\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 173, 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-2012/173\hfil Approximate solutions] {Approximate solutions to neutral type finite difference equations} \author[D. B. Pachpatte \hfil EJDE-2012/173\hfilneg] {Deepak B. Pachpatte} % in alphabetical order \address{Deepak B. Pachpatte \newline Department of Mathematics, Dr. Babasaheb Ambedekar Marathwada University, Aurangabad, Maharashtra 431004, India} \email{pachpatte@gmail.com} \thanks{Submitted June 4, 2012. Published October 12, 2012.} \subjclass[2000]{26D10, 39A10, 65R20, 45G10} \keywords{Neutral type; finite difference; approximate solution; explicit estimate} \begin{abstract} In this article, we study the approximate solutions and the dependency of solutions on parameters to a neutral type finite difference equation, under a given initial condition. A fundamental finite difference inequality, with explicit estimate, is used to establish the results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $\mathbb{R}$ denote the set of real numbers $\mathbb{R}_ += [ {0,\infty })$, and $N_0 = \{ 0,1,2,\dots \}$. Let $D({S_1 ,S_2 } )$ denote the class of discrete functions from the set ${S_1 }$ to the set ${S_2 }$. For any function $z \in D(N_0 ,\mathbb{R} )$ we define the operator $\Delta$ by $\Delta z(n ) = z({n + 1} ) - z(n )$. We use the conventions that empty sums and products are taken to be $0$ and $1$ respectively. In the present article we consider the neutral type finite difference equation $$\Delta x(n ) = f({n,x(n),\Delta x(n )} ), \label{e1.1}$$ with the initial condition $$x(0 ) = x_0 ,\label{e1.2}$$ for $n \in N_0$, where $f \in D({N_0 \times\mathbb{R}^2,\mathbb{R}} )$ is a given function, $x_0$ is a real constant and $x$ is an function to be found. The continuous analogue of equation \eqref{e1.1} is often referred to as a neutral differential equation, see for example \cite[p. 155]{b1} and \cite{b2}. It is easy to observe that the equation \eqref{e1.1} contains as special case the discrete analogue of the well known Clairaut's differential equation, see \cite[pp 117-118]{a1} and \cite{m2}. In general the solutions to \eqref{e1.1} cannot be found analytically and thus will need more insight to study the qualitative properties of its solutions. The problem of existence of solutions for \eqref{e1.1} with \eqref{e1.2} (IVP \eqref{e1.1}-\eqref{e1.2}, for short) can be dealt with the method employed in \cite[Theorem 1]{s1} with suitable modifications, see also \cite{a1,a2,k1,k2,m1}. In this article we offer the conditions for the error evaluation of approximate solutions of \eqref{e1.1}-\eqref{e1.2} by establishing some new bounds on solutions of approximate problems. We also study the dependency of solutions of \eqref{e1.1}-\eqref{e1.2} on parameters. The main tool employed in the analysis is based on the application of a certain basic finite difference inequality with explicit estimate given in \cite{p1} (see also \cite[Theorem 4.1.1]{a1} and \cite[Theorem 1.2.3]{p1}). Results on two independent variables are also given as a generalization of \eqref{e1.1}-\eqref{e1.2}. A particular feature of our approach is to present in a simple and unified way conditions some of the important qualitative properties of solutions of \eqref{e1.1}-\eqref{e1.2}. \section{Main Results} Let $x_i (n ) \in D({N_0 ,\mathbb{R}} )$ $({i = 1,2} )$ be functions such that $\Delta x_i (n )$ exist for $n \in N_0$ and satisfy the inequalities $$| {\Delta x_i (n ) - f({n,x_i (n ),\Delta x_i (n )} )} | \le \varepsilon _i , \label{e2.1}$$ for given constants $\varepsilon _i \ge 0$ where it is supposed that the initial conditions $$x_i (0 ) = x_i , \label{e2.2}$$ are fulfilled. Then we call $x_i (n )$ $({i = 1,2} )$ the $\varepsilon _i$-approximate solutions of \eqref{e1.1}-\eqref{e1.2}. We use the following finite difference inequality established in \cite{p1} (see also \cite{a1,p2}). \begin{lemma} \label{lem1} Let $u,a,b,p \in D({N_0 ,\mathbb{R}_ + } )$ and $$u(n ) \le a(n ) + p(n )\sum_{s = 0}^{n - 1} b(s )u(s ),\label{e2.3}$$ for ${n \in N_0 }$. Then $$u(n ) \le a(n ) + p(n )\sum_{s = 0}^{n - 1} {a(s )b(s ) \prod_{\sigma = s + 1}^{n - 1} {[ {1 + b(\sigma )p(\sigma )} ]} ,} \label{e2.4}$$ for ${n \in N_0 }$. \end{lemma} The following theorem estimates the difference between the two approximate solutions of \eqref{e1.1}-\eqref{e1.2}. \begin{theorem} \label{thm1} Suppose that the function $f$ in \eqref{e1.1} satisfies the condition $$| {f({n,x,y} ) - f({n,\bar x,\bar y} )} | \le p(n ) [ {| {x - \bar x} | + | {y - \bar y} |} ], \label{e2.5}$$ where $p \in D({N_0 ,\mathbb{R}_ + } )$ and $p(n ) < 1$ for ${n \in N_0 }$. Let $x_i (n )$ $({i = 1,2} )$ be $\varepsilon _i$-approxi\-mate solutions of equation \eqref{e1.1} with \eqref{e2.2} on ${N_0 }$ such that $$| {x_1 - x_2 } | \le \delta ,\label{e2.6}$$ where $\delta \ge 0$ is a constant. Then $$\begin{split} &| {x_1 (n ) - x_2 (n )} | + | {\Delta x_1 (n ) - \Delta x_2 (n )} |\\ & \le A(n ) + B(n )\sum_{s = 0}^{n - 1} {A(s )p(s ) \prod_{\sigma = s + 1}^{n - 1} {[ {1 + p(\sigma )B(\sigma )} ]} ,} \end{split}\label{e2.7}$$ for ${n \in N_0 }$, where $$A(n ) = \frac{{({\varepsilon _1 + \varepsilon _2 } )({n + 1} ) + \delta }}{{1 - p(n )}},\quad B(n ) = \frac{1}{{1 - p(n )}}. \label{e2.8}$$ \end{theorem} \begin{proof} Since $x_i (n )$ $({i = 1,2} )$ for ${n \in N_0 }$ are $\varepsilon _i$-approximate solutions of \eqref{e1.1} with \eqref{e2.2}, we have \eqref{e2.1}. By taking $n=s$ in \eqref{e2.1} and summing up both sides over $s$ from $0$ to $n-1$, we have $$\begin{split} \varepsilon _i n &\ge \sum_{s = 0}^{n - 1} {| {\Delta x_i (s ) - f({s,x_i (s ),\Delta x_i (s )} )} |}\\ &\ge | {\sum_{s = 0}^{n - 1} {\{ {\Delta x_i (s ) - f({s,x_i (s ),\Delta x_i (s )} )} \}} } | \\ &= \Big| {x_i (n ) - x_i (0 ) - \sum_{s = 0}^{n - 1} {f({s,x_i (s ),\Delta x_i (s )} ) } } \Big|, \end{split}\label{e2.9}$$ for $i=1,2$. From \eqref{e2.9} and using the elementary inequalities $$| {v - z} | \le | v | + | z |,\quad | v | - | z | \le | {v - z} |, \label{e2.10}$$ we observe that $$\begin{split} ({\varepsilon _1 + \varepsilon _2 } )n & \ge \Big| {x_1 (n ) - x_1 (0 ) - \sum_{s = 0}^{n - 1} {f({s,x_1 (s ),\Delta x_1 (s )} )} } \Big| \\ &\quad + \Big| {x_2 (n ) - x_2 (0 ) - \sum_{s = 0}^{n - 1} {f({s,x_2 (s ),\Delta x_2 (s )} )} } \Big| \\ &\ge \Big| {x_1 (n ) - x_1 (0 ) - \sum_{s = 0}^{n - 1} {f({s,x_1 (s ),\Delta x_1 (s )} )} } \\ &\quad - \Big( {x_2 (n ) - x_2 (0 ) - \sum_{s = 0}^{n - 1} {f({s,x_2 (s ),\Delta x_2 (s )} )} } \Big)\Big| \\ & \ge | {x_1 (n ) - x_2 (n )} | - | {x_1 (0 ) - x_2 (0 )} | \\ &\quad - \Big| {\sum_{s = 0}^{n - 1} {f({s,x_1 (s ),\Delta x_1 (s )} )} - \sum_{s = 0}^{n - 1} {f({s,x_2 (s ),\Delta x_2 (s )} )} } \Big|. \end{split}\label{e2.11}$$ Moreover, from \eqref{e2.1} and using the elementary inequalities in \eqref{e2.10}, we observe that $$\begin{split} &\varepsilon _1 + \varepsilon _2\\ &\ge | {\Delta x_1 (n ) - f({n,x_1 (n ),\Delta x_1 (n )} )} | + | {\Delta x_2 (n ) - f({n,x_2 (n ),\Delta x_2 (n )} )} | \\ & \ge | {\{ {\Delta x_1 (n ) - f({n,x_1 (n ),\Delta x_1 (n )} )} \} - \{ {\Delta x_2 (n ) - f({n,x_2 (n ),\Delta x_2 (n )} )} \}} | \\ &\ge | {\Delta x_1 (n ) - \Delta x_2 (n )} | - | {f({n,x_1 (n ),\Delta x_1 (n )} ) - f({n,x_2 (n ),\Delta x_2 (n )} )} |. \end{split}\label{e2.12}$$ Let ${u(n ) = x_1 (n ) - x_2 (n )}$ for $n \in N_0$. From \eqref{e2.11}, \eqref{e2.12} and using the hypotheses, we observe that \begin{align*} &| {u(n )} | + | {\Delta u(n )} |\\ &\le ({\varepsilon _1 + \varepsilon _2 } )n + | {u(0 )} | + \sum_{s = 0}^{n - 1} {| {f({s,x_1 (s ),\Delta x_1 (s )} ) - f({s,x_2 (s ),\Delta x_2 (s )} )} |} \\ &\quad + ({\varepsilon _1 + \varepsilon _2 } ) + | {f({n,x_1 (n ),\Delta x_1 (n )} ) - f({n,x_2 (n ),\Delta x_2 (n )} )} | \\ &\le ({\varepsilon _1 + \varepsilon _2 } )({n + 1} ) + \delta + \sum_{s = 0}^{n - 1} {p(s )[ {| {u(s )} | + | {\Delta u(s )} |} ]} + p(n )[ {| {u(n )} | + | {\Delta u(n )} |} ]. \end{align*} %\label{e2.13} From this inequality, we obtain $$| {u(n )} | + | {\Delta u(n )} | \le A(n ) + B(n )\sum_{s = 0}^{n - 1} {p(s )[ {| {u(s )} | + | {\Delta u(s )} |} ]} , \label{e2.14}$$ for $n \in N_0$, where $A(n), B(n)$ are given as in \eqref{e2.8}. Now an application of Lemma \ref{lem1} to \eqref{e2.14} yields \eqref{e2.7}. \end{proof} \begin{remark} \label{rmk1}\rm We note that, if ${x_1 (n )}$ is a solution of \eqref{e1.1}-\eqref{e1.2} with ${x_1 (0 ) = x_1 }$, then we have ${\varepsilon _1 = 0}$ and from \eqref{e2.7} we see that ${x_1 (n ) \to x_2 (n )}$ as ${\varepsilon _2 \to 0}$ and $\delta \to 0$. Moreover, if we put: (i) $\varepsilon _1 = \varepsilon _2 = 0$, $x_1 = x_2$ in \eqref{e2.7}, then the uniqueness of solutions of \eqref{e1.1} is established, and (ii) $\varepsilon _1 = \varepsilon _2 = 0$ in \eqref{e2.7}, then we obtain the bound which shows the dependency of solutions of \eqref{e1.1} on given initial values. For the estimate on the difference between the two approximate solutions of special version of \eqref{e1.1} when the function $f$ in \eqref{e1.1} is independent of ${\Delta x(n )}$, by using comparison theorem, we refer the interested reader to \cite[Theorem 1.5]{s1}. \end{remark} Consider \eqref{e1.1}-\eqref{e1.2} together with the initial-value problem \begin{gather} \Delta y(n ) = g({n,y(n ),\Delta y(n )} ), \label{e2.15}\\ y(0 ) = y_0 , \label{e2.16} \end{gather} for $n \in N_0$, where $g \in D({N_0 \times\mathbb{R}^2,R} )$ and $y_0$ is a real constant. Next, we shall prove the following theorem concerning the closeness of the solutions of \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16}. \begin{theorem} \label{thm2} Suppose that the function $f$ in \eqref{e1.1} satisfies \eqref{e2.5}, and that there exist constants $\bar \varepsilon \ge 0$, $\bar \delta \ge 0$ such that \begin{gather} | {f({n,u,v} ) - g({n,u,v} )} | \le \bar \varepsilon , \label{e2.17}\\ | {x_0 - y_0 } | \le \bar \delta ,\label{e2.18} \end{gather} where $f,x_0$ and $g,y_0$ are as in \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16}. Let $x(n)$ and $y(n)$ be respectively, solutions of \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16} on $N_0$. Then $$\begin{split} &| {x(n ) - y(n )} | + | {\Delta x(n ) - \Delta y(n )} |\\ & \le \bar A(n ) + B(n )\sum_{s = 0}^{n - 1} {\bar A(s )p(s ) \prod_{\sigma = s + 1}^{n - 1} {[ {1 + p(\sigma )B(\sigma )} ]} } , \end{split} \label{e2.19}$$ for $n \in N_0$, where $$\bar A(n ) = \frac{{\bar \varepsilon ({n + 1} ) + \bar \delta }}{{1 - p(n )}}, \label{e2.20}$$ and $B(n)$ is as in \eqref{e2.8}. \end{theorem} \begin{proof} Let $v(n)=x(n)-y(n)$ for $n \in N_0$. Using the facts that $x(n)$ and $y(n)$ are the solutions of \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16} and hypotheses, we observe that \begin{aligned} | {v(n )} | + | {\Delta v(n )} | &\le | {x_0 - y_0 } | + \sum_{s = 0}^{n - 1} {| {f({s,x(s ),\Delta x(s )} ) - f({s,y(s ),\Delta y(s )} )} |} \\ &\quad + \sum_{s = 0}^{n - 1} {| {f({s,y(s ),\Delta y(s )} ) - g({s,y(s ),\Delta y(s )} )} |} \\ &\quad + | {f({n,x(n ),\Delta x(n )} ) - f({n,y(n ),\Delta y(n )} )} | \\ &\quad + | {f({n,y(n ),\Delta y(n )} ) - g({n,y(n ),\Delta y(n )} )} | \\ & \le \bar \delta + \sum_{s = 0}^{n - 1} {p(s )[ {| {v(s )} | + | {\Delta v(s )} |} ] + \sum_{s = 0}^{n - 1} {\bar \varepsilon } }\\ &\quad+ p(n )[ {| {v(n )} | + | {\Delta v(n )} |} ] + \bar \varepsilon . \end{aligned}\label{e2.21} From \eqref{e2.21}, we obtain $$| {v(n )} | + | {\Delta v(n )} | \le \bar A(n ) + B(n )\sum_{s = 0}^{n - 1} {p(s )[ {| {v(s )} | + | {\Delta v(s )} |} ]} . \label{e2.22}$$ Now an application of Lemma \ref{lem1} to \eqref{e2.22} yields \eqref{e2.19}. \end{proof} \begin{remark} \label{rmk2} \rm We note that the result given in Theorem \ref{thm2} relates the solutions of \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16} in the sense that if $f$ is close to $g$ and $x_0$ is close to $y_0$, then the solutions of \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16} are also close together. For further results on the qualitative properties of solutions of various types of finite difference equations, see \cite{a1,a2,k1,k2,m1,p1,p2,p3,s1}. \end{remark} A slight variant of Theorem \ref{thm2} is embodied in the following theorem. \begin{theorem} \label{thm3} Suppose that the functions $f$ and $g$ in \eqref{e1.1} and \eqref{e2.15} satisfy the condition $$| {f({n,u,v} ) - g({n,\bar u,\bar v} )} | \le q(n )[ {| {u - \bar u} | + | {v - \bar v} |} ], \label{e2.23}$$ where $q \in D({N_0 ,\mathbb{R}_ + } )$ and $q(n)<1$ for $n \in N_0$ and \eqref{e2.18} holds. Let $x(n)$ and $y(n)$ be respectively, solutions of \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16} on $N_0$. Then $$\begin{split} &| {x(n ) - y(n )} | + | {\Delta x(n ) - \Delta y(n )} |\\ &\le A_0 (n ) + B_0 (n )\sum_{s = 0}^{n - 1} {A_0 (s )q(s ) \prod_{\sigma = s + 1}^{n - 1} {[ {1 + q(\sigma )B_0 (\sigma )} ]} } , \end{split} \label{e2.24}$$ for $n \in N_0$, where $$A_0 (n ) = \frac{{\bar \delta }}{{1 - q(n )}},\quad B_0 (n ) = \frac{1}{{1 - q(n )}}.\label{e2.25}$$ \end{theorem} \begin{proof} Let $w(n)=x(n)-y(n)$ for $n \in N_0$. Using the facts that $x(n)$ and $y(n)$ are respectively, solutions of \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.15}-\eqref{e2.16}, and the hypotheses, we observe that \begin{aligned} &| {w(n )} | + | {\Delta w(n )} |\\ &\le | {x_0 - y_0 } | + \sum_{s = 0}^{n - 1} {| {f({s,x(s ),\Delta x(s )} ) - g({s,y(s ),\Delta y(s )} )} |} \\ &\quad + | {f({n,x(n ),\Delta x(n )} ) - g({n,y(n ),\Delta y(n )} )} | \\ & \le \bar \delta + \sum_{s = 0}^{n - 1} {q(s )[ {| {w(s )} | + | {\Delta w(s )} |} ]} + q(n )[ {| {w(n )} | + | {\Delta w(n )} |} ]. \end{aligned} \label{e2.26} From \eqref{e2.26}, we obtain $$| {w(n )} | + | {\Delta w(n )} | \le A_0 (n ) + B_0 (n )\sum_{s = 0}^{n - 1} {q(s )[ {| {w(s )} | + | {\Delta w(s )} |} ]} . \label{e2.27}$$ Now an application of Lemma \ref{lem1} to \eqref{e2.27} yields \eqref{e2.24}. \end{proof} We consider the following