New Fixed Point Theorems of Mixed Monotone Operators and Applications to Singular Boundary Value Problems on Time Scales

Boundary Value Problems20102011:567054

DOI: 10.1155/2011/567054

Received: 3 July 2010

Accepted: 13 December 2010

Published: 21 December 2010

Abstract

Some new existence and uniqueness theorems of fixed points of mixed monotone operators are obtained, and then they are applied to a nonlinear singular second-order three-point boundary value problem on time scales. We prove the existence and uniqueness of a positive solution for the above problem which cannot be solved by using previously available methods.

1. Introduction

The study of mixed monotone operators has been a matter of discussion since they were introduced by Guo and Lakshmikantham [1] in 1987, because it has not only important theoretical meaning but also wide applications in microeconomics, the nuclear industry, and so on (see [14]). Recently, some new and interesting results about these kinds of operators have emerged, and they are used extensively in nonlinear differential and integral equations (see [59]).

In this paper, we extend the main results of [9] to mixed monotone operators. Without demanding compactness and continuity conditions and the existence of upper and lower solutions, we study the existence, uniqueness, and iterative convergence of fixed points of a class of mixed monotone operators. Then, we apply these results to the following singular second-order three-point boundary value problem on time scales:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ1_HTML.gif
(1.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq1_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq3_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq4_HTML.gif . The functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq6_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq7_HTML.gif ) are continuous. Our nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq8_HTML.gif may have singularity at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq9_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq11_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq12_HTML.gif ) may have singularity at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq13_HTML.gif .

To understand the notations used in (1.1), we recall that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq14_HTML.gif is a time scales, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq14_HTML.gif is an arbitrary nonempty closed subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq16_HTML.gif . For each interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq17_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq16_HTML.gif , we define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq19_HTML.gif . For more details on time scales, one can refer to [1012].

In recent years, there is much attention paid to the existence of positive solutions for nonlocal boundary value problems on time scales, see [1318] and references therein. Dynamic equations have been applied in the study of insect population models, stock market and heat transfer and so on [1922]. Time scales can be used in microeconomics models to study behavior which is sometimes continuous and sometimes discrete. A simple example of this continuous-discrete behavior is seen in suppliers short-run decisions and long-run decisions. Unifying both continuous and discrete model can avoid repeat research and has the capacity to get some different types of models which neither continuous models nor discrete models can effectively describe.

On the other hand, singular boundary value problems on time scales have also been investigated extensively (see [2327]). We would like to mention some results of DaCunha et al. [23], Hao et al. [25], Luo [26], and Hu [27], which motivated us to consider problem (1.1).

In [23], DaCunha et al. considered the following singular second-order three-point dynamic boundary value problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ2_HTML.gif
(1.2)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq20_HTML.gif is fixed and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq21_HTML.gif is singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq22_HTML.gif and possible at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq24_HTML.gif . The authors claimed that "we note that this is the first work (to our knowledge) that deals with singular boundary value problems in a general time scales setting." The results on existence of positive solutions were obtained by means of a fixed point theorem due to Gatica, Oliker and Waltman for mappings that are decreasing with respect to a cone.

In [25], Hao et al. were concerned with the following singular boundary value problem of nonlinear dynamic equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ3_HTML.gif
(1.3)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq25_HTML.gif is rl-continuous and may be singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq26_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq27_HTML.gif . With suitable growth and limit conditions, an existence theorem of positive solutions was established by using the Krasnoselskii fixed point theorem.

In [26], Luo studied the following singular http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq28_HTML.gif -point dynamic eigenvalue problem with mixed derivatives:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ4_HTML.gif
(1.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq29_HTML.gif is singular at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq31_HTML.gif . The author obtained eigenvalue intervals in which there exists at least one positive solution of problem (1.4) by making use of the fixed point index theory.

In [27], Hu were concerned with the following singular third-order three-point boundary value problem on time scales:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ5_HTML.gif
(1.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq32_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq33_HTML.gif are continuous. The nonlinearity http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq34_HTML.gif may have singularity at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq35_HTML.gif and/or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq37_HTML.gif may have singularity at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq38_HTML.gif . With the aid of the fixed point theorem of cone expansion and compression type, results on the existence of positive solutions to (1.5) were obtained in the bounded set.

