Iterative Solutions of Singular Boundary Value Problems of Third-Order Differential Equation

Boundary Value Problems20112011:483057

DOI: 10.1155/2011/483057

Received: 19 January 2011

Accepted: 6 March 2011

Published: 15 March 2011

Abstract

By using the cone theory and the Banach contraction mapping principle, the existence and uniqueness results are established for singular third-order boundary value problems. The theorems obtained are very general and complement previous known results.

1. Introduction

Third-order differential equations arise in a variety of different areas of applied mathematics and physics, such as the deflection of a curved beam having a constant or varying cross section, three-layer beam, electromagnetic waves, or gravity-driven flows [1]. Recently, third-order boundary value problems have been studied extensively in the literature (see, e.g., [213], and their references). In this paper, we consider the following third-order boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq2_HTML.gif .

Three-point boundary value problems (BVPs for short) have been also widely studied because of both practical and theoretical aspects. There have been many papers investigating the solutions of three-point BVPs, see [25, 10, 12] and references therein. Recently, the existence of solutions of third-order three-point BVP (1.1) has been studied in [2, 3]. Guo et al. [2] show the existence of positive solutions for BVP (1.1) when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq3_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq4_HTML.gif is separable by using cone expansion-compression fixed point theorem. In [3], the singular third-order three-point BVP (1.1) is considered under some conditions concerning the first eigenvalues corresponding to the relevant linear operators, where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq6_HTML.gif is separable and is not necessary to be nonnegative, and the existence results of nontrivial solutions and positive solutions are given by means of the topological degree theory. Motivated by the above works, we consider the singular third-order three-point BVP (1.1). Here, we give the unique solution of BVP (1.1) under the conditions that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq8_HTML.gif is mixed nonmonotone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq9_HTML.gif and does not need to be separable by using the cone theory and the Banach contraction mapping principle.

2. Preliminaries

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq11_HTML.gif . By [2, Lemma  2.1], we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq12_HTML.gif is a solution of (1.1) if and only if
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ2_HTML.gif
(2.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ3_HTML.gif
(2.2)

It is shown in [2] that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq13_HTML.gif is the Green's function to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq15_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq16_HTML.gif .

Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ4_HTML.gif
(2.3)

It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq17_HTML.gif .

Lemma 2.1 (Guo [14, 15]).

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq18_HTML.gif is generating if and only if there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq19_HTML.gif such that every element http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq20_HTML.gif can be represented in the form http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq21_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq22_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq23_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq24_HTML.gif

3. Singular Third-Order Boundary Value Problem

This section discusses singular third-order boundary value problem (1.1).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq25_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq26_HTML.gif is a normal solid cone of Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq27_HTML.gif ; by [16, Lemma  2.1.2], we have that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq28_HTML.gif is a generating cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq29_HTML.gif .

Theorem 3.1.

Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq30_HTML.gif , and there exist two positive linear bounded operators http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq32_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq33_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq37_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ5_HTML.gif
(3.1)
and there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq38_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ6_HTML.gif
(3.2)
Then (1.1) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq39_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq40_HTML.gif . And moreover, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq41_HTML.gif , the iterative sequence
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ7_HTML.gif
(3.3)

converges to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq42_HTML.gif .

Remark 3.2.

Recently, in the study of BVP (1.1), almost all the papers have supposed that the Green's function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq43_HTML.gif is nonnegative. However, the scope of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq44_HTML.gif is not limited to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq45_HTML.gif in Theorem 3.1, so, we do not need to suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq46_HTML.gif is nonnegative.

Remark 3.3.

The function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq47_HTML.gif in Theorem 3.1 is not monotone or convex; the conclusions and the proof used in this paper are different from the known papers in essence.

Proof.

