Open Access

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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq1_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq3_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq4_HTML.gif . The functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq6_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq7_HTML.gif ) are continuous. Our nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq8_HTML.gif may have singularity at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq9_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq11_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq12_HTML.gif ) may have singularity at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq14_HTML.gif is a time scales, that is, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq16_HTML.gif . For each interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq17_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq16_HTML.gif , we define https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq20_HTML.gif is fixed and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq21_HTML.gif is singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq22_HTML.gif and possible at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq23_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ3_HTML.gif
(1.3)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq26_HTML.gif and/or https://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 https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq29_HTML.gif is singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq30_HTML.gif and https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq32_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq33_HTML.gif are continuous. The nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq34_HTML.gif may have singularity at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq35_HTML.gif and/or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq37_HTML.gif may have singularity at https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq39_HTML.gif at https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq43_HTML.gif be partially ordered by a cone https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq44_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq45_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq46_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq47_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq49_HTML.gif is increasing in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq50_HTML.gif and decreasing in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq51_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq52_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq53_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq54_HTML.gif implies https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq55_HTML.gif . Element https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq56_HTML.gif is called a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq57_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq58_HTML.gif .

Recall that cone https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq60_HTML.gif is nonempty and we denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq61_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq62_HTML.gif . https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq64_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq65_HTML.gif , the smallest https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq66_HTML.gif is called the normal constant of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq67_HTML.gif . For all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq68_HTML.gif , the notation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq69_HTML.gif means that there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq71_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq72_HTML.gif . Clearly, ~ is an equivalence relation. Given https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq73_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq75_HTML.gif ), we denote by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq76_HTML.gif the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq77_HTML.gif . It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq78_HTML.gif is convex and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq79_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq80_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq81_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq82_HTML.gif , it is clear that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq84_HTML.gif - https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq86_HTML.gif be a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq87_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq88_HTML.gif . We say an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq89_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq90_HTML.gif - https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq92_HTML.gif on interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq93_HTML.gif such that

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

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

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq100_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq101_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq103_HTML.gif be a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq104_HTML.gif be a normal cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq105_HTML.gif . Suppose that an operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq106_HTML.gif is increasing and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq107_HTML.gif - https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq109_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq110_HTML.gif . Then

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

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

(iii)for any initial https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq118_HTML.gif , constructing successively the sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq119_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq120_HTML.gif , we have https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq123_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq125_HTML.gif , there exist two positive-valued functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq126_HTML.gif on interval https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq127_HTML.gif such that

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

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

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

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

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

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

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

Corollary 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq153_HTML.gif be a real Banach space, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq154_HTML.gif a normal, solid cone in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq155_HTML.gif . Suppose https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq158_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq159_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq160_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq161_HTML.gif ;

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

(iii)for any initial https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq165_HTML.gif , constructing successively the sequences https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq166_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq167_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq168_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq169_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq170_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq172_HTML.gif with https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq176_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ6_HTML.gif
(2.1)
According to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq177_HTML.gif , we can obtain that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq178_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq179_HTML.gif , thus
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ7_HTML.gif
(2.2)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq180_HTML.gif , we can find a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq181_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ8_HTML.gif
(2.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq182_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq183_HTML.gif , and construct successively the sequences
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ9_HTML.gif
(2.4)

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

It follows from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq189_HTML.gif , (2.2), and (2.3) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ10_HTML.gif
(2.5)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq190_HTML.gif , we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ12_HTML.gif
(2.7)
Thus, we obtain
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ14_HTML.gif
(2.9)
Take any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq191_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq193_HTML.gif . So we can know that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ15_HTML.gif
(2.10)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ16_HTML.gif
(2.11)
Thus, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq195_HTML.gif , and then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ17_HTML.gif
(2.12)
Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq196_HTML.gif ; that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ18_HTML.gif
(2.13)
Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq197_HTML.gif , we will show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq198_HTML.gif . In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq199_HTML.gif , by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq200_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq201_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq203_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq204_HTML.gif . In this case, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq205_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq206_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq207_HTML.gif hold. Hence
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ19_HTML.gif
    (2.14)
     
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq208_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq209_HTML.gif . Then, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq210_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq211_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq212_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq213_HTML.gif . Hence
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ21_HTML.gif
    (2.16)
     
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq214_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ22_HTML.gif
(2.17)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq215_HTML.gif , we have
https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq217_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq218_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq219_HTML.gif is the fixed point of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq220_HTML.gif .

In the following, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq221_HTML.gif is the unique fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq222_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq223_HTML.gif . In fact, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq224_HTML.gif is another fixed point of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq225_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ24_HTML.gif
(2.19)
Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq226_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq227_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq228_HTML.gif , according to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq229_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq230_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq231_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ25_HTML.gif
(2.20)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ26_HTML.gif
(2.21)
Hence, https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq233_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq234_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq235_HTML.gif has a unique fixed point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq236_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq237_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq238_HTML.gif , so we know that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq239_HTML.gif is the unique fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq240_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq241_HTML.gif . For any initial https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq242_HTML.gif , we can choose a small number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq243_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ27_HTML.gif
(2.22)
From https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq244_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq245_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq246_HTML.gif , thus
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq247_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ29_HTML.gif
(2.24)
Take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq248_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq249_HTML.gif . We can find that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ30_HTML.gif
(2.25)
constructing successively the sequences
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq250_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq251_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq252_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq253_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq254_HTML.gif . Taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq255_HTML.gif is normal, we deduce that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq258_HTML.gif ) function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq259_HTML.gif defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq260_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq261_HTML.gif .

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

It is clear that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ35_HTML.gif
(3.2)
is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ36_HTML.gif
(3.3)
where
https://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:
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq263_HTML.gif , we have
https://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

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

we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq278_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq279_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq281_HTML.gif
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq283_HTML.gif is a fixed point of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq284_HTML.gif . Clearly, we can know that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq287_HTML.gif , according to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq288_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ44_HTML.gif
(3.11)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ45_HTML.gif
(3.12)
In addition, from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq289_HTML.gif , we know that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ46_HTML.gif
(3.13)

Thus https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ47_HTML.gif
(3.14)

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

For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq300_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq301_HTML.gif , it is easy to check that
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ49_HTML.gif
(3.16)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq302_HTML.gif , since
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ50_HTML.gif
(3.17)
We choose https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq304_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq305_HTML.gif . For any initial https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq306_HTML.gif , constructing successively the sequences
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_Equ52_HTML.gif
(3.19)

we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq307_HTML.gif as https://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 https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq309_HTML.gif such that https://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 " https://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 https://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 " https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq313_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq314_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F567054/MediaObjects/13661_2010_Article_45_IEq315_HTML.gif )" is not satisfied, therefore, the condition https://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 https://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.