Analysis of the Error in a Numerical Method Used to Solve Nonlinear Mixed Fredholm-Volterra-Hammerstein Integral Equations

Abstract

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: 𝑥(𝑡)=𝑦₀(𝑡)+_𝛼^𝛼+𝛽𝑘₁(𝑡,𝑠)𝑔₁(𝑠,𝑥(𝑠))𝑑𝑠+^𝑡_𝛼𝑘₂(𝑡,𝑠)𝑔₂(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝛼,𝛼+𝛽, (1.1) where 𝑦₀∶[𝛼,𝛼+𝛽]→ℝ, 𝑔₁,𝑔₂∶[𝛼,𝛼+𝛽]×ℝ→ℝ and the kernels 𝑘₁,𝑘₂∶[𝛼,𝛼+𝛽]²→ℝ are assumed to be known continuous functions, and 𝑥∶[𝛼,𝛼+𝛽]→ℝ 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., [1–4]). Several numerical methods for approximating the solution of integral, and integrodifferential equations are known (see, e.g., [5–8]). 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 [11–13]. 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 [16–23]).

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 𝐶([𝛼,𝛼+𝛽]) and 𝐶([𝛼,𝛼+𝛽]²) be the Banach spaces of all continuous and real-valued functions on [𝛼,𝛼+𝛽] and [𝛼,𝛼+𝛽]² endowed with their usual supnorms. Throughout this paper we will make the following assumptions on 𝑘_𝑖 and 𝑔_𝑖 for 𝑖∈{1,2}. (i)Since 𝑘_𝑖∈𝐶([𝛼,𝛼+𝛽]²), there exists 𝑀_𝑘𝑖≥0 such that |𝑘_𝑖(𝑡,𝑠)|≤𝑀_𝑘𝑖 for all (𝑡,𝑠)∈[𝛼,𝛼+𝛽]².(ii)𝑔_𝑖∶[𝛼,𝛼+𝛽]×ℝ→ℝ are functions such that there exists 𝐿_𝑔𝑖>0 such that |𝑔_𝑖(𝑠,𝑦)−𝑔_𝑖(𝑠,𝑧)|≤𝐿_𝑔𝑖|𝑦−𝑧| for 𝑠∈[𝛼,𝛼+𝛽] and for all 𝑦,𝑧∈ℝ. (iii)𝛽∑²_𝑖=1𝑀_𝑘𝑖𝐿_𝑔𝑖<1.

We organize this paper as follows. In Section 2, we reformulate (1.1) in terms of a convenient integral operator 𝑇 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 𝑇∶𝐶([𝛼,𝛼+𝛽])→𝐶([𝛼,𝛼+𝛽]) defined by (𝑇𝑥)(𝑡)∶=𝑦₀(𝑡)+_𝛼^𝛼+𝛽𝑘₁(𝑡,𝑠)𝑔₁((𝑠+𝑠,𝑥))𝑑𝑠_𝑡𝛼𝑘₂(𝑡,𝑠)𝑔₂(𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈𝛼,𝛼+𝛽,𝑥∈𝐶𝛼,𝛼+𝛽. (2.1)

A direct calculation over 𝑇 leads to 𝑇𝑦₁−𝑇𝑦₂≤𝑀𝑦₁−𝑦₂ (2.2) for all 𝑦₁,𝑦₂∈𝐶([𝛼,𝛼+𝛽]), where we denote 𝑀∶=𝛽∑²_𝑖=1𝑀_𝑘𝑖𝐿_𝑔𝑖. As the operator 𝑇 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 𝑥∈𝐶([𝛼,𝛼+𝛽]) for 𝑇 that is the unique solution of (1.1). In addition, for each 𝑥∈𝐶([𝛼,𝛼+𝛽]), we have𝑇^𝑚≤𝑀𝑥−𝑥^𝑚‖𝑇𝑥‖1−𝑀𝑥− (2.3) and in particular 𝑥=lim_𝑚𝑇^𝑚𝑥.

But it is not possible, in an explicit way, to calculate the sequence of iterations {𝑇^𝑚}_𝑚≥1, to obtain the unique sequence 𝑥 of (1.1), for which reason a numerical method is needed in order to approximate the fixed point of 𝑇.

