\alpha^2}\}[2\mu(m - j)\lambda](-1), \\ & = \sum_{j = 1}^{m}\gamma^{n}\alpha^{-1} \{- 2(n\mu)(m\lambda) + j[2n\mu \lambda + \frac{2\mu}{\alpha}(m\lambda)] - j^2(\frac{2\mu \lambda}{\alpha})\}; \end{align*} where the negative terms associated with $ \sum_{i = 0}^{j - 1}$ were dropped; \begin{align*} &\sum_{j = 1}^{m}(\gamma \alpha)^{j - 1} (E_2 - \gamma \beta)^{j *}\{\gamma^{n}\}(m - j)^2\lambda^2 \\ &= \sum_{j = 1}^{m}(\gamma \alpha)^{j - 1} \{\gamma^{n - j}[\frac{1}{\alpha^{j}} - \frac{1}{\alpha^{j}} \sum_{i = 0}^{j - 1}\binom{n}{i}\beta^{n - i}\alpha^{i}]\} (m - j)^2\lambda^2 \\ & \le \sum_{j = 1}^{m}\gamma^{n}\alpha^{-1}(m^2 - 2mj + j^2)\lambda^2, \end{align*} where the negative terms associated with $ \sum_{i = 0}^{j - 1} $ were dropped. Adding up the right sides of the above three inequalities and grouping them as a polynomial in $ j $ of degree two, we have the following: The term involving $ j^{0} = 1 $ has the factor \[ \mu \frac{1}{\alpha}\sum_{j = 1}^{m}[n^2\mu^2 - 2 (n\mu)(m\lambda)+(m\lambda)^2] = (m\lambda)(n\mu - m\lambda)^2; \] the term involving $ j^2 $ has the factor \[ \frac{\mu^2}{\alpha^{3}} - \frac{2\mu \lambda}{\alpha^2} + \frac{\lambda^2}{\alpha} = 0; \] the term involving $ j $ has two parts, one of which has the factor \[ \frac{2n\mu \lambda}{\alpha} + \frac{2\mu m \lambda}{\alpha^2} - \frac{2m\lambda^2}{\alpha} - \frac{2n\mu^2}{\alpha^2} = 0, \] and the other of which has the factor \[ \mu \sum_{j = 1}^{m}(\frac{1 + \beta}{\alpha^{3}} - \frac{1}{\alpha^{3}})j\mu^2 = (\lambda - \mu)\frac{m(m + 1)}{2} \lambda^2. \] The proof is complete. \end{proof} \begin{remark} \label{rmk7} \rm The results in Proposition \ref{P:VaryB} are true for $ n, m \ge 0 $, but a similar result in the \cite[Proposition 4, page 236]{Lin1} has the restriction $ n\mu - m\lambda \ge 0 $ which is not suitable for a mathematical induction proof. \end{remark} \subsection*{Acknowledgments} The author wishes to thank very much Professor Jerome A. Goldstein at University of Memphis, for his teaching and training, which helps the author in many ways. \begin{thebibliography}{00} \bibitem{Ba} V. Barbu; \emph{Semigroups and Differential Equations in Banach Spaces}, Leyden : Noordhoff, 1976. \bibitem{Br} H. Brezis; \emph{On a problem of T. Kato}, Comm. Pure Appli. Math., \textbf{ 24} (1971), 1-6. \bibitem{Bre} H. Brezis; \emph{Semi-groupes non Lineaires et applications}, Symposium sur les problemes d'evolution, Instituto Nazionale di Alta Mathematica, Rome, May 1970. \bibitem{Bro} F. E. Browder; \emph{Nonlinear equations of evolution and nonlinear accretive operators in Banach spaces}, Bull. Amer. Math. Soc., \textbf{73} (1967), 867-874. \bibitem{Co} E. A. Coddington and N. Levinson; \emph{Theory of Ordinary Differential Equations}, McGraw-Hill Book Company Inc., New York, 1955. \bibitem{Cr} M. G. Crandall and T. M. Liggett; \emph{Generation of semigroups of nonlinear transformations on general Banach spaces}, Amer. J. Math., \textbf{93} (1971), 256-298. \bibitem{Crand} M. G. Crandall; \emph{A generalized domain for semigroup Generators}, Proceedings of the AMS, \textbf{2}, (1973), 435-440. \bibitem{Cra} M. G. Crandall and A. Pazy; \emph{Semi-groups of nonlinear contractions and dissipative sets}, J. Functional Analysis, \textbf{3} (1969), 376-418. \bibitem{Cran} M. G. Crandall and A. Pazy; \emph{Nonlinear evolution equations in Banach spaces}, Israel J. Math., \textbf{11} (1972), 57-94. \bibitem{En} K. Engel and R. Nagel; \emph{One-Parameter Semigroups for Linear Evolution Equations}, Springer-Verlag, New York, 1999. \bibitem{Eng} K. Engel and R. Nagel; \emph{A Short Course on Operator semigroups}, Springer-Verlag, New York, 2006. \bibitem{Ka} J. Kacur; \emph{Method of Rothe in Evolution Equations}, Teubner Texte Zur