 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
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
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 .
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 realvalued 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 wellknown 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.
where .
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 3.1.









































Example 3.2.
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. MTM200612533 and by Junta de Andalucía Grant FQM359.
Authors’ Affiliations
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