Now we recall the concrete Schauder bases in the spaces 𝐶([𝛼,𝛼+𝛽]) and 𝐶([𝛼,𝛼+𝛽]²). Let {𝑡_𝑛}_𝑛≥1 be a dense sequence of distinct points in [𝛼,𝛼+𝛽] such that 𝑡₁=𝛼 and 𝑡₂=𝛼+𝛽. We set 𝑏₁(𝑡)∶=1 for 𝑡∈[𝛼,𝛼+𝛽], and for 𝑛≥1, and we let 𝑏_𝑛 be a piecewise linear continuous function on [𝛼,𝛼+𝛽] with nodes at {𝑡_𝑗∶1≤𝑗≤𝑛}, uniquely determined by the relations 𝑏_𝑛(𝑡_𝑛)=1 and 𝑏_𝑛(𝑡_𝑘)=0 for 𝑘<𝑛. We denote by {𝑃_𝑛}_𝑛≥1 the sequence of associated projections and {𝑏^∗_𝑛}_𝑛≥1 the coordinate functionals. It is easy to check that {𝑏_𝑛}_𝑛≥1 is a Schauder basis in 𝐶([𝛼,𝛼+𝛽]) (see [24]).

From the Schauder basis {𝑏_𝑛}_𝑛≥1 in 𝐶([𝛼,𝛼+𝛽]), we can build another Schauder basis {𝐵_𝑛}_𝑛≥1 of 𝐶([𝛼,𝛼+𝛽]²) (see [25, 26]). It is sufficient to consider 𝐵_𝑛(𝑡,𝑠)∶=𝑏_𝑖(𝑡)𝑏_𝑗(𝑠) for all 𝑡,𝑠∈[𝛼,𝛼+𝛽], with 𝜏(𝑛)=(𝑖,𝑗), where for a real number 𝑝, [𝑝] will denote its integer part and 𝜏=(𝜏₁,𝜏₂)∶ℕ→ℕ×ℕ is the bijective mapping defined by 𝜏(𝑛)√∶=𝑛,√𝑛,if√𝑛=√√𝑛,𝑛−𝑛²,√𝑛+1,if0<𝑛−√𝑛²≤√𝑛,√𝑛+1,𝑛−√𝑛²−√𝑛,if√𝑛<𝑛−√𝑛². (2.4)

We denote by {𝑄_𝑛}_𝑛≥1 the sequence of associated projections and by {𝐵^∗_𝑛}_𝑛≥1 the coordinate functionals. The Schauder basis {𝐵_𝑛}_𝑛≥1 of 𝐶([𝛼,𝛼+𝛽]²) 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 {𝑡_𝑖}_𝑖≥1 of distinct points in [𝛼,𝛼+𝛽], 𝑇_𝑛 as the set {𝑡₁,…,𝑡_𝑛} ordered in an increasing way for 𝑛≥2, and Δ𝑇_𝑛 denotes the maximum distance between two consecutive points of 𝑇_𝑛.

Table 1 

Properties of the univariate and bivariate Schauder bases.

Let us consider the continuous integral operator 𝑇∶𝐶([𝛼,𝛼+𝛽])→𝐶([𝛼,𝛼+𝛽]) defined in (2.1). Let 𝑥∈𝐶([𝛼,𝛼+𝛽]), and the functions 𝜙₁,𝜙₂∈𝐶([𝛼,𝛼+𝛽]²), defined for 𝜙₁(𝑡,𝑠)=𝑘₁(𝑡,𝑠)𝑔₁(𝑠,𝑥(𝑠)), 𝜙₂(𝑡,𝑠)=𝑘₂(𝑡,𝑠)𝑔₂(𝑠,𝑥(𝑠)). Let {𝜆_𝑛}_𝑛≥1 and {𝜇_𝑛}_𝑛≥1 be the sequences of scalars satisfying 𝜙₁=∑_𝑛≥1𝜆_𝑛𝐵_𝑛, 𝜙₂=∑_𝑛≥1𝜇_𝑛𝐵_𝑛. Then for all 𝑡∈[𝛼,𝛼+𝛽], we have that (𝑇𝑥)(𝑡)=𝑦₀(𝑡)+_𝑛≥1𝜆_𝑛𝛼^𝛼+𝛽𝐵_𝑛(𝑡,𝑠)𝑑𝑠+_𝑛≥1𝜇_𝑛^𝑡_𝛼𝐵_𝑛(𝑡,𝑠)𝑑𝑠. (2.5)

