Open Access

Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation

  • M. I. Berenguer1,
  • M. V. Fernández Muñoz1,
  • A. I. Garralda Guillem1 and
  • M. Ruiz Galán1Email author
Fixed Point Theory and Applications20092009:735638

DOI: 10.1155/2009/735638

Received: 15 May 2009

Accepted: 8 July 2009

Published: 4 August 2009

Abstract

The authors present a method of numerical approximation of the fixed point of an operator, specifically the integral one associated with a nonlinear Fredholm integral equation, that uses strongly the properties of a classical Schauder basis in the Banach space .

1. Introduction

Let us consider the nonlinear Fredholm integral equation of the second kind
(1.1)
where and and are continuous functions. By defining in the Banach space of those continuous and real-valued functions defined on (usual sup norm) the integral operator as
(1.2)

then the Banach fixed point, theorem guarantees that, under certain assumptions, has a unique fixed point; that is, the Fredholm integral equation has exactly one solution. Indeed, assume in addition that is a Lipschitz function at its third variable with Lipschitz constant and that then the operator is contractive with contraction number , and thus has a unique fixed point . Moreover, where is any continuous function on Since in general it is not possible to calculate explicitly from a the sequence of functions we define in this work a new sequence of functions, denoted by obtained recursively making use of certain Schauder basis in (Banach space of those continuous real-valued functions on endowed with its usual sup norm). More concretely, we get from , approximating by means of the sequence of projections of such Schauder basis.

2. Numerical Approximation of the Solution

We start by recalling certain aspects about some well-known Schauder bases in the Banach spaces and .

Let us consider the usual Schauder basis in that is, for a dense sequence of distinct points in with and , we define
(2.1)
and for is the piecewise linear and continuous function with nodes satisfying for all and From this Schauder basis we define the usual Schauder basis for We consider the bijective mapping ( denotes integer part) given by
(2.2)
and take, for each with ,
(2.3)
The sequence is the usual Schauder basis in (see [1]). We will denote by and , respectively, the sequences of biorthogonal functionals and projections associated with such basis, that is, given the (continuous) functionals verify
(2.4)
and the (continuous) projections are defined by
(2.5)

Let us now introduce some notational conventions. For each the definition of projection just needs the first points of the sequence ordered in an increasing way that will be denoted by , and in addition we will write

We now describe idea of the numerical method proposed. The beginning point is the operator formulation of the integral Fredholm equation; from an initial function and since in general we cannot calculate explicitly we approximate this function in the following way: let
(2.6)
so
(2.7)

where is an adequate integer. We denote the last function by and repeat the same construction. Then we define recursively for each and ,

(2.8)
Now we state some technical results in order to study the error . In the first of them we give a bound for the distance between a continuous function and its projections. It is not difficult to prove it as a consequence of the Mean Value Theorem and the following interpolation property satisfied by the sequence of projections (see [1]): whenever and then
(2.9)

Lemma 2.1.

Let , let , and let be the sequence of projections associated with the basis , then it holds that
(2.10)
Let us introduce some notation, useful in what follows: given we write
(2.11)

Lemma 2.2.

Suppose that , and is the operator given by
(2.12)
Then, maintaining the preceding notation, we have that for all ,
(2.13)

Proof.

Since and is a Schauder basis for the Banach space then
(2.14)
On the other hand, taking into account the definition of , we have that
(2.15)
Finally, in view of Lemma 2.1 we arrive at
(2.16)

Finally we arrive at the following estimation of the error.

Theorem 2.3.

Assume that , , is a lipschitzian function at its third variable with Lipschitz constant with and that is the unique fixed point of the integral operator defined by
(2.17)
Suppose in addition that and that satisfy
(2.18)
Then, with the previous notation, it is satisfied that
(2.19)

where .

Proof.

We begin with the triangular inequality
(2.20)
In order to obtain a bound for the first right-hand side term we observe that operator is contractive, with contraction constant . Hence the Banach fixed point Theorem gives that
(2.21)
(2.22)
For deducing a bound for the second right-hand side term of (2.20), we use Lemma 2.2 and the assumption in the following chain of inequalities:
(2.23)
Once again, in view of Lemma 2.2 it follows that
(2.24)
Therefore, inequalities (2.20), (2.22), and (2.24) allow us to conclude that
(2.25)

as announced.

Remark 2.4 s.

(?1) The linear case was previously stated in [2]. For a general overview of the classical methods, see [3, 4].

(?2) The use of Schauder bases in the numerical study of integral and differential equations has been previously considered in [57] or [8].

(?3) For other approximating methods in Hilbert or Banach spaces, we refer to [9, 10].

