Nonlinear Volterra Integral Equation of the Second Kind and Biorthogonal Systems

Abstract

We obtain an approximation of the solution of the nonlinear Volterra integral equation of the second kind, by means of a new method for its numerical resolution. The main tools used to establish it are the properties of a biorthogonal system in a Banach space and the Banach fixed point theorem.

1. Introduction

This paper puts forth a new method in order to numerically solve the nonlinear Volterra integral equation of the second kind 𝑥(𝑡)=𝑦₀(𝑡)+^𝑡_𝛼𝐾(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈[𝛼,𝛼+𝛽],(1.1) where 𝑦₀ : [𝛼,𝛼+𝛽]→ℝ and the kernel 𝐾 : [𝛼,𝛼+𝛽]²×ℝ→ℝ are assumed to be known continuous functions, and the unknown function to be determined is 𝑥 : [𝛼,𝛼+𝛽]→ℝ.

Modeling many problems of science, engineering, physics, and other disciplines leads to linear and nonlinear Volterra integral equations of the second kind. These are usually difficult to solve analytically and in many cases the solution must be approximated. Therefore, in recent years several numerical approaches have been proposed (see, e.g., [4–8]). The numerical methods usually transform the integral equation into a linear or nonlinear system that can be solved by direct or iterative methods. In a recent work [9], the authors use a new technique for solving the linear Volterra integral equation. The method is based on two classical analytical tools: the Geometric Series theorem and Schauder bases in a Banach space. The purpose of this paper is to develop, and generalize to the nonlinear case, an effective method for approximating the solution using biorthogonal systems and another classical tool in analysis: the Banach fixed point theorem.

The work is structured in three parts: in Section 2, we will recall one well-known result and some useful definitions needed later. In Section 3, we define the approximating functions and we study the error. Finally, the numerical results given in Section 4 show the high accuracy of the method.

2. Preliminaries

Let 𝐶([𝛼,𝛼+𝛽]) be the Banach space of all continuous and real-valued functions on [𝛼,𝛼+𝛽], endowed with its usual supnorm. 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])(2.1) defined by(𝑇𝑥)(𝑡)∶=𝑦₀(𝑡)+^𝑡_𝛼𝐾(𝑡,𝑠,𝑥(𝑠))𝑑𝑠,𝑡∈[𝛼,𝛼+𝛽],𝑥∈𝐶([𝛼,𝛼+𝛽]).(2.2) To establish the existence of fixed points of (2.2), we will use the version of the Banach fixed-point theorem (see [10]) which we enunciate below: let (𝑋,‖⋅‖) be a Banach space, let 𝐹 : 𝑋→𝑋 and let {𝜇_𝑛}_𝑛≥1 be a sequence of nonnegative real numbers such that the series ∑_𝑛≥1𝜇_𝑛 is convergent and for all 𝑥,𝑦∈𝑋 and for all 𝑛≥1, ‖𝐹^𝑛𝑥−𝐹^𝑛𝑦‖≤𝜇_𝑛‖𝑥−𝑦‖. Then 𝐹 has a unique fixed point 𝑢∈𝑋. Moreover, if 𝑥 is an element in 𝑋, then we have that for all 𝑛≥1, ‖𝐹^𝑛𝑥−𝑢‖≤(^∞_𝑖=𝑛𝜇_𝑖)‖𝐹𝑥−𝑥‖.(2.3) In particular, 𝑢=lim_𝑛𝐹^𝑛(𝑥).

On the other hand, we recall briefly some definitions on the theory of Schauder bases and biorthogonal systems in general (see [11]), which are central areas of research, and also some important tools in Functional Analysis. The use of Schauder bases in the numerical study of integral and differential equations has been previously considered in [12–14].

Let us start by recalling the notion of biorthogonal system of a Banach space. Let 𝑋 be a Banach space and 𝑋^∗ its topological dual space. A system {𝑥_𝑛,𝑓_𝑛}_𝑛≥1, where 𝑥_𝑛∈𝑋, 𝑓_𝑛∈𝑋^∗, and 𝑓_𝑛(𝑥_𝑚)=𝛿_𝑛𝑚 (𝛿 is Kronecker's delta), is called a biorthogonal system in 𝑋. We say that the system is a fundamental biorthogonal system if span{𝑥_𝑛}_𝑛≥1=𝑋.