From the above research, we note that there is no result on the uniqueness of solutions and convergence of the iterative sequences for singular boundary value problems on time scales. As we know, completely continuity condition is crucial for the above discussion. However, it is difficult to verify for singular problems on time scales, in particular, in order to remove the singularity in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq39_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq40_HTML.gif , more restricted conditions are required. For instance, condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq41_HTML.gif of Theorem 2.3 in [23] and condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq42_HTML.gif of Theorem 3.1 in [27]. In our abstract results on mixed monotone operators, since the compactness and continuity conditions are not required, they can be directly applied to singular boundary value problem (1.1).

The purpose of this paper is to present some conditions for problem (1.1) that have a unique solution, the iterative sequences yielding approximate solutions are also given. Our main result generalizes and improves Theorem 2.3 in [18].

2. Preliminaries and Abstract Theorems

Let the real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq43_HTML.gif be partially ordered by a cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq44_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq45_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq46_HTML.gif if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq47_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq48_HTML.gif is said to be a mixed monotone operator if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq49_HTML.gif is increasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq50_HTML.gif and decreasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq51_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq52_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq54_HTML.gif implies http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq55_HTML.gif . Element http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq56_HTML.gif is called a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq57_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq58_HTML.gif .

Recall that cone http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq59_HTML.gif is said to be solid if the interior http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq60_HTML.gif is nonempty and we denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq61_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq62_HTML.gif . http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq63_HTML.gif is said to be normal if there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq64_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq65_HTML.gif , the smallest http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq66_HTML.gif is called the normal constant of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq67_HTML.gif . For all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq68_HTML.gif , the notation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq69_HTML.gif means that there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq71_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq72_HTML.gif . Clearly, ~ is an equivalence relation. Given http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq73_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq75_HTML.gif ), we denote by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq76_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq77_HTML.gif . It is easy to see that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq78_HTML.gif is convex and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq79_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq80_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq81_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq82_HTML.gif , it is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq83_HTML.gif .

All the concepts discussed above can be found in [1, 2, 4]. For more results about mixed monotone operators and other related concepts, the reader is referred to [3, 59] and some of the references therein.

In [9], Zhai and Cao introduced the following definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq84_HTML.gif - http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq85_HTML.gif -concave operators.

Definition 2.1 (see [9]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq86_HTML.gif be a real Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq87_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq88_HTML.gif . We say an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq89_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq90_HTML.gif - http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq91_HTML.gif -concave if there exist two positive-valued functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq92_HTML.gif on interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq93_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq95_HTML.gif is a surjection;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq97_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq98_HTML.gif ;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq100_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq101_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq102_HTML.gif .

They obtained the following result.

Theorem 2.2 (see [9]).

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq103_HTML.gif be a real Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq104_HTML.gif be a normal cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq105_HTML.gif . Suppose that an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq106_HTML.gif is increasing and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq107_HTML.gif - http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq108_HTML.gif -concave. In addition, suppose that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq109_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq110_HTML.gif . Then

(i)there are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq111_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq112_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq113_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq114_HTML.gif ;

(ii)operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq115_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq116_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq117_HTML.gif ;

(iii)for any initial http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq118_HTML.gif , constructing successively the sequence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq119_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq120_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq121_HTML.gif .

We can extend Theorem 2.2 to mixed monotone operators, our main results can be stated as follows.

Theorem 2.3.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq122_HTML.gif be a normal cone in a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq123_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq124_HTML.gif a mixed monotone operator. Assume that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq125_HTML.gif , there exist two positive-valued functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq126_HTML.gif on interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq127_HTML.gif such that

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq129_HTML.gif is a surjection;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq131_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq132_HTML.gif ;

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq134_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq135_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq136_HTML.gif .

In addition, suppose that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq137_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq138_HTML.gif . Then

(i)there are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq140_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq141_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq142_HTML.gif ;

(ii)operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq143_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq144_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq145_HTML.gif ;

(iii)for any initial http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq146_HTML.gif , constructing successively the sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq148_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq149_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq150_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq151_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq152_HTML.gif .

