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Approximate method for boundary value problems of anti-periodic type for differential equations with ‘maxima’

Snezhana Hristova*, Angel Golev and Kremena Stefanova

Author Affiliations

Faculty of Mathematics and Informatics, Plovdiv University, Plovdiv, 4000, Bulgaria

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Boundary Value Problems 2013, 2013:12  doi:10.1186/1687-2770-2013-12


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/12


Received:16 October 2012
Accepted:9 January 2013
Published:25 January 2013

© 2013 Hristova 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

An algorithm for constructing two sequences of successive approximations of a solution of the nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’ is given. The case of a boundary condition of anti-periodic type is investigated. This algorithm is based on the monotone iterative technique. Two sequences of successive approximations are constructed. It is proved both sequences are monotonically convergent. Each term of the constructed sequences is a solution of an initial value problem for a linear differential equation with ‘maxima’ and it is a lower/upper solution of the given problem. A computer realization of the algorithm is suggested and it is illustrated on a particular example.

MSC: 34K10, 34K25, 34B15.

Keywords:
differential equations with ‘maxima’; nonlinear boundary value problem; approximate solution; computer realization

1 Introduction

Differential equations with ‘maxima’ are adequate models of real world problems, in which the present state depends significantly on its maximum value on a past time interval (see [1-4], monograph [5]).

Note that usually differential equations with ‘maxima’ are not possible to be solved in an explicit form and that requires the application of approximate methods. In the current paper, the monotone iterative technique [6,7], based on the method of lower and upper solutions, is theoretically proved to a boundary value problem for a nonlinear differential equation with ‘maxima’. The case when the nonlinear boundary function is a nondecreasing one with respect to its second argument is studied. This type of the boundary function covers the case of an anti-periodic boundary condition. An improved algorithm of monotone-iterative techniques is suggested. The main advantage of this scheme is connected with the construction of the initial conditions.

2 Preliminary notes and definitions

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M1">View MathML</a> be a given fixed point and h be a positive constant. Consider the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M2">View MathML</a>

Consider the following nonlinear differential equation with ‘maxima’:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M3">View MathML</a>

(1)

with a boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M4">View MathML</a>

(2)

and an initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M5">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M8">View MathML</a>.

In this paper, we study boundary condition (2) in the case when the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M9">View MathML</a> is nondecreasing with respect to its second argument y. So, the anti-periodic boundary value problem is a partial case of boundary condition (2). Note that similar problems are investigated for ordinary differential equations [8], delay differential equations [9] and impulsive differential equations [10], and some approximate methods are suggested. The presence of the maximum of the unknown function requires additionally some new comparison results, existence results as well as a new algorithm for constructing successive approximations to the exact unknown solution.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M10">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M11">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M12">View MathML</a>. Define the following sets:

Definition 1 The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M14">View MathML</a> is said to be from the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M15">View MathML</a> if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M16">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M17">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M18">View MathML</a>, the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M19">View MathML</a> holds.

Definition 2 The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M14">View MathML</a> is said to be quasi-nondecreasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M21">View MathML</a> if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M22">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M23">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M24">View MathML</a>, the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M25">View MathML</a> holds.

In connection with the construction of successive approximations, we will introduce a couple of quasi-solutions of boundary value problem (1)-(3).

Definition 3 We will say that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M26">View MathML</a> form a couple of quasi-solutions of boundary value problem (1)-(3), if they satisfy the equations <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M27">View MathML</a>, (1) and (3).

Definition 4 We will say that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M26">View MathML</a> form a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3), if

(4)

and

(5)

In the proof of our main results, we will use the following lemma.

Lemma 1 (Comparison result)

Let the following conditions be fulfilled:

1. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M31">View MathML</a>satisfy the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M32">View MathML</a>

(6)

2. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M33">View MathML</a>satisfies the inequalities

(7)

(8)

Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M36">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>.

Proof Assume the statement of Lemma 1 is not true. Consider the following two cases.

Case 1: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M38">View MathML</a>. According to the assumption, it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M39">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M40">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M41">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M42">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M43">View MathML</a>.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M44">View MathML</a>, where λ is a positive constant. Let the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M45">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M46">View MathML</a>.

According to the mean value theorem, it follows that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M47">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M48">View MathML</a>

(9)

From inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M50">View MathML</a> and (7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M51">View MathML</a>

(10)

Inequality (10) contradicts (6).

Case 2: Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M52">View MathML</a>. Define a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M53">View MathML</a> by the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M54">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M55">View MathML</a> is a small enough constant.

