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Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

Andrei Rontó1, Miklos Rontó2, Gabriela Holubová3 and Petr Nečesal3*

Author Affiliations

1 Institute of Mathematics, Academy of Sciences of the Czech Republic, Brno, Czech Republic

2 Department of Analysis, University of Miskolc, Egyetemvaros, Hungary

3 Department of Mathematics, University of West Bohemia, Pilsen, Czech Republic

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Boundary Value Problems 2011, 2011:58  doi:10.1186/1687-2770-2011-58


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/58


Received:26 May 2011
Accepted:28 December 2011
Published:28 December 2011

© 2011 Rontó et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic method based upon successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples.

2000 Mathematics Subject Classification: 34B15; 65L10.

Keywords:
nonlinear boundary value problem; numerical-analytic method; Chebyshev interpolation polynomials

1 Introduction

In studies of solutions of various types of nonlinear boundary value problems for ordinary differential equations side by side with numerical methods, it is often used an appropriate technique based upon some types of successive approximations constructed in analytic form. This class of methods includes, in particular, the approach suggested at first in [1,2] for investigation of periodic solutions. Later, appropriate versions of this method were developed for handling more general types of nonlinear boundary value problems for ordinary and functional-differential equations. We refer, e.g., to the books [3-5], the articles [6-12], and the series of survey articles [13] for the related references.

According to the basic idea, the given boundary value problem is replaced by the Cauchy problem for a suitably modified system of integro-differential equations containing some artificially introduced parameters. The solution of the perturbed problem is searched in analytic form by successive iterations. The perturbation term, which depends on the original differential equation, on the introduced parameters and on the boundary conditions, yields a system of algebraic or transcendental determining equations. These equations enable us to determine the values of the introduced parameters for which the original and the perturbed problems coincide. Moreover, studying solvability of the approximate determining systems, we can establish existence results for the original boundary value problem.

In this article, we introduce the Chebyshev polynomial version of the known numerical-analytic method based on successive iterations. At the beginning, we follow the ideas presented by Rontó and Rontó [14] and by Rontó and Shchobak [15], which contains existence results for a system of two nonlinear differential equations with separated boundary conditions. In order to avoid some technical difficulties, we deal in this article, for simplicity, with nonlinear differential equations with homogeneous Dirichlet boundary conditions. On the other hand, our basic recurrence relation has the same general form as it is presented in [15]. In Section 2, we state the studied problem and the corresponding setting. Sections 3 and 4 contain the construction of the sequence of successive approximations, its convergence analysis, the properties of the limit function, and its correspondence to the solution of the original boundary value problem. The existence questions are discussed as well. Main results of the article are in Section 5, which contains a justification of the Chebyshev polynomial version of the introduced method with corresponding convergence analysis and error estimates. Results in Section 5 allow us to construct the Chebyshev polynomial approximations of the solution of the nonlinear boundary value problem, which essentially simplify the computations of successive approximations in analytic form and simplify also the form of the determining equation. In Section 6, we illustrate the applicability of our approach to three Dirichlet boundary value problems: the linear one, the semilinear one, and the quasi-linear one containing the p-Laplace operator. Finally, let us note that presented polynomial version of the numerical-analytic method in this article can be extended to more general nonlinear boundary value problems studied in [14].

2 Problem setting and preliminaries

We consider the following system of two nonlinear differential equations with Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M1">View MathML</a>

(1)

In the vector form, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M2">View MathML</a>

(2)

where x = col(x1, x2), f(t, x) = col(f1(t, x), f2(t, x)) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M3">View MathML</a>

Let the function f(t, x) be defined and continuous in the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M4">View MathML</a>

(3)

To avoid dealing with singular matrix C1 in (2), which does not enable us to express explicitly x(T), it is useful to carry out the following parametrization

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M5">View MathML</a>

(4)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M6">View MathML</a>

(5)

Thus, instead of (2) we use the equivalent problem with two-point parameterized boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M7">View MathML</a>

(6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M8">View MathML</a>

The two-point parameterized boundary conditions in (6) allow us to write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M9">View MathML</a>

which will be used in the sequel for the construction of the iterative scheme.

Throughout the text, C([0, T], ℝ2) is the Banach space of vector functions with continuous components and L1([0, T], ℝ2) is the usual Banach space of vector functions with Lebesgue integrable components.