The equality (2.5) enables us to determine, in an elemental way, the image of any continuous function under the operator 𝑇. 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 𝑥∈𝐶([𝛼,𝛼+𝛽]) and consider 𝑥₀(𝑡)∶=𝑥(𝑡)∈𝐶𝛼,𝛼+𝛽, (3.1) and for 𝑚∈ℕ, define inductively for 𝑟∈{1,…,𝑚} the following functions: 𝜎_𝑟−1(𝑡,𝑠)∶=𝑘₁(𝑡,𝑠)𝑔₁𝑠,𝑥_𝑟−1(𝑠),𝜓_𝑟−1(𝑡,𝑠)∶=𝑘₂(𝑡,𝑠)𝑔₂𝑠,𝑥_𝑟−1(𝑠),𝑥_𝑟(𝑡)∶=𝑦₀(𝑡)+_𝛼^𝛼+𝛽𝑄_𝑛2𝑟𝜎_𝑟−1(𝑡,𝑠)𝑑𝑠+^𝑡_𝛼𝑄_𝑛2𝑟𝜓_𝑟−1(𝑡,𝑠)𝑑𝑠, (3.2)(3.3)(3.4) where 𝑡,𝑠∈[𝛼,𝛼+𝛽] and 𝑛_𝑟∈ℕ.

Proposition 3.1. The sequence {𝑥_𝑟}_𝑟≥1 is uniformly bounded.

Proof. Let 𝑅=max{|𝑔₁(𝑠,0)|∶𝑠∈[𝛼,𝛼+𝛽]},  𝑆=max{|𝑔₂(𝑠,0)|∶𝑠∈[𝛼,𝛼+𝛽]}, and we have for all 𝑟≥1 and (𝑡,𝑠)∈[𝛼,𝛼+𝛽]²𝜎_𝑟−1(𝑡,𝑠)=𝑘₁(𝑡,𝑠)𝑔₁𝑠,𝑥_𝑟−1(𝑠)≤𝑀_𝑘1𝑔₁𝑠,𝑥_𝑟−1(𝑠)−𝑔₁(𝑠,0)+𝑔₁(𝑠,0)≤𝑀_𝑘1𝐿_𝑔1𝑥_𝑟−1(𝑠,𝜓)+𝑅_𝑟−1(𝑡,𝑠)=𝑘₂(𝑡,𝑠)𝑔₂𝑠,𝑥_𝑟−1(𝑠)≤𝑀_𝑘2𝑔₂𝑠,𝑥_𝑟−1(𝑠)−𝑔₂(𝑠,0)+𝑔₂(𝑠,0)≤𝑀_𝑘2𝐿_𝑔2𝑥_𝑟−1(𝑠)+𝑆. (3.5)
For the monotonicity of the Schauder basis, we have 𝑥_𝑟(𝑡)≤𝑦₀(𝑡)+_𝛼^𝛼+𝛽𝑄_𝑛2𝑟𝜎_𝑟−1(𝑡,𝑠)𝑑𝑠+^𝑡_𝛼𝑄_𝑛2𝑟𝜓_𝑟−1()≤𝑦𝑡,𝑠𝑑𝑠₀(𝑡)+_𝛼^𝛼+𝛽𝜎_𝑟−1𝑑𝑠+^𝑡_𝛼𝜓_𝑟−1≤𝑦𝑑𝑠₀(𝑡)+𝛽𝑀_𝑘1𝑅+𝑀_𝑘2𝑆+𝑀_𝑘1𝐿_𝑔1𝛼^𝛼+𝛽𝑥_𝑟−1𝑑𝑠+𝑀_𝑘2𝐿_𝑔2^𝑡_𝛼𝑥_𝑟−1𝑑𝑠. (3.6) Therefore, 𝑥_𝑟≤𝑦₀+𝛽𝑀_𝑘1𝑅+𝑀_𝑘2𝑆+𝑀𝑥_𝑟−1. (3.7)
Applying recursively this process we get 𝑥_𝑟≤𝑦₀+𝛽𝑀_𝑘1𝑅+𝑀_𝑘2𝑆1+𝑀+⋯+𝑀^𝑟−1+𝑀^𝑟𝑥₀≤𝑦₀+𝛽𝑀_𝑘1𝑅+𝑀_𝑘2𝑆1−𝑀^𝑟1−𝑀+𝑀^𝑟𝑥₀ (3.8) for all 𝑟≥1. Then {𝑥_𝑟}_𝑟≥1 is uniformly bounded.

Remark 3.2. For 𝑖∈{1,2}, the sequence {𝑔_𝑖(⋅,𝑥_𝑟(⋅)}_𝑟≥1 is uniformly bounded, as it follows Proposition 3.1 and the fact that 𝑔_𝑖 for 𝑖∈{1,2} is Lipschitz in its second variable.

