Open Access

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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq1_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq3_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq5_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq7_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq8_HTML.gif is mixed nonmonotone in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq10_HTML.gif , https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ2_HTML.gif
(2.1)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq13_HTML.gif is the Green's function to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq15_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq16_HTML.gif .

Let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq17_HTML.gif .

Lemma 2.1 (Guo [14, 15]).

https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq19_HTML.gif such that every element https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq20_HTML.gif can be represented in the form https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq21_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq22_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq23_HTML.gif https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq25_HTML.gif . Obviously, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq28_HTML.gif is a generating cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq29_HTML.gif .

Theorem 3.1.

Suppose that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq32_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq33_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq37_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ5_HTML.gif
(3.1)
and there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq38_HTML.gif , such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq39_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq40_HTML.gif . And moreover, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq41_HTML.gif , the iterative sequence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ7_HTML.gif
(3.3)

converges to https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq43_HTML.gif is nonnegative. However, the scope of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq44_HTML.gif is not limited to https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq48_HTML.gif , https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ8_HTML.gif
(3.4)
For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq51_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq52_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq53_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq54_HTML.gif , by (3.1), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ9_HTML.gif
(3.5)
Hence
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ11_HTML.gif
(3.7)
thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ12_HTML.gif
(3.8)
Similarly, by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq56_HTML.gif being converged, we have that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ13_HTML.gif
(3.9)
Define the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq57_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ14_HTML.gif
(3.10)
Then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq59_HTML.gif . Let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq60_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq62_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ16_HTML.gif
(3.12)
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq63_HTML.gif , which satisfies https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq65_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ18_HTML.gif
(3.14)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq66_HTML.gif is a generating cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq67_HTML.gif , from Lemma 2.1, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq68_HTML.gif such that every element https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq69_HTML.gif can be represented in
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ19_HTML.gif
(3.15)
This implies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ20_HTML.gif
(3.16)
Let
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq70_HTML.gif is well defined for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq71_HTML.gif . It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq72_HTML.gif is a norm in https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq74_HTML.gif which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq75_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq76_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq77_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq78_HTML.gif denotes the normal constant of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq79_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq80_HTML.gif is arbitrary, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq82_HTML.gif are equivalent.

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

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

It follows from (3.12) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ25_HTML.gif
(3.21)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ26_HTML.gif
(3.22)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ28_HTML.gif
(3.24)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq91_HTML.gif ; then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ29_HTML.gif
(3.25)
As https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq92_HTML.gif and https://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, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq96_HTML.gif in (3.14), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ30_HTML.gif
(3.26)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq97_HTML.gif , we see that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq98_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ32_HTML.gif
(3.28)

By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq99_HTML.gif , we have https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq101_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq102_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq103_HTML.gif , and so https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq104_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq105_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq106_HTML.gif ; by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq107_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq108_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq109_HTML.gif , that is, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq111_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq112_HTML.gif ; we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq113_HTML.gif . By the equivalence of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq115_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq116_HTML.gif again, we get https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ33_HTML.gif
(3.29)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq119_HTML.gif and there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq120_HTML.gif , such that for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq121_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq122_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq123_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq124_HTML.gif provided https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq125_HTML.gif . And moreover, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq126_HTML.gif , the iterative sequence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ35_HTML.gif
(3.31)

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

To see that, we put
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq128_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq129_HTML.gif ,   https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq130_HTML.gif ,  and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq131_HTML.gif .

In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq132_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ37_HTML.gif
(3.33)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq133_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ38_HTML.gif
(3.34)
Similarly,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ39_HTML.gif
(3.35)
Next, for any https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq135_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ42_HTML.gif
(3.38)
thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ43_HTML.gif
(3.39)
so
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ44_HTML.gif
(3.40)
then we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_Equ45_HTML.gif
(3.41)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F483057/MediaObjects/13661_2011_Article_42_IEq136_HTML.gif ; then
https://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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