Moreover, the signs |·|, ≤, ≥, max, and min operations will be everywhere understood componentwise. Let us define the vector

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M10">View MathML</a>

(7)

for which the following estimate is true (cf. [5,16])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M11">View MathML</a>

(8)

For z ∈ ℝ2 of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M12">View MathML</a>

(9)

and λ ∈ Λ we define the vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M13">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M14">View MathML</a>

(10)

where I2 is the unit matrix of order 2. In the sequel, we use the following assumptions.

(A1) The function f : [0, T] × D → ℝ2 is continuous.

(A2) The function f satisfies the following Lipschitz condition: there exists a nonnegative constant square matrix K of order 2 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M15">View MathML</a>

(A3) The subset

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M16">View MathML</a>

is non-empty, where B(z, γ(z, λ)) := {u ∈ ℝ2 : |u - z| ≤ γ(z, λ)}.

(A4) The greatest eigenvalue r(Q) of the non-negative matrix

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M17">View MathML</a>

satisfies the inequality r(Q) < 1.

Remark 1. The history and possible improvements of the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M18">View MathML</a> in the definition of Q can be found in [5,17,18].

We will use the auxiliary sequence {αm} of continuous functions αm = αm(t), t ∈ [0, T], defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M19">View MathML</a>

(11)

It is obvious that, in particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M20">View MathML</a>

According to [[16], Lemma 4] or [[5], Lemma 2.4], we have the following estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M21">View MathML</a>

(12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M22">View MathML</a>

(13)

3 Successive approximations and convergence analysis

To investigate the solution of the parameterized boundary value problem (6) let us introduce the sequence of functions defined by the recurrence relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M23">View MathML</a>

(14)

where d(λ) = col(λ, 0) and x0(t, z, λ) = z, z Dγ. Let us note that for m = 0, 1, 2, ..., we have xm(t, z, λ) = col (xm,1(t, z, λ), xm,2(t, z, λ)). Moreover, all the functions xm = xm(t, z, λ) are continuously differentiable and satisfy the initial condition xm(0, z, λ) = z as well as the boundary conditions in (6).

Let us establish the uniform convergence of the sequence (14) and the relation of the limit function to the solution of some additively modified boundary value problem.

Theorem 2. Let the assumptions (A1)-(A4) be satisfied. Then for all z Dγ and λ ∈ Λ, the following statements hold

1. The sequence {xm} converges uniformly in t ∈ [0, T] to the limit function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M24">View MathML</a>

which satisfies the initial condition x*(0, z, λ) = z and the boundary conditions in (6).

2. For all t ∈ [0, T], the limit function x* satisfies the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M25">View MathML</a>

(15)

Moreover, x* is continuously differentiable and it is a unique solution of the Cauchy problem for the additively modified differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M26">View MathML</a>

(16)

3. The following error estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M27">View MathML</a>

(17)

Remark 3. We emphasize that the first component of the vector z is fixed and coincide with the value of x1(0) in the first boundary condition in (1), while its second component z2 is considered as free parameter. Thus, the expression "for all z", actually means "for all z2".

Proof (of Theorem 2). First, we show that for all (t, z, λ) ∈ [0, T] × Dγ × Λ and m ∈ ℕ, all functions xm = xm(t, z, λ) belong to D. Indeed, using the estimate in [[19], Lemma 2], an arbitrary continuous function u : [0, T] → ℝ satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M28">View MathML</a>

(18)

Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M29">View MathML</a>

(19)

Therefore, we conclude that x1(t, z, λ) ∈ D, whenever (t, z, λ) ∈ [0, T] × Dγ × Λ. By induction, we obtain that for all m ∈ ℕ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M30">View MathML</a>

i.e., all functions xm are also contained in D.

For m = 0, 1, 2, ..., let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M31">View MathML</a>

Due to the assumption (A2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M32">View MathML</a>

(20)

Relation (19) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M33">View MathML</a>

and thus using (20), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M34">View MathML</a>

By induction, we obtain for m = 1, 2, ... that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M35">View MathML</a>

Using (12) and (13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M36">View MathML</a>

and thus, for all (t, z, λ) ∈ [0, T] × Dγ × Λ and j, m ∈ ℕ, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M37">View MathML</a>

(21)

Due to the assumption (A4), the sequence {Qm} converges to the zero matrix for m → +∞. Hence, (21) implies that {xm} is a Cauchy sequence in the Banach space C([0, T], ℝ2) and therefore, the limit function x* = x*(t, z, λ) exists. Passing to the limit for j → +∞ in (21), we obtain the final error estimate (17).