Proposition 3.3. Let 𝑦₀∈𝐶¹([𝛼,𝛼+𝛽]), and for 𝑖∈{1,2}, 𝑘_𝑖∈𝐶¹([𝛼,𝛼+𝛽]²),  𝑔_𝑖∈𝐶¹([𝛼,𝛼+𝛽]×ℝ) such that 𝜕𝑔_𝑖/𝜕𝑠 and 𝜕𝑔_𝑖/𝜕𝑥 satisfy a global Lipschitz condition in the last variable. Let 𝑥₀(𝑡)∶=𝑥(𝑡)∈𝐶¹([𝛼,𝛼+𝛽]), and define inductively as in (3.2), (3.3), and (3.4) the functions 𝜎_𝑟−1, 𝜓_𝑟−1 and 𝑥_𝑟, respectively. Then 𝜕𝜎_𝑟−1𝜕𝑡_𝑟≥1,𝜕𝜎_𝑟−1𝜕𝑠_𝑟≥1,𝜕𝜓_𝑟−1𝜕𝑡_𝑟≥1,𝜕𝜓_𝑟−1𝜕𝑠_𝑟≥1 (3.9) are uniformly bounded.

Proof. From (3.2) and (3.3), we have, respectively, that for all 𝑟≥1, (𝜕𝜎_𝑟−1/𝜕𝑡)(𝑡,𝑠)=(𝜕𝑘₁/𝜕𝑡)(𝑡,𝑠)𝑔₁(𝑠,𝑥_𝑟−1(𝑠)), (𝜕𝜓_𝑟−1/𝜕𝑡)(𝑡,𝑠)=(𝜕𝑘₂/𝜕𝑡)(𝑡,𝑠)𝑔₂(𝑠,𝑥_𝑟−1(𝑠)), and therefore by the conditions over 𝑘₁,  𝑘₂, and Remark 3.2, {𝜕𝜎_𝑟−1/𝜕𝑡}_𝑟≥1, {𝜕𝜓_𝑟−1/𝜕𝑡}_𝑟≥1 are uniformly bounded.
Observe that 𝑥^′_𝑟(𝑡)≤𝑦^′₀(𝑡)+_𝛼^𝛼+𝛽𝜕𝑄𝜕𝑡^𝑛2𝑟𝜎_𝑟−1()+𝑄𝑡,𝑠𝑑𝑠_𝑛2𝑟𝜓_𝑟−1(𝑡,𝑡)+^𝑡_𝛼𝜕𝑄𝜕𝑡^𝑛2𝑟𝜓_𝑟−1(𝑡,𝑠)𝑑𝑠. (3.10) In view of the monotonicity of the Schauder basis, we have 𝑥^′_𝑟≤𝑦^′₀+𝜓_𝑟−1+𝛽𝜕𝜎_𝑟−1+𝜕𝑡𝜕𝜓_𝑟−1,𝜕𝑡 (3.11) and hence the sequence {𝑥^′_𝑟}_𝑟≥1 is uniformly bounded.
On the other hand from (3.2) and (3.3), respectively, we have 𝜕𝜎_𝑟−1()=𝜕𝑠𝑡,𝑠𝜕𝑘₁()𝑔𝜕𝑠𝑡,𝑠₁𝑠,𝑥_𝑟−1(𝑠)+𝑘₁(𝑡,𝑠)𝜕𝑔₁𝜕𝑠𝑠,𝑥_𝑟−1(𝑠)+𝜕𝑔₁𝜕𝑥𝑠,𝑥_𝑟−1(𝑠)𝑥^′_𝑟−1(𝑠),𝜕𝜓_𝑟−1()=𝜕𝑠𝑡,𝑠𝜕𝑘₂()𝑔𝜕𝑠𝑡,𝑠₂𝑠,𝑥_𝑟−1(𝑠)+𝑘₂(𝑡,𝑠)𝜕𝑔₂𝜕𝑠𝑠,𝑥_𝑟−1(𝑠)+𝜕𝑔₂𝜕𝑥𝑠,𝑥_𝑟−1(𝑠)𝑥^′_𝑟−1(𝑠). (3.12)
For 𝑖∈{1,2}, let 𝑈=max{|(𝜕𝑔_𝑖/𝜕𝑠)(𝑠,0)|∶𝑠∈[𝛼,𝛼+𝛽]}, and we have for all 𝑟≥1 and 𝑠∈[𝛼,𝛼+𝛽]𝜕𝑔_𝑖𝜕𝑠𝑠,𝑥_𝑟−1(𝑠)≤𝜕𝑔_𝑖𝜕𝑠𝑠,𝑥_𝑟−1(𝑠)−𝜕𝑔_𝑖(+𝜕𝑠𝑠,0)𝜕𝑔_𝑖(𝜕𝑠𝑠,0)≤𝑙_𝑔𝑖𝑥_𝑟−1(𝑠)+𝑈 (3.13) with 𝑙_𝑔𝑖 as the Lipschitz constant of 𝜕𝑔_𝑖/𝜕𝑠 in the last variable.
By repeating the previous argument, we have 𝜕𝑔_𝑖𝜕𝑥𝑠,𝑥_𝑟−1(𝑠)≤𝑞_𝑔𝑖𝑥_𝑟−1(𝑠)+𝑉, (3.14) where 𝑉=max{|(𝜕𝑔_𝑖/𝜕𝑥)(𝑠,0)|∶𝑠∈[𝛼,𝛼+𝛽]}, and 𝑞_𝑔𝑖 is the Lipschitz constant of 𝜕𝑔_𝑖/𝜕𝑥 in the last variable.
Therefore by the conditions over 𝑘₁,  𝑘₂, Proposition 3.1, Remark 3.2, and (3.11), 𝜕𝜎_𝑟−1𝜕𝑠_𝑟≥1,𝜕𝜓_𝑟−1𝜕𝑠_𝑟≥1 (3.15) are uniformly bounded.

