Bul. Ştiinţ. Univ, Baia Mare, Ser. B,

Matematică-Informatică, Vol. XVIII (2002) Nr. 1, 69–72

Dedicated to Costică Mustăţa on his 60^th anniversary

On the approximation of solutions to nonlinear operators between metric spaces

Ion Păvăloiu

Abstract.

A Gauss-Seidel-type method for the solution of linear systems, based on the decomposition of the system matrix into four matrices blocks, has been proposed by R. Varga in [3]. The convergence of this method was studied in [1] and [2].

In this paper we shall extend the ideas contained in the above quoted works to the case of nonlinear system equations.

In the paper [3], R. Varga proposes a Gauss-Seidel type method for solving linear systems, which is based on decomposing the matrix of the system in four submatrix blocks. The convergence of this method has been proved in [1] and [2].

We shall extend these idees to the case of nonlinear systems.

Let $\left(X_{i},\rho_{i}\right),i=1,2$ , two complete metric spaces, and $X=X_{1}\times X_{2}$ , $F:X\rightarrow X_{1}$ , $G:X\rightarrow X_{2}$ two mappings. We are interested in studying the existence and uniqueness of the solution of the system

(1)

\left\{\begin{array}[c]{l}u=F\left(u,v\right)\\ v=G\left(u,v\right),\qquad\left(u,v\in X\right).\end{array}\right.

In this sense, we shall consider the sequences $\left(u_{n}\right)_{n\geq 0},\left(v_{n}\right)_{n\geq 0}$ generated by the Gauss-Seidel method, i.e.,

(2)

\left\{\begin{array}[c]{l}u_{n+1}=F\left(u_{n},v_{n}\right)\\ v_{n+1}=G\left(u_{n+1},v_{n}\right),\qquad n=0,1,2,\ldots,\left(u_{0},v_{0}% \right)\in X.\end{array}\right.

Let $D_{i}\subset X_{i},i=1,2$ and $D=D_{1}\times D_{2}$ . We shall assume that $F$ and $G$ verify Lipschitz-type conditions on $D$ , i.e., there exist $\alpha,\beta,a,b\geq 0$ such that

(3)

\left\{\begin{array}[c]{c}\rho_{1}\left(F\left(x_{1},y_{1}\right),F\left(x_{2}% ,y_{2}\right)\right)\leq\alpha\rho_{1}\left(x_{1},x_{2}\right)+\beta\rho_{2}% \left(y_{1},y_{2}\right)\\ \rho_{2}\left(G\left(x_{1},y_{1}\right),G\left(x_{2},y_{2}\right)\right)\leq% \alpha\rho_{1}\left(x_{1},x_{2}\right)+b\rho_{2}\left(y_{1},y_{2}\right)\end{% array}\right.

for all $\left(x_{i},y_{i}\right)\subseteq D,i=1,2.$

For the study of the convergence of (2) we consider two sequences of real numbers $\left(f_{n}\right)_{n\geq 0},\left(g_{n}\right)_{n\geq 0}$ with nonnegative terms, obeying the following system of difference inequalities:

(4)

\left\{\begin{array}[c]{l}f_{n}\leq\alpha f_{n-1}+\beta g_{n-1}\\ g_{n}\leq\alpha f_{n}+bg_{n-1},\qquad n=1,2,\ldots,\end{array}\right.

where $\alpha,\beta,a,b$ are given in (3).

We associate to (4) the following system in the unknowns $h, k$ :

(5)

\left\{\begin{array}[c]{c}\alpha+\beta h=hk\\ ah+b=hk\end{array}\right.

It was shown in [1] that if $\alpha,\beta,a,b$ obey

(6)

\left\{\begin{array}[c]{c}\alpha+b+a\beta<2\\ \left(1-\alpha\right)\left(1-b\right)-a\beta>0\\ \alpha>0,b>0,\end{array}\right.

then the system (5) has two real solutions $\left(h_{i},k_{i}\right)$ , $i=1,2$ such that $0<h_{i}$ , $k_{i}<1$ , $i=1,2$ , and one of these solutions has both the components positive. Denote by $\left(h_{1},k_{1}\right)$ this solution, i.e., $h_{1}>0,k_{1}>0$ , so that the elements of the sequences $\left(f_{n}\right)_{n\geq 0}$ and $\left(g_{n}\right)_{n\geq 0}$ obey

(7)

\left\{\begin{array}[c]{l}f_{n}\leq Ch_{1}^{n-1}k_{1}^{n-1}\\ g_{n}\leq Ch_{1}^{n}k_{1}^{n-1},\qquad n=1,2,\ldots\end{array}\right.

where $C=\max\left\{\alpha f_{0}+\beta g_{0},\left(af_{1}+bg_{0}\right)/h_{1}\right\}.$

