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Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 604105, 7 pages

http://dx.doi.org/10.1155/2013/604105

## The Solutions of Mixed Monotone Fredholm-Type Integral Equations in Banach Spaces

School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan, Shandong 250014, China

Received 15 January 2013; Accepted 23 February 2013

Academic Editor: Yanbin Sang

Copyright © 2013 Hua Su. 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.

#### Abstract

By introducing new definitions of convex and concave quasioperator and quasilower and quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new to convex and concave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces.

#### 1. Introduction

In this paper, we will consider the following nonlinear Fredholm integral equation: where and , is a real Banach space with the norm , and there exists a function such that for any

Guo and Lakshmikantham [1] introduced the definition of mixed monotone operator and coupled fixed point; there are many good results (see [1–13]). In the special case where is nondecreasing in for fixed , Guo [2] established an existence theorem of the maximal and minimal solutions for (1) in the ordered Banach spaces by means of monotone iterative techniques. Recently, Jingxian and Lishan [3] and Lishan [4] obtained iterative sequences that converge uniformly to solutions and coupled minimal and maximal quasisolutions of the nonlinear Fredholm integral equations in ordered Banach spaces by using the Möuch fixed point theorem and establishing new comparison results. But these all required the compactness conditions and the monotone conditions in the above papers, and furthermore they did not obtain the unique solutions. In addition, extensive studies have also been carried out to study the global or iterative solutions of initial value problems [8–13].

In this paper, by introducing new definitions of convex and concave quasioperator and quasilower and quasiupper, by means of the monotone iterative techniques without any compactness conditions which are of the essence in [2–4, 7, 8, 14], we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces and then apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations.

#### 2. Preliminaries and Definitions

Let be a cone in , that is, a closed convex subset such that for any and . By means of , a partial order is defined as if and only if . A cone is said to be normal if there exists a constant such that , implies , where denotes the zero element of (see [2, 14]), and we call the smallest number the normal constant of and denote . The cone is normal if and only if every ordered interval is bounded.

Let , where denotes the Banach space of all the continuous mapping with the norm . It is clear that is a cone of space , and so it defines a partial ordering in . Obviously, the normality of implies the normality of and the normal constants of , and are the same.

Let . Then, are said to be coupled lower and upper quasi-solutions of (1) if If the equality in (3) holds, then are said to be coupled quasi-solutions of (1).

We will always assume in this paper that is a normal cone of . For any such that , we define the ordered interval .

Next, we will give the new definition of convex and concave quasi operator and quasi-lower and quasi-upper.

*Definition 1. *Suppose that, . Then is called convex and concave quasi operator, if there exist functions
such that(1), for all , (2), for all .

*Definition 2. *Suppose that , . Then, is called quasi-upper, if for any , such that .

*Definition 3. *Suppose that , . Then, is called quasi-lower, if for any , such that .

Let us list the following assumption for convenience. is uniformly continuous on , and is convex and concave quasi operator. is nondecreasing in for fixed . is nonincreasing in for fixed . are all increasing in , decreasing in , and , and for , ,

#### 3. The Main Result

The main results of this paper are the following three theorems.

Theorem 4. *Let be a normal cone of , let be coupled lower and upper quasi-solutions of (1). Assume that conditions and hold and** There exists such that , and for of such that , . **
Then, (1) has a unique solution , and for any initial , one has
**
where are defined as
*

*Proof. *We first define the operator by the formula
It follows from the assumption that is a mixed monotone operator, that is, is nondecreasing in and nonincreasing in , and , .

By (7), we have , and set for initial in , and we also define that
Since is a mixed monotone operator, it is easy to see that

Obviously, by induction, it is easy to see that
where , .

In fact, by the assumption , we have that inequality (11) holds as . Suppose that inequality (11) holds as , that is, , . Then, as , by the assumption , we have
Then, it is easy to show by induction that inequality (11) holds.

For inequality (12), by and the above discussion, we have . Obviously, it follows from the assumption that , . Suppose that , , so it is easy to show by that
that is, , . Then, it is easy to show by induction that inequality (12) holds.

Then, it follows from the inequality (12) that there exist limits of the sequences . Suppose that there exist such that , , and , and by , we also have
they , and taking limits in the above inequality as , we have .