The limit function x* satisfies the initial condition x*(0, z, λ) = z as well as the boundary conditions in (6), since these conditions are satisfied by all functions xm = xm(t, z, λ) of the sequence {xm}. Passing to the limit in the recurrence relation (14) for xm, we show that the limit function x* satisfies the identity (15). If we differentiate this identity, we obtain that x* is a unique solution of the Cauchy problem (16).

Let us find a relation of the limit function x* = x*(t, z, λ) of the sequence {xm} and the solution of the parameterized boundary value problem (6). For this purpose, let us define the function Δ : ℝ2 → ℝ2

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M38">View MathML</a>

Theorem 4. Let the assumptions (A1)-(A4) be satisfied. The limit function x* of the sequence {xm} is a solution of the boundary value problem (6) if and only if the value of the vector parameters z Dγ and λ ∈ Λ are such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M39">View MathML</a>

Proof. It is sufficient to apply Theorem 2 and notice that the equation in (16) coincides with the original equation in (6) if and only if the relation Δ(z, λ) = 0 holds.

Remark 5. The function Δ = Δ(z, λ) is called the determining function and the equation Δ(z, λ) = 0 is called the determining equation, because it determines the values of the unknown parameters z ∈ Dγ and λ ∈ Λ involved in the recurrence relation (14).

4 Properties of the limit function and the existence theorem

Let us investigate some properties of the limit function x* of the sequence {xm} and the determining function Δ.

Lemma 6. Under the assumptions (A1)-(A4), the limit function x* satisfies the following Lipschitz condition for all t ∈ [0, T], all z, y Dγ and λ ∈ Λ

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M40">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M41">View MathML</a>.

Proof. Using the assumption (A2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M42">View MathML</a>

Similarly, using the above estimate, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M43">View MathML</a>

and by induction, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M44">View MathML</a>

Using the estimates in (13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M45">View MathML</a>

and passing to the limit for m → +∞, due to the assumption (A4), we obtain the final inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M46">View MathML</a>

Lemma 7. Under the assumptions (A1)-(A4), the determining function Δ is well defined and bounded in Dγ × Λ. Furthermore, it satisfies the following Lipschitz condition for all z, y Dγ and λ ∈ Λ

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M47">View MathML</a>

Proof. It follows from Theorem 2 that the limit function x* of the sequence {xm} exists and is continuously differentiable in Dγ × Λ. Therefore, Δ is bounded and the assumption (A2) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M48">View MathML</a>

Using Lemma 6 and taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M49">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M50">View MathML</a>

Let us define the mth approximate determining function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M51">View MathML</a>

which has the following property.

Lemma 8. Let the assumptions (A1)-(A4) be satisfied. Then for any z Dγ, λ ∈ Λ and m ∈ ℕ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M52">View MathML</a>

(22)

Proof. Using the assumption (A2) and (17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M53">View MathML</a>

Let us introduce the relation f1 Ω f2 of two vector functions f1 = f1(x) and f2 = f2(x), which means that for all x ∈ Ω, at least one component of f1(x) is greater then the corresponding component of f2(x).

Definition 9. Let Ω be an arbitrary non-empty set. For any pair of functions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M54">View MathML</a>

we write f1 Ω f2 if and only if there exists a function k : Ω → {1, 2} such that for all x ∈ Ω

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M55">View MathML</a>

The following statement provides sufficient conditions for the solvability of the boundary value problem (6).

Theorem 10. Let the assumptions (A1)-(A4) be satisfied. Moreover, let there exist m ∈ ℕ and non-empty set Ω ⊂ Dγ × Λ such that the approximate determining function Δm satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M56">View MathML</a>

(23)

and the Brouwer degree of Δm over Ω with respect to 0 satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M57">View MathML</a>

(24)

Then, there exists a pair (z*, λ*) ∈ Ω such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M58">View MathML</a>

and the corresponding limit function x* = x*(t, z*, λ*) of the sequence {xm} solves the boundary value problem (6).

