Research Article | Open Access

# Fixed Point Theorems of Binary Contraction Comparable Operators and an Application

**Academic Editor:**Juan R. Torregrosa

#### Abstract

The aim of this paper is to present the concept of binary comparable operators in partially ordered Banach spaces and prove several fixed point theorems under some contractive conditions. The results of this paper can be used to investigate a large class of nonlinear problems. As an application, we study the existence of solution of a nonlinear integral equation.

#### 1. Introduction

The Banach contraction principle [1] as a popular tool for solving problems in nonlinear analysis was invented in 1922. Since then, the study of fixed points of mappings with contractive property has been at the center of various research activities. For example, Liu and Zhu [2] studied the solvability of a binary operator equation satisfying certain contractive conditions; Romaguera [3] obtained several fixed point theorems of mappings satisfying some generalized contractive conditions. For more details on fixed point results of contractive type mappings and applications, we refer to Yan et al. [4], Mukherjea [5], Ran and Reurings [6], Hussain et al. [7], Amini-Harandi [8], Sintunavarat and Kumam [9], Nieto and Rodríguez-López [10, 11], and O’Regan and Saadati [12].

One of the common properties of the above results is that the involved operators must satisfy the monotone or mixed monotone conditions. In 2005, Zhang [13] studied an ordinal -ordering symmetric contraction operator without the mixed monotone property and proved some coupled fixed point theorems. On the other hand, Qiao [14] investigated the fixed point theorems of the ordered contractive operators with the comparable property.

In the present paper, we introduce the concept of binary comparable operators, which can be seen as a generalization of the concept of mixed monotone operators and the concept of antimixed monotone operators. Using the iterative techniques ([15, 16]), we obtain several fixed point theorems for such operators under some contractive conditions. The results of this paper generalize several classical results in the literature. As an application, the existence of solution of an integral equation is presented.

For the sake of convenience, let us recall the following definitions and lemmas (see [14, 17, 18] for more details and recent results).

Let be a real Banach space. A subset of is called a cone whenever the following conditions hold:(i)is closed, nonempty, and ;(ii), and implies ;(iii).

Given a cone , we define a partial ordering with respect to by if and only if .

Let be a normed Banach space, which is partially ordered by a cone . The cone is said to be normal if there exists a constant , such that, for all , implies , where denotes the zero element of .

*Definition 1 (see [14]). *For some , if or holds, and are said to be comparable. Moreover, one will write to indicate that , while stand for .

Lemma 2 (see [14]). *If are comparable, then and are comparable, and*

Lemma 3 (see [14]). *For some , if any two of them are comparable, then *

Lemma 4 (see [14]). *If, for each positive integer , and are comparable and , then and are comparable.*

Lemma 5 (see [14]). *If, for each positive integer , and are comparable and , , then and are comparable.*

*Definition 6. *A binary operator is said to be comparable if, for all comparable pairs and , and are comparable.

*Definition 7. *A comparable operator is said to be -contractive, if, for all comparable pairs and , there exists a constant such that

#### 2. Main Results

Theorem 8. *Let be a real Banach space, a normal cone of with the normal constant , and a partial order with respect to . Let be demicontinuous. Suppose that the following two conditions are satisfied:*(i)* is -contractive and comparable, where ;*(ii)*there exists a comparable pair in , such that and , and are comparable.**Then has a fixed point in ; that is, . Moreover, the iterative sequences and converge to , and*

*Proof . *Consider the iterative sequencesSince and are comparable and is a comparable operator, it is easy to verify that and are comparable. By inductions, we can prove that and are comparable for any positive integer .

Since is a -contractive comparable operator, we have If we continue in the same way, for each , we haveBy the normality of , we haveSince is a comparable pair in and, moreover, and , and are comparable, we know that and , and are comparable, which implies that and , and are comparable. By induction, we know that and are all comparable pairs for each .

Furthermore, for each , we have By induction we can prove thatSince , we can prove that and are Cauchy sequences in . Then there exist two points , such thatSince is demicontinuous, converges to weakly and converges to weakly and thus , ; that is, is a coupled fixed point of .

