\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 104, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/104\hfil Error evaluation of approximate solutions] {Error evaluation of approximate solutions for sum-difference equations in two variables} \author[B. G. Pachpatte\hfil EJDE-2009/104\hfilneg] {Baburao G. Pachpatte} \address{Baburao G. Pachpatte\newline 57 Shri Niketan Colony, Near Abhinay Talkies\\ Aurangabad 431 001 (Maharashtra) India} \email{bgpachpatte@gmail.com} \thanks{Submitted June 8, 2009. Published August 27, 2009.} \subjclass[2000]{26D10, 39A10, 65R20, 45G10} \keywords{Approximate solutions; sum-difference equation; \hfill\break\indent finite difference inequality} \begin{abstract} This article presents estimates for the approximation of solutions of certain sum-difference equations in two independent variables with given initial conditions. A fundamental finite difference inequality with explicit estimate is used to establish our results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} The method of approximation provides a very useful and important technique in the study of qualitative properties of solutions for mathematical models of various dynamic phenomena (see \cite{h1,k1,l1}). This paper focuses on the study of initial value problem (IVP, for short) \begin{equation} \Delta _2 \Delta _1 u(m,n) = f({m,n,u(m,n),Gu(m,n)}), \label{e1.1} \end{equation} with \begin{equation} \begin{gathered} u(m,0) = \alpha (m),\quad u(0,n) = \beta (m),\quad n \in \mathbb{N}_0 \\ \alpha (0) = \beta (0), \end{gathered}\label{e1.2} \end{equation} where \begin{equation} Gu(m,n): = \sum_{\sigma = 0}^{m - 1} \sum_{\tau = 0}^{n - 1} g({m,n,\sigma ,\tau ,u({\sigma ,\tau })}), \label{e1.3} \end{equation} $f,g$ are given functions and $u$ is the unknown function to be found. Let $\mathbb{R}$ denote the set of real numbers, $\mathbb{R}_+ = [{0,\infty })$, $\mathbb{N}_0 = \{{0,1,2,\dots }\}$ be the given subsets of $\mathbb{R}$. For the functions $z(m),w(m,n)$, $m,n \in \mathbb{N}_0$, we define the operators \begin{gather*} \Delta z(m) = z(m + 1) - z(m),\quad \Delta _1 w(m,n) = w(m + 1,n) - w(m,n),\\ \Delta _2 w(m,n) = w(m,n + 1) - w(m,n),\quad \Delta _2 \Delta _1 w(m,n) = \Delta _2 (\Delta _1 w(m,n)). \end{gather*} We denote by $D(S_1 ,S_2 )$ the class of discrete functions from the set ${S_1 }$ to the set ${S_2 }$ and use the usual conventions that empty sums and products are taken to be $0$ and $1$ respectively. We assume that $f \in D({\mathbb{N}_0^2 \times \mathbb{R}^2 ,\mathbb{R}})$, $g \in D({\mathbb{N}_0^4 \times \mathbb{R},\mathbb{R}})$, $\alpha ,\beta \in D({\mathbb{N}_0 ,\mathbb{R}})$. When dealing with the discrete event dynamical systems the basic questions to be answered are: (i) under what conditions the systems under considerations have solutions? (ii) how can we find the solutions or closely approximate them? (iii) what are their nature?. The study of such questions gives rise to new results and need a fresh outlook for handling such problems for \eqref{e1.1}-\eqref{e1.2}. We note that the equation \eqref{e1.1} under the conditions \eqref{e1.2} admits a unique solution. In this paper we offer the conditions for the error evaluation of approximate solutions of equation \eqref{e1.1} by establishing some new bounds and convergence properties on solutions of approximate problems. We also study the dependency of solutions of equations of the forms \eqref{e1.1} on parameters. The main tool employed in the analysis is based on the application of a certain finite difference inequality with explicit estimate given in \cite{p2}. \section{Main Results} The following is a variation of the finite difference inequality established in \cite[Theorem 5.3.2]{p2} and \cite[Theorem 4.3.2]{p1}, and is crucial in the proof of our main results. \begin{lemma} \label{lem1} Let $u,a,p \in D({\mathbb{N}_0^2 ,\mathbb{R}_+})$, $q,\Delta _1 q,\Delta _2 q,\Delta _2 \Delta _1 q \in D({\mathbb{N}_0^4,\mathbb{R}_ +})$. If $a(m,n)$ is nondecreasing in each variable $m,n \in \mathbb{N}_0$, and \begin{equation} u(m,n) \leq a(m,n) + \sum_{s = 0}^{m - 1} \sum_{t = 0}^{n - 1} {p({s,t}) \Big[{u({s,t}) + \sum_{\sigma = 0}^{s - 1} {\sum_{\tau = 0}^{t - 1} {q({s,t,\sigma ,\tau }) u({\sigma ,\tau })} } } \Big],} \label{e2.1} \end{equation} for $m,n \in \mathbb{N}_0$, then \begin{equation} u(m,n) \leq a(m,n)\Big[{1 + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})\prod_{\xi = 0}^{s - 1} {\Big[{1 + \sum_{\eta = 0}^{t - 1} {[{ p({\xi ,\eta }) + Tq({\xi ,\eta })} ]} } \Big]} } } } \Big], \label{e2.2} \end{equation} for $m,n \in \mathbb{N}_0$, where \begin{equation} \begin{aligned} Tq(m,n) &: = q({m + 1,n + 1,m,n}) + \sum_{\sigma = 0}^{m - 1} \Delta _1 q({m,n + 1,\sigma ,n})\\ &\quad + \sum_{\tau = 0}^{n - 1} \Delta _2 q({m + 1,n,m,\tau }) + \sum_{\sigma = 0}^{m - 1} \sum_{\tau = 0}^{n - 1} \Delta _2 \Delta _1 q({m,n,\sigma ,\tau }) . \end{aligned}\label{e2.3} \end{equation} \end{lemma} Let $u \in D({\mathbb{N}_0^2 ,\mathbb{R}})$ and $\Delta _2 \Delta _1 u(m,n)({m,n \in \mathbb{N}_0 })$ exist and satisfy the inequality \[ | {\Delta _2 \Delta _1 u(m,n) - f({m,n,u(m,n),Gu(m,n)})}| \leq \varepsilon , \] for a given constant $\varepsilon \geq 0$, where it is assumed that \eqref{e1.2} holds. Then we call $u(m,n)$ an $\varepsilon$-approximate solution of \eqref{e1.1}. Our main result estimates the difference between the two approximate solutions of \eqref{e1.1}. \begin{theorem} \label{thm1} Suppose that $f,g$ in \eqref{e1.1} satisfy the conditions \begin{gather} | {f({m,n,u,v}) - f({m,n,\bar u,\bar v})} | \leq p(m,n)[{| {u - \bar u}| + | {v - \bar v} |} ],\label{e2.4}\\ | {g({m,n,\sigma ,\tau ,u}) - g({m,n,\sigma ,\tau ,\bar u})} | \leq q({m,n,\sigma ,\tau }) | {u - \bar u}|, \label{e2.5} \end{gather} where $p \in D({\mathbb{N}_0^2 ,\mathbb{R}_ + })$, $q \in D({\mathbb{N}_0^4,\mathbb{R}_ + })$ with $\Delta _1 q,\Delta _2 q,\Delta _2 \Delta _1 q \in D({\mathbb{N}_0^4,\mathbb{R}_ + })$. For $i=1,2$, let $u_i (m,n)$ $({m,n \in \mathbb{N}_0 })$ be respectively $ \varepsilon _i$-approximate solutions of \eqref{e1.1} with \begin{equation} \begin{aligned} u_i (m,0) = \alpha _i (m),\quad u_i (0,n) = \beta _i (n),\quad n \in \mathbb{N}_0, \\ \alpha _i (0) = \beta _i (0), \end{aligned}\label{e2.6} \end{equation} where $\alpha _i ,\beta _i$ are in $D({\mathbb{N}_0 ,\mathbb{R}})$ satisfy \begin{equation} | {\alpha _1 (m) - \alpha _2 (m) + \beta _1 (n) - \beta _2 (n)} | \leq \delta , \label{e2.7} \end{equation} in which $\delta \geq 0$ is a constant. Then \begin{equation} \begin{aligned} &|{u_1 (m,n) - u_2 (m,n)}| \\ &\leq ({({\varepsilon _1 + \varepsilon _2 })mn + \delta }) \Big[{1 + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})} } } {\prod_{\xi = 0}^{s - 1} {\Big[{1 + \sum_{\eta = 0}^{t - 1} {[{p({\xi ,\eta }) + Tq({\xi ,\eta })} ]} } \Big]} } \Big], \end{aligned} \label{e2.8} \end{equation} for ${m,n \in \mathbb{N}_0 }$, where $Tq(m,n)$ is given by \eqref{e2.3}. \end{theorem} \begin{proof} Since $u_i (m,n)$ $({i = 1,2})$ for ${m,n \in \mathbb{N}_0 }$ are respectively $\varepsilon _i$-approximate solutions of equation \eqref{e1.1} with \eqref{e2.6} we have \begin{equation} |{\Delta _2 \Delta _1 u_i (m,n) - f({m,n,u_i (m,n),Gu_i(m,n)})}| \leq \varepsilon _i .\label{e2.9} \end{equation} Now keeping $m$ fixed in \eqref{e2.9}, 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{e2.6}, we observe that \begin{equation} \begin{aligned} \varepsilon _i mn &\geq \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}),Gu_i({s,t})})}|} } \\ & \geq |{\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}), Gu_i({s,t})})}\}} } }| \\ & = \Big|{\big\{{u_i (m,n) - [{\alpha _i (m) + \beta _i (n)} ] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {f({s,t,u_i ({s,t}), Gu_i({s,t})})} } }\big\}}\Big|. \end{aligned} \label{e2.10} \end{equation} From this inequality and using the elementary inequalities $|{v - z}| \leq |v| + |z|,|v| - |z| \leq |{v - z}|$, for $v,z \in \mathbb{R}$, we observe that \begin{equation} \begin{aligned} &({\varepsilon _1 + \varepsilon _2 })mn \\ & \geq \Big|{\{{u_1 (m,n) - [{\alpha _1 (m) + \beta _1 (n)}] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {f({s,t,u_1 ({s,t}), Gu_1({s,t})})} } }\}}\Big| \\ &\quad + \Big|{\{{u_2 (m,n) - [{\alpha _2 (m) + \beta _2 (n)}] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {f({s,t,u_2 ({s,t}), Gu_2({s,t})})} } }\}}\Big| \\ &\geq \Big|{\{{u_1 (m,n) - [{\alpha _1 (m) + \beta _1 (n)}] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {f({s,t,u_1 ({s,t}), Gu_1({s,t})})} } }\}} \\ &\quad - \big\{{u_2 (m,n) - [{\alpha _2 (m) + \beta _2 (n)} ] - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {f({s,t,u_2 ({s,t}), Gu_2({s,t})})} } }\big\}\Big| \\ &\geq |{u_1 (m,n) - u_2 (m,n)}| - |{\alpha _1 (m) + \beta _1 (n) - \{{\alpha _2 (m) + \beta _2 (n)}\}}| \\ &\quad - \Big|{\sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {f({s,t,u_1 ({s,t}),Gu_1({s,t})}) - \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {f({s,t,u_2 ({s,t}), Gu_2({s,t})})} } } } }\Big| \end{aligned}\label{e2.11} \end{equation} Let $u(m,n) = |{u_1 (m,n) - u_2 (m,n)}|$ for ${m,n \in \mathbb{N}_0 }$. From the above inequality and using the hypotheses, we