Proof. Let us introduce the mapping P : [0, 1] × Ω → ℝ2

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M59">View MathML</a>

Since the mappings (z, λ) ↦ Δ(z, λ) and (z, λ) ↦ Δm (z, λ) are continuous due to the continuity of xm, x* and f, the mapping P is continuous as well. Moreover, using Lemma 8 and (23), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M60">View MathML</a>

for all Θ ∈ [0, 1] and (z, λ) ∈ ∂Ω. Thus, the mapping P is the admissible homotopy connecting Δm and Δ and the Brouwer degrees deg(Δ, Ω, 0) and deg (Δm, Ω, 0) are well defined. The invariance property of the Brouwer degree under homotopy implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M61">View MathML</a>

The assumption (24) then guarantees the existence of (z*, λ*) ∈ Ω such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M62">View MathML</a>

Applying Theorem 4, we obtain that the limit function x* = x* (t, z*, λ*) of the sequence {xm} is the solution of the boundary value problem (6).

5 Polynomial successive approximations

In order to make the computations of xm possible or more easier, we give a justification of a polynomial version of the iterative scheme (14). At first, we recall some results of the theory of approximations in [20].

We denote by Hq a set of all polynomials of degree not higher than q and by Eq(f, Pq) the deviation of the function f from the polynomial Pq Hq

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M63">View MathML</a>

There exists a unique polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M64">View MathML</a> for which

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M65">View MathML</a>

This polynomial <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M66">View MathML</a> is said to be a polynomial of the best uniform approximation of f in Hq and the number Eq(f) is called the error of the best uniform approximation. It is known that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M67">View MathML</a>

Definition 11. Let f : [0, T] → ℝ be a uniformly continuous function and δ be a positive real number. Then we define the modulus of continuity of f as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M68">View MathML</a>

where the supremum is taken over all t, s ∈ [0, T] for which |t - s| ≤ δ.

Let us note that the modulus of continuity ω(f, δ) is a continuous non-decreasing function of the variable δ, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M69">View MathML</a>

Definition 12. We say that the function f : [0, T] → ℝ satisfies the Dini condition if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M70">View MathML</a>

Let us note that, e.g., all α-Hölder continuous functions on [0, T] with 0 < α ≤ 1 satisfy the Dini condition.

For a given function f, let us denote by fq the interpolation Chebyshev polynomial of degree q, q ∈ ℕ, which satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M71">View MathML</a>

where ti are the Chebyshev interpolation nodes in the interval [0, T]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M72">View MathML</a>

Lemma 13 (see [20]). If the function f satisfies the Dini condition, then the sequence {fq} of the corresponding interpolation Chebyshev polynomials converges uniformly on [0, T] to f and the following estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M73">View MathML</a>

(25)

for all t ∈ [0, T].

Let us introduce the sufficiency for the Dini condition for a composite function F(t) = f(t, x(t)).

Lemma 14. Let the function f = f(t, x) satisfy the Dini condition with respect to t ∈ [0, T] and the Lipschitz condition with respect to x D. Then for any continuously differentiable function x = x(t), t ∈ [0, T], with the values in D, the composite function F(t) = f(t, x(t)) satisfies the Dini condition in the interval [0, T].

Proof. Taking into account the Lipschitz condition, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M74">View MathML</a>

Since f and x both satisfy the Dini condition, we conclude that F satisfies the Dini condition as well.

For a given function f = f(t, z), let us define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M75">View MathML</a>

Using Lemmas 13 and 14, the function F(t) = f(t, x(t)) and its interpolation Chebyshev polynomial Fq(t) = fq(t, x(t)) satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M76">View MathML</a>

(26)

Let us note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M77">View MathML</a> and the sequence {Fq} uniformly converges to the function F on the interval [0, T].

Remark 15. In the case of vector functions f, the error of the best uniform approximation Eq(f), the modulus of continuity ω(f, δ) and Lq are vectors as well. The Dini condition and the construction of the corresponding Chebyshev polynomials are understood componentwise.

Let us return again to the boundary value problem (6) considered in the domain [0, T] × D × Λ. To investigate the solution of the parameterized boundary value problem (6), instead of (14), we introduce the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78">View MathML</a> of vector polynomials <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M79">View MathML</a> of degree (q + 1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M81">View MathML</a> is the vector of interpolation Chebyshev polynomial of degree q corresponding to f . Let us point out that the coefficients of the interpolation polynomials depend on the parameters z and λ. Moreover, all the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M82">View MathML</a> are continuously differentiable and satisfy the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M83">View MathML</a> as well as the boundary conditions in (6).

Let us define the domain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M84">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M85">View MathML</a>

(27)

Theorem 16. Let the assumptions (A1)-(A4) be satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M86">View MathML</a> instead of Dγ. Then for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M87">View MathML</a>, λ ∈ Λ, the following statements hold

1. The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78">View MathML</a>converges uniformly in t ∈ [0, T] to the limit function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M88">View MathML</a>

which satisfies the initial condition x*(0, z, λ) = z and the boundary conditions in (6).

