Abstract
A Gauss-Seidel method for linear systems, based on decomposing the matrix system into four submatrices blocks, has been proposed by Varga, and subsequently studied in two papers. Here we extend these ideas to the case of nonlinear systems of equations.
Authors
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
nonlinear systems of equations; Gauss-Seidel method; convergence.
Cite this paper as:
I. Păvăloiu, On the approximation of solutions to nonlinear operators between metric spaces, Bul. Ştiinţ. Univ. Baia Mare, 18 (2002) no. 1, pp. 69-72.
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On the approximation of solutions to nonlinear operators between metric spaces
Bul. Ştiinţ. Univ, Baia Mare, Ser. B,
Matematică-Informatică, Vol. XVIII (2002) Nr. 1, 69–72
Dedicated to Costică Mustăţa on his 60th anniversary
On the approximation of solutions to nonlinear operators between metric spaces
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 , two complete metric spaces, and , , two mappings. We are interested in studying the existence and uniqueness of the solution of the system
| (1) |
In this sense, we shall consider the sequences generated by the Gauss-Seidel method, i.e.,
| (2) |
Let and . We shall assume that and verify Lipschitz-type conditions on , i.e., there exist such that
| (3) |
for all
For the study of the convergence of (2) we consider two sequences of real numbers with nonnegative terms, obeying the following system of difference inequalities:
| (4) |
where are given in (3).
We associate to (4) the following system in the unknowns :
| (5) |
It was shown in [1] that if obey
| (6) |
then the system (5) has two real solutions , such that , , , and one of these solutions has both the components positive. Denote by this solution, i.e., , so that the elements of the sequences and obey
| (7) |
where
Let and be a positive number such that the sets
| (8) | ||||
verify
Denoting and taking into account the above relations we obtain the following result.
Theorem 1.
If the mappings and verify conditions (3) on the set , , the numbers verify (6) and are such that , then the following statements hold:
-
a)
the sequences converge, and denoting , , then is the unique solution of (1) in the set ;
-
b)
the following inequalities are true
(9)
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 be a complete metric space and the chartesian products.
If , , we define the metric in such a space in the following way: let , and put
| (10) |
Consider the mappings , and the following system of equations:
| (11) |
and define the mapping resp. in the following way. If and then
| (12) | ||||
For solving of (12) we consider the following iterations.
| (13) |
Assuming that the mappings verify the Lipschitz type conditions, i.e., such that it follows
| (14) |
Denoting
| (15) | ||||
then it can be seen that the mappings and obey
It is clear that if in Theorem 1 we set then obey , where
verify the relations , and assuming that the assumptions of Theorem 1 are satisfied, we get the same conclusions regarding the solution of (12).
References
- [1] ††margin: clickable 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 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