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This article is part of the series Proceedings of the International Congress in Honour of Professor Hari M. Srivastava.

Open Access Research

Expanding the applicability of Lavrentiev regularization methods for ill-posed problems

Ioannis K Argyros1, Yeol Je Cho2* and Santhosh George3

Author Affiliations

1 Department of Mathematical Sciences, Cameron University, Lawton, OK, 73505, USA

2 Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju, 660-701, Korea

3 Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Karnataka, 757 025, India

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Boundary Value Problems 2013, 2013:114  doi:10.1186/1687-2770-2013-114


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/114


Received:29 January 2013
Accepted:18 April 2013
Published:7 May 2013

© 2013 Argyros 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

In this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009).

MSC: 65F22, 65J15, 65J22, 65M30, 47A52.

Keywords:
Lavrentiev regularization method; Hilbert space; ill-posed problems; stopping index; Fréchet-derivative; source function; boundary value problem

1 Introduction

In this paper, we are interested in obtaining a stable approximate solution for a nonlinear ill-posed operator equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M2">View MathML</a> is a monotone operator and X is a Hilbert space. We denote the inner product and the corresponding norm on a Hilbert space by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M4">View MathML</a>, respectively. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M5">View MathML</a> stand for the open ball in X with center <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M6">View MathML</a> and radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M7">View MathML</a>. Note that F is a monotone operator if it satisfies the relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M8">View MathML</a>

(1.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M9">View MathML</a>.

We assume, throughout this paper, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M10">View MathML</a> is the available noisy data with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M11">View MathML</a>

(1.3)

and (1.1) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M12">View MathML</a>. Since (1.1) is ill-posed, its solution need not depend continuously on the data, i.e., small perturbation in the data can cause large deviations in the solution. So, the regularization methods are used [1-8]. Since F is monotone, the Lavrentiev regularization is used to obtain a stable approximate solution of (1.1). In the Lavrentiev regularization, the approximate solution is obtained as a solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M13">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M14">View MathML</a> is the regularization parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M15">View MathML</a> is an initial guess for the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M12">View MathML</a>.

In [9], Bakushinskii and Smirnova proposed an iterative method

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M17">View MathML</a>

(1.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M18">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M19">View MathML</a> is a sequence of positive real numbers satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M20">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M21">View MathML</a>. It is important to stop the iteration at an appropriate step, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M22">View MathML</a>, and show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23">View MathML</a> is well defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26">View MathML</a> (see [10]).

In [9,11,12], Bakushinskii and Smirnova chose the stopping index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27">View MathML</a> by requiring it to satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M28">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M30">View MathML</a>. In fact, they showed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26">View MathML</a> under the following assumptions:

(1) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M33">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M34">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M35">View MathML</a>;

(2) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M36">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M37">View MathML</a>

(1.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M38">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M39">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M40">View MathML</a>

However, no error estimate was given in [9] (see [10]).

In [10], Mahale and Nair, motivated by the work of Qi-Nian Jin [13] for an iteratively regularized Gauss-Newton method, considered an alternate stopping criterion which not only ensures the convergence, but also derives an order optimal error estimate under a general source condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M41">View MathML</a>. Moreover, the condition that they imposed on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42">View MathML</a> is weaker than (1.6).

In the present paper, we are motivated by [10]. In particular, we expand the applicability of the method (1.5) by weakening one of the major hypotheses in [10] (see Assumption 2.1(2) in the next section).

In Section 2, we consider some basic assumptions required throughout the paper. Section 3 deals with the stopping rule and the result that establishes the existence of the stopping index. In Section 4, we prove results for the iterations based on the exact data and, in Section 5, the error analysis for the noisy data case is proved. The main order optimal result using the a posteriori stopping rule is provided in Section 6.

2 Basic assumptions and some preliminary results

We use the following assumptions to prove the results in this paper.

Assumption 2.1

(1) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M7">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M44">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M45">View MathML</a> is Fréchet differentiable.

(2) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M46">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49">View MathML</a>, there exists an element, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M50">View MathML</a>, satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M51">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49">View MathML</a>.

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M54">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M52">View MathML</a>.

(4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M56">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M52">View MathML</a>.

The condition (2) in Assumption 2.1 weakens the popular hypotheses given in [10,14] and [15].

Assumption 2.2 There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M58">View MathML</a> such that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49">View MathML</a>, there exists an element denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M61">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M62">View MathML</a>

Clearly, Assumption 2.2 implies Assumption 2.1(2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M63">View MathML</a>, but not necessarily vice versa. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M64">View MathML</a> holds in general and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M65">View MathML</a> can be arbitrarily large [16-20]. Indeed, there are many classes of operators satisfying Assumption 2.1(2), but not Assumption 2.2 (see the numerical examples at the end of this study). Moreover, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M66">View MathML</a> is sufficiently smaller than K, which can happen since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M67">View MathML</a> can be arbitrarily large, then the results obtained in this study provide a tighter error analysis than the one in [10].