We will work with a particular type of fundamental biorthogonal systems. Let us recall that a sequence {𝑥_𝑛}_𝑛≥1 of elements of a Banach space 𝑋 is called a Schauder basis of 𝑋 if for every 𝑧∈𝑋 there is a unique sequence {𝜆_𝑛}_𝑛≥1 of scalars such that 𝑧=∑_𝑛≥1𝜆_𝑛𝑥_𝑛. A Schauder basis gives rise to the canonical sequence of (continuous and linear) finite dimensional projections 𝑃_𝑛∶𝑋→𝑋, 𝑃_𝑛(∑_𝑛≥1𝜆_𝑛𝑥_𝑛)=∑^𝑛_𝑘=1𝜆_𝑘𝑥_𝑘 and the associated sequence of (continuous and linear) coordinate functionals {𝑥^∗_𝑛}_𝑛≥1 in 𝑋^∗ is given by 𝑥^∗_𝑛(∑_𝑛≥1𝜆_𝑛𝑥_𝑛)=𝜆_𝑛. Note that a Schauder basis is always a fundamental biorthogonal system, under the interpretation of the coordinate functionals as biorthogonal functionals.

3. Main Results

We begin this section making use of a Schauder basis in the Banach space 𝐶([𝛼,𝛼+𝛽]²) endowed with its usual supnorm. To construct such a basis, we recall that a usual Schauder basis {𝑏_𝑛}_𝑛≥1 in 𝐶([𝛼,𝛼+𝛽]) can be obtained from a dense sequence {𝑡_𝑛}_𝑛≥1 of distinct points from [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, 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. For this basis, the sequence of biorthogonal functionals {𝑏^∗_𝑛}_𝑛≥1 satisfies (see [1]) for all 𝑦∈𝐶([𝛼,𝛼+𝛽])𝑏^∗₁(𝑦)=𝑦(𝑡₁),𝑏^∗_𝑛(𝑦)=𝑦(𝑡_𝑛)−^𝑛−1_𝑘=1𝑏^∗_𝑘(𝑦)𝑏_𝑘(𝑡_𝑛),for𝑛≥2.(3.1) In addition, the sequence of associated projections {𝑃_𝑛}_𝑛≥1 satisfies for all 𝑦∈𝐶([𝛼,𝛼+𝛽]), for all 𝑛≥1 and for all 𝑘≤𝑛 such that 𝑃_𝑛(𝑦)(𝑡_𝑘)=𝑦(𝑡_𝑘).

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 [1, 2]). It is sufficient to consider 𝐵_𝑛(𝑡,𝑠)∶=𝑏_𝑖(𝑡)𝑏_𝑗(𝑠) for all 𝑡,𝑠∈[𝛼,𝛼+𝛽], with 𝜎(𝑛)=(𝑖,𝑗), where for a real number 𝑎, [𝑎] will denote its integer part and 𝜎=(𝜎₁,𝜎₂)∶ℕ→ℕ×ℕ is the bijective mapping defined by (√𝜎(𝑛)∶=𝑛,√𝑛)if[√𝑛]=√√𝑛,(𝑛−[𝑛]²,[√𝑛]+1)if0<𝑛−[√𝑛]²≤[√√𝑛],([𝑛]+1,𝑛−[√𝑛]²−[√𝑛])if[√𝑛]<𝑛−[√𝑛]².(3.2)

Remark 3.1. The Schauder basis {𝐵_𝑛}_𝑛≥1 of 𝐶([𝛼,𝛼+𝛽]²) has similar properties to the ones for the one-dimensional case.
(a) For all 𝑡,𝑠∈[𝛼,𝛼+𝛽], 𝐵₁(𝑡,𝑠)=1 and for 𝑛≥2, 𝐵_𝑛(𝑡_𝑖,𝑡_j)=1if𝜎(𝑛)=(𝑖,𝑗),0if𝜎⁻¹(𝑖,𝑗)<𝑛.(3.3)
(b) If 𝑧∈𝐶([𝛼,𝛼+𝛽]²), then 𝐵^∗₁(𝑧)=𝑧(𝑡₁,𝑡₁), and for all 𝑛≥2, if 𝜎(𝑛)=(𝑖,𝑗), 𝐵^∗_𝑛(𝑧)=𝑧(𝑡_𝑖,𝑡_𝑗)−∑^𝑛−1_𝑘=1𝐵^∗_𝑘(𝑧)𝐵_𝑘(𝑡_𝑖,𝑡_𝑗). 
(c)The sequence of associated projections {𝑄_𝑛}_𝑛≥1 satisfies 𝑄_𝑛(𝑧)(𝑡_𝑖,𝑡_𝑗)=𝑧(𝑡_𝑖,𝑡_𝑗), whenever 𝑛,𝑖,𝑗∈ℕ and 𝜎⁻¹(𝑖,𝑗)≤𝑛. 
(d) This Schauder basis is monotone, that is, sup{‖𝑄_𝑛‖}_𝑛∈ℕ=1.

