Research Article | Open Access

# Fixed Point Theorems on Nonlinear Binary Operator Equations with Applications

**Academic Editor:**Krzysztof Ciepliński

#### Abstract

The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone and partial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some corresponding results are obtained. Finally, the applications of our results are given.

#### 1. Introduction

In recent years, more and more scholars have studied binary operator equations and have obtained many conclusions; see [1–6]. In this paper, we will discuss solutions for these equations which associated with an ordinal symmetric contraction operator and obtain some results which generalized and improved those of [3–6]. Finally, we apply our conclusions to two-point boundary value problem with two-degree super-linear ordinary differential equations.

In the following, let always be a real* Banach* space which is partially ordered by a cone , let be a normal cone of , is normal constant of , partial order ≤ is determined by and denotes zero element of . Let denote an ordering interval of *.*

For the concepts of normal cone and partially order, mixed monotone operator, coupled solutions of operator equations, and so forth see [1, 5].

*Definition 1. *Let be a binary operator. is said to be -ordering symmetric contraction operator if there exists a bounded linear and positive operator , where spectral radius such that for any , , where is called a contraction operator of .

#### 2. Main Results

Theorem 2. *Let be -ordering symmetric contraction operator, and there exists a such that
**
If condition (H _{1}) or (H_{2}) holds, then the following statements hold.*

*(C*

_{1}) has a unique solution , and for any coupled solutions , .*(C*

_{2}) For any , we construct symmetric iterative sequences:*Then , , and for any , there exists a natural number ; and if , we get error estimates for iterative sequences (2):*

*Proof. *Set , and if condition (H_{1}) or (H_{2}) holds, then it is obvious that
By (1), we easily prove that is mixed monotone operator, and for any , ,
where is a bounded linear and positive operator and , is identical operator.

By the mathematical induction, we easily prove that
where , , .

By the character of normal cone , it is shown that

For any , since , there exists a natural number , and if , we have , and . Considering mixed monotone operator and constant , has a unique solution and for any coupled solution , such that by Theorem 3 in [3].

From , and the uniqueness of solution with , then we have and .

We take note of that and have the same coupled solution; therefore, a coupled solution for must be a coupled solution for ; consequently, (C_{1}) has been proved.

Considering iterative sequence (2), we construct iterative sequences:
where , , it is obvious that
by the mathematical induction and characterization of mixed monotone of ; then
Hence,
Moreover, if , we get
Consequently, .

*Remark 3. *When , Theorem 1 in [4] is a special case of this paper Theorem 2 under condition (H_{1}) or (H_{2}).

Corollary 4. *Let be -ordering symmetric contraction operator; if there exists a such that satisfies condition of Theorem 2, the following statement holds.**(C _{3}) For any and , we make iterative sequences:
*

*or*

*where , .*

Thus, , and there exists a natural number , and if , we have error estimates for iterative sequences (13) or (14):

*Proof. *By the character of mixed monotone of , then (1) and (C_{1}), (C_{2}) [in (1), (C_{2}) where ] hold.

In the following, we will prove (C_{3}).

Consider iterative sequence (13); since , we get
By the mathematical induction, we easily prove , , hence

It is clear that

For any , , since
there exists a natural number , if , such that

Moreover,
Consequently, , , .

Similarly, we can prove (14).

Theorem 5. *Let be a -ordering symmetric contraction operator; if there exists a such that , then the following statements hold.**(C _{4}) Operator equation has a unique solution , and for any coupled solutions , .*

*(C*

_{5}) For any , we make symmetric iterative sequences*Then , , , , and for any , there exists a natural number , and if , then we have error estimates for iterative sequences (22) and (23), respectively,*

*Proof. *Set or ; we can prove that this theorem imitates proof of Theorem 2.

Similarly, we can prove the following theorems.

Theorem 6. *Let be -ordering symmetric contraction operator; if there exists a such that , , then the following statements hold.**(C _{6}) Equation has a unique solution , and for any coupled solutions .*

*(C*

_{7}) For any , we make symmetric iterative sequence:*Then , ; moreover, , and there exists natural number , and if , then we have error estimates for iterative sequence (25):*

*(C*

_{8}) For any , we make symmetry iterative sequence , ; then , , and there exists a natural number , and if , we have error estimates for iterative sequence (24).*Remark 7. *When , Corollary 2 in [4] is a special case of this paper Theorems 2–6.

*Remark 8. *The contraction constant of operator in [5] is expand into the contraction operator of this paper.

*Remark 9. *Operator of this paper does not need character of mixed monotone as operator in [6].

#### 3. Application

We consider that two-point boundary value problem for two-degree super linear ordinary differential equations:

Let be* Green *function with boundary value problem (23); that is,

Then the solution with boundary value problem (23) and solution for nonlinear integral equation with type of* Hammerstein*
are equivalent, where .

Theorem 10. *Let , be nonnegative continuous function in , , . If , then boundary value problem (23) has a unique solution such that . Moreover, for any initial function , such that
**
we make iterative sequence:
**
Then and are all uniformly converge to on , and we have error estimates:
*

*Proof. *Let , , denote norm of; then has become space, is normal cone of , and its normal constant . It is obvious that integral Equation (24) transforms to operator equation , where

Set , ; then denote ordering interval of , is mixed monotone operator, and .

Set

Then is bounded linear operator, its spectral radius , and for any such that , is -ordering symmetric contraction operator, by Theorem 2 (where ); then Theorem 10 has been proved.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the NSF of Henan Education Bureau (2000110019) and by the NSF of Shangqiu (200211125).

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#### Copyright

Copyright © 2014 Baomin Qiao. 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.