Next, we will show that the sequences are all Cauchy sequences on .

In fact, by (10) and (11), for any natural number , we know that
By the normality of and (15), we have
where is a normal constant. So are all Cauchy sequences on , and then there exists such that , .

It is easy to know by (10) and (11) that
so by the normality of , we have
and taking limits in the above inequality as , we have , and for any natural number , we also have , .

Then, by the mixed monotone quality of we have
and taking limits in above inequality as , we know that
that is, is the fixed point of ; thus, is the solution of (1) on .

Furthermore, we will show that the solution is unique. Suppose that satisfy . Then, by the mixed monotone quality of and induction, for any natural number , it is easy to have that . Then, by the normality of and taking limits in the above inequality as and the above discussion, we have .

For any initial , by (7) and (8), the mixed monotone quality of and induction, for any natural number , we have , , . Then, the normality of and (19) imply that
Thus, the sequence all converges uniformly to on . This completes the proof of Theorem 4.

Theorem 5. *Let be a normal cone of , let be coupled lower and upper quasi-solutions of (1). Assume that conditions , and hold.** is quasi-upper, and there exists such that , and there exist , .**
Then, (1) has a unique solution , and for any initial , one has
**
where are defined as
*

* Proof. *We first define the operator by the formula
It follows from the assumption that is a mixed monotone operator, that is, is nondecreasing in and nonincreasing in and , . By (7), we have , , and set , , and we also define
Since is a mixed monotone operator, it is easy to see that

Because is quasi-upper and , we have
So for any natural number , by induction, we know that .

It is easy to see by induction that
where , .

In fact, by the assumptions and and the above discussion, as , we have
By the above two inequalities and assumption , we have
Suppose that for we have , , and . Then, for , by the assumption , we have
By the above two inequalities and assumption , we have
Then, it is easy to show by induction that inequalities and hold.

The following proof is similar to that of Theorem 4. This completes the proof of Theorem 5.

By a similar argument to that of Theorem 5, we obtain the following results.

Theorem 6. *Let be a normal cone of , and let be coupled lower and upper quasi-solutions of (1). Assume that condition , and hold.** is quasi-lower, and there exists such that , and there exist , . **
Then, (1) has a unique solution , and for any initial , one has
**
where are defined as
*

#### 4. Applications

Consider the following two-point BVP in the Banach space: where , is a cone in a real Banach space . Suppose that there exists a mapping such that , and that satisfies the following conditions: is uniformly continuous on , and is convex and concave quasi operator, is nondecreasing in for fixed , and is nonincreasing in , for fixed , there exist the bounded nonnegative Lebesgue integrable functions , , , and satisfying , such that

It is well known that is a solution of BVP(34) in if and only if is a solution of the following integral equation: where

Lemma 7. *If assumption holds, then there exists such that
*

*Proof. *In fact, let
Obviously, by assumption , we can get that , then the equation has a unique solution

Similarly, the equation has a unique solution
Thus, by assumption , for any , we have
that is, (38) holds.

Theorem 8. *Let be a normal cone of . Assume that and hold,** there exists and in (38) of Lemma 7 such that , and also there exists such that , ,** are all increasing in , decreasing in and , , for , ,
**
Then, (34) has a unique solution , and for any initial , one has
**
where are defined as
*

* Proof. *It is easy to see by conditions and that satisfy the conditions and of Theorem 4. By and (38), we have as coupled lower and upper quasi-solutions of (34).

Thus, the assumption of Theorem 4 is satisfied from the assumption of Theorem 8. The conclusion of Theorem 8 follows from Theorem 4.

*Example 9. *In fact, we can construct the function in Theorem 8.

Let then Thus, is convex and concave quasi operator and thus satisfies .

It is easy to check that is nondecreasing in for fixed and is nonincreasing in for fixed and thus satisfies .

There exist , , , and satisfying such that Thus, holds.

There exist Choose such that , and also there exist , such that Thus, is satisfied.

are all increasing in and nondecreasing in , and for , , we have Thus, also holds.

#### Acknowledgment

The author was supported financially by Natural Science Foundation of China (11071141) and the Project of Shandong Province Higher Educational Science and Technology Program (J13LI12, J10LA62).

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