Approximation of Solutions for Second-Order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq1_HTML.gif -Point Nonlocal Boundary Value Problems via the Method of Generalized Quasilinearization

Boundary Value Problems20102011:929061

DOI: 10.1155/2011/929061

Received: 11 May 2010

Accepted: 2 October 2010

Published: 4 October 2010

Abstract

We discuss the existence and uniqueness of the solutions of a second-order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq2_HTML.gif -point nonlocal boundary value problem by applying a generalized quasilinearization technique. A monotone sequence of solutions converging uniformly and quadratically to a unique solution of the problem is presented.

1. Introduction

The monotone iterative technique coupled with the method of upper and lower solutions [17] manifests itself as an effective and flexible mechanism that offers theoretical as well as constructive existence results in a closed set, generated by the lower and upper solutions. In general, the convergence of the sequence of approximate solutions given by the monotone iterative technique is at most linear [8, 9]. To obtain a sequence of approximate solutions converging quadratically, we use the method of quasilinearization [10]. This method has been developed for a variety of problems [1120]. In view of its diverse applications, this approach is quite an elegant and easier for application algorithms.

The subject of multipoint nonlocal boundary conditions, initiated by Bicadze and Samarskiĭ [21], has been addressed by many authors, for instance, [2232]. The multipoint boundary conditions appear in certain problems of thermodynamics, elasticity and wave propagation, see [23] and the references therein. The multipoint boundary conditions may be understood in the sense that the controllers at the endpoints dissipate or add energy according to censors located at intermediate positions.

In this paper, we develop the method of generalized quasilinearization to obtain a sequence of approximate solutions converging monotonically and quadratically to a unique solution of the following second-order http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq3_HTML.gif point nonlocal boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ1_HTML.gif
(1.1)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq4_HTML.gif is continuous and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq5_HTML.gif are nonnegative real constants such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq8_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq9_HTML.gif

Here we remark that [26] studies (1.1) with the boundary conditions of the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ3_HTML.gif
(1.3)
A perturbed integral equation equivalent to the problem (1.1) and (1.3) considered in [26] is
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ4_HTML.gif
(1.4)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ5_HTML.gif
(1.5)
It can readily be verified that the solution given by (1.4) does not satisfy (1.1). On the other hand, by Green's function method, a unique solution of the problem (1.1) and (1.3) is
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ6_HTML.gif
(1.6)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq10_HTML.gif is given by (1.5). Thus, (1.6) represents the correct form of the solution for the problem (1.1) and (1.3).

2. Preliminaries

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq11_HTML.gif we define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq12_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq13_HTML.gif It can easily be verified that the homogeneous problem associated with (1.1)-(1.2) has only the trivial solution. Therefore, by Green's function method, the solution of (1.1)-(1.2) can be written as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ7_HTML.gif
(2.1)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq14_HTML.gif is the Green's function and is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ8_HTML.gif
(2.2)

Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq15_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq16_HTML.gif

We say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq17_HTML.gif is a lower solution of the boundary value problem (1.1) and (1.2) if
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ9_HTML.gif
(2.3)
and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq18_HTML.gif is an upper solution of (1.1) and (1.2) if
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ10_HTML.gif
(2.4)

Definition 2.1.

A continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq19_HTML.gif is called a Nagumo function if
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ11_HTML.gif
(2.5)

for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq20_HTML.gif . We say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq21_HTML.gif satisfies a Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq22_HTML.gif relative to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq23_HTML.gif if for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq25_HTML.gif there exists a Nagumo function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq26_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq27_HTML.gif

We need the following result [33] to establish the main result.

Theorem 2.2.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq28_HTML.gif be a continuous function satisfying the Nagumo condition on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq29_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq30_HTML.gif are continuous functions such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq31_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq32_HTML.gif Then there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq33_HTML.gif (depending only on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq34_HTML.gif the Nagumo function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq35_HTML.gif ) such that every solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq36_HTML.gif of (1.1)-(1.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq38_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq39_HTML.gif

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq40_HTML.gif are assumed to be lower and upper solutions of (1.1)-(1.2), respectively, in the statement of Theorem 2.2, then there exists a solution, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq41_HTML.gif of (1.1) and (1.2) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq43_HTML.gif

Theorem 2.3.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq44_HTML.gif are, respectively, lower and upper solutions of (1.1)-(1.2). If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq45_HTML.gif is decreasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq46_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq47_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq48_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq49_HTML.gif

Proof.

Let us define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq50_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq51_HTML.gif and satisfies the boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ12_HTML.gif
(2.6)
For the sake of contradiction, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq52_HTML.gif have a positive maximum at some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq53_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq54_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq56_HTML.gif On the other hand, in view of the decreasing property of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq57_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq58_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ13_HTML.gif
(2.7)
which is a contradiction. If we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq59_HTML.gif has a positive maximum at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq60_HTML.gif , then it follows from the first of boundary conditions (2.6) that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ14_HTML.gif
(2.8)

which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq61_HTML.gif Now as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq63_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq65_HTML.gif therefore we obtain a contradiction. We have a similar contradiction at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq66_HTML.gif Thus, we conclude that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq68_HTML.gif

3. Main Results

Theorem 3.1.