Let $p_{1}=h_{1}k_{1}$ and $d_{1}>0$ be a positive number such that the sets

(8)		$\displaystyle S_{1}$	$\displaystyle=\big\{x\in X_{1}:\rho_{1}\left(x,u_{0}\right)\leq\tfrac{d_{1}}{1% -p_{1}}\big\};$
	$\displaystyle S_{2}$	$\displaystyle=\big\{x\in X_{2}:\rho_{2}\left(x,v_{0}\right)\leq\tfrac{d_{1}h_{% 1}}{1-p_{1}}\big\},$

verify $S_{i}\subseteq D_{i},\ i=1,2.$

Denoting $f_{n}=\rho_{1}\left(u_{n},u_{n-1}\right),g_{n}=\rho_{2}\left(v_{n},v_{n-1}% \right),n=1,2,\ldots$ and taking into account the above relations we obtain the following result.

Theorem 1.

If the mappings $F$ and $G$ verify conditions (3) on the set $D$ , $S_{1}\times S_{2}\subseteq D$ , the numbers $\alpha,\beta,a,b$ verify (6) and $u_{1}=F\left(u_{0},v_{0}\right),v_{1}=G\left(u_{1},v_{0}\right)$ are such that $\rho_{1}\left(u_{1},u_{0}\right)\leq d_{1},\rho_{2}\left(u_{1},v_{0}\right)% \leq d_{1}h_{1}$ , then the following statements hold:

a)

the sequences $\left(u_{n}\right)_{n\geq 0},\left(v_{n}\right)_{n\geq 0}$ converge, and denoting $\lim u_{n}=\bar{u}$ , $\lim v_{n}=\bar{v}$ , then $\left(\bar{u},\bar{v}\right)$ is the unique solution of (1) in the set $S=S_{1}\times S_{2}$ ;
b)

the following inequalities are true

(9) $\left\{\begin{array}[c]{l}\rho_{1}\left(\bar{u},u_{n}\right)\leq\frac{d_{1}p_{% 1}^{n}}{1-p_{1}}\\ \rho_{2}\left(\bar{v},v_{n}\right)\leq\frac{d_{1}h_{1}p_{1}^{n}}{1-p_{1}},% \qquad n=0,1,\ldots\end{array}\right.$

This theorem is proved using (3) and inequalities (4). We shall apply this Theorem to the study of a Gauss-Seidel type method for solving nonlinear operator equations.

Let $\left(X,\rho\right)$ be a complete metric space and $X^{m},X^{s},X^{m-s},1\leq s\leq m-1$ the chartesian products.

If $u,v\in X^{i}$ , $i=\left\{m,s,m-s\right\}$ , we define the metric in such a space in the following way: let $u=\left(u_{1},\ldots,u_{i}\right)$ , $v=\left(v_{1},\ldots,v_{i}\right)$ and put

(10)

\rho_{i}\left(u,v\right)=\max\limits_{1\leq j\leq i}\left\{\rho\left(u_{j},v_{% j}\right)\right\},\qquad i\in\left\{m,s,m-s\right\}.

Consider the mappings $\varphi_{k}:X^{m}\rightarrow X,k=\overline{1,m}$ , and the following system of equations:

(11)

x_{k}=\varphi_{k}\left(x_{1},x_{2},\ldots,x_{m}\right),\qquad k=\overline{1,m}.

and define the mapping $\bar{F}:X^{s}\times X^{m-s}\rightarrow X^{s}$ resp. $\bar{G}:X^{s}\times X^{m-s}\rightarrow X^{m-s}$ in the following way. If $u=\left(u_{1},\ldots,u_{s}\right)\in X^{s}$ and $v=\left(v_{1},\ldots,v_{m-s}\right)\in X^{m-s}$ then

(12)		$\displaystyle\bar{F}\left(u,v\right)$	$\displaystyle=\left(\varphi_{1}\left(u_{1},u_{2},\ldots,u_{s},v_{1},v_{2},% \ldots,v_{m-s}\right),\ldots,\varphi_{s}\left(u_{1},u_{2},\ldots,u_{s},v_{1},v% _{2},\ldots,v_{m-s}\right)\right)$
	$\displaystyle\bar{G}\left(u,v\right)$	$\displaystyle=\left(\varphi_{s+1}\left(u_{1},u_{2},\ldots,u_{s},v_{1},v_{2},% \ldots,v_{m-s}\right),\ldots,\varphi_{m}\left(u_{1},u_{2},\ldots,u_{s},v_{1},v% _{2},\ldots,v_{m-s}\right)\right).$

For solving of (12) we consider the following iterations.

(13)

\left\{\begin{array}[c]{l}u_{n+1}=\bar{F}\left(u_{n},v_{n}\right)\\ v_{n+1}=\bar{G}\left(u_{n+1},v_{n}\right),\left(u_{0},v_{0}\right)\in X^{s}% \times X^{m-s},\qquad n=0,1,\ldots\end{array}\right.