observe that \begin{equation} \begin{aligned} u(m,n) &\leq ({\varepsilon _1 + \varepsilon _2 })mn + \delta \\ &\quad + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {|{f({s,t,u_1 ({s,t}), Gu_1({s,t})}) - f({s,t,u_2 ({s,t}),Gu_2({s,t})})}|} } \\ & \leq ({\varepsilon _1 + \varepsilon _2 })mn + \delta + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t}) \Big[{u({s,t}) + \sum_{\sigma = 0}^{s - 1} {\sum_{\tau = 0}^{t - 1} {q({s,t,\sigma ,\tau })u({\sigma ,\tau })} } } \Big]} } . \end{aligned}\label{e2.12} \end{equation} Now an application of Lemma \ref{lem1} yields \eqref{e2.8}. Consider the initial-value problem \eqref{e1.1}-\eqref{e1.2} together with \begin{gather} \Delta _2 \Delta _1 v(m,n) = \bar f({m,n,v(m,n),Gv(m,n)}), \label{e2.13}\\ \begin{gathered} v(m,0) = \bar \alpha (m),\quad v(0,n) = \bar \beta (n),\quad n \in \mathbb{N}_0 , \\ \bar \alpha (0) = \bar \beta (0), \end{gathered} \label{e2.14} \end{gather} where $G$ is given by \eqref{e1.3} and $\bar f \in D({\mathbb{N}_0^2 \times \mathbb{R}^2 ,\mathbb{R}}), \bar \alpha ,\bar \beta \in D({\mathbb{N}_0 ,\mathbb{R}})$. \end{proof} The following theorem concerns the closeness of solutions of \eqref{e1.1}-\eqref{e1.2} and of \eqref{e2.13}-\eqref{e2.14}. \begin{theorem} \label{thm2} Suppose that $f,g$ in \eqref{e1.1} satisfy \eqref{e2.4}, \eqref{e2.5} and there exist constants $\bar \varepsilon \geq 0$, $\bar\delta \geq 0$ such that \begin{gather} |{f({m,n,u,w}) - \bar f({m,n,u,w})}| \leq \bar \varepsilon , \label{e2.15}\\ |{\alpha (m) - \bar \alpha (m) + \beta (n) - \bar \beta (n)}| \leq \bar \delta , \label{e2.16} \end{gather} where $f,\alpha ,\beta $ and $\bar f,\bar \alpha ,\bar \beta$ are as in \eqref{e1.1}-\eqref{e1.2} and \eqref{e2.13}-\eqref{e2.14}. Let $u(m,n)$ and $v(m,n)$ be respectively the solutions of \eqref{e1.1}-\eqref{e1.2} and of \eqref{e2.13}-\eqref{e2.14} for $m,n \in \mathbb{N}_0$. Then \begin{equation} \begin{aligned} &|{u(m,n) - v(m,n)}|\\ & \leq ({\bar \varepsilon mn + \bar \delta }) \Big[{1 + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})} } } \prod_{\xi = 0}^{s - 1} \Big[{1 + \sum_{\eta = 0}^{t - 1} {[{p({\xi ,\eta }) + Tq({\xi ,\eta })}]} } \Big] \Big], \end{aligned} \label{e2.17} \end{equation} for $m,n \in \mathbb{N}_0$. \end{theorem} \begin{proof} Let $e(m,n) = |{u(m,n) - v(m,n)}|$ for $m,n \in \mathbb{N}_0$. Using the fact that $u(m,n),v(m,n)$ are the solutions of \eqref{e1.1}-\eqref{e1.2}, and of \eqref{e2.13}-\eqref{e2.14}, and the hypotheses, we observe that \begin{equation} \begin{aligned} e(m,n) &\leq |{\alpha (m) - \bar \alpha (m) + \beta (n) - \bar \beta (n)}|\\ &\quad + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {|{f({s,t,u({s,t}),Gu({s,t})}) - f({s,t,v({s,t}),Gv({s,t})})}|} } \\ &\quad + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {|{f({s,t,v({s,t}), Gv({s,t})}) - \bar f({s,t,v({s,t}),Gv({s,t})})}|} } \\ & \leq ({\bar \varepsilon mn + \bar \delta }) + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})} } \Big[{e({s,t}) + \sum_{\sigma = 0}^{s - 1} {\sum_{\tau = 0}^{t - 1} {q({s,t,\sigma ,\tau })e({\sigma ,\tau })} } } \Big]. \end{aligned}\label{e2.18} \end{equation} Now an application of Lemma \ref{lem1} yields \eqref{e2.17}. \end{proof} \begin{remark} \label{rmk1}\rm The result given in Theorem \ref{thm2} relates the solutions of \eqref{e1.1}-\eqref{e1.2} and of \eqref{e2.13}-\eqref{e2.14} in the sense that if $f$ is close to $\bar f$, $\alpha$ is close to $\bar \alpha $, $\beta $ is close to $\bar \beta$, then the solutions of \eqref{e1.1}-\eqref{e1.2} and of \eqref{e2.13}-\eqref{e2.14} are also close to each other. \end{remark} Now we consider \eqref{e1.1}-\eqref{e1.2} and sequence of initial-value problems \begin{gather} \Delta _2 \Delta _1 w(m,n) = f_k ({m,n,w(m,n),Gw(m,n)}), \label{e2.19}\\ \begin{gathered} w(m,0) = \alpha _k (m),\quad w(0,n) = \beta _k (n),\quad n \in \mathbb{N}_0, \\ \alpha _k (0) = \beta _k (0), \end{gathered} \label{e2.20} \end{gather} for $m,n \in \mathbb{N}_0$, $(k = 1,2,\dots)$ where $G$ is given by \eqref{e1.3} and $f_k \in D({\mathbb{N}_0^2 \times \mathbb{R}^2 ,\mathbb{R}})$, $\alpha _k ,\beta _k \in D({\mathbb{N}_0 ,\mathbb{R}})$. As an immediate consequence of Theorem \ref{thm2}, we have the following corollary. \begin{corollary} \label{coro1} Suppose that $f,g$ in \eqref{e1.1} satisfy \eqref{e2.4}, \eqref{e2.5} and \begin{gather} |{f({m,n,u,v}) - f_k ({m,n, u, v})}| \leq \varepsilon _k , \label{e2.21}\\ |{\alpha (m) - \alpha _k (m) + \beta (n) - \beta _k (n)}| \leq \delta _k , \label{e2.22} \end{gather} with $\varepsilon _k \to 0$ and $\delta _k \to 0$ as $k \to \infty$, where $f,\alpha ,\beta$ and $f_k ,\alpha _k ,\beta _k$ are as in \eqref{e1.1}-\eqref{e1.2} and in \eqref{e2.19}-\eqref{e2.20}. If $ w_k (m,n)$ $({k = 1,2,\dots })$ and $u(m,n)$ are respectively the solutions of \eqref{e2.19}-\eqref{e2.20} and of \eqref{e1.1}-\eqref{e1.2} for $m,n \in \mathbb{N}_0$, then $w_k (m,n) \to u(m,n)$ as $k \to \infty$. \end{corollary} \begin{proof} For $k=1,2,\dots$, the conditions of Theorem \ref{thm2} hold. An application of Theorem \ref{thm2} yields \begin{equation} \begin{aligned} &|{w_k (m,n) - u(m,n)}| \\ &\leq ({\varepsilon _k mn + \delta _k }) \Big[{1 + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})} } } \prod_{\xi = 0}^{s - 1} {\Big[{1 + \sum_{\eta = 0}^{t - 1} {[{p({\xi ,\eta }) + Tq({\xi ,\eta })}]} } \Big]} \Big], \end{aligned} \label{e2.23} \end{equation} for $m,n \in \mathbb{N}_0$ and $k=1,2,\dots $. The required result follows from \eqref{e2.23}. \end{proof} \begin{remark} \label{rmk2}\rm We note that the result obtained in Corollary \ref{coro1} provides sufficient conditions that ensures, solutions of \eqref{e2.19}-\eqref{e2.20} will converge to the solutions to \eqref{e1.1}-\eqref{e1.2}. \end{remark} \section{Dependency on parameters} In this section, we present results on the dependency of solutions of equation \eqref{e1.1} and its variants on given initial conditions and pure parameters. The following theorem shows the dependency of solutions of \eqref{e1.1} on given initial conditions. \begin{theorem} \label{thm3} Suppose that $f,g$ in \eqref{e1.1} satisfy \eqref{e2.4}, \eqref{e2.5}. Let $u(m,n)$ and $z(m,n)$ be respectively the solutions of \eqref{e1.1} with the initial conditions \eqref{e1.2} and \begin{equation} \begin{gathered} z(m,0) = \alpha _0 (m),\quad z(0,n) = \beta _0 (n),\quad n \in \mathbb{N}_0 \\ \alpha _0 (0) = \beta _0 (0), \end{gathered} \label{e3.1} \end{equation} where $\alpha _0 ,\beta _0 \in D({\mathbb{N}_0 ,\mathbb{R}})$ and \begin{equation} |{\alpha (m) - \alpha _0 (m) + \beta (n) - \beta _0 (n)}| \leq k, \label{e3.2} \end{equation} in which $k \geq 0$ is a constant. Then \begin{equation} |{u(m,n) - z(m,n)}| \leq k \Big[{1 + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})} } } \prod_{\xi = 0}^{s - 1} {\Big[{1 + \sum_{\eta = 0}^{t - 1} {[{p({\xi ,\eta }) + Tq({\xi ,\eta })}]} } \Big]} \Big], \label{e3.3} \end{equation} for $m,n \in \mathbb{N}_0$. \end{theorem} \begin{proof} Using the facts that $u(m,n)$ and $z(m,n)$ are respectively the solutions of \eqref{e1.1}-\eqref{e1.2} and of \eqref{e1.1}-\eqref{e3.1} and the hypotheses, we have \begin{equation} \begin{aligned} &|{u(m,n) - z(m,n)}| \\ &\leq |{\alpha (m) - \alpha _0 (m) + \beta (n) - \beta _0 (n)}|\\ &\quad + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {|{f({s,t,u({s,t}), Gu({s,t})}) - f({s,t,z({s,t}),Gz({s,t})})}|} } \\ &\leq k + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})} } \Big[{|{u({s,t}) - z({s,t})}| + \sum_{\sigma = 0}^{s - 1} {\sum_{\tau = 0}^{t - 1} {q({s,t,\sigma ,\tau })|{u({\sigma ,\tau }) - z({\sigma ,\tau })}|} } } \Big]. \end{aligned}\label{e3.4} \end{equation} Now a suitable application of Lemma \ref{lem1} yields \eqref{e3.3}, which shows the dependency of solutions of \eqref{e1.1} on the given initial values. \end{proof} We now consider the sum-difference equations \begin{gather} \Delta _2 \Delta _1 u(m,n) = f({m,n,u(m,n),Gu(m,n),\mu }),\label{e3.5}\\ \Delta _2 \Delta _1 u(m,n) = f({m,n,u(m,n),Gu(m,n),\mu _0 }), \label{e3.6} \end{gather} with the initial conditions \eqref{e1.2}, where $G$ is given by \eqref{e1.3}, $f \in D({\mathbb{N}_0^2 \times \mathbb{R}^2 \times \mathbb{R},\mathbb{R}})$ and ${\mu ,\mu _0 }$ are parameters. The next theorem shows the dependency of solutions of \eqref{e3.5}-\eqref{e1.2} and of \eqref{e3.6}-\eqref{e1.2} on the parameters ${\mu ,\mu _0 }$. \begin{theorem} \label{thm4} Suppose that $g,f$ in \eqref{e3.5}, \eqref{e3.6} satisfy respectively \eqref{e2.5} and \begin{gather} |{f({m,n,u,v,\mu }) - f({m,n,\bar u,\bar v,\mu })}| \leq p(m,n)[{|{u - \bar u}| + |{v - \bar v}|} ],\label{e3.7}\\ |{f({m,n,u,v,\mu }) - f({m,n,u,v,\mu _0 })}| \leq r(m,n)|{\mu - \mu _0 }|, \label{e3.8} \end{gather} where $p,r \in D({\mathbb{N}_0^2 ,\mathbb{R}_ + })$. Let $u_1 (m,n)$ and $u_2 (m,n)$ be the solutions of \eqref{e3.5}-\eqref{e1.2} and of \eqref{e3.6}-\eqref{e1.2} respectively. Then \begin{equation} \begin{aligned} &|{u_1 (m,n) - u_2 (m,n)}|\\ & \leq \bar a(m,n)\Big[{1 + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})} } } \prod_{\xi = 0}^{s - 1} {\Big[{1 + \sum_{\eta = 0}^{t - 1} {[{p({\xi ,\eta }) + Tq({\xi ,\eta })} ]} } \Big]} \Big], \label{e3.9} \end{aligned} \end{equation} for ${m,n \in \mathbb{N}_0 }$, where \begin{equation} \bar a(m,n) = |{\mu - \mu _0 }|\sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {r({s,t})} } ,\label{e3.10} \end{equation} for ${m,n \in \mathbb{N}_0 }$. \end{theorem} \begin{proof} Let $h(m,n) = |{u_1 (m,n) - u_2 (m,n)}|$ for ${m,n \in \mathbb{N}_0 }$. Using the facts that ${u_1 (m,n)}$ and ${u_2 (m,n)}$ are respectively the solutions of \eqref{e3.5}-\eqref{e1.2} and of \eqref{e3.6}-\eqref{e1.2} and the hypotheses, we observe that \begin{equation} \begin{aligned} h(m,n) &\leq \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {|{f({s,t,u_1 ({s,t}),Gu_1({s,t}),\mu })} } } { - f({s,t,u_2 ({s,t}),Gu_2({s,t}),\mu })}| \\ &\quad + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {|{f({s,t,u_2 ({s,t}),Gu_2({s,t}),\mu }) { - f({s,t,u_2 ({s,t}),Gu_2({s,t}),\mu _0 })}|} } } \\ &\leq \bar a(m,n) + \sum_{s = 0}^{m - 1} {\sum_{t = 0}^{n - 1} {p({s,t})\Big[{h({s,t}) + \sum_{\sigma = 0}^{s - 1} {\sum_{\tau = 0}^{t - 1} {q({s,t,\sigma ,\tau })h({\sigma ,\tau })} } } \Big]} }. \end{aligned}\label{e3.11} \end{equation} Now an application of Lemma \ref{lem1} yields \eqref{e3.9}, which shows the dependency of solutions of \eqref{e3.5}-\eqref{e1.2} and of \eqref{e3.6}-\eqref{e1.2} on the parameters ${\mu ,\mu _0 }$. \end{proof} \begin{remark} \label{rmk3} \rm We note that the results given in this paper can be extended very easily to study the sum-difference equation \begin{equation} \begin{aligned} &\Delta _2 \Delta _1 u(m,n) + \Delta _2 ({b(m,n)u(m,n)}) \\ &= f({m,n,u(m,n),Gu(m,n),Hu(m,n)}), \end{aligned} \label{e3.12} \end{equation} with the given initial conditions in \eqref{e1.2}, where $G$ is given by \eqref{e1.3} and $H$ is given by \[ Hu(m,n): = \sum_{\sigma = 0}^\infty {\sum_{\tau = 0}^\infty {h({m,n,\sigma ,\tau ,u({\sigma ,\tau })})} } , %\label{e3.13} \] under some suitable conditions on $b,f,g,h$ involved in \eqref{e3.12} by making use of the finite difference inequality given in \cite[Theorem 5.2.3]{p2}. For further results on the qualitative properties of solutions of various finite difference equations, see \cite{a1,m1,p1,p2,p3}. \end{remark} \subsection*{Acknowledgements} The author is grateful to the anonymous referee and Professor Julio G. Dix whose comments and suggestions helped to improve this article. \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal and P. J. Y. Wong; \emph{Advanced Topics in Difference Equations}, Kluwer Academic Publishers, Dordrecht, 1997. \bibitem{h1} P. Henrici; \emph{Discrete variable methods in Ordinary Differential Equations}, John Wiley, New York, 1962. \bibitem{h2} F. B. Hildbrand; \emph{Finite Difference Equations and Simulations}, Prentice-Hall, Englewood Cliffs, 1968. \bibitem{k1} M. Kwapisz; \emph{On the error evaluation of approximate solutions of discrete equations}, Nonlinear Analysis TMA 13 (1989), 1317-1327. \bibitem{l1} M. Lees; \emph{Approximate solutions of parabolic equations}, J. Soc. Ind. Appl. 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