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.
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 RodríguezLópez 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).
Let be a partially ordered set and a metric on such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on . Assume that there exists a with
If there exist such that and , then, there exist , such that and .
Let be a topological vector space (t.v.s. for short) with its zero vector . A nonempty subset of is called a convex cone if and for . A convex cone is said to be if . For a given proper, pointed, and convex cone in , we can define a partial ordering with respect to by
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 vectorvalued 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]).
Let be a cone metric space. Then defined by is a metric, where is defined by
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
Let be a nonempty set and "" a quasiorder (preorder or pseudoorder, i.e., a reflexive and transitive relation) on . Then is called a quasiordered set. A sequence is called nondecreasing (resp., nonincreasing) if (resp., ) for each . In this paper, we endow the product space with the following quasiorder :
Recall that the nonlinear scalarization function is defined by
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 .
Remark 2.2.
(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 .
For any cone metric space , we can define the map by
It is obvious that is also a cone metric on , and if and as , then (i.e., TVScone converges to .
By Theorem 1.3, we know that is a metric on . Hence the function : , defined by
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]).
Let be a quasiordered set and a map. one says that has the mixed monotone property on if is monotone nondecreasing in and is monotone nonincreasing in , that is, for any , ,
Definition 2.4 . (see [2, 4]).
Let be a nonempty set and a map. One calls an element a coupled fixed point of if
Definition 2.5.
Let be a cone metric space with a quasiorder ( 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 MizoguchiTakahashi'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]).
One says that is a function of contractive factor if for any strictly decreasing sequence in , one has
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.
Let be a function, and let be a strictly decreasing sequence in . Then exists. Since is a function, there exist and such that for all . On the other hand, there exists , such that
for all with . Hence for all . Let
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.
One says that is a function of strong contractive factor if for any strictly decreasing sequence in , one has
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.
Let be a function of contractive factor. Define , . We claim that is a function of strong contractive factor. Clearly, for all . Let be a strictly decreasing sequence in . Since is a function of contractive factor, . Thus it follows that
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.
To see (i), since the set is open in and is a convex cone, we have
Since and , it follows that
which means that . The proofs of conclusions (ii) and(iii) are similar to (i).
Lemma 3.4 (see [4]).
Let be a quasiordered set and a multivalued map having the mixed monotone property on . Let . Define two sequences and by
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.
Let be a cone sequentially complete metric space, a map having the mixed monotone property on , and . Assume that there exists a function of contractive factor such that for any with ,
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.
Since is a locally convex Hausdorff t.v.s. with its zero vector , let denote the topology of and let be the base at consisting of all absolutely convex neighborhood of . Let
Then is a family of seminorms on . For each , let
and let
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]).
By Lemma 3.2, we can define a function of strong contractive factor by . Then for all . For any , let and . Then, by Lemma 3.4, is nondecreasing and is nonincreasing. So and for each . By (3.6), we obtain
By (3.10) and Theorem 2.1,
Similarly, by (3.11) and Theorem 2.1, we also have
Combining (3.12) and (3.13), we get
For each , let . Then . By induction, we can obtain the following. For each ,
Since for all , the sequence is strictly decreasing in from (3.19). Since is a function of strong contractive factor, we have
So for all . We want to prove that is a nondecreasing cone Cauchy sequence and is a nonincreasing cone Cauchy sequence in . For each , by (3.15), we have
Similarly, by (3.16), we obtain
From (3.21) and (3.22), we get
Hence it follows from (3.21), (3.22), and (3.23) that
Therefore, for with , we have
Given with (i.e., , there exists a neighborhood of such that . Therefore, there exists with such that , where
for some and , . Let
If , since each is a seminorm, we have and
for all and all . If , since , , and hence there exists such that for all . So, for each and any , we obtain
Therefore for any , for all , and hence . So we obtain
or
for all . For with , by (3.25), (3.26), (3.32), and Lemma 3.3, we obtain
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 .
On the other hand, applying Theorem 1.4, we have the following:
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.
Finally, in order to complete the proof of conclusion (b), we need to verify that is a coupled fixed point of . Let be given. Since is continuous on and , is continuous at . So there exists such that
whenever with . Since and as , for there exists such that
So, for each with , by (3.39),
and hence