Proposition 3.4. With the previous notation and the same hypothesis as in Proposition 3.3, there is 𝜌₁,𝜌₂>0 such that for all 𝑟≥1 and 𝑛_𝑟≥2, we have 𝜎_𝑟−1−𝑄_𝑛2𝑟𝜎_𝑟−1≤𝜌₁Δ𝑇_𝑛𝑟,𝜓_𝑟−1−𝑄_𝑛2𝑟𝜓_𝑟−1≤𝜌₂Δ𝑇_𝑛𝑟. (3.16)

Proof. In the last property in Table 1, take 𝜌₁=4max{‖𝜕𝜎_𝑟−1/𝜕𝑡‖,‖𝜕𝜎_𝑟−1/𝜕𝑠‖}_𝑟≥1 and 𝜌₂=4max{‖𝜕𝜓_𝑟−1/𝜕𝑡‖,‖𝜕𝜓_𝑟−1/𝜕𝑠‖}_𝑟≥1.

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 𝑚∈ℕ, 𝑛_𝑟∈ℕ, 𝑛_𝑟≥2, and {𝜀₁,…,𝜀_𝑚} be a set of positive numbers such that for all 𝑟∈{1,…,𝑚} we have Δ𝑇_𝑛𝑟≤𝜀_𝑟𝛽𝜌₁+𝜌₂. (3.17) Then, 𝑇𝑥_𝑟−1−𝑥_𝑟≤𝜀_𝑟. (3.18) Moreover, if 𝑥 is the exact solution of the integral equation (1.1), then the error ‖𝑥−𝑥_𝑚‖ is given by 𝑥−𝑥_𝑚≤𝑀^𝑚‖𝑇𝑥‖+1−𝑀𝑥−^𝑚_𝑟=1𝑀^𝑚−𝑟𝜀_𝑟. (3.19)