Assume that

the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq70_HTML.gif are, respectively, lower and upper solutions of (1.1)-(1.2) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq71_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq72_HTML.gif

the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq74_HTML.gif satisfies a Nagumo condition relative to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq76_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq77_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq78_HTML.gif is a positive constant depending on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq79_HTML.gif and the Nagumo function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq80_HTML.gif . Further, there exists a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq81_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq82_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq83_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq84_HTML.gif where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ15_HTML.gif
(3.1)

Then, there exists a monotone sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq85_HTML.gif of approximate solutions converging uniformly to a unique solution of the problems (1.1)-(1.2).

Proof.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq86_HTML.gif we define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq87_HTML.gif and consider the following modified http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq88_HTML.gif -point BVP
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ16_HTML.gif
(3.2)
We note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq89_HTML.gif are, respectively, lower and upper solutions of (3.2) and for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq90_HTML.gif we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ17_HTML.gif
(3.3)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq91_HTML.gif As
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ18_HTML.gif
(3.4)
so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq92_HTML.gif is a Nagumo function. Furthermore, there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq93_HTML.gif depending on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq94_HTML.gif , and Nagumo function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq95_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ19_HTML.gif
(3.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq96_HTML.gif . Thus, any solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq97_HTML.gif of (3.2) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq98_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq99_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq100_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq101_HTML.gif and hence it is a solution of (1.1)-(1.2).

Let us define a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq102_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ20_HTML.gif
(3.6)
In view of the assumption http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq103_HTML.gif it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq104_HTML.gif and satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq105_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq106_HTML.gif Therefore, by Taylor's theorem, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ21_HTML.gif
(3.7)
We set
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ22_HTML.gif
(3.8)
and observe that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ23_HTML.gif
(3.9)
By the mean value theorem, we can find http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq108_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq109_HTML.gif depend on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq110_HTML.gif , resp.), such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ24_HTML.gif
(3.10)
Letting
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ25_HTML.gif
(3.11)
we note that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ26_HTML.gif
(3.12)
Let us define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq111_HTML.gif as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ27_HTML.gif
(3.13)
Clearly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq112_HTML.gif is continuous and bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq113_HTML.gif and satisfies a Nagumo condition relative to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq114_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq116_HTML.gif , we consider the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq117_HTML.gif -point BVP
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ28_HTML.gif
(3.14)
Using (3.9), (3.12) and (3.13), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ29_HTML.gif
(3.15)

Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq118_HTML.gif are lower and upper solutions of (3.14), respectively. Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq119_HTML.gif satisfies a Nagumo condition, there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq120_HTML.gif (depending on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq121_HTML.gif and a Nagumo function) such that any solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq122_HTML.gif of (3.14) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq123_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq124_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq125_HTML.gif

Now, we choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq126_HTML.gif and consider the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ30_HTML.gif
(3.16)
Using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq127_HTML.gif , (3.9), (3.12) and (3.13), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ31_HTML.gif
(3.17)
which imply that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq128_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq129_HTML.gif are lower and upper solutions of (3.16). Hence by Theorems 2.2 and 2.3, there exists a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq130_HTML.gif of (3.16) such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ32_HTML.gif
(3.18)
Note that the uniqueness of the solution follows by Theorem 2.3. Using (3.9) and (3.13) together with the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq131_HTML.gif is solution of (3.16), we find that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq132_HTML.gif is a lower solution of (3.2), that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ33_HTML.gif
(3.19)
In a similar manner, it can be shown by using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq133_HTML.gif , (3.12), (3.13), and (3.19) that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq135_HTML.gif are lower and upper solutions of the following http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq136_HTML.gif -point BVP
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ34_HTML.gif
(3.20)
Again, by Theorems 2.2 and 2.3, there exists a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq137_HTML.gif of (3.20) such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ35_HTML.gif
(3.21)
Continuing this process successively, we obtain a bounded monotone sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq138_HTML.gif of solutions satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ36_HTML.gif
(3.22)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq139_HTML.gif is a solution of the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ37_HTML.gif
(3.23)
and is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ38_HTML.gif
(3.24)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq140_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq141_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq142_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq143_HTML.gif therefore it follows that the sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq144_HTML.gif are uniformly bounded and equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq145_HTML.gif Hence, by Ascoli-Arzela theorem, there exist the subsequences and a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq146_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq147_HTML.gif uniformly on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq148_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq149_HTML.gif Taking the limit http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq150_HTML.gif we find that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq151_HTML.gif which consequently yields
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ39_HTML.gif
(3.25)

This proves that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq152_HTML.gif is a solution of (3.2).