Remark 3.2. We have chosen the Schauder basis above for simplicity in the exposition, although the method to be presented also works considering any fundamental biorthogonal system in 𝐶([𝛼,𝛼+𝛽]²).
With the previous notation, our first result enables us to obtain the image under operator 𝑇 defined in (2.2) of any continuous function in terms of certain sequences of scalars, sequences which are obtained just by evaluating some functions at adequate points.

Proposition 3.3. Let 𝑇 : 𝐶([𝛼,𝛼+𝛽])→𝐶([𝛼,𝛼+𝛽]) be the continuous integral operator defined in (2.2). Let 𝑥∈𝐶[𝛼,𝛼+𝛽], and let us consider the function Φ∈𝐶([𝛼,𝛼+𝛽]²), defined by Φ(𝑡,𝑠)=𝐾(𝑡,𝑠,𝑥(𝑠)). Let {𝜆_𝑛}_𝑛≥1 be the sequences of scalars satisfying Φ=∑_𝑛≥1𝜆_𝑛𝐵_𝑛. Then for all 𝑡∈[𝛼,𝛼+𝛽], we have that (𝑇𝑥)(𝑡)=𝑦₀(𝑡)+_𝑛≥1𝜆_𝑛^𝑡_𝛼𝐵_𝑛(𝑡,𝑠)𝑑𝑠,(3.4) where 𝜆₁=Φ(𝑡₁,𝑡₁)(3.5) and for 𝑛≥2,𝜆_𝑛=Φ(𝑡_𝑖,𝑡_𝑗)−^𝑛−1_𝑘=1𝐵^∗_𝑘(Φ)𝐵_𝑘(𝑡_𝑖,𝑡_𝑗)with𝜎(𝑛)=(𝑖,𝑗).(3.6)

Proof. The result follows directly from the expression Φ(𝑡,𝑠)=_𝑛≥1𝐵^∗_𝑛(Φ)𝐵_𝑛(𝑡,𝑠)(3.7) in the integral appearing in the definition of 𝑇.

On the other hand, in order to discuss the application of Banach's fixed point theorem to find a fixed point for the operator 𝑇 defined in (2.2), we establish the following result.

Proposition 3.4. Assume that in (1.1) the kernel 𝐾 satisfies a Lipschitz condition in its third variable: |𝐾(𝑡,𝑠,𝑥)−𝐾(𝑡,𝑠,𝑦)|≤𝑀|𝑥−𝑦|∀𝑡,𝑠∈[𝛼,𝛼+𝛽],𝑥,𝑦∈ℝ(3.8) for some constant 𝑀>0. Then the integral equation (1.1) has a unique solution 𝑥∈𝐶([𝛼,𝛼+𝛽]). In addition, for each 𝑥∈𝐶([𝛼,𝛼+𝛽]), the sequence {𝑇^𝑛𝑥}_𝑛≥1 in 𝐶([𝛼,𝛼+𝛽]) converges uniformly to the unique solution 𝑥 and for all 𝑛≥1, ‖𝑇^𝑛𝑥−𝑥‖≤(𝑀𝛽)^𝑛𝑒𝑛!^𝑀𝛽‖𝑇𝑥−𝑥‖.(3.9)

Proof. For all 𝑥,𝑦∈𝐶([𝛼,𝛼+𝛽]) and for all 𝑛≥1, we obtain by a mathematical induction (see [3, Theorem 5.2.3]) that |(𝑇^𝑛𝑥)(𝑡)−(𝑇^𝑛𝑦)(𝑡)|≤(𝑀^𝑛/𝑛!)‖𝑥−𝑦‖(𝑡−𝛼)^𝑛 for all 𝑡∈[𝛼,𝛼+𝛽]. In particular, ‖𝑇^𝑛𝑥−𝑇^𝑛𝑦‖≤(𝑀𝛽)^𝑛𝑛!‖𝑥−𝑦‖.(3.10) Since ∑_𝑛≥1((𝑀𝛽)^𝑛/𝑛!) converges for any 𝛽 and 𝑀, by Banach's fixed point theorem, we will derive the existence and uniqueness of a solution of the integral equation (1.1). From (3.10) and (2.3), we deduce (3.9).

