This work presents an analysis of the error that is committed upon having obtained the approximate solution of the nonlinear Fredholm-Volterra-Hammerstein integral equation by means of a method for its numerical resolution. The main tools used in the study of the error are the properties of Schauder bases in a Banach space.

1. Introduction

In this paper we consider the following nonlinear mixed Fredholm-Volterra-Hammerstein integral equation: where , and the kernels are assumed to be known continuous functions, and is the unknown function to be determined.

Equation (1.1) arises in a variety of applications in many fields, including continuum mechanics, potential theory, electricity and magnetism, three-dimensional contact problems, and fluid mechanics, and so forth (see, e.g., [14]). Several numerical methods for approximating the solution of integral, and integrodifferential equations are known (see, e.g., [58]). For Fredholm-Volterra-Hammerstein integral equations, the classical method of successive approximations was introduced in [9]. An optimal control problem method was presented in [10], and a collocation-type method was developed in [1113]. Computational methods based on Bernstein operational matrices and the Chebyshev approximation method were presented in [14, 15], respectively.

The use of fixed point techniques and Schauder bases, in the field of numerical resolution of differential, integral and integro-differential equations, allows for the development of new methods providing significant improvements upon other known methods (see [1623]).

In this work we make an analysis of the error committed upon having obtained the approximate solution of the nonlinear Fredholm-Volterra-Hammerstein integral equation, using the theorem of Banach fixed point and Schauder bases (see [21], for a detailed description of the numerical method used in a more general equation).

In order to recall the aforementioned numerical method, let and be the Banach spaces of all continuous and real-valued functions on and endowed with their usual supnorms. Throughout this paper we will make the following assumptions on and for . (i)Since , there exists such that for all .(ii) are functions such that there exists such that for and for all . (iii).

We organize this paper as follows. In Section 2, we reformulate (1.1) in terms of a convenient integral operator and we describe the numerical method used. The study of the error is described in Section 3. Finally, in Section 4 we show some illustrative examples.

2. Analytical Preliminaries

In this section we recall, in a summarized form, the concepts and results relative to the numerical method used for the study of the error that we carried out.

Let us start by observing that (1.1) is equivalent to the problem of finding fixed points of the operator defined by

A direct calculation over leads to for all , where we denote . As the operator defined in (2.1) satisfies (2.2), under condition (iii) and from the Banach fixed-point theorem, it follows that there exists a unique fixed point for that is the unique solution of (1.1). In addition, for each , we have and in particular .

But it is not possible, in an explicit way, to calculate the sequence of iterations , to obtain the unique sequence of (1.1), for which reason a numerical method is needed in order to approximate the fixed point of .

Now we recall the concrete Schauder bases in the spaces and . Let be a dense sequence of distinct points in such that and . We set for , and for , and we let be a piecewise linear continuous function on with nodes at , uniquely determined by the relations and for . We denote by the sequence of associated projections and the coordinate functionals. It is easy to check that is a Schauder basis in (see [24]).

From the Schauder basis in , we can build another Schauder basis of (see [25, 26]). It is sufficient to consider for all , with , where for a real number , will denote its integer part and is the bijective mapping defined by

We denote by the sequence of associated projections and by the coordinate functionals. The Schauder basis of has similar properties to the ones for the one-dimensional case. See Table 1 and note under some weak conditions (see the last row, which is derived easily from the third row of Table 1, resp., and the Mean-Value theorems for one and two variables) we can estimate the rate of the convergence of the sequence of projections in the one and two-dimensional cases, where we consider the dense subset of distinct points in , as the set ordered in an increasing way for , and denotes the maximum distance between two consecutive points of .

Let us consider the continuous integral operator defined in (2.1). Let , and the functions , defined for , . Let and be the sequences of scalars satisfying , . Then for all , we have that

The equality (2.5) enables us to determine, in an elemental way, the image of any continuous function under the operator . However, it does not seem to be a usable expression due to the two infinite sums appearing in it. For this reason, the aforementioned sums are truncated.