Finally, note that the computation of constant K is more expensive than the computation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M68">View MathML</a>.

We need the auxiliary results based on Assumption 2.1.

Proposition 2.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M69">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M70">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M71">View MathML</a>

Proof Using the fundamental theorem of integration, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M69">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M73">View MathML</a>

Hence, by Assumption 2.2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M74">View MathML</a>

Then, by (2), (3) in Assumption 2.1 and the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M75">View MathML</a>, we obtain in turn

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M76">View MathML</a>

This completes the proof. □

Proposition 2.4For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M69">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M70">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M79">View MathML</a>

(2.1)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M80">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49">View MathML</a>. Then we have, by Assumption 2.2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M82">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M49">View MathML</a>. This completes the proof. □

Assumption 2.5 There exists a continuous and strictly monotonically increasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M84">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M85">View MathML</a> satisfying

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M86">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M87">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M88">View MathML</a>;

(3) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M89">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M90">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M91">View MathML</a>

(2.2)

Next, we assume a condition on the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42">View MathML</a> considered in (1.5).

Assumption 2.6 ([10], Assumption 2.6)

The sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M93">View MathML</a> of positive real numbers is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M94">View MathML</a>

(2.3)

for a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M95">View MathML</a>.

Note that the condition (2.3) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42">View MathML</a> is weaker than (1.6) considered by Bakushinskii and Smirnova [9] (see [10]). In fact, if (1.6) is satisfied, then it also satisfies (2.3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M97">View MathML</a>, but the converse need not be true (see [10]). Further, note that for these choices of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M42">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M99">View MathML</a> is bounded, whereas <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M100">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M101">View MathML</a>. (2) in Assumption 2.1 is used in the literature for regularization of many nonlinear ill-posed problems (see [4,7,8,13,21]).

3 Stopping rule

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M102">View MathML</a> and choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27">View MathML</a> to be the first non-negative integer such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M104">View MathML</a> in (1.5) is defined for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M105">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M106">View MathML</a>

(3.1)

In the following, we establish the existence of such a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27">View MathML</a>. First, we consider the positive integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M108">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M109">View MathML</a>

(3.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M110">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M112">View MathML</a>.

The following technical lemma from [10] is used to prove some of the results of this paper.

Lemma 3.1 ([10], Lemma 3.1)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M113">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M114">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M115">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M116">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M117">View MathML</a>be non-negative real numbers such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M118">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M119">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M120">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M121">View MathML</a>.

The rest of the results in this paper can be proved along the same lines as those of the proof in [10]. In order for us to make the paper as self-contained as possible, we present the proof of one of them, and for the proof of the rest, we refer the reader to [10].

Theorem 3.2 ([10], Theorem 3.2)

Let (1.2), (1.3), (2.3) and Assumption 2.1 be satisfied. LetNbe as in (3.2) for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M111">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M123">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M104">View MathML</a>is defined iteratively for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M126">View MathML</a>

(3.3)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>. In particular, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M128">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M129">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M131">View MathML</a>

(3.4)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M132">View MathML</a>.

Proof We show (3.3) by induction. It is obvious that (3.3) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M133">View MathML</a>. Now, assume that (3.3) holds for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>. Then it follows from (1.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M135">View MathML</a>

(3.5)

Using (1.3), the estimates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M136">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M137">View MathML</a> and Proposition 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M138">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M139">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M140">View MathML</a>

But, by (3.2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M141">View MathML</a> and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M142">View MathML</a>

which leads to the recurrence relation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M143">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M144">View MathML</a>

From the hypothesis of the theorem, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M145">View MathML</a>. It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M146">View MathML</a>

Hence, by Lemma 3.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M147">View MathML</a>

(3.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>. In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M149">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M150">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>.

Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M152">View MathML</a>. Then, using the estimates

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M153">View MathML</a>

and Proposition 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M154">View MathML</a>

(3.7)

Therefore, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M155">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M132">View MathML</a>. This completes the proof. □

4 Error bound for the case of noise-free data

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M157">View MathML</a>

(4.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M158">View MathML</a>.

We show that each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23">View MathML</a> is well defined and belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M160">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M161">View MathML</a>. For this, we make use of the following lemma.

Lemma 4.1 ([10], Lemma 4.1)

Let Assumption 2.1 hold. Suppose that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23">View MathML</a>in (4.1) is well defined and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M164">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M165">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M166">View MathML</a>

(4.2)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M167">View MathML</a>.