In view of Propositions 3.3 and 3.4, (3.4) gives the unique solution 𝑥(𝑡) of (1.1). The problem is that generally this expression cannot be calculated explicitly. The idea of the proposed method is to truncate to calculate approximately a sequence of iterations and projections that converge to the solution. More specifically, let 𝑥 : [𝛼,𝛼+𝛽]→ℝ be a continuous function, and 𝑛₁,𝑛₂,𝑛₃,…,∈ℕ. Consider the continuous functions𝑧₀(𝑡)∶=𝑥(𝑡),𝑡∈[𝛼,𝛼+𝛽],(3.11) and for 𝑟∈ℕ, we define𝐿_𝑟−1(𝑡,𝑠)∶=𝐾(𝑡,𝑠,𝑧_𝑟−1𝑧(𝑠))(𝑡,𝑠∈[𝛼,𝛼+𝛽]),(3.12)_𝑟(𝑡)∶=𝑦₀(𝑡)+^𝑡_𝛼𝑄_𝑛2𝑟(𝐿_𝑟−1(𝑡,𝑠))𝑑𝑠(𝑡∈[𝛼,𝛼+𝛽]).(3.13)

In order to obtain the convergence of the sequence {𝑧_𝑟}_𝑟≥1 to the unique solution of (1.1), we need, under some weak condition, to uniformly estimate the rate of the convergence of the sequence of projections {𝑄_𝑛}_𝑛≥1 in the bidimensional case. To this end, we introduce the following notation that will be used in the next results: if {𝑡_𝑛}_𝑛≥1 is the dense subset of distinct points in [𝛼,𝛼+𝛽], we considered to define the Schauder basis, let 𝑇_𝑛 be the set {𝑡_𝑗,1≤𝑗≤𝑛} ordered in an increasing way for 𝑛≥2. Let Δ𝑇_𝑛 denote the maximum distance between two consecutive points of 𝑇_𝑛.

Proposition 3.5. Let 𝐾∈𝐶¹([𝛼,𝛼+𝛽]²×ℝ) such that 𝐾,𝜕𝐾/𝜕𝑡, 𝜕𝐾/𝜕𝑠, 𝜕𝐾/𝜕𝑥 satisfy a global Lipschitz condition in the third variable. Then {𝜕𝐿_𝑟−1/𝜕𝑡}_𝑟≥1 and {𝜕𝐿_𝑟−1/𝜕𝑠}_𝑟≥1 are uniformly bounded.