3. Study of the Error

In this section we realize a new study of the error, obtaining one bound of it. Supposing conditions of regularity in the functions data, we improve and complete the study realized in [21].

Let and consider and for , define inductively for the following functions: where and .

Proposition 3.1. The sequence is uniformly bounded.

Proof. Let ,  , and we have for all and
For the monotonicity of the Schauder basis, we have Therefore,
Applying recursively this process we get for all . Then is uniformly bounded.

Remark 3.2. For , the sequence is uniformly bounded, as it follows Proposition 3.1 and the fact that for is Lipschitz in its second variable.

Proposition 3.3. Let , and for , ,   such that and satisfy a global Lipschitz condition in the last variable. Let , and define inductively as in (3.2), (3.3), and (3.4) the functions , and , respectively. Then are uniformly bounded.

Proof. From (3.2) and (3.3), we have, respectively, that for all , , , and therefore by the conditions over ,  , and Remark 3.2, , are uniformly bounded.
Observe that In view of the monotonicity of the Schauder basis, we have and hence the sequence is uniformly bounded.
On the other hand from (3.2) and (3.3), respectively, we have
For , let , and we have for all and with as the Lipschitz constant of in the last variable.
By repeating the previous argument, we have where , and is the Lipschitz constant of in the last variable.
Therefore by the conditions over ,  , Proposition 3.1, Remark 3.2, and (3.11), are uniformly bounded.

Proposition 3.4. With the previous notation and the same hypothesis as in Proposition 3.3, there is such that for all and , we have

Proof. In the last property in Table 1, take and .

In the result below we show that the sequence defined in (3.4) approximates the exact solution of (1.1) as well as giving an upper bound of the error committed.

Theorem 3.5. With the previous notation and the same hypothesis as in Proposition 3.3, let , , , and be a set of positive numbers such that for all we have Then, Moreover, if is the exact solution of the integral equation (1.1), then the error is given by

Proof. First we deal with proving (3.18). For all and , Proposition 3.4 gives
To conclude the proof, we derive (3.19). From (2.3), we have and in addition, on the other hand, applying recursively (2.2) and (3.18), we obtain
Then we use the triangular inequality and the proof is complete in view of (3.21) and (3.22).

Remark 3.6. Under the hypotheses of Theorem 3.5, let us observe that by the inequality (3.19) we have
The first sumand on the right hand side approximates zero when increases; with respect to the second sumand, since the points of the partition can be chosen in such a way that becomes so close to zero as we desire, the can become so small as we desire, arriving in this way at an explicit control of the error committed.
Therefore, given , there exists such that when choosing sufficiently small.

4. Numerical Examples

In this last section we illustrate the results previously developed, stressing the significance of inequality (3.19) in Theorem 3.5, as mentioned in Remark 3.6.

First of all, we show how the numerical method works, because we use it later in the estimation of the error. For solving the numerical example, Mathematica 7 is used, and to construct the Schauder basis in , we considered the particular choice , and for , if where are integers. To define the sequence , we take and (for all ). In Tables 2 and 3, we exhibit, for , , and , the absolute errors committed in eight representative points of when we approximate the exact solution by the iteration . Its numerical results are also given in Figures 1 and 2, respectively.

Example 4.1. We solve (1.1) with , , , , and with the exact solution .

Example 4.2. We solve (1.1) with , , , , and with the exact solution .

Now we realize that the choice of a particular , determining the dyadic partition of the interval from the first nodes, and in such a way that the error is less than a fixed positive , that is, , can be easily determined practically: it suffices to compute, once again by means of Mathematica 7, the error. To this end, since it is measured in terms of the supnorm, we consider the nodes 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1 and maximum of the absolute values of the differences between the values of the exact solution and the approximation obtained for the third iteration (). The numerical tests are given in Table 4 and correspond to the nonlinear mixed Fredhol-Volterra-Hammerstein equations considered in Examples 4.1 and 4.2, respectively.


This paper is partially supported by Junta de Andaluca Grant FQM359.