- Research Article
- Open Access
Nonlinear Contractive Conditions for Coupled Cone Fixed Point Theorems
- Wei-Shih Du^{1}Email author
https://doi.org/10.1155/2010/190606
© Wei-Shih Du. 2010
- Received: 19 April 2010
- Accepted: 5 July 2010
- Published: 21 July 2010
Abstract
We establish some new coupled fixed point theorems for various types of nonlinear contractive maps in the setting of quasiordered cone metric spaces which not only obtain several coupled fixed point theorems announced by many authors but also generalize them under weaker assumptions.
Keywords
- Fixed Point Theorem
- Convex Cone
- Topological Vector Space
- Nonnegative Number
- Couple Fixed Point
1. Introduction
The existence of fixed point in partially ordered sets has been studied and investigated recently in [1–13] and references therein. Since the various contractive conditions are important in metric fixed point theory, there is a trend to weaken the requirement on contractions. Nieto and Rodríguez-López in [8, 10] used Tarski's theorem to show the existence of solutions for fuzzy equations and fuzzy differential equations, respectively. The existence of solutions for matrix equations or ordinary differential equations by applying fixed point theorems are presented in [2, 6, 9, 11, 12]. In [3, 13], the authors proved some fixed point theorems for a mixed monotone mapping in a metric space endowed with partial order and applied their results to problems of existence and uniqueness of solutions for some boundary value problems.
In 2006, Bhaskar and Lakshmikantham [2] first proved the following interesting coupled fixed point theorem in partially ordered metric spaces.
Theorem BL. (Bhaskar and Lakshmikantham).
If there exist such that and , then, there exist , such that and .
will stand for and while will stand for , where denotes the interior of .
In the following, unless otherwise specified, we always assume that is a locally convex Hausdorff t.v.s. with its zero vector , a proper, closed, convex, and pointed cone in with , a partial ordering with respect to , and .
Very recently, Du [14] first introduced the concepts of -cone metric and -cone metric space to improve and extend the concept of cone metric space in the sense of Huang and Zhang [15].
Definition 1.1 . (see [14]).
Let be a nonempty set. A vector-valued function is said to be a -cone metric if the following conditions hold:
(C1) for all and if and only if ;
(C2) for all ;
(C3) for all .
The pair is then called a -cone metric space.
Definition 1.2 . (see [14]).
Let be a -cone metric space, , and a sequence in .
(i) is said to -cone converge to if for every with there exists a natural number such that for all . We denote this by cone- or as and call the -cone limit of .
(ii) is said to be a -cone Cauchy sequence if for every with there is a natural number such that for all , .
(iii) is said to be -cone complete if every -cone Cauchy sequence in is -cone convergent in .
In [14], the author proved the following important results.
Theorem 1.3 . (see [14]).
Theorem 1.4 . (see [14]).
Let be a -cone metric space, , and a sequence in . Then the following statements hold:
(a)if -cone converges to (i.e., as , then as (i.e., as ;
(b)if is a -cone Cauchy sequence in , then is a Cauchy sequence (in usual sense) in .
In this paper, we establish some new coupled fixed point theorems for various types of nonlinear contractive maps in the setting of quasiordered cone metric spaces. Our results generalize and improve some results in [2, 4, 9, 11] and references therein.
2. Preliminaries
Theorem 2.1 . (see [14, 16, 17]).
For each and , the following statements are satisfied:
(i) ;
(ii) ;
(iii) ;
(iv) ;
(v) is positively homogeneous and continuous on ;
(vi)if , then ;
(vii) for all .
- (a)
Clearly, .
- (b)
The reverse statement of (vi) in Theorem 2.1 (i.e., ) does not hold in general. For example, let , and . Then is a proper, closed, convex, and pointed cone in with and . For , it is easy to see that , and . By applying (iii) and (iv) of Theorem 2.1, we have but indeed .
It is obvious that is also a -cone metric on , and if and as , then (i.e., TVS-cone converges to .
is a metric on .
A map is said to be -continuous at if any sequence with implies that . is said to be -continuous on if is continuous at every point of .
Definition 2.3 . (see [2, 4]).
Definition 2.4 . (see [2, 4]).
Definition 2.5.
Let be a -cone metric space with a quasi-order ( for short). A nonempty subset of is said to be
(i) -cone sequentially - if every -nondecreasing -cone Cauchy sequence in converges,
(ii) -cone sequentially - if every -nonincreasing -cone Cauchy sequence in converges,
(iii) -cone sequentially - if it is both -cone sequentially -complete and -cone sequentially - .
Definition 2.6 . (see [4, 18]).