It is easy to see that, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq48_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq49_HTML.gif can be divided into finite partitioned monotone and bounded function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq50_HTML.gif , and then by (3.2), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ8_HTML.gif
(3.4)
For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq51_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq52_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq53_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq54_HTML.gif , by (3.1), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ9_HTML.gif
(3.5)
Hence
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ10_HTML.gif
(3.6)
Following the former inequality, we can easily have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ11_HTML.gif
(3.7)
thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ12_HTML.gif
(3.8)
Similarly, by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq55_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq56_HTML.gif being converged, we have that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ13_HTML.gif
(3.9)
Define the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq57_HTML.gif by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ14_HTML.gif
(3.10)
Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq58_HTML.gif is the solution of BVP (1.1) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq59_HTML.gif . Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ15_HTML.gif
(3.11)
By (3.1) and (3.10), for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq60_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq62_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ16_HTML.gif
(3.12)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ17_HTML.gif
(3.13)
so we can choose an http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq63_HTML.gif , which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq64_HTML.gif , and so there exists a positive integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq65_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ18_HTML.gif
(3.14)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq66_HTML.gif is a generating cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq67_HTML.gif , from Lemma 2.1, there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq68_HTML.gif such that every element http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq69_HTML.gif can be represented in
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ19_HTML.gif
(3.15)
This implies
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ20_HTML.gif
(3.16)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ21_HTML.gif
(3.17)
By (3.16), we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq70_HTML.gif is well defined for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq71_HTML.gif . It is easy to verify that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq72_HTML.gif is a norm in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq73_HTML.gif . By (3.15)–(3.17), we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ22_HTML.gif
(3.18)
On the other hand, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq74_HTML.gif which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq75_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq76_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq77_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq78_HTML.gif denotes the normal constant of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq79_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq80_HTML.gif is arbitrary, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ23_HTML.gif
(3.19)

It follows from (3.18) and (3.19) that the norms http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq82_HTML.gif are equivalent.

Now, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq84_HTML.gif which satisfies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq85_HTML.gif , let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ24_HTML.gif
(3.20)

then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq86_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq87_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq88_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq89_HTML.gif ,   and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq90_HTML.gif .

It follows from (3.12) that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ25_HTML.gif
(3.21)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ26_HTML.gif
(3.22)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ27_HTML.gif
(3.23)
Subtracting (3.22) from (3.21) + (3.23), we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ28_HTML.gif
(3.24)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq91_HTML.gif ; then we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ29_HTML.gif
(3.25)
As http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq93_HTML.gif are both positive linear bounded operators, so, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq94_HTML.gif is a positive linear bounded operator, and therefore http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq95_HTML.gif . Hence, by mathematical induction, it is easy to know that for natural number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq96_HTML.gif in (3.14), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ30_HTML.gif
(3.26)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq97_HTML.gif , we see that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ31_HTML.gif
(3.27)
which implies by virtue of the arbitrariness of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq98_HTML.gif that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ32_HTML.gif
(3.28)

By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq99_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq100_HTML.gif . Thus the Banach contraction mapping principle implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq101_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq102_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq103_HTML.gif , and so http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq104_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq105_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq106_HTML.gif ; by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq107_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq108_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq109_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq110_HTML.gif is the unique solution of (1.1). And, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq111_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq112_HTML.gif ; we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq113_HTML.gif . By the equivalence of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq116_HTML.gif again, we get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq113_HTML.gif . This completes the proof.

Example 3.4.

In this paper, the results apply to a very wide range of functions, we are following only one example to illustrate.

Consider the following singular third-order boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ33_HTML.gif
(3.29)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq119_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq120_HTML.gif , such that for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq121_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq122_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq123_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ34_HTML.gif
(3.30)
Applying Theorem 3.1, we can find that (3.29) has a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq124_HTML.gif provided http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq125_HTML.gif . And moreover, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq126_HTML.gif , the iterative sequence
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ35_HTML.gif
(3.31)

converges to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq127_HTML.gif .

To see that, we put
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ36_HTML.gif
(3.32)

Then (3.1) is satisfied for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq128_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq129_HTML.gif ,   http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq130_HTML.gif ,  and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq131_HTML.gif .

In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq132_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ37_HTML.gif
(3.33)
If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq133_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ38_HTML.gif
(3.34)
Similarly,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ39_HTML.gif
(3.35)
Next, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq134_HTML.gif , by (3.30) and (3.32), we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ40_HTML.gif
(3.36)
Then, from (3.32) and (3.36), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ41_HTML.gif
(3.37)
so it is easy to know by induction, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq135_HTML.gif , we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ42_HTML.gif
(3.38)
thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ43_HTML.gif
(3.39)
so
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ44_HTML.gif
(3.40)
then we get
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ45_HTML.gif
(3.41)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq136_HTML.gif ; then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ46_HTML.gif
(3.42)

Thus all conditions in Theorem 3.1 are satisfied.

Declarations

Acknowledgment

The author is grateful to the referees for valuable suggestions and comments.

Authors’ Affiliations

(1)
Department of Elementary Education, Heze University

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© Peiguo Zhang. 2011

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