Proof. From (3.12), we have that for all 𝑟≥1, 𝜕𝐿_𝑟−1𝜕𝑡(𝑡,𝑠)=𝜕𝐾𝜕𝑡(𝑡,𝑠,𝑧_𝑟−1(𝑠)),𝜕𝐿_𝑟−1𝜕𝑠(𝑡,𝑠)=𝜕𝐾𝜕𝑠(𝑡,𝑠,𝑧_𝑟−1(𝑠))+𝜕𝐾𝜕𝑥(𝑡,𝑠,𝑧_𝑟−1(𝑠))𝑧^′_𝑟−1(𝑠).(3.14) Let 𝑅=max_{(𝑡,𝑠)∈[𝛼,𝛼+𝛽]}²‖𝐾(𝑡,𝑠,0)‖, and we have for all 𝑟≥1 and (𝑡,𝑠)∈[𝛼,𝛼+𝛽]²,‖𝐿_𝑟−1(𝑡,𝑠)‖=‖𝐾(𝑡,𝑠,𝑧_𝑟−1(𝑠))‖≤‖𝐾(𝑡,𝑠,𝑧_𝑟−1(𝑠))−𝐾(𝑡,𝑠,0)‖+‖𝐾(𝑡,𝑠,0)‖≤𝑀‖𝑧_𝑟−1(𝑠)‖+𝑅(3.15) with 𝑀 as the Lipschitz constant of 𝐾.
As a consequence of the monotonicity of the Schauder basis, we have ‖𝑧_𝑟(𝑡)‖≤‖𝑦₀‖+^𝑡_𝛼‖𝐿_𝑟−1(𝑡,𝑡₁)‖𝑑𝑡₁.(3.16) If one applies (3.15) and by repeating the previous argument, ‖𝑧_𝑟(𝑡)‖≤‖𝑦₀‖+^𝑡_𝛼(𝑀‖𝑧_𝑟−1(𝑡₁)‖+𝑅)𝑑𝑡₁≤‖𝑦₀‖+^𝑡_𝛼(𝑀(‖𝑦₀‖+^𝑡1_𝛼‖𝐿_𝑟−2(𝑡₁,𝑡₂)‖𝑑𝑡₂)+𝑅)𝑑𝑡₁=‖𝑦₀‖(1+𝑀^𝑡_𝛼𝑑𝑡₁)+𝑅^𝑡_𝛼𝑑𝑡₁+𝑀^𝑡_𝛼^𝑡1_𝛼‖𝐿_𝑟−2(𝑡₁,𝑡₂)‖𝑑𝑡₂𝑑𝑡₁.(3.17) Applying recursively this process and the Fubini theorem, we get ‖𝑧_𝑟(𝑡)‖≤‖𝑦₀‖+‖𝑦₀‖^𝑟−1_𝑘=1𝑀^𝑘(𝑡−𝛼)^𝑘𝑘!+𝑅^𝑟−1_𝑘=1𝑀^𝑘−1(𝑡−𝛼)^𝑘𝑘!+𝑀^𝑟−1(𝑡−𝛼)^𝑟𝑟!‖𝐿₀(𝑡_𝑟−1,𝑡_𝑟)‖.(3.18) Thus for all 𝑟≥1 and (𝑡,𝑠)∈[𝛼,𝛼+𝛽]²,‖𝑧_𝑟‖≤‖𝑦₀‖+(‖𝑦₀𝑅‖+𝑀)^𝑟−1_𝑘=1(𝛽𝑀)^𝑘+𝑘!‖𝐿₀‖𝑀(𝛽𝑀)^𝑟.𝑟!(3.19) Hence the sequence {𝑧_𝑟}_𝑟≥1 is uniformly bounded.
Meanwhile 𝑧^′_𝑟(𝑡)=𝑦^′₀(𝑡)+𝑄_𝑛2𝑟(𝐿_𝑟−1(𝑡,𝑡))+∫^𝑡_𝛼(𝜕𝑄_𝑛2𝑟/𝜕𝑡)(𝐿_𝑟−1(𝑡,𝑠))𝑑𝑠, and in view of the monotonicity of the Schauder basis {𝐵_𝑛}_𝑛≥1 and the fact that for all 𝑧∈𝐶¹([𝛼,𝛼+𝛽]²) and 𝑛≥2 we have that |(𝜕𝑄_𝑛2(𝑧)/𝜕𝑡)(𝑡,𝑠)|≤‖𝜕𝑧/𝜕𝑡‖, the boundedness of {𝑧^′_𝑟}_𝑟≥1 follows from that of (3.15) and of {𝜕𝐿_𝑟/𝜕𝑡}_𝑟≥1, which is done below.
Then |𝜕𝐾𝜕𝑡(𝑡,𝑠,𝑧_𝑟−1(𝑠))|≤|𝜕𝐾𝜕𝑡(𝑡,𝑠,𝑧_𝑟−1(𝑠))−𝜕𝐾𝜕𝑡(𝑡,𝑠,0)|+|𝜕𝐾𝜕𝑡(𝑡,𝑠,0)|≤𝑀₁|𝑧_𝑟−1(𝑠)|+𝑈(3.20) with 𝑈=max_{(𝑡,𝑠)∈[𝛼,𝛼+𝛽]}²|(𝜕𝐾/𝜕𝑡)(𝑡,𝑠,0)| and 𝑀₁ being the Lipschitz constant of (𝜕𝐾/𝜕𝑡)(𝑡,𝑠,𝑧_𝑟−1(𝑠)).
Similarly, |𝜕𝐾𝜕𝑠(𝑡,𝑠,𝑧_𝑟−1(𝑠))|≤𝑀₂|𝑧_𝑟−1(𝑠)|+𝑉(3.21) with 𝑉=max_{(𝑡,𝑠)∈[𝛼,𝛼+𝛽]}²|(𝜕𝐾/𝜕𝑠)(𝑡,𝑠,0)| and 𝑀₂ as the Lipschitz constant of (𝜕𝐾/𝜕𝑠)(𝑡,𝑠,𝑧_𝑟−1(𝑠)).
Finally |𝜕𝐾𝜕𝑥(𝑡,𝑠,𝑧_𝑟−1(𝑠))𝑧^′_𝑟−1(𝑠)|≤(𝑀₃|𝑧_𝑟−1(𝑠)|+𝑊)|𝑧^′_𝑟−1(𝑠)|(3.22) with 𝑊=max_{(𝑡,𝑠)∈[𝛼,𝛼+𝛽]}²|(𝜕𝐾/𝜕𝑥)(𝑡,𝑠,0)| and 𝑀₃ as the Lipschitz constant of (𝜕𝐾/𝜕𝑥)(𝑡,𝑠,𝑧_𝑟−1(𝑠)).
Hence, {𝜕𝐿_𝑟−1/𝜕𝑡}_𝑟≥1 and {𝜕𝐿_𝑟−1/𝜕𝑠}_𝑟≥1 are uniformly bounded.