Taking limit in (8) as , we getThis means thatHence is a fixed point of .

Moreover, By a similar method, we can proveThen we complete the proof of Theorem 8.

Theorem 9. *Let be a real Banach space, a normal cone of with the normal constant , and the partial order with respect to . Suppose that the following two conditions are satisfied:*(i)* is -contractive comparable one, where ;*(ii)*there exists a comparable pair in , such that and , and are comparable for each .**Then has a fixed point in ; that is, . Moreover, the iterative sequences and converge to , and*

*Proof. *By a similar approach as in the proof of Theorem 8, we can prove that and are Cauchy sequences in , and there exist two points , such thatFurthermore We now prove that is a fixed point of . For some , assume that ; then by condition (ii), we know that and , and are comparable; hence and , and are comparable. If we continue in the same way, we can prove that and , and are comparable. As , by Lemma 4, we know that, for each , and , and are comparable.

Hence and , and are comparable; furthermoreFrom the normality of , we haveThose imply thatTaking limit in (21) as , we get , ; that is, is a coupled fixed point of .

By (18), we can prove that is a fixed point of .

Using the same argument as that in the proof of Theorem 8, we can obtainThen we complete the proof of Theorem 9.

Theorem 10. *Let be a Banach space and a normal cone in with the normal constant , with , and an order interval. Suppose that is -contractive and comparable, where ; then has a unique fixed point in . Moreover, for any initial , the iterative sequences and converge to .*

*Proof. *Consider the iterative sequencesSince and are comparable and is a comparable operator, it is easy to verify that and are comparable. By inductions, we can prove that and are comparable for any positive integer .

Since is a -contractive comparable operator, we have If we continue in the same way, for each , we haveBy the normality of , we haveSince , we know that and , and are comparable; thus and , and are comparable, which implies that and , and are comparable. By induction, we know that and are all comparable pairs for each .

Furthermore, for each , we have By the normality of , we haveFor , we can prove that and are Cauchy sequences.

Since , there exist two points in , such thatTaking limit in (26) as , we getWe now prove that is a fixed point of . For some , assume that ; since , we know that and , and are comparable; hence and , and are comparable. If we continue in the same way, we can prove that and , and are comparable. As , by Lemma 4, we know that, for each , and , and are comparable.

Hence and , and are comparable; furthermoreFrom the normality of , we haveThis implies thatTaking limit in (33) as , we get , ; that is, is a coupled fixed point of .

By (30), we can prove that is a fixed point of .

For any initial , we construct iterative sequences and , . Since , that is, and , and are comparable, we know that and , and are comparable, which mean that and , and are comparable. By inductions, we know that and , and are comparable for each . Moreover,From the normality of , we haveThis implies thatFor the uniqueness, we assume that there exists a point , such that . Construct the iterative sequence as , , and , and we know that . On the other hand, for , , and hence .

Then we complete the proof of Theorem 10.

*Remark 11. *It is easy to verify that mixed monotone operators and antimixed monotone operators are precisely comparable operators. However, a comparable operator is not necessarily a mixed monotone operator or an antimixed monotone operator. Thus, the fixed point theorems in this work generalize and extend the fixed point theorems of mixed monotone operators and antimixed monotone operators.

#### 3. Applications

As an application, we consider the nonlinear Hammerstein integral equation of the following form.

Let be the closed unit interval in . Consider the following integral equation: Suppose that is continuous about and bounded measurable about and, moreover, nonnegative on and ; then (37) has a unique solution .

*Proof. *Let denote the Banach space of all real valued continuous functions on , with , and let . Clearly, is a normal cone of with the normal constant .

We transform integral equation (37) to the formSince is nonnegative on , we get that is a binary operator from to , and apparently is comparable.

For , we get Then we obtain that is -contractive. According to Theorem 10, we can prove that the integral equation (37) has a unique solution.

