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 .
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1. Introduction
Let us consider the nonlinear Fredholm integral equation of the second kind
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ3_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ4_HTML.gif)
and take, for each with
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ6_HTML.gif)
and the (continuous) projections are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ8_HTML.gif)
so
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ9_HTML.gif)
where is an adequate integer. We denote the last function by
and repeat the same construction. Then we define recursively for each
and
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ10_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ11_HTML.gif)
Lemma 2.1.
Let , let
, and let
be the sequence of projections associated with the basis
, then it holds that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ12_HTML.gif)
Let us introduce some notation, useful in what follows: given we write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ13_HTML.gif)
Lemma 2.2.
Suppose that ,
and
is the operator given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ14_HTML.gif)
Then, maintaining the preceding notation, we have that for all ,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ15_HTML.gif)
Proof.
Since and
is a Schauder basis for the Banach space
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ16_HTML.gif)
On the other hand, taking into account the definition of , we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ17_HTML.gif)
Finally, in view of Lemma 2.1 we arrive at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ19_HTML.gif)
Suppose in addition that and that
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ20_HTML.gif)
Then, with the previous notation, it is satisfied that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ21_HTML.gif)
where .
Proof.
We begin with the triangular inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ22_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ23_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ24_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ25_HTML.gif)
Once again, in view of Lemma 2.2 it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ26_HTML.gif)
Therefore, inequalities (2.20), (2.22), and (2.24) allow us to conclude that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ27_HTML.gif)
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
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ28_HTML.gif)
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 :
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ29_HTML.gif)
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
.
Example 3.2.
Now we consider the following Fredholm integral equation appearing in [5, Example (11.2.1)]:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ30_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ31_HTML.gif)
where are the nodes of the collocation method. Now write
for the error
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ32_HTML.gif)
with being the approximation obtained with our method, with
for
and choosing
in such a way that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ33_HTML.gif)
In Table 2 we show the errors for both methods.
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ34_HTML.gif)
while for all if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F735638/MediaObjects/13663_2009_Article_1172_Equ35_HTML.gif)
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.
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Acknowledgments
This research partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533 and by Junta de Andalucía Grant FQM359.
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Berenguer, M.I., Fernández Muñoz, M.V., Garralda Guillem, A.I. et al. Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation. Fixed Point Theory Appl 2009, 735638 (2009). https://doi.org/10.1155/2009/735638
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DOI: https://doi.org/10.1155/2009/735638