Theorem 3.6. With the previous notation, let 𝑥∈𝐶([𝛼,𝛼+𝛽]), 𝑦₀∈𝐶¹([𝛼,𝛼+𝛽]), and 𝐾∈𝐶¹([𝛼,𝛼+𝛽]²×ℝ) with 𝐾, 𝜕𝐾/𝜕𝑡, 𝜕𝐾/𝜕𝑠, 𝜕𝐾/𝜕𝑥 satisfying the Lipschitz global condition of the third variable. Then, there is 𝜌>0 such that for all 𝑟≥1 and 𝑛_𝑟≥2,‖𝐿_𝑟−1−𝑄_𝑛𝑟2(𝐿_𝑟−1)‖≤𝜌Δ𝑇_𝑛𝑟.(3.23)

Proof. We use Proposition 3.5. and the inequality resulting from Remark 3.1.(c) and the Mean Value Theorem to get ‖𝑧−𝑄_𝑛2(𝑧)‖≤4max{‖𝜕𝑧𝜕𝑡‖,‖𝜕𝑧𝜕𝑠‖}Δ𝑇_𝑛(3.24) for 𝑧∈𝐶¹([𝛼,𝛼+𝛽]²) and 𝑛≥2.

The main result that establishes that the sequence defined in (3.11) and (3.13) approximates the solution of (1.1) as well as giving an upper bond of the error committed is given below.

Theorem 3.7. Let 𝐾∈𝐶([𝛼,𝛼+𝛽]²×ℝ) such that 𝐾 satisfies a global Lipschitz condition in the third variable and let 𝑥∈𝐶([𝛼,𝛼+𝛽]). Let 𝑚∈ℕ, and assume that certain positive numbers 𝜀₁,…,𝜀_𝑚 satisfy ‖𝑇𝑧_𝑟−1−𝑧_𝑟‖<𝜀_𝑟,𝑟=1,…,𝑚(3.25) and let 𝑥 be the exact solution of the integral equation (1.1). Then ‖𝑥−𝑧_𝑚‖≤(𝑀𝛽)^𝑚𝑒𝑚!^𝑀𝛽‖𝑇𝑥−𝑥‖+^𝑚_𝑟=1𝜀_𝑟(𝑀𝛽)^𝑚−𝑟,(𝑚−𝑟)!(3.26) where 𝑀 is the Lipschitz constant of 𝐾.