*Remark 12. *The operator defined by (37) is a comparable operator, but does not satisfy the mixed monotone or antimixed monotone condition. However, by Theorem 10 of this work, we can easily get the conclusion. Thus, from this application, it is shown that some of the results in this work generalize and extend the corresponding results of mixed monotone operators and antimixed monotone operators again.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the Fundamental Research Funds for the Central Universities, China (no. JBK1307050), and the National Natural Science Foundation of China (no. 11361042).

#### References

- S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,”
*Fundamenta Mathematicae*, vol. 3, no. 1, pp. 133–181, 1922. View at: Google Scholar - Z. Liu and C. X. Zhu, “Solvability theorems with applications of an operator equation in partial order space,”
*Acta Mathematica Scientia A*, vol. 52, no. 6, pp. 1181–1188, 2009. View at: Google Scholar - S. Romaguera, “Fixed point theorems for generalized contractions on partial metric spaces,”
*Topology and Its Applications*, vol. 159, no. 1, pp. 194–199, 2012. View at: Publisher Site | Google Scholar - F. Yan, Y. Su, and Q. Feng, “A new contraction mapping principle in partially ordered metric spaces and applications to ordinary differential equations,”
*Fixed Point Theory and Applications*, vol. 2012, article 152, 2012. View at: Publisher Site | Google Scholar - A. Mukherjea, “Contractions and completely continuous mappings,”
*Nonlinear Analysis—Theory, Methods & Applications*, vol. 1, no. 3, pp. 235–247, 1977. View at: Publisher Site | Google Scholar - A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,”
*Proceedings of the American Mathematical Society*, vol. 132, no. 5, pp. 1435–1443, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH - N. Hussain, V. Berinde, and N. Shafqat, “Common fixed point and approximation results for generalized $\varphi $-contractions,”
*Fixed Point Theory*, vol. 10, no. 1, pp. 111–124, 2009. View at: Google Scholar - A. Amini-Harandi, “Coupled and tripled fixed point theory in partially ordered metric spaces with application to initial value problem,”
*Mathematical and Computer Modelling*, vol. 57, no. 9-10, pp. 2343–2348, 2013. View at: Publisher Site | Google Scholar - W. Sintunavarat and P. Kumam, “Coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces,”
*Thai Journal of Mathematics*, vol. 10, no. 3, pp. 551–563, 2012. View at: Google Scholar | Zentralblatt MATH - J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,”
*Order*, vol. 22, no. 3, pp. 223–239, 2005. View at: Publisher Site | Google Scholar - J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,”
*Acta Mathematica Sinica—English Series*, vol. 23, no. 12, pp. 2205–2212, 2007. View at: Publisher Site | Google Scholar - D. O'Regan and R. Saadati, “Nonlinear contraction theorems in probabilistic spaces,”
*Applied Mathematics and Computation*, vol. 195, no. 1, pp. 86–93, 2008. View at: Publisher Site | Google Scholar - X. Zhang, “Fixed point of ordered contractive maps,”
*Acta Mathematica Scientia A*, vol. 48, no. 5, pp. 973–978, 2005. View at: Google Scholar - B. Qiao, “Fixed point theorems on nonlinear binary operator equations with applications,”
*Discrete Dynamics in Nature and Society*, vol. 2014, Article ID 241942, 4 pages, 2014. View at: Publisher Site | Google Scholar - J. F. Traub,
*Iterative Methods for the Solution Of Equations*, Prentice-Hall, Englewood Cliffs, New Jersey, NJ, USA, 1964. View at: MathSciNet - D. K. Lika, “On the solution of nonlinear operator equations by a class of iterative processes,”
*Siberian Mathematical Journal*, vol. 11, no. 4, pp. 709–713, 1970. View at: Publisher Site | Google Scholar - D. J. Guo,
*Nonlinear Functional Analysis*, Shandong Science Technology Publishing House, Shandong, China, 1985 (Chinese). - H. H. Scharfer,
*Topological Vector Space*, Springer, New York, NY, USA, 1971.

#### Copyright

Copyright © 2015 Zhan Liu and Chuanxi Zhu. 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.