two neutral type difference equations \begin{gather} \Delta z(n ) = h({n,z(n ),\Delta z(n ),\mu } ), \label{e2.28}\\ \Delta z(n ) = h({n,z(n ),\Delta z(n ),\mu _0 } ), \label{e2.29} \end{gather} with the initial condition $$z(0 ) = z_0 ,\label{e2.30}$$ for $n \in N_0$, where $h \in D({N_0 \times\mathbb{R}^3,R} )$, $z_0$ is a real constant and ${\mu ,\mu _0 }$ are parameters. The following theorem deals with the dependency of solutions of \eqref{e2.28}-\eqref{e2.30} and \eqref{e2.29}-\eqref{e2.30} on parameters. \begin{theorem} \label{thm4} Suppose that the function $h$ in \eqref{e2.28}, \eqref{e2.29} satisfy the conditions \begin{gather} | {h({n,x,y,\mu } ) - h({n,\bar x,\bar y,\mu } )} | \le a(n )[ {| {x - \bar x} | + | {y - \bar y} |} ],\label{e2.31} \\ | {h({n,x,y,\mu } ) - h({n,x,y,\mu _0 } )} | \le b(n )| {\mu - \mu _0 } |, \label{e2.32} \end{gather} where $a,b \in D({N_0 ,\mathbb{R}_ + } )$ and $a(n ) < 1$ for $n \in N_0$. Let $z_1 (n )$ and $z_2 (n )$ be the solutions of \eqref{e2.28}-\eqref{e2.30} and \eqref{e2.29}-\eqref{e2.30} respectively. Then $$\begin{split} &| {z_1 (n ) - z_2 (n )} | + | {\Delta z_1 (n ) - \Delta z_2 (n )} | \\ & \le C(n ) + D(n )\sum_{s = 0}^{n - 1} {C(s )a(s ) \prod_{\sigma = s + 1}^{n - 1} {[ {1 + a(\sigma )D(\sigma )} ]} } , \end{split}\label{e2.33}$$ for $n \in N_0$, where $$C(n ) = \frac{{| {\mu - \mu _0 } |}}{{1 - a(n )}}\Big[ {b(n ) + \sum_{s = 0}^{n - 1} {b(s )} } \big],\quad D(n ) = \frac{1}{{1 - a(n )}}. \label{e2.34}$$ \end{theorem} \begin{proof} Let ${e(n ) = z_1 (n ) - z_2 (n )}$ for $n \in N_0$. Using the facts that ${z_1 (n )}$ and ${z_2 (n )}$ are the solutions of \eqref{e2.28}-\eqref{e2.30} and \eqref{e2.29}-\eqref{e2.30} respectively, we observe that $$\begin{split} | {e(n )} | + | {\Delta e(n )} | &\le \sum_{s = 0}^{n - 1} {| {h({s,z_1 (s ),\Delta z_1 (s ),\mu } ) - h({s,z_2 (s ),\Delta z_2 (s ),\mu } )} |} \\ &\quad + \sum_{s = 0}^{n - 1} {| {h({s,z_2 (s ),\Delta z_2 (s ),\mu } ) - h({s,z_2 (s ),\Delta z_2 (s ),\mu _0 } )} |} \\ &\quad + | {h({n,z_1 (n ),\Delta z_1 (n ),\mu } ) - h({n,z_2 (n ),\Delta z_2 (n ),\mu } )} | \\ &\quad + | {h({n,z_2 (n ),\Delta z_2 (n ),\mu } ) - h({n,z_2 (n ),\Delta z_2 (n ),\mu _0 } )} |. \end{split}\label{e2.35}$$ The rest of the proof can be completed by closely looking at the proofs of the above theorems and hence we omit it here. \end{proof} \section{Two independent variable generalization} Our approach here allow us to deal with the following initial boundary value problem (IBVP, for short) for neutral type finite difference equation in two independent variables $$\Delta _2 \Delta _1 u({m,n} ) = F({m,n,u({m,n} ),\Delta _2 \Delta _1 u({m,n} )} ), \label{e3.1}$$ with the initial boundary conditions $$u({m,0} ) = \alpha (m ),\quad u({0,n} ) = \beta (n ),\quad u({0,0} ) = 0,\label{e3.2}$$ for $m,n \in N_0$, where $u \in D(N_0^2 ,\mathbb{R} )$, $\alpha ,\beta \in D(N_0 ,\mathbb{R} )$, $F \in D(N_0^2 \times\mathbb{R}^2 ,\mathbb{R} )$ and the operators $\Delta _1 ,\Delta _2 ,\Delta _2 \Delta _1$ are as defined in \cite[p. 3]{p2}. In this section, we formulate in brief the results analogues to Theorems \ref{thm1} and \ref{thm2} related to the solution of \eqref{e3.1}-\eqref{e3.2} only. One can formulate results similar to those in Theorems \ref{thm3} and \ref{thm4} with suitable modifications. We require the following finite difference inequality presented in \cite[Corollary 4.2.1]{p2}. \begin{lemma} \label{lem2} Let $u,a,b,c \in D({N_0^2 ,\mathbb{R}_ + } )$. If $$u({m,n} ) \le