Mathematik, Band bf80, BSB B. G. Teubner Verlagsgessellschaft, Leipzig, 1985. \bibitem{Gau} L. Gaul, M. Kogl, and M. Wagner; \emph{Boundary Element Methods for Engineers And Scientists: An Introductory Course with Advanced Topics}, Springer-Verlag, New York, 2003. \bibitem{Gi2} D. Gilbarg and N. S. Trudinger; \emph{Elliptic Partial Differential Equations of Second Order}, Second Edition, Springer-Verlag, New York, 1983. \bibitem{Gol} J. A. Goldstein; \emph{Semigroups of Linear Operators and Applications}, Oxford University Press, New York, 1985. \bibitem{Hi} E. Hille and R. S. Phillips; \emph{Functional Analysis and Semi-groups}, Amer. Math. Soc. Coll. Publ., Vol. 31, Providence, R. I., 1957. \bibitem{Kat} T. Kato; \emph{Nonlinear semi-groups and evolution equations}, J. Math. Soc. Japan, \textbf{19} (1967), 508-520. \bibitem{Kato} T. Kato; \emph{Accretive operators and nonlinear evolution equations in Banach spaces}, Proc. Symp. in Pure Math. \textbf{18}, Part I, Amer. Math. Soc., Providence, R. I., 138-161. \bibitem{Lie} G. M. Lieberman; \emph{Second Order Parabolic Differential Equations}, World Scientic, Singapore, 1996. \bibitem{Li} C. -Y. Lin; \emph{Cauchy problems and applications}, Topological Methods in Nonlinear Analysis, \textbf{15}(2000), 359-368. \bibitem{Lin2} C. -Y. Lin; \emph{Time-dependent nonlinear evolution equations}, Diff. and Int. Equations, \textbf{ 15} (2002), 257-270. \bibitem{Lin0} C. -Y. Lin; \emph{On generation of $ C_0 $ semigroups and nonlinear operator semigroups}, Semigroup Forum, \textbf{66} (2003), 110-120. \bibitem{Lin1} C. -Y. Lin; \emph{On generation of nonlinear operator semigroups and nonlinear evolution operators}, Semigroup Forum, \textbf{67} (2003), 226-246. \bibitem{Lin4} C. -Y. Lin; \emph{Nonlinear evolution equations}, Electronic Journal of Differential Equations, Vol. 2005(2005), No. 42, pp. 1-42. \bibitem{LinA} C. -Y. Lin; \emph{Theory and Examples of Ordinary Differential Equations}, published by World Scientific, Singapore, 2011. \bibitem{Mic} R. E. Mickens; \emph{Difference Equations, Theory and Applications}, Second Edition, Van Mostrand Reinhold, New York, 1990. \bibitem{Mi} I. Miyadera; \emph{Nonlinear Semigroups}, Translations of Mathematical Monographs, vol. 109, American Mathematical Society, 1992. \bibitem{Miy} I. Miyadera; \emph{Some remarks on semigroups of nonlinear operators}, Tohoku Math. J., \textbf{23} (1971), 245-258. \bibitem{Oh} S. Oharu; \emph{On the generation of semigroups of nonlinear contractions}, J. Math. Soc. Japan, \textbf{22} (1970), 526-550. \bibitem{Pa} A. Pazy; \emph{Semigroups of nonlinear operators in Hilbert spaces}, Problems in Non-linear Analysis, C. I. M. E. session \textbf{4} (1970), Edizioni Cremonese, Rome, 343-430. \bibitem{Paz} A. Pazy; \emph{Semigroups of Linear Operators and Applications in Partial Differential Equations}, Springer-Verlag, New York, 1983. \bibitem{Ro} E. Rothe; \emph{Zweidimensionale parabolische Randvertaufgaben als Grenfall eindimensionale Renvertaufgaben}, Math. Ann., \textbf{102} (1930), 650-670. \bibitem{Roy} H. L. Royden; \emph{Real Analysis}, Macmillan Publishing Company, New York, 1989. \bibitem{Sch} A. Schatz, V. Thomee, and W. Wendland; \emph{Mathematical Theory of Finite and Boundary Element Methods}, Birkhauser, Basel, Boston, 1990. \bibitem{Tr} G. M. Troianiello; \emph{Elliptic Differential Equations and Obstacle Problems}, Plenum Press, New York, 1987. \bibitem{We} G. F. Webb; \emph{Nonlinear evolution equations and product stable operators in Banach spaces}, Trans, Amer. Math. Soc., \textbf{155} (1971), 409-426. \bibitem{Wes} U. Westphal; \emph{Sur la saturation pour des semi-groups ono lineaires}, C. R. Acad. Sc. Paris \textbf{274} (1972), 1351-1353. \end{thebibliography} \end{document}