- Research Article
- Open Access

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

- M. I. Berenguer
^{1}, - M. V. Fernández Muñoz
^{1}, - A. I. Garralda Guillem
^{1}and - M. Ruiz Galán
^{1}Email author

**2009**:735638

https://doi.org/10.1155/2009/735638

© M. I. Berenguer et al. 2009

**Received:**15 May 2009**Accepted:**8 July 2009**Published:**4 August 2009

## Abstract

## Keywords

- Banach Space
- Integral Operator
- Fixed Point Theorem
- Collocation Method
- Fredholm Integral Equation

## 1. Introduction

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 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

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

Lemma 2.1.

Lemma 2.2.

Proof.

Finally we arrive at the following estimation of the error.

Theorem 2.3.

Proof.

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 [5–7] or [8].

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

## 3. Numerical Examples

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.

Example 3.2.

Remark 3.3.

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

## References

- Semadeni Z:
*Schauder Bases in Banach Spaces of Continuous Functions, Lecture Notes in Mathematics*.*Volume 918*. Springer, Berlin, Germany; 1982:v+136.Google Scholar - 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 - 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 - Atkinson KE, Han W:
*Theoretical Numerical Analysis*. 2nd edition. Springer, New York, NY, USA; 2005.View ArticleMATHGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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

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