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M57">View MathML</a> satisfies inequality (7). From case 1 it follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M58">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59">View MathML</a>. Take a limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M60">View MathML</a> and obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M36">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>. □

In our further investigations, we will use the following result for differential equations with ‘maxima’ which is a partial case of Theorem 3.1.1 [5].

Lemma 2 (Existence and uniqueness)

Let the following conditions be fulfilled:

1. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M63">View MathML</a>.

2. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M64">View MathML</a>and satisfy inequality (6).

Then the initial value problem for a linear differential equation with ‘maxima’

has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M66">View MathML</a>.

3 Monotone-iterative method

We will give an algorithm for obtaining an approximate solution of the boundary value problem for a nonlinear differential equation with ‘maxima’ (1)-(3).

Theorem 1Let the following conditions be fulfilled:

1. The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M67">View MathML</a>form a couple of quasi-lower and quasi-upper solutions of (1)-(3) such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M68">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>.

2. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M70">View MathML</a>is quasi-nondecreasing in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M71">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M72">View MathML</a>.

3. The function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M73">View MathML</a>and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M74">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M24">View MathML</a>the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M77">View MathML</a>

holds, where the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M31">View MathML</a>satisfy inequality (6).

Then there exist two sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M79">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M80">View MathML</a>such that

(a) The functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M81">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M82">View MathML</a>) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M83">View MathML</a>is a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3).

(b) The sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M84">View MathML</a>is nondecreasing.

(c) The sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M85">View MathML</a>is nonincreasing.

(d) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>the inequalities

(11)

hold.

(e) Both sequences are uniformly convergent on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M12">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M89">View MathML</a>is a couple of quasi-solutions of boundary value problem (1)-(3) in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M90">View MathML</a>.

(f) If additionally the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M91">View MathML</a>is Lipschitz in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M92">View MathML</a>, then there exists a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M93">View MathML</a>of boundary value problem (1)-(3) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M94">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>.

Proof We will give an algorithm for construction of successive approximations to the unknown exact solution of nonlinear boundary value problem (1)-(3).

Assume the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M96">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M98">View MathML</a>, are constructed. Then consider both initial value problems for the linear differential equations with ‘maxima’

(12)

(13)

and

(14)

(15)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M103">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M104">View MathML</a>

According to Lemma 2, initial value problems (12), (13) and (14), (15) have unique solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M105">View MathML</a>.

So, step by step we can construct two sequences of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M106">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M107">View MathML</a>.

Now, we will prove by induction that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M108">View MathML</a> ,

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M110">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M112">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M114">View MathML</a> is a couple of quasi-lower and quasi-upper solutions of boundary value problem (1)-(3).

Assume the claims (H1)-(H3) are satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M115">View MathML</a>.

We will prove (H1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116">View MathML</a>.

Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M117">View MathML</a> by the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M118">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119">View MathML</a>. Then according to condition 2 of Theorem 1, the inductive assumption and the definition of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M122">View MathML</a>

(16)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a>. From (H1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M124">View MathML</a>, condition 3 of Theorem 1, the definition of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121">View MathML</a> and (12), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M127">View MathML</a>

(17)

Note that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a> the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M129">View MathML</a>

(18)

From inequalities (17) and (18) it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M130">View MathML</a>

According to Lemma 1, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M131">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M133">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>.

Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M135">View MathML</a> by the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M136">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M138">View MathML</a>

(19)

From equation (14), the inductive assumption, the definition of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140">View MathML</a> and condition 3 of Theorem 1, it follows the validity of the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M141">View MathML</a>

According to Lemma 1, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M142">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59">View MathML</a>, i.e., the claim (H1) is true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116">View MathML</a>.

Define the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M145">View MathML</a> by the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M146">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119">View MathML</a>. From condition 2 of Theorem 1, the inductive assumption and the definition of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M150">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a>. According to the choice of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140">View MathML</a>, condition 3 of Theorem 1 and inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M154">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M155">View MathML</a>

According to Lemma 1, it follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M156">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M59">View MathML</a>. Therefore, the claim (H2) is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116">View MathML</a>.

Now, we will prove the claim (H3) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M119">View MathML</a>. Then from (13) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M161">View MathML</a>

(20)

From (H1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116">View MathML</a>, condition 2 of Theorem 1 and the choice of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M121">View MathML</a>, we obtain

(21)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a>. From condition 3 of Theorem 1, inequalities (18) and (H1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M166">View MathML</a>

(22)

Similarly, we prove the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M140">View MathML</a> satisfies inequalities (5). Therefore, the claim (H3) is true for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M116">View MathML</a>. Furthermore, the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M169">View MathML</a>.