2. The following error estimate holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M89">View MathML</a>

(28)

Proof. We show that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M90">View MathML</a> and m ∈ ℕ, all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M91">View MathML</a> belong to D. Similarly as in the proof of Theorem 2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M92">View MathML</a>

Therefore, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M93">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M94">View MathML</a>. By induction, we obtain that for all m ∈ ℕ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M95">View MathML</a>

i.e., all functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M96">View MathML</a> are also contained in D.

For j = 1, 2, ..., m and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M97">View MathML</a>, we estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M98">View MathML</a>

Taking into account that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M99">View MathML</a>

and using the assumption (A2) and the estimate (26), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M100">View MathML</a>

In particular, for j = 1 and j = 2 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M101">View MathML</a>

and by induction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M102">View MathML</a>

Using (12) and (13), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M103">View MathML</a>

and due to the assumption (A4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M104">View MathML</a>

By Theorem 2, we can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M105">View MathML</a>

Recall that the sequence {Qm} converges to the zero matrix for m → +∞ and Lq tends to the zero vector for q → +∞, which implies immediately that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78">View MathML</a> converges uniformly to x* on [0, T].

Let us define the mth approximate polynomial determining function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M106">View MathML</a>

(29)

Lemma 17. Let the assumptions (A1)-(A4) be satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M86">View MathML</a> instead of Dγ. Then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M87">View MathML</a>, λ ∈ Λ and m ∈ ℕ,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M107">View MathML</a>

Proof. Due to the assumption (A2), (26) and the error estimate (28), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M108">View MathML</a>

Theorem 18. Let the assumptions (A1)-(A4) be satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M86">View MathML</a>instead of Dγ. Moreover, let there exist m ∈ ℕ and nonempty set Ωq Dγ × Λ such that the approximate polynomial determining function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M109">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M110">View MathML</a>

(30)

and the Brouwer degree of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M109">View MathML</a>over Ωq with respect to 0 satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M111">View MathML</a>

Then there exists a pair (z*, λ*) ∈ Ωq such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M112">View MathML</a>

and the corresponding limit function x* = x*(t, z*, λ*) of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78">View MathML</a>solves the boundary value problem (6).

Proof. Using the same steps as in the proof of Theorem 10, we construct the admissible homotopy Pq : [0, 1] × Ωq → ℝ2

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M113">View MathML</a>

and we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M114">View MathML</a>

The assumption (31) then guarantees the existence of (z*, λ*) ∈ Ωq such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M115">View MathML</a>

Applying Theorem 4, we obtain that the limit function x* = x* (t, z*, λ*) of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M78">View MathML</a> is the solution of the boundary value problem (6).

6 Examples

In this section, we introduce three particular boundary value problems in the form of the system (1). The first problem is a linear one and enables us to build the sequence {xm} directly by the recurrence relation (14). The second problem is nonlinear and it is impossible to integrate in (14) in a closed form. Thus, we use the Chebyshev interpolation of the integrand to construct the sequence of successive approximations also in this case. In the last example, we use again the polynomial version of presented method in order to approximate a solution of the nonlinear Dirichlet problem containing p-Laplacian.

Example 1. Let us consider the following linear problem with the Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M116">View MathML</a>

(32)

which has a unique solution given in a closed form. Let us denote this solution by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M117">View MathML</a>.

All the assumptions (A1)-(A4) are satisfied. Let us take x0(t, z, λ) = col(0, z2) and construct the successive approximations Xm of the exact solution x° for m = 0, 1, 2, ... in the following way:

1. using the recurrence relation (14), evaluate xm+1(t, z, λ),

2. solve the (m+1)-th approximate determining equation Δm+1(z, λ) = 0 (system of two linear equations) and denote its solution by (zm+1, λm+1),

3. define Xm+1(t) := xm+1(t, zm+1, λm+1).

Figure 1 contains both components Xm,1 and Xm,2 of approximations Xm for m = 1, 2, 11 and also their differences from the exact solution x°. Let us point out that the maxima of both components of |X11(t) - x°(t)| for t ∈ 〈0, 1〉 are both less then 10-9.

thumbnailFigure 1. The approximations X1, X2 and X11 of the exact solution x° of (32).