Theorem 4.2 ([10], Theorem 4.2)

Let Assumption 2.1 hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M168">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M161">View MathML</a>, then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170">View MathML</a>, the iterates<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M23">View MathML</a>in (4.1) are well defined and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M172">View MathML</a>

(4.3)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170">View MathML</a>.

Lemma 4.3 ([10], Lemma 4.3)

Let Assumptions 2.1 and 2.6 hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M161">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M175">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M176">View MathML</a>for someηwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M177">View MathML</a>. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M179">View MathML</a>

(4.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M180">View MathML</a>

(4.5)

The following corollary follows from Lemma 4.3 by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M181">View MathML</a>. We show that this particular case of Lemma 4.3 is better suited for our later results.

Corollary 4.4 ([10], Corollary 4.4)

Let Assumptions 2.1 and 2.6 hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M182">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M183">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M184">View MathML</a>. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M170">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M186">View MathML</a>

(4.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M187">View MathML</a>

Theorem 4.5 ([10], Theorem 4.5)

Let the assumptions of Lemma 4.3 hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M15">View MathML</a>is chosen such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M189">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M190">View MathML</a>.

Lemma 4.6 ([10], Lemma 4.6)

Let the assumptions of Lemma 4.3 hold forηsatisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M191">View MathML</a>

(4.7)

Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M192">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M193">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M194">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M195">View MathML</a>

Remark 4.7 ([10], Remark 4.7)

It can be seen that (4.7) is satisfied if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M196">View MathML</a>.

Now, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M181">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M198">View MathML</a> in Lemma 4.6, then it takes the following form.

Lemma 4.8 ([10], Lemma 4.8)

Let the assumptions of Lemma 4.3 hold with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M181">View MathML</a>. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M200">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M201">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M202">View MathML</a>

5 Error analysis with noisy data

The first result in this section gives an error estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M203">View MathML</a> under Assumption 2.5, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M204">View MathML</a>.

Lemma 5.1 ([10], Lemma 5.1)

Let Assumption 2.1 hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M205">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M206">View MathML</a>, andNbe the integer satisfying (3.2) with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M207">View MathML</a>

Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M209">View MathML</a>

(5.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M210">View MathML</a>

If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M211">View MathML</a> in Lemma 5.1, then we get the following corollary as a particular case of Lemma 5.1. We make use of it in the following error analysis.

Corollary 5.2 ([10], Corollary 5.2)

Let Assumption 2.1 hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M212">View MathML</a>. LetNbe the integer defined by (3.2) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M213">View MathML</a>. Then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M125">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M215">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M216">View MathML</a>

Lemma 5.3 ([10], Lemma 5.3)

Let the assumptions of Lemma 5.1 hold. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M217">View MathML</a>

Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M218">View MathML</a>, then, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M219">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M220">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M221">View MathML</a>

with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M222">View MathML</a>andκas in Lemma 5.1.

Theorem 5.4 ([10], Theorem 5.4)

Let Assumptions 2.1 and 2.6 hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M223">View MathML</a>and the integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27">View MathML</a>is chosen according to stopping rule (3.1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M225">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M226">View MathML</a>

(5.2)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M227">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M228">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M229">View MathML</a>andκas in Lemma 4.8 and Corollary 5.2, respectively, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M230">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M231">View MathML</a>as in Lemma 5.3.

6 Order optimal result with an a posteriori stopping rule

In this section, we show the convergence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M233">View MathML</a> and also give an optimal error estimate for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M234">View MathML</a>.

Theorem 6.1 ([10], Theorem 6.1)

Let the assumptions of Theorem 5.4 hold and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M27">View MathML</a>be the integer chosen by (3.1). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M15">View MathML</a>is chosen such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M189">View MathML</a>, then we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M238">View MathML</a>. Moreover, if Assumption 2.5 is satisfied, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M239">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M240">View MathML</a>withξas in Theorem 5.4 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M241">View MathML</a>is defined as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M243">View MathML</a>.

Proof From (4.6) and (5.2), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M244">View MathML</a>

(6.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M245">View MathML</a>. Now, we choose an integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M246">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M247">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M248">View MathML</a>

(6.2)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M249">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M250">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26">View MathML</a>. Therefore by (6.2) to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M25">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26">View MathML</a>, it is enough to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M254">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26">View MathML</a>. Observe that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M256">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M257">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M258">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M259">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26">View MathML</a>. Now since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M261">View MathML</a> is a dense subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M262">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M263">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M26">View MathML</a>. Using Assumption 2.5, we get that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M265">View MathML</a>

(6.3)

So, by (6.2) and (6.3), we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M266">View MathML</a>

(6.4)

Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M267">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M268">View MathML</a>

(6.5)

This also implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M269">View MathML</a>

(6.6)

From (6.4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M270">View MathML</a>. Now, using (6.5) and (6.6), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M271">View MathML</a>. This completes the proof. □

7 Numerical examples

We provide two numerical examples, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M272">View MathML</a>.

