Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation
© M. I. Berenguer et al. 2009
Received: 15 May 2009
Accepted: 8 July 2009
Published: 4 August 2009
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 .
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 ,
Finally we arrive at the following estimation of the error.
Remark 2.4 s.
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
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.
This research partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533 and by Junta de Andalucía Grant FQM359.
- 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
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.