a({m,n} ) + b({m,n} )\sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {c({s,t} )u({s,t} ),} }\label{e3.3}$$ for $m,n \in N_0$, then $$u({m,n} ) \le a({m,n} ) + b({m,n} ) \Big({\sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {c({s,t} )a({s,t} )} } } \Big) \prod_{s = 0}^{m - 1} {[ {1 + \sum_{t = 0}^{n - 1} {c({s,t} )b({s,t} )} } ],} \label{e3.4}$$ for $m,n\in N_0$. \end{lemma} Let $u_i ({m,n} ) \in D(N_0^2 ,\mathbb{R} )$ $({i = 1,2} )$, and $\Delta _2 \Delta _1 u_i ({m,n} )$ $({m,n \in N_0 } )$ exist and satisfy the inequalities $$| {\Delta _2 \Delta _1 u_i ({m,n} ) - F({m,n,u_i ({m,n} ),\Delta _2 \Delta _1 u_i ({m,n} )} )} | \le \varepsilon _i , \label{e3.5}$$ for given constants $\varepsilon _i \ge 0$ $(i=1,2)$, where it is supposed that the initial boundary conditions $$u_i ({m,0} ) = \alpha _i (m ),\quad u_i ({0,n} ) = \beta _i (n ),\quad u_i ({0,0} ) = 0, \label{e3.6}$$ are fulfilled and $\alpha _i ,\beta _i \in D(N_0 ,\mathbb{R} )$. Then we call $u_i ({m,n} )$ the $\varepsilon _i$-approximate solutions with respect to \eqref{e3.1}-\eqref{e3.2}. \begin{theorem} \label{thm5} Suppose that the function $F$ in \eqref{e3.1} satisfies the condition $$| {F({m,n,u,v} ) - F({m,n,\bar u,\bar v} )} | \le p({m,n} ) [ {| {u - \bar u} | + | {v - \bar v} |} ], \label{e3.7}$$ where $p \in D({N_0^2 ,\mathbb{R}_ + } )$ and ${p({m,n} ) < 1}$ for $m,n \in N_0$. For $i=1,2$, let $u_i ({m,n} )$ $({m,n \in N_0 } )$ be $\varepsilon _i$-approximate solutions of \eqref{e3.1} with \eqref{e3.6} such that $$| {\alpha _1 (m ) - \alpha _2 (m ) + \beta _1 (n ) - \beta _2 (n )} | \le \delta , \label{e3.8}$$ where $\delta \ge 0$ is a constant. Then $$\begin{split} &| {u_1 ({m,n} ) - u_2 ({m,n} )} | + | {\Delta _2 \Delta _1 u_1 ({m,n} ) - \Delta _2 \Delta _1 u_2 ({m,n} )} | \\ & \le L({m,n} ) + E({m,n} )({\sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t} )L({s,t} )} } } )\prod_{s = 0}^{m - 1} \Big[ {1 + \sum_{t = 0}^{n - 1} {p({s,t} )E({s,t} )} } \Big], \end{split}\label{e3.9}$$ for $m,n \in N_0$, where $$L({m,n} ) = \frac{{({\varepsilon _1 + \varepsilon _2 } )({mn + 1} ) + \delta }}{{1 - p({m,n} )}},\quad E({m,n} ) = \frac{1}{{1 - p({m,n} )}}.\label{e3.10}$$ \end{theorem} \begin{proof} Since $u_i ({m,n} )$ $({i = 1,2} )$ for $m,n \in N_0$ are respectively $\varepsilon _i$-approximate solutions of \eqref{e3.1} with \eqref{e3.6}, we have \eqref{e3.5}. Keeping $m$ fixed in \eqref{e3.5}, setting $n=t$ and taking sum on both sides over $t$ from $0$ to $n-1$, then keeping $n$ fixed in the resulting inequality and setting $m=s$ and taking sum over $s$ from $0$ to $m-1$ and using \eqref{e3.6}, we observe that \begin{align*} \varepsilon _i mn &\ge \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {| {\Delta _2 \Delta _1 u_i ({s,t} ) - F({s,t,u_i ({s,t} ),\Delta _2 \Delta _1 u_i ({s,t} )} )} |} } \\ &\ge \Big| {\sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {\{ {\Delta _2 \Delta _1 u_i ({s,t} ) - F({s,t,u_i ({s,t} ), \Delta _2 \Delta _1 u_i ({s,t} )} )} \}} } } \big| \\ &= \Big| \Big\{ {u_i ({s,t} ) - [ {\alpha _i (m ) + \beta _i (n )} ] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {F({s,t,u_i ({s,t} ), \Delta _2 \Delta _1 u_i ({s,t} )} )} } } \Big\} \Big|. \end{align*} %\label{e3.11} From this inequality and using the elementary inequalities in \eqref{e2.10}, we observe that \begin{align*} &({\varepsilon _1 + \varepsilon _2 } )mn\\ &\ge \Big| {\{ {u_1 ({s,t} ) - [ {\alpha _1 (m ) + \beta _1 (n )} ] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {F({s,t,u_1 ({s,t} ), \Delta _2 \Delta _1 u_1 ({s,t} )} )} } } \}} \Big| \\ &+ \Big| \Big\{ {u_2 ({s,t} ) - [ {\alpha _2 (m ) + \beta _2 (n )} ] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {F({s,t,u_2 ({s,t} ), \Delta _2 \Delta _1 u_2 ({s,t} )} )} } } \Big\} \Big| \\ &\ge \Big| \Big\{ {u_1 ({s,t} ) - [ {\alpha _1 (m ) + \beta _1 (n )} ] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {F({s,t,u_1 ({s,t} ), \Delta _2 \Delta _1 u_1 ({s,t} )} )} } } \Big\} \\ &\quad - \Big\{ {u_2 ({s,t} ) - [ {\alpha _2 (m ) + \beta _2 (n )} ] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {F({s,t,u_2 ({s,t} ), \Delta _2 \Delta _1 u_2 ({s,t} )} )} } } \Big\} \Big| \\ &\ge \Big| {u_1 ({s,t} ) - u_2 ({s,t} )} | - | {[ {\alpha _1 (m ) + \beta _1 (n )} ] - [ {\alpha _2 (m ) + \beta _2 (n )} ]} \Big| \\ &\quad- \Big| \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {F({s,t,u_1 ({s,t} ), \Delta _2 \Delta _1 u_1 ({s,t} )} )} }\\ &\quad - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {F({s,t,u_2 ({s,t} ), \Delta _2 \Delta _1 u_2 ({s,t} )} )} } \Big| \end{align*} %\label{e3.12} Furthermore, from \eqref{e3.5} and using the elementary inequalities in \eqref{e2.10}, we observe that \begin{align*} &\varepsilon _1 + \varepsilon _2\\ &\ge | {\Delta _2 \Delta _1 u_1 ({m,n} ) - F({m,n,u_1 ({m,n} ), \Delta _2 \Delta _1 u_1 ({m,n} )} )} | \\ &\quad+ | {\Delta _2 \Delta _1 u_2 ({m,n} ) - F({m,n,u_2 ({m,n} ), \Delta _2 \Delta _1 u_2 ({m,n} )} )} | \\ &\ge | {\{ {\Delta _2 \Delta _1 u_1 ({m,n} ) - F({m,n,u_1 ({m,n} ), \Delta _2 \Delta _1 u_1 ({m,n} )} )} \}} \\ &\quad - \{ {\Delta _2 \Delta _1 u_2 ({m,n} ) - F({m,n,u_2 ({m,n} ), \Delta _2 \Delta _1 u_2 ({m,n} )} )} \} | \\ &\ge | {\Delta _2 \Delta _1 u_1 ({m,n} ) - \Delta _2 \Delta _1 u_2 ({m,n} )} | \\ &\quad - | {F({m,n,u_1 ({m,n} ),\Delta _2 \Delta _1 u_1 ({m,n} )} ) - F({m,n,u_2 ({m,n} ),\Delta _2 \Delta _1 u_2 ({m,n} )} )} |. \end{align*} %\label{e3.13} The remaining proof can be completed by following the proof of Theorem \ref{thm1} with suitable modifications and using Lemma \ref{lem2}. We omit the further details. \end{proof} Consider \eqref{e3.1}-\eqref{e3.2} with the IBVP \begin{gather} \Delta _2 \Delta _1 v({m,n} ) = G({m,n,v({m,n} ),\Delta _2 \Delta _1 v({m,n} )} ), \label{e3.14}\\ v({m,0} ) = \bar \alpha (m ),\quad v({0,n} ) = \bar \beta (n ),\quad v({0,0} ) = 0, \label{e3.15} \end{gather} for $m,n \in N_0$, where $v \in D({N_0^2 ,\mathbb{R}} )$, $\bar \alpha ,\bar \beta \in D({N_0 ,\mathbb{R}} )$, $G \in D({N_0^2 \times\mathbb{R}^2 ,\mathbb{R}} )$. \begin{theorem} \label{thm6} Suppose that the function $F$ in \eqref{e3.1} satisfies \eqref{e3.7} and that there exist constants $\bar \varepsilon \ge 0,\bar \delta \ge 0$ such that \begin{gather} | {F({m,n,u,v} ) - G({m,n,u,v} )} | \le \bar \varepsilon ,\label{e3.16}\\ | {\alpha (m ) - \bar \alpha (m ) + \beta (n ) - \bar \beta (n )} | \le \bar \delta , \label{e3.17} \end{gather} where $F,\alpha ,\beta$ and $G,\bar \alpha ,\bar \beta$ are as in \eqref{e3.1}-\eqref{e3.2} and \eqref{e3.14}-\eqref{e3.15}. Let $u(m,n)$ and $v(m,n)$ be respectively the solutions of \eqref{e3.1}-\eqref{e3.2} and \eqref{e3.14}-\eqref{e3.15} for $m,n \in N_0$. Then $$\begin{split} &| {u({m,n} ) - v({m,n} )} | + | {\Delta _2 \Delta _1 u({m,n} ) - \Delta _2 \Delta _1 v({m,n} )} | \\ & \le \bar L({m,n} ) + E({m,n} )\Big({\sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t} )\bar L({s,t} )} } } \Big) \prod_{s = 0}^{m - 1} {\Big[ {1 + \sum_{t = 0}^{n - 