Proof. First we deal with proving (3.18). For all 𝑟∈{1,…,𝑚} and 𝑡∈[𝛼,𝛼+𝛽], Proposition 3.4 gives 𝑇𝑥_𝑟−1(𝑡)−𝑥_𝑟(𝑡)≤_𝛼^𝛼+𝛽𝜎_𝑟−1(𝑡,𝑠)−𝑄_𝑛2𝑟𝜎_𝑟−1()+𝑡,𝑠𝑑𝑠_𝑡𝛼𝜓_𝑟−1(𝑡,𝑠)−𝑄_𝑛2𝑟𝜓_𝑟−1(𝑡,𝑠)𝑑𝑠≤𝜌₁Δ𝑇_𝑛𝑟𝛽+𝜌₂Δ𝑇_𝑛𝑟𝛽=Δ𝑇_𝑛𝑟𝛽𝜌₁+𝜌₂≤𝜀_𝑟. (3.20)
To conclude the proof, we derive (3.19). From (2.3), we have 𝑥−𝑇^𝑚𝑥≤𝑀^𝑚‖𝑇𝑥‖,1−𝑀𝑥− (3.21) and in addition, on the other hand, applying recursively (2.2) and (3.18), we obtain 𝑇^𝑚𝑥−𝑥_𝑚≤^𝑚_𝑟=1𝑇^{𝑚−𝑟+1}𝑥_𝑟−1−𝑇^𝑚−𝑟𝑥_𝑟=^𝑚_𝑟=1𝑇^𝑚−𝑟𝑇𝑥_𝑟−1−𝑇^𝑚−𝑟𝑥_𝑟≤^𝑚_𝑟=1𝑀^𝑚−𝑟𝑇𝑥_𝑟−1−𝑥_𝑟≤^𝑚_𝑟=1𝑀^𝑚−𝑟𝜀_𝑟. (3.22)
Then we use the triangular inequality 𝑥−𝑥_𝑚≤𝑥−𝑇^𝑚𝑥+𝑇^𝑚𝑥−𝑥_𝑚, (3.23) 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 𝑥−𝑥_𝑚≤𝑀^𝑚‖𝑇𝑥‖+1−𝑀𝑥−1−𝑀^𝑚1−𝑀max_𝑟≥1𝜀_𝑟. (3.24)
The first sumand on the right hand side approximates zero when 𝑚 increases; with respect to the second sumand, since the points of the partition can be chosen in such a way that Δ𝑇_𝑛𝑟 becomes so close to zero as we desire, the 𝜀^′_𝑟𝑠 can become so small as we desire, arriving in this way at an explicit control of the error committed.
Therefore, given 𝜀>0, there exists 𝑚≥1 such that ‖𝑥−𝑥_𝑚‖<𝜀 when choosing 𝜀_𝑟 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 𝐶([0,1]²), we considered the particular choice 𝑡₁=0, 𝑡₂=1 and for 𝑛∈ℕ∪{0}, 𝑡_𝑖+1=(2𝑘+1)/2^𝑛+1 if 𝑖=2^𝑛+𝑘+1 where 0≤𝑘<2^𝑛 are integers. To define the sequence {𝑥_𝑟}_𝑟≥1, we take 𝑥₀(𝑡)=𝑦₀(𝑡) and 𝑛_𝑟=𝑗 (for all 𝑟≥1). In Tables 2 and 3, we exhibit, for 𝑗=9, 17, and 33, the absolute errors committed in eight representative points of [0,1] when we approximate the exact solution 𝑥 by the iteration 𝑥₄. Its numerical results are also given in Figures 1 and 2, respectively.

Table 2 

Absolute errors for Example 4.1.

Table 3 

Absolute errors for Example 4.2.

Figure 1 

The plot of absolute errors for Example 4.1.

Figure 2 

The plot of absolute errors for Example 4.2.

Example 4.1. We solve (1.1) with 𝑘₁(𝑡,𝑠)=𝑡𝑠/5, 𝑔₁(𝑠,𝑥(𝑠))=cos(𝑥(𝑠)), 𝑘₂(𝑡,𝑠)=𝑠/3, 𝑔₂(𝑠,𝑥(𝑠))=sin(𝑥(𝑠)), and 𝑦₀(𝑡)=1+𝑡−(𝑡/5)(cos(2)−cos(1)+sin(2))+(1/3)(𝑡cos(1+𝑡)−sin(1+𝑡)+sin(1)) with the exact solution 𝑥(𝑡)=1+𝑡.

Example 4.2. We solve (1.1) with 𝑘₁(𝑡,𝑠)=(1/4)(1−𝑡)³, 𝑔₁(𝑠,𝑥(𝑠))=arctan(𝑥(𝑠)), 𝑘₂(𝑡,𝑠)=1/8, 𝑔₂(𝑠,𝑥(𝑠))=𝑥(𝑠), and 𝑦₀(𝑡)=𝑡−(𝑡²/16)−((𝜋−ln(4))/16)(𝑡−1)³ with the exact solution 𝑥(𝑡)=𝑡.

Now we realize that the choice of a particular 𝑗, determining the dyadic partition of the interval [0,1] from the first 2^𝑗+1 nodes, and in such a way that the error is less than a fixed positive 𝜀, that is, ‖𝑥−𝑥_𝑚‖<𝜀, 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 (𝑚=3). 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.

Table 4 

Number of nodes (𝑗) from error (𝜀) and for 𝑚=3.

Acknowledgment

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