A function is said to be a - if it satisfies Mizoguchi-Takahashi's condition (i.e., for all ).
Clearly, if is a nondecreasing function, then is a -function. Notice that is a -function if and only if for each there exist and such that for all ; for more detail, see [4, Remark (iii)].
Very recently, Du and Wu [5] introduced and studied the concept of functions of contractive factor.
Definition 2.7 . (see [5]).
The following result tells us the relationship between -functions and functions of contractive factor.
Theorem 2.8.
Any -function is a function of contractive factor.
Proof.
Then for all , and hence . Therefore is a function of contractive factor.
3. Coupled Fixed Point Theorems for Various Types of Nonlinear Contractive Maps
Definition 3.1.
It is quite obvious that if is a function of strong contractive factor, then is a function of contractive factor but the reverse is not always true.
The following results are crucial to our proofs in this paper.
Lemma 3.2.
A function of strong contractive factor can be structured by a function of contractive factor.
Proof.
Hence is a function of strong contractive factor.
Lemma 3.3.
Let be a t.v.s., a convex cone with in , and . Then the following statements hold.
(i)If and , then ;
(ii)If and , then ;
(iii)If and , then .
Proof.
which means that . The proofs of conclusions (ii) and(iii) are similar to (i).
Lemma 3.4 (see [4]).
for each . If and , then is -nondecreasing and is -nonincreasing.
In this section, we first present the following new coupled fixed point theorem for functions of contractive factor in quasiordered cone metric spaces which is one of the main results of this paper.
Theorem 3.5.
and there exist such that and . Define the iterative sequence in by and for . Then the following statements hold.
(a)There exists a nonempty subset of , such that is a complete metric space.
(b)There exists a nonempty subset of , such that is a complete metric space, where for any . Moreover, if is -continuous on , then -cone converges to a coupled fixed point in of .
Proof.
Then is a base at , and the topology generated by is the weakest topology for such that all seminorms in are continuous and . Moreover, given any neighborhood of , there exists such that (see, e.g., [19, Theorem in II.12, Page 113]).
Hence is a -nondecreasing -cone Cauchy sequence and is a -nonincreasing -cone Cauchy sequence in . By the -cone sequential -completeness of , there exist such that -cone converges to and -cone converges to . Therefore -cone converges to .
Since for all , by (3.36) and (3.37), we have as . Let , , and . Then , , and are also complete metric spaces. Hence conclusion (a) holds.
Since is arbitrary, or . Similarly, we can also prove that . So is a coupled fixed point of . The proof is finished.
The following conclusions are immediate from Theorems 2.8 and 3.5.
Theorem 3.6.
and there exist such that and . Define the iterative sequence in by and for . Then the following statements hold.
(a)There exists a nonempty subset of , such that is a complete metric space.
(b)There exists a nonempty subset of , such that is a complete metric space. Moreover, if is -continuous on , then -cone converges to a coupled fixed point in of .
Theorem 3.7.
and there exist such that and . Define the iterative sequence in by and for . Then the following statements hold.
(a)There exists a nonempty subset of , such that is a complete metric space.
(b)There exists a nonempty subset of , such that is a complete metric space. Moreover, if is -continuous on , then -cone converges to a coupled fixed point in of .
- (a)
- (b)
Theorems 3.5–3.7 all generalize Bhaskar-Lakshmikantham's coupled fixed points theorem (i.e., Theorem BL).
Finally, we focus our research on -cone metric spaces.
Theorem 3.9.
Let . Define the iterative sequence in by and for . Then the following statements hold.
(a)There exists a nonempty subset of , such that is a complete metric space.
(b)There exists a nonempty subset of , such that is a complete metric space.
(c) has a unique coupled fixed point in . Moreover, -cone converges to the coupled fixed point of .
Proof.
which leads to a contradiction. The proof is completed.
The following results are immediate from Theorem 3.9.
Theorem 3.10.
Let . Define the iterative sequence in by and for . Then the following statements hold.
(a)There exists a nonempty subset of , such that is a complete metric space.
(b)There exists a nonempty subset of , such that is a complete metric space.
(c) has a unique coupled fixed point in . Moreover, -cone converges to the coupled fixed point of .
Theorem 3.11.
Let . Define the iterative sequence in by and for . Then the following statements hold.
- (b)
There exists a nonempty subset of , such that is a complete metric space.
(c) has a unique coupled fixed point in . Moreover, -cone converges to the coupled fixed point of .
Declarations
Acknowledgment
This research was supported by the National Science Council of the Republic of China.
Authors’ Affiliations
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