Open Access

Approximation of Solutions for Second-Order https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq3_HTML.gif point nonlocal boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq4_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq5_HTML.gif are nonnegative real constants such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq8_HTML.gif with https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ4_HTML.gif
(1.4)
where
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ6_HTML.gif
(1.6)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq11_HTML.gif we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq12_HTML.gif where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ7_HTML.gif
(2.1)
where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ8_HTML.gif
(2.2)

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

We say that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ9_HTML.gif
(2.3)
and https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq19_HTML.gif is called a Nagumo function if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ11_HTML.gif
(2.5)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq20_HTML.gif . We say that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq21_HTML.gif satisfies a Nagumo condition on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq22_HTML.gif relative to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq23_HTML.gif if for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq24_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq25_HTML.gif there exists a Nagumo function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq26_HTML.gif such that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq29_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq30_HTML.gif are continuous functions such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq31_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq32_HTML.gif Then there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq33_HTML.gif (depending only on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq34_HTML.gif the Nagumo function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq35_HTML.gif ) such that every solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq36_HTML.gif of (1.1)-(1.2) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq38_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq39_HTML.gif

If https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq43_HTML.gif

Theorem 2.3.

Assume that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq45_HTML.gif is decreasing in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq46_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq47_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq48_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq49_HTML.gif

Proof.

Let us define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq50_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq51_HTML.gif and satisfies the boundary conditions
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq52_HTML.gif have a positive maximum at some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq53_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq54_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq55_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq57_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq58_HTML.gif we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq59_HTML.gif has a positive maximum at https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ14_HTML.gif
(2.8)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq61_HTML.gif Now as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq64_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq66_HTML.gif Thus, we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq67_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq71_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq72_HTML.gif

the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq74_HTML.gif satisfies a Nagumo condition relative to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq75_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq76_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq77_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq78_HTML.gif is a positive constant depending on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq79_HTML.gif and the Nagumo function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq80_HTML.gif . Further, there exists a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq81_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq82_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq83_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq84_HTML.gif where
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq86_HTML.gif we define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq87_HTML.gif and consider the following modified https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq88_HTML.gif -point BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ16_HTML.gif
(3.2)
We note that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq90_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ17_HTML.gif
(3.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq91_HTML.gif As
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ18_HTML.gif
(3.4)
so https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq93_HTML.gif depending on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq94_HTML.gif , and Nagumo function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq95_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ19_HTML.gif
(3.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq96_HTML.gif . Thus, any solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq97_HTML.gif of (3.2) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq98_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq99_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq100_HTML.gif on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq102_HTML.gif by
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq103_HTML.gif it follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq104_HTML.gif and satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq105_HTML.gif on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ21_HTML.gif
(3.7)
We set
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ22_HTML.gif
(3.8)
and observe that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq108_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq109_HTML.gif depend on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq110_HTML.gif , resp.), such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ24_HTML.gif
(3.10)
Letting
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ25_HTML.gif
(3.11)
we note that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ26_HTML.gif
(3.12)
Let us define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq111_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ27_HTML.gif
(3.13)
Clearly https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq112_HTML.gif is continuous and bounded on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq114_HTML.gif . For every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq116_HTML.gif , we consider the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq117_HTML.gif -point BVP
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ29_HTML.gif
(3.15)

Thus, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq120_HTML.gif (depending on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq122_HTML.gif of (3.14) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq123_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq124_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq125_HTML.gif

Now, we choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq126_HTML.gif and consider the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ30_HTML.gif
(3.16)
Using https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ31_HTML.gif
(3.17)
which imply that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq128_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq130_HTML.gif of (3.16) such that
https://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 https://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 https://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,
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq134_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq136_HTML.gif -point BVP
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq137_HTML.gif of (3.20) such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq138_HTML.gif of solutions satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ36_HTML.gif
(3.22)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq139_HTML.gif is a solution of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ37_HTML.gif
(3.23)
and is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ38_HTML.gif
(3.24)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq140_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq141_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq142_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq143_HTML.gif therefore it follows that the sequences https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq144_HTML.gif are uniformly bounded and equicontinuous on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq146_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq147_HTML.gif uniformly on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq148_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq149_HTML.gif Taking the limit https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq150_HTML.gif we find that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq151_HTML.gif which consequently yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ39_HTML.gif
(3.25)

This proves that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq153_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq156_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq157_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq158_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq160_HTML.gif

Then, the convergence of the sequence https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq162_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq163_HTML.gif satisfies the boundary conditions
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq164_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq165_HTML.gif it follows that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ42_HTML.gif
(3.28)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq166_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq167_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq169_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq170_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq171_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq173_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq174_HTML.gif satisfying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq175_HTML.gif on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ43_HTML.gif
(3.29)

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

Now we show that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq181_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq182_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ44_HTML.gif
(3.30)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq183_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq184_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq185_HTML.gif and (3.30) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ45_HTML.gif
(3.31)
In particular, taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq186_HTML.gif and using (3.27), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ46_HTML.gif
(3.32)
which contradicts that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq187_HTML.gif Similarly, letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq188_HTML.gif for some https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq190_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq191_HTML.gif , which implies that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ47_HTML.gif
(3.33)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq193_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq194_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq195_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq196_HTML.gif is a solution of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ49_HTML.gif
(3.35)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq197_HTML.gif is continuous and bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq198_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq199_HTML.gif (independent of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq200_HTML.gif ) such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq201_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq202_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq203_HTML.gif on https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ51_HTML.gif
(3.37)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ52_HTML.gif
(3.38)
Introducing the integrating factor https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq205_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq206_HTML.gif (3.34) takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ53_HTML.gif
(3.39)
Integrating (3.39) from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq207_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq208_HTML.gif and using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq209_HTML.gif we obtain
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ55_HTML.gif
(3.41)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq210_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq211_HTML.gif . Using the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq212_HTML.gif together with (3.41) yields
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ57_HTML.gif
(3.43)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ58_HTML.gif
(3.44)
Taking the maximum over https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq213_HTML.gif and then solving (3.43) for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq214_HTML.gif we obtain
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ60_HTML.gif
(3.46)
Integrating (3.46) from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq215_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq216_HTML.gif and using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq217_HTML.gif (from the boundary condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq218_HTML.gif we obtain
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq219_HTML.gif and (3.45), yields
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ62_HTML.gif
(3.48)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ63_HTML.gif
(3.49)
As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq220_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq221_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ64_HTML.gif
(3.50)
Integrating (3.46) from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq222_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq223_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq224_HTML.gif ) and using (3.50), we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ66_HTML.gif
(3.52)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq225_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq226_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq227_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq228_HTML.gif we can choose https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq229_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq230_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq231_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq232_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ67_HTML.gif
(3.53)
Integrating (3.53) from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq234_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq235_HTML.gif ( https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ68_HTML.gif
(3.54)
Letting
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ71_HTML.gif
(3.57)
where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ72_HTML.gif
(3.58)
we obtain
https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_Equ74_HTML.gif
(3.60)

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq239_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F929061/MediaObjects/13661_2010_Article_67_IEq241_HTML.gif and https://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 https://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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