Proof. On one hand, Proposition 3.4 gives ‖𝑥−𝑇^𝑚𝑥‖≤(𝑀𝛽)^𝑚𝑒𝑚!^𝑀𝛽‖𝑇𝑥−𝑥‖.(3.27) On the other hand, in view of (3.10) for all 𝑟=1,…,𝑚, we have that ‖𝑇^{𝑚−𝑟+1}𝑧_𝑟−1−𝑇^𝑚−𝑟𝑧_𝑟‖=‖𝑇^𝑚−𝑟𝑇𝑧_𝑟−1−𝑇^𝑚−𝑟𝑧_𝑟‖≤(𝑀𝛽)^𝑚−𝑟(𝑚−𝑟)!‖𝑇𝑧_𝑟−1−𝑧_𝑟‖.(3.28) Hence ‖𝑇^𝑚𝑥−𝑧_𝑚‖=‖𝑇^𝑚𝑧₀−𝑧_𝑚‖≤^𝑚_𝑟=1‖𝑇^{𝑚−𝑟+1}𝑧_𝑟−1−𝑇^𝑚−𝑟𝑧_𝑟‖≤^𝑚_𝑟=1𝜀_𝑟(𝑀𝛽)^𝑚−𝑟.(𝑚−𝑟)!(3.29) Then we use the triangular inequality ‖𝑥−𝑧_𝑚‖≤‖𝑥−𝑇^𝑚𝑥‖+‖𝑇^𝑚𝑥−𝑧_𝑚‖,(3.30) and the proof is complete in view of (3.27) and (3.29).

Remark 3.8. Under the hypothesis of Theorem 3.6, ‖𝑇𝑧_𝑟−1−𝑧_𝑟‖ can be estimated as follows: there is 𝜌>0 such that for all 𝑟≥1 and 𝑛_𝑟≥2, ‖𝑇𝑧_𝑟−1−𝑧_𝑟‖≤𝛽‖𝐿_𝑟−1−𝑄_𝑛2𝑟(𝐿_𝑟−1)‖≤𝛽𝜌Δ𝑇_𝑛𝑟.(3.31) Hence, given certain 𝜀₁,…,𝜀_𝑚>0, we can find 𝑚 positive integers 𝑛₁,…,𝑛_𝑚 such that ‖𝑇𝑧_𝑟−1−𝑧_𝑟‖<𝜀_𝑟, and by Theorem 3.7, we can state the convergence of {𝑧_𝑟}_𝑟≥1 and an estimation of the error.

4. Some Examples

The behaviour of the method introduced above will be illustrated with the following three examples.

Example 4.1. The equation 1𝑥(𝑡)=2𝑡(2+𝑡)−2𝑡arctg(𝑡)+ln(1+𝑡²)+^𝑡₀(−𝑥(𝑠)+2arctg(𝑥(𝑠)))𝑑𝑠(𝑡∈[0,1]),𝑥(0)=0(4.1) has the exact solution 𝑥(𝑡)=𝑡.

Example 4.2. Consider the equation 1𝑥(𝑡)=3𝑡cos(𝑡³)+𝑡³−𝑡3+^𝑡₀𝑡𝑠²sin(𝑥(𝑠))𝑑𝑠(𝑡∈[0,1]),𝑥(0)=0,(4.2) whose exact solution is 𝑥(𝑡)=𝑡³.

Example 4.3. Consider the equation 1𝑥(𝑠)=2(3𝑡−(1+𝑡²)arctg(𝑡))+^𝑡₀𝑠arctg(𝑥(𝑠))𝑑𝑠(𝑡∈[0,1]),𝑥(0)=0,(4.3) whose exact solution is 𝑥(𝑡)=𝑡.

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 {𝑧_𝑟}_𝑟≥1, we take 𝑧₀(𝑡)=1 and 𝑛_𝑟=𝑗 (for all 𝑟≥1). In Tables 1, 2, and 3, we exhibit, for 𝑗=9,17,33,65, and 129, the absolute errors committed in eight representative points (𝑡_𝑖) of [0,1] when we approximate the exact solution 𝑥 by the iteration 𝑧₂. The computations associated with the examples were performed using Mathematica 7.

Table 1 

Absolute errors for Example 4.1.

Table 2 

Absolute errors for Example 4.2.

Table 3 

Absolute errors for Example 4.3.

5. Conclusions

In this paper, we introduce a new numerical method which approximates the solution of the nonlinear Volterra integral equation of the second kind (1.1). Unlike what happens in the classical methods, as in the collocation one, we do not need to solve high-order nonlinear systems of algebraical equations: for our method we just calculate linear combinations of scalar obtained by evaluating adequate functions. This is done due to the properties of the Schauder basis {𝐵_𝑛}_𝑛≥1 considered in its development.

Acknowledgments

The research is partially supported by M.E.C. (Spain) and FEDER project no. MTM2006-12533, and by Junta de Andaluca Grant FQM359.