Theorem 3.2.

Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq153_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq154_HTML.gif hold. Further, one assumes that

the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq156_HTML.gif satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq157_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq158_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq160_HTML.gif

Then, the convergence of the sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq161_HTML.gif of approximate solutions (obtained in Theorem 3.1) is quadratic.

Proof.

Let us set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq162_HTML.gif so that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq163_HTML.gif satisfies the boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ40_HTML.gif
(3.26)
In view of the assumption http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq164_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq165_HTML.gif it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ41_HTML.gif
(3.27)
Now, by Taylor's theorem, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ42_HTML.gif
(3.28)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq166_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq167_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq168_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq169_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq170_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq173_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq174_HTML.gif satisfying http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq175_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq176_HTML.gif Also, in view of (3.13), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ43_HTML.gif
(3.29)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq177_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq179_HTML.gif

Now we show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq180_HTML.gif By the mean value theorem, for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq181_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq182_HTML.gif we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ44_HTML.gif
(3.30)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq183_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq184_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq185_HTML.gif and (3.30) becomes
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ45_HTML.gif
(3.31)
In particular, taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq186_HTML.gif and using (3.27), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ46_HTML.gif
(3.32)
which contradicts that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq187_HTML.gif Similarly, letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq188_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq189_HTML.gif we get a contradiction. Thus, it follows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq190_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq191_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq192_HTML.gif and consequently, (3.28) and (3.29) take the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ47_HTML.gif
(3.33)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq193_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ48_HTML.gif
(3.34)
Now, by a comparison principle, we can obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq194_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq195_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq196_HTML.gif is a solution of the problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ49_HTML.gif
(3.35)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq197_HTML.gif is continuous and bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq198_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq199_HTML.gif (independent of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq200_HTML.gif ) such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq201_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq202_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq203_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq204_HTML.gif so we can rewrite (3.35) as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ50_HTML.gif
(3.36)
whose solution is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ51_HTML.gif
(3.37)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ52_HTML.gif
(3.38)
Introducing the integrating factor http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq205_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq206_HTML.gif (3.34) takes the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ53_HTML.gif
(3.39)
Integrating (3.39) from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq207_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq208_HTML.gif and using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq209_HTML.gif we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ54_HTML.gif
(3.40)
which can alternatively be written as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ55_HTML.gif
(3.41)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq210_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq211_HTML.gif . Using the fact that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq212_HTML.gif together with (3.41) yields
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ56_HTML.gif
(3.42)
which, on substitutingin (3.37), yields
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ57_HTML.gif
(3.43)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ58_HTML.gif
(3.44)
Taking the maximum over http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq213_HTML.gif and then solving (3.43) for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq214_HTML.gif we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ59_HTML.gif
(3.45)
Also, it follows from (3.33) that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ60_HTML.gif
(3.46)
Integrating (3.46) from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq215_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq216_HTML.gif and using http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq217_HTML.gif (from the boundary condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq218_HTML.gif we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ61_HTML.gif
(3.47)
which, in view of the fact http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq219_HTML.gif and (3.45), yields
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ62_HTML.gif
(3.48)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ63_HTML.gif
(3.49)
As http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq220_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq221_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ64_HTML.gif
(3.50)
Integrating (3.46) from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq222_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq223_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq224_HTML.gif ) and using (3.50), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ65_HTML.gif
(3.51)
Using (3.45) in (3.34), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ66_HTML.gif
(3.52)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq225_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq226_HTML.gif is bounded on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq227_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq228_HTML.gif we can choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq229_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq230_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq231_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq232_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq233_HTML.gif so that (3.52) takes the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ67_HTML.gif
(3.53)
Integrating (3.53) from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq234_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq235_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq236_HTML.gif ), and using (3.51), we find that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ68_HTML.gif
(3.54)
Letting
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ69_HTML.gif
(3.55)
it follows from (3.51) and (3.54) that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ70_HTML.gif
(3.56)
Hence, from (3.48) and (3.56), it follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ71_HTML.gif
(3.57)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq237_HTML.gif From (3.45) and (3.57) with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ72_HTML.gif
(3.58)
we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ73_HTML.gif
(3.59)

This proves the quadratic convergence in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq238_HTML.gif norm.

Example 3.3.

Consider the boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ74_HTML.gif
(3.60)

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq239_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq240_HTML.gif be, respectively, lower and upper solutions of (3.60). Clearly http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq241_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq242_HTML.gif are not the solutions of (3.60) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq243_HTML.gif Also, the assumptions of Theorem 3.1 are satisfied. Thus, the conclusion of Theorem 3.1 applies to the problem (3.60).

Declarations

Acknowledgment

The author is grateful to the referees and professor G. Infante for their valuable suggestions and comments that led to the improvement of the original paper.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University

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© Ahmed Alsaedi. 2011

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