Assuming that the mappings $\varphi_{k},k=\overline{1,m}$ verify the Lipschitz type conditions, i.e., $\exists a_{kl}\geq 0,k,l=\overline{1,m}$ such that $\forall\left(x_{1},\ldots,x_{m}\right),\left(y_{1},\ldots,y_{m}\right)\in D% \leq X^{m}$ it follows

(14)

\rho\left(\varphi_{k}\left(x_{1},x_{2},\ldots,x_{m}\right),\varphi_{k}\left(y_% {1},y_{2},\ldots,y_{m}\right)\right)\leq\sum\limits_{l=1}^{m}a_{kl}\rho\left(x% _{e},y_{e}\right),\qquad k=\overline{1,m}

Denoting

(15)		$\displaystyle\bar{\alpha}$	$\displaystyle=\max\limits_{1\leq k\leq s}\Big\{{\textstyle\sum\limits_{l=1}^{m% }}a_{kl}\Big\},\quad\bar{\beta}=\max\limits_{1\leq k\leq s}\Big\{{\textstyle% \sum\limits_{l=s+1}^{m}}a_{kl}\Big\}$
	$\displaystyle\bar{a}$	$\displaystyle=\max\limits_{s+1\leq k\leq m}\Big\{{\textstyle\sum\limits_{l=1}^% {s}}a_{kl}\Big\},\quad\bar{b}=\max\limits_{s+1\leq k\leq m}\Big\{{\textstyle% \sum\limits_{l=s+1}^{m}}a_{kl}\Big\}$

then it can be seen that the mappings $\bar{F}$ and $\bar{G}$ obey

\left\{\begin{array}[c]{c}\rho_{s}\left(\bar{F}\left(u,v\right),\bar{F}\left(x% ,y\right)\right)\leq\bar{\alpha}\rho_{s}\left(u,x\right)+\bar{\beta}\rho_{n-s}% \left(v,y\right)\\ \rho_{n-s}\left(\bar{G}\left(u,v\right),\bar{G}\left(x,y\right)\right)\leq\bar% {\alpha}\rho_{s}\left(u,x\right)+\bar{b}\rho_{n-s}\left(v,y\right)\end{array}\right.

$\forall\left(u,v\right),\left(x,y\right)\in D=D^{s}\times D^{m-s}.$

It is clear that if in Theorem 1 we set $X_{1}=X^{s},X_{2}=X^{m-s},\rho_{1}=\rho_{s},\rho_{2}=\rho_{m-s},\alpha=\bar{% \alpha},\beta=\bar{\beta},a=\bar{a},b=\bar{b}$ then $\left(u_{0},v_{0}\right),\left(u_{1},v_{1}\right)$ obey $\rho\left(u_{0},u_{1}\right)\leq d_{1},\rho\left(v_{0},v_{1}\right)\leq d_{1}h% _{1},\bar{S}_{1}=D^{s},\bar{S}_{2}=D^{m-s}$ , where

	$\displaystyle\bar{S}_{1}$	$\displaystyle=\left\{x\in X^{s}:\rho_{s}\left(x,u_{0}\right)\leq\tfrac{d_{1}}{% 1-p_{1}}\right\},$
	$\displaystyle\bar{S}_{2}$	$\displaystyle=\left\{x\in X^{m-s}:\rho_{m-s}\left(x,v_{0}\right)\leq\tfrac{d_{% 1}h_{1}}{1-p_{1}}\right\}$

verify the relations $S_{1}\subseteq D^{s},S_{2}\subseteq D^{m-s}$ , and assuming that the assumptions of Theorem 1 are satisfied, we get the same conclusions regarding the solution of (12).

References

[1] ^†^†margin: clickable $\rightarrow$ Păvăloiu, I., La resolution de systemes d’equations operationnelles a l’aide des methodes ite- ratives, Mathematica, vol. 11 (34) pp.137–141 (1969).
[2] Păvăloiu, I., ^†^†margin: available soon,
refresh and click here $\rightarrow$ Introducere în teoria aproximării soluţiilor ecuaţiilor, Ed. Dacia (1976).
[3] Varga, R.S., Matrix Iterative Analysis, Englewood Cliffs H.I. Prentice Hall (1962).

Received: 5.05.2002 ”T. Popoviciu” Institute of Numerical Analysis

P.O. Box 68-1, 3400 Cluj-Napoca, Romania

e-mail: pavaloiu@ictp.acad.ro

North University Baia Mare

Department of Mathematics and Computer Science

Victoriei 76, 4800 Baia Mare, Romania

On the approximation of solutions to nonlinear operators between metric spaces

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