Example 7.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M273">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M274">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275">View MathML</a> and define a function F on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M276">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M277">View MathML</a>

(7.1)

Then, using (7.1) and Assumptions 2.1(2) and 2.2, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M278">View MathML</a>

Example 7.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M279">View MathML</a> (: the space of continuous functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M280">View MathML</a> equipped with the max norm) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M274">View MathML</a>. Define an operator F on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M276">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M283">View MathML</a>

(7.2)

Then the Fréchet-derivative is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M284">View MathML</a>

(7.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M258">View MathML</a>. Using (7.2), (7.3), Assumptions 2.1(2), 2.2 for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M287">View MathML</a>.

Next, we provide an example where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M67">View MathML</a> can be arbitrarily large.

Example 7.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275">View MathML</a> and define a function F on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M276">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M292">View MathML</a>

(7.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M293">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M294">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M295">View MathML</a> are the given parameters. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M296">View MathML</a>. Then it can easily be seen that, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M295">View MathML</a> sufficiently large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M294">View MathML</a> sufficiently small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M67">View MathML</a> can be arbitrarily large.

We now present two examples where Assumption 2.2 is not satisfied, but Assumption 2.1(2) is satisfied.

Example 7.4 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M275">View MathML</a> and define a function F on D by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M302">View MathML</a>

(7.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M230">View MathML</a> is a real parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M304">View MathML</a> is an integer. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M305">View MathML</a> is not Lipschitz on D. Hence Assumption 2.2 is not satisfied. However, the central Lipschitz condition in Assumption 2.2(2) holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M306">View MathML</a>. We also have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M307">View MathML</a>. Indeed, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M308">View MathML</a>

and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M309">View MathML</a>

Example 7.5 We consider the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M310">View MathML</a>

(7.6)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M311">View MathML</a>, where f is a given continuous function satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M312">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M313">View MathML</a>, λ is a real number and the kernel G is continuous and positive in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M314">View MathML</a>.

For example, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M315">View MathML</a> is the Green kernel, the corresponding integral equation is equivalent to the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M316">View MathML</a>

These types of problems have been considered in [16-20]. The equation of the form (7.6) generalizes the equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M317">View MathML</a>

(7.7)

which was studied in [16-20]. Instead of (7.6), we can try to solve the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M318">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M319">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M320">View MathML</a>

The norm we consider is the max-norm. The derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M321">View MathML</a> is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M322">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M323">View MathML</a>. First of all, we notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M321">View MathML</a> does not satisfy the Lipschitz-type condition in Ω. Let us consider, for instance, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M325">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M326">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M327">View MathML</a>. Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M328">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M329">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M321">View MathML</a> were the Lipschitz function, then we had

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M331">View MathML</a>

or, equivalently, the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M332">View MathML</a>

(7.8)

would hold for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M333">View MathML</a> and for a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M334">View MathML</a>. But this is not true. Consider, for example, the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M335">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M336">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M337">View MathML</a>. If these are substituted into (7.7), then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M338">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M336">View MathML</a>. This inequality is not true when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M340">View MathML</a>. Therefore, Assumption 2.2 is not satisfied in this case. However, Assumption 2.1(2) holds. To show this, suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M341">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M342">View MathML</a>. Then, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M323">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M344">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M345">View MathML</a>. Hence it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M346">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M347">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M348">View MathML</a>. Then Assumption 2.1(2) holds for sufficiently small λ.

In the following remarks, we compare our results with the corresponding ones in [10].

Remark 7.6 Note that the results in [10] were shown using Assumption 2.2, whereas we used weaker Assumption 2.1(2) in this paper. Next, our result, Proposition 2.3, was shown with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M349">View MathML</a> replacing K. Therefore, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M350">View MathML</a> (see Example 7.3), then our result is tighter. Proposition 2.4 was shown with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M68">View MathML</a> replacing K. Then, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M272">View MathML</a>, then our result is tighter. Theorem 3.2 was shown with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M353">View MathML</a> replacing 2K. Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M350">View MathML</a>, our result is tighter. Similar favorable to us observations are made for Lemma 4.1, Theorem 4.2 and the rest of the results in [10].

Remark 7.7 The results obtained here can also be realized for the operators F satisfying an autonomous differential equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M355">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M356">View MathML</a> is a known continuous operator. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M357">View MathML</a>, we can compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M68">View MathML</a> in Assumption 2.1(2) without actually knowing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M12">View MathML</a>. Returning back to Example 7.1, we see that we can set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/114/mathml/M360">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This paper was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

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