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>, the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M171">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M172">View MathML</a> are nondecreasing and nonincreasing, respectively, and they are bounded by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M174">View MathML</a>.

Therefore, both sequences converge pointwisely and monotonically. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M175">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M176">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>. According to Dini’s theorem, both sequences converge uniformly and the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M178">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M179">View MathML</a> are continuous. Additionally, the claims (H1), (H2) prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M180">View MathML</a>.

Now, we will prove that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a> the following equality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M182">View MathML</a>

(23)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a>, we introduce the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M184">View MathML</a>. From condition (H1) it follows that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M185">View MathML</a> the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M186">View MathML</a> hold and thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M82">View MathML</a> , i.e., the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M189">View MathML</a> is monotone nondecreasing and bounded from above by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M190">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>. Therefore, there exists the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M192">View MathML</a>.

From the monotonicity of the sequence of the quasi-lower solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120">View MathML</a>, we get that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M185">View MathML</a> the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M195">View MathML</a> holds. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M196">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M197">View MathML</a>.

Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M198">View MathML</a>. Then there exists a natural number N such that the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M199">View MathML</a> hold. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M200">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M201">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M202">View MathML</a>. The obtained contradiction proves the assumption is not valid.

Assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M203">View MathML</a>. According to the definition of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M178">View MathML</a>, it follows that for the fixed number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M205">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M206">View MathML</a>. Then there exists a natural number N such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M207">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M208">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M209">View MathML</a>. The obtained contradiction proves the assumption is not valid.

Therefore, the required equality (23) is fulfilled.

In a similar way, we can prove that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a> the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M211">View MathML</a>

(24)

holds.

Take a limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M212">View MathML</a> in (13) and get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M213">View MathML</a>

(25)

From (25) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M214">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M215">View MathML</a>.

Taking a limit in the integral equation equivalent to (12), we obtain the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M178">View MathML</a> satisfies equation (1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a>.

In a similar way, we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M179">View MathML</a> satisfies equation (1) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M123">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M220">View MathML</a>. Therefore, the couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M89">View MathML</a> is a couple of quasi-solutions of (1)-(3) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M90">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M223">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>.

Let the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M91">View MathML</a> be Lipschitz. Then if (1) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M93">View MathML</a>, it is unique (see [11]). In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M227">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M37">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M229">View MathML</a>

 □

4 Applications

We will apply the given above algorithm for approximate solving of a nonlinear boundary value problem.

Example

Consider the following nonlinear boundary value problem for a nonlinear differential equation with ‘maxima’:

(26)

(27)

Boundary value problem (26), (27) is of type (1)-(3), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M235">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M237">View MathML</a>. The couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M238">View MathML</a> is a couple of quasi-lower and quasi-upper solutions of boundary value problem (26), (27).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M241">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M242">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M244">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M245">View MathML</a>. Thus, condition 3 of Theorem 1 holds.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M246">View MathML</a> is quasi-nondecreasing with respect to y and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M248">View MathML</a>.

The above given problem has a zero solution. We will apply the procedure given in Theorem 1 to obtain two sequences, which are monotonically convergent to 0.

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M250">View MathML</a>, is a solution of problem (12), (13), which is reduced to the following linear initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M251">View MathML</a>

(28)

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M250">View MathML</a>, is a solution of problem (14), (15), which is reduced to the following linear initial value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M254">View MathML</a>

(29)

According to Lemma 2, initial value problems (28) and (29) have unique solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139">View MathML</a>, respectively. Because of the presence of the maximum of the unknown function over a past time interval, there is no explicit formula for the exact solutions of (28) and (29). We use a computer program based on a modified numerical method to solve these problems (see [12]).

Also, by a computer realization of the scheme given in Theorem 1 and applied to problems (28) and (29), we obtain the values in Table 1.

Table 1. Values of the successive approximations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M120">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M139">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M259">View MathML</a>

From Table 1 and Figure 1, it is obvious that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M270">View MathML</a> is increasing and the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M271">View MathML</a> is decreasing and both monotonically converge to the unique solution 0 of nonlinear boundary value problem (26), (27).

thumbnailFigure 1. Graphic of the successive approximations<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M272">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M273">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/12/mathml/M274">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors SH, AG and KS contributed to each part of the work equally and read and proved the final version of the manuscript.

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