Example 2. Let us investigate the nonlinear Dirichlet boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M118">View MathML</a>

(33)

which has a solution in the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M119">View MathML</a>. In this case, it is not possible to construct the sequence {Xm} of approximations using the iterative scheme as in the previous Example 1. Thus, we use the following polynomial version of the iterative scheme. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M120">View MathML</a> and for m = 0, 1, 2, ..., proceed the steps:

1. define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M121">View MathML</a> and realize the Chebyshev interpolation,

2. define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M122">View MathML</a>

3. define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M123">View MathML</a> and realize the Chebyshev interpolation with parameters z and λ

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M124">View MathML</a>

4. solve the (m + 1)-th approximate polynomial determining equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M125">View MathML</a>

and denote its solution by (zm+1, λm+1),

5. define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M126">View MathML</a>.

In Figure 2, it is possible to compare polynomial approximations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M127">View MathML</a> for q = 15, m = 1, 2, 11 and the corresponding differences from the exact solution x°. Let us note that the maximum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M128">View MathML</a> for t ∈ 〈0, 1〉 is less then 10-7 for j = 1 and less then 2·10-7 for j = 2.

thumbnailFigure 2. The polynomial approximations of the exact solution x° of (33) for q = 15.

Example 3. Let us consider the following nonlinear problem with the Dirichlet boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M129">View MathML</a>

(34)

where p > 1, sinp is the generalized sine function (see [21] for the definition),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M130">View MathML</a>

Let us recall that sinp is 2πp-periodic function on ℝ, which coincide with the sin function for p = 2. Moreover, the pair (sinp(πpt), (πp cosp(πpt))p-1) is a solution of the following initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M131">View MathML</a>

For ε = 0, the problem (34) reads as the Dirichlet boundary value problem with p-Laplacian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M132">View MathML</a>

For the problem (34), all the assumptions (A1)-(A4) are satisfied in the linear case p = 2. If p ≠ 2, then there exist bounded domains D = 〈a1, a2〉 × 〈b1, b2〉 for which the second assumption (A2) concerning the Lipschitz condition of f is not satisfied. Thus, we have to take into account the following additional assumptions on D in order to satisfy the assumption (A2):

1. for 1 < p < 2, we have to ensure that -ε < a1 < a2 or a1 < a2 < -ε,

2. for 2 < p, we have to ensure that 0 < b1 < b2 or b1 < b2 < 0.

Let us note that for p > 2, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M133">View MathML</a>, which appears in the first component of f, is not Lipschitz continuous on any interval containing zero. On the other hand, in the case of 1 < p < 2, the function gε in the second component of f is not Lipschitz continuous on any interval containing -ε.

Thus, all the assumptions (A1)-(A4) are satisfied if we take, e.g., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M136">View MathML</a>. The polynomial version of the iterative scheme from the previous Example 2 is applicable in this case. Figure 3 shows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M127">View MathML</a> for q = 15, m = 1, 2, 11. Their differences from the exact solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M137">View MathML</a> of the problem (34) are also available. Let us note that the maximum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M138">View MathML</a> for t ∈ 〈0, 1〉 is less then 7·10-4 for j = 1 and less then 6·10-4 for j = 2.

thumbnailFigure 3. The approximations of the exact solution x° of (34) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M141">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M142">View MathML</a> and q = 15.

Last Figure 4 shows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M127">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M139">View MathML</a>, ε = 0 and q = 15, m = 1, 2, 11. In spite of the fact that the assumption (A2) is not satisfied in this case, we obtain the polynomial approximation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M140">View MathML</a>, which differs from the exact solution less than 4·10-3 in both components.

thumbnailFigure 4. The approximations of the exact solution x° of (34) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/58/mathml/M143">View MathML</a>, ε = 0 and q = 15.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors contributed to each part of this study equally, and also read and approved the final manuscript.

Acknowledgements

The authors were partially supported by the Ministry of Education, Youth and Sports of the Czech Republic, grant no. ME09109 (Program KONTAKT) and by MSM 4977751301 (G. Holubová, P. Nečesal), and by the Hungarian Scientific Research Fund OTKA throught grant no. 68311 (M. Rontó). This research was carried out as part of the TAMOP-4.2.1.B-10/2/KONV-2010-0001 project with support by the European Union, co-financed by the European Social Fund. Finally, the research was supported in part by the Academy of Sciences of the Czech Republic through the Institutional Research Plan No.~AV0Z10190503 (A.~Ront\'o).

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