1} {p({s,t} )E({s,t} )} } \Big],} \end{split}\label{e3.18}$$ for $m,n \in N_0$, where $$\bar L({m,n} ) = \frac{{\bar \varepsilon ({mn + 1} ) + \bar \delta }}{{1 - p({m,n} )}}, \label{e3.19}$$ and $E(m,n)$ is as in \eqref{e3.10}. \end{theorem} The proof follows by the similar argument as in the proof of Theorem \ref{thm2} given above with suitable modifications. Here, we omit the details. \begin{remark} \label{rmk3} \rm We note that the idea used in this paper can be extended very easily to establish similar results as given above for the following finite difference equations \begin{gather} \Delta _2 \Delta _1 u({m,n} ) = F({m,n,v({m,n} ),\Delta _1 u({m,n} )} ), \label{e3.20}\\ \Delta _2 \Delta _1 u({m,n} ) = F({m,n,v({m,n} ),\Delta _2 u({m,n} )} ), \label{e3.21} \end{gather} with the given initial boundary conditions in \eqref{e3.2} under some suitable conditions and by making use of a suitable variant of the inequality in Lemma \ref{lem2} (see also \cite{p3}). \end{remark} In concluding we note that, in the study of convergence of finite element approximations to solutions of various types of dynamic equations, the dependence of the error bounds on certain derivatives (or differences) of the exact solution will become apparent in the course of analysis (see, for example \cite{t1}). Here, it is to be noted that our analysis yields explicit bounds not only on the solutions of the problems but also on the forward differences of the solutions. We hope that our approach here will revel as a model for future investigations. \begin{thebibliography}{99} \bibitem{a1} R. P. Agarwal; \emph{Difference Equations and Inequalities}, Marcel Dekker, Inc., New York, 1992. \bibitem{a2} R. P. Agarwal, P.J.Y. Wong; \emph{Advanced Topics in Difference Equations}, Kluwer Academic Publishers, Dordrecht, 1997. \bibitem{b1} H. Brunner; \emph{Collection methods for Volterra integral and related functional differential equations}, Cambridge University Press, Cambridge, 2004. \bibitem{b2} M. Buhmann and A. Iserles; \emph{On the dynamics of a discretized neutral equation}, IAM J. Numer. Anal. 12(1992), 339-363. \bibitem{k1} W. G. Kelley, A.C. Peterson; \emph{Difference Equations: An Introduction with Applications}, Academic Press, San Diego, 1991. \bibitem{k2} M. Kwapisz; \emph{Some existence and uniqueness results for boundary value problems for difference equations}, Applicable Analysis 37(1990), 169-182. \bibitem{m1} R. E. Mickens; \emph{ Difference Equations, Van Nostrand Comp.}, New York, 1987. \bibitem{m2} L.M. Milne Thomson; \emph{ The Calculus of Finite Differences}, Macmillan, London, 1960. \bibitem{p1} B. G. Pachpatte; \emph{On the discrete generalizations of Gronwall's inequality}, J. Indian Math. Soc. 37(1973), 147-156. \bibitem{p2} B. G. Pachpatte; \emph{Inequalities for Finite Difference Equations}, Marcel Dekker, Inc., New York, 2002. \bibitem{p3} B. G. Pachpatte; \emph{On a certain finite difference equation in two independent variables}, Bul. Inst. Polit. Iasi Mat. Meca. Theor. Fiz. LII(LVII)(2007), 35-44. \bibitem{p4} B.G. Pachpatte; \emph{On a certain hyperbolic partial differential equation}, Tamsui Oxford Jour. Math. Sci. 25(1) (2009), 39-54. \bibitem{s1} S. Sugiyama; \emph{Comparison theorems on difference equations}, Bull. Sci. Engr. Research Lab., Waseda Univ. 47(1970), 77-82. \bibitem{t1} V. Thom\'{e}e; \emph{Galerkin Finite Element Methods for Parabolic Problems}, Lecture Notes in Math. Vol. 1054, Springer-Verlag, Berlin, New York, 1984. \end{thebibliography} \end{document}