3. Numerical Examples

We finally illustrate the numerical method proposed above by means of the two following examples. In both of them we choose the dense subset of
(3.1)

to construct the Schauder bases in and . To define the sequence of approximating functions we have taken an initial function and for all with different values of of the form with For such a choice, the value appearing in Lemma 2.2 is for all

Example 3.1.

Let us consider the nonlinear Fredholm integral equation of the second kind in :
(3.2)
whose analytical solution is the function In Table 1 we exhibit the absolute errors committed in nine points in when we approximate the exact solution by the iteration , by considering different values of .
Table 1

Example 3.1.

Example 3.2.

Now we consider the following Fredholm integral equation appearing in [5, Example (11.2.1)]:
(3.3)
where is defined in such a way that is the exact solution. We denote by the approximation of the exact solution given by the collocation method and by the error:
(3.4)
where are the nodes of the collocation method. Now write for the error
(3.5)
with being the approximation obtained with our method, with for and choosing in such a way that
(3.6)
In Table 2 we show the errors for both methods.
Table 2

Example 3.2.

Remark 3.3.

Although the errors obtained in the preceding example by our algorithm are similar to those derived from the collocation method, the computational cost is quite different: in order to apply the collocation method we need to solve high-order linear systems of algebraical equations, but for our method we just calculate linear combinations of scalar obtained by evaluating adequate functions. Indeed, the sequence of biorthogonal functionals satisfies the following easy property (see [1]): for all ,
(3.7)
while for all if
(3.8)

Obviously, this easy way of determining the biorthogonal functionals and consequently the approximating functions (integrals of a piecewise linear function) is equally valid in the general nonlinear case.

Declarations

Acknowledgments

This research partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533 and by Junta de Andalucía Grant FQM359.

Authors’ Affiliations

(1)
Departamento de Matemática Aplicada, Universidad de Granada, E.U. Arquitectura Técnica

References

  1. Semadeni Z: Schauder Bases in Banach Spaces of Continuous Functions, Lecture Notes in Mathematics. Volume 918. Springer, Berlin, Germany; 1982:v+136.Google Scholar
  2. Domingo Montesinos M, Garralda Guillem AI, Ruiz Galán M: Fredholm integral equations and Schauder bases. In VIII Journées Zaragoza-Pau de Mathématiques Appliquées et de Statistiques, Monografías del Seminario Matemático García de Galdeano. Volume 31. Zaragoza University Press, Zaragoza, Spain; 2004:121–128.Google Scholar
  3. Golberg MA: Numerical Solution of Integral Equations, Mathematical Concepts and Methods in Science and Engineering. Volume 42. Plenum Press, New York, NY, USA; 1990:xiv+417.View ArticleGoogle Scholar
  4. Atkinson KE, Han W: Theoretical Numerical Analysis. 2nd edition. Springer, New York, NY, USA; 2005.View ArticleMATHGoogle Scholar
  5. Berenguer MI, Fortes MA, Garralda Guillem AI, Ruiz Galán M: Linear Volterra integro-differential equation and Schauder bases. Applied Mathematics and Computation 2004,159(2):495–507. 10.1016/j.amc.2003.08.132MathSciNetView ArticleMATHGoogle Scholar
  6. Gámez D, Garralda Guillem AI, Ruiz Galán M: Nonlinear initial-value problems and Schauder bases. Nonlinear Analysis: Theory, Methods & Applications 2005,63(1):97–105. 10.1016/j.na.2005.05.005MathSciNetView ArticleMATHGoogle Scholar
  7. Gámez D, Garralda Guillem AI, Ruiz Galán M: High-order nonlinear initial-value problems countably determined. Journal of Computational and Applied Mathematics 2009,228(1):77–82. 10.1016/j.cam.2008.08.039MathSciNetView ArticleMATHGoogle Scholar
  8. Palomares A, Galán MRuiz: Isomorphisms, Schauder bases in Banach spaces, and numerical solution of integral and differential equations. Numerical Functional Analysis and Optimization 2005,26(1):129–137. 10.1081/NFA-200051625MathSciNetView ArticleMATHGoogle Scholar
  9. Gu Z, Li Y: Approximation methods for common fixed points of mean nonexpansive mapping in Banach spaces. Fixed Point Theory and Applications 2008, 2008:-7.Google Scholar
  10. Yao Y, Chen R: Iterative algorithm for approximating solutions of maximal monotone operators in Hilbert spaces. Fixed Point Theory and Applications 2007, 2007:-8.Google Scholar

Copyright

© M. I. Berenguer et al. 2009

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