Corollary 2.4.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq153_HTML.gif be a real Banach space, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq154_HTML.gif a normal, solid cone in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq155_HTML.gif . Suppose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq156_HTML.gif is a mixed monotone operator and satisfies the conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq157_HTML.gif of Theorem 2.3. Then

(i)there are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq158_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq159_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq160_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq161_HTML.gif ;

(ii)operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq162_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq163_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq164_HTML.gif ;

(iii)for any initial http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq165_HTML.gif , constructing successively the sequences http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq166_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq167_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq168_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq169_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq170_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq171_HTML.gif .

Remark 2.5.

In Theorem 2.3, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq172_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq173_HTML.gif is a solid cone, we can know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq174_HTML.gif is automatically satisfied. Therefore, we can deduce that Corollary 2.4 holds from Theorem 2.3. For simplicity, we only present the proof of Theorem 2.3.

Proof of Theorem 2.3.

Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq175_HTML.gif , we can find a sufficiently small number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq176_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ6_HTML.gif
(2.1)
According to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq177_HTML.gif , we can obtain that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq178_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq179_HTML.gif , thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ7_HTML.gif
(2.2)
Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq180_HTML.gif , we can find a positive integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq181_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ8_HTML.gif
(2.3)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq182_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq183_HTML.gif , and construct successively the sequences
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ9_HTML.gif
(2.4)

It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq184_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq185_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq186_HTML.gif . In general, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq187_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq188_HTML.gif .

It follows from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq189_HTML.gif , (2.2), and (2.3) that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ10_HTML.gif
(2.5)
From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq190_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ11_HTML.gif
(2.6)
Combining (2.2) with (2.3) and (2.6), we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ12_HTML.gif
(2.7)
Thus, we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ13_HTML.gif
(2.8)
By induction, it is easy to obtain that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ14_HTML.gif
(2.9)
Take any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq191_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq192_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq193_HTML.gif . So we can know that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ15_HTML.gif
(2.10)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ16_HTML.gif
(2.11)
Thus, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq194_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq195_HTML.gif , and then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ17_HTML.gif
(2.12)
Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq196_HTML.gif ; that is,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ18_HTML.gif
(2.13)
Set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq197_HTML.gif , we will show that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq198_HTML.gif . In fact, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq199_HTML.gif , by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq200_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq201_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq202_HTML.gif . Consider the following two cases.
  1. (i)
    There exists an integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq203_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq204_HTML.gif . In this case, we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq205_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq206_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq207_HTML.gif hold. Hence
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ19_HTML.gif
    (2.14)
     
By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq208_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ20_HTML.gif
(2.15)
which is a contradiction.
  1. (ii)
    For all integers http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq209_HTML.gif . Then, we obtain http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq210_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq211_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq212_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq213_HTML.gif . Hence
    http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ21_HTML.gif
    (2.16)
     
By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq214_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ22_HTML.gif
(2.17)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq215_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ23_HTML.gif
(2.18)

which is also a contradiction. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq216_HTML.gif .

Furthermore, similarly to the proof of Theorem 2.1 in [9], there exits http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq217_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq218_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq219_HTML.gif is the fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq220_HTML.gif .

In the following, we prove that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq221_HTML.gif is the unique fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq222_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq223_HTML.gif . In fact, suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq224_HTML.gif is another fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq225_HTML.gif . Let
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ24_HTML.gif
(2.19)
Clearly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq226_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq227_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq228_HTML.gif , according to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq229_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq230_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq231_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ25_HTML.gif
(2.20)
It follows that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ26_HTML.gif
(2.21)
Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq232_HTML.gif , which is a contradiction. Thus we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq233_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq234_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq235_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq236_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq237_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq238_HTML.gif , so we know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq239_HTML.gif is the unique fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq240_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq241_HTML.gif . For any initial http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq242_HTML.gif , we can choose a small number http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq243_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ27_HTML.gif
(2.22)
From http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq244_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq245_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq246_HTML.gif , thus
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ28_HTML.gif
(2.23)
We can choose a sufficiently large positive integer http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq247_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ29_HTML.gif
(2.24)
Take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq248_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq249_HTML.gif . We can find that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ30_HTML.gif
(2.25)
constructing successively the sequences
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ31_HTML.gif
(2.26)
By using the mixed monotone properties of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq250_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ32_HTML.gif
(2.27)
Similarly to the above proof, we can know that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq251_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ33_HTML.gif
(2.28)

By the uniqueness of fixed points of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq252_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq253_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq254_HTML.gif . Taking into account that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq255_HTML.gif is normal, we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq256_HTML.gif . This completes the proof.

