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.