3. Applications to Singular BVP (1.1) on Time Scales

A Banach space http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq257_HTML.gif is the set of real-valued continuous (in the topology of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq258_HTML.gif ) function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq259_HTML.gif defined on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq260_HTML.gif with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq261_HTML.gif .

Define a cone by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ34_HTML.gif
(3.1)

It is clear that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq262_HTML.gif is a normal cone of which the normality constant is 1.

In order to obtain our main result, we need the following lemmas.

Lemma 3.1 (see [18]).

The Green function corresponding to the following problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ35_HTML.gif
(3.2)
is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ36_HTML.gif
(3.3)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ37_HTML.gif
(3.4)
is Green's function for the BVP:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ38_HTML.gif
(3.5)

Lemma 3.2 (see [18]).

For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq263_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ39_HTML.gif
(3.6)

Our main result is the following theorem.

Theorem 3.3.

Assume that

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq265_HTML.gif is nondecreasing, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq266_HTML.gif is nonincreasing and there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq267_HTML.gif on interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq268_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq269_HTML.gif is a surjection and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq270_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq271_HTML.gif which satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ40_HTML.gif
(3.7)
there exist two constants http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq273_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq274_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ41_HTML.gif
(3.8)
Then problem (1.1) has a unique positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq275_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq276_HTML.gif . Moreover, for any initial http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq277_HTML.gif , constructing successively the sequences
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ42_HTML.gif
(3.9)

we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq278_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq279_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq280_HTML.gif .

Proof of Theorem 3.3.

Define an operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq281_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ43_HTML.gif
(3.10)
It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq282_HTML.gif is a solution of problem (1.1) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq283_HTML.gif is a fixed point of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq284_HTML.gif . Clearly, we can know that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq285_HTML.gif is a mixed monotone operator. For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq286_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq287_HTML.gif , according to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq288_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ44_HTML.gif
(3.11)
Hence,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ45_HTML.gif
(3.12)
In addition, from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq289_HTML.gif , we know that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ46_HTML.gif
(3.13)

Thus http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq290_HTML.gif . Therefore, all the conditions of Theorem 2.3 are satisfied. By Theorem 2.3, we can obtain the conclusions of Theorem 3.3.

Now, let us end this paper by the following example.

Example 3.4.

Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq291_HTML.gif , consider the following BVP on time scales
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ47_HTML.gif
(3.14)

Set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq292_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq293_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq294_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq295_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq296_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq297_HTML.gif is a surjection and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq298_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq299_HTML.gif .

For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq300_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq301_HTML.gif , it is easy to check that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ48_HTML.gif
(3.15)
It follows from Lemma 3.1 that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ49_HTML.gif
(3.16)
Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq302_HTML.gif , since
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ50_HTML.gif
(3.17)
We choose http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq303_HTML.gif , according to Lemma 3.2, we have
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ51_HTML.gif
(3.18)
By Theorem 3.3, problem (3.14) has a unique positive solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq304_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq305_HTML.gif . For any initial http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq306_HTML.gif , constructing successively the sequences
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ52_HTML.gif
(3.19)

we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq307_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq308_HTML.gif .

Remark 3.5.

Example 3.4 indicates that Theorem 3.3 generalizes and complements Theorem 2.3 in [18] at the following aspects. Firstly, in our proof, we only need to check the conditions "there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq309_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq310_HTML.gif ", in fact, the author has shown that " http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq311_HTML.gif " in the proof of Theorem 2.3 in [18]. It is clear that our hypotheses are weaker than those imposed in Theorem 2.3 in [18]. According to Lemma 3.2, we can know that the condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq312_HTML.gif is automatically satisfied. Secondly, we have considered the case that the condition " http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq313_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq314_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq315_HTML.gif )" is not satisfied, therefore, the condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq316_HTML.gif incorporates the more comprehensive functions than the condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq317_HTML.gif in Theorem 2.3 in [18]. Thirdly, the more general conditions are imposed on our nonlinear term, they can be the sum of nondecreasing functions and nonincreasing functions.

Declarations

Acknowledgment

H. Xu was supported financially by the Science Foundation of North University of China.

Authors’ Affiliations

(1)
College of Economics and Management, North University of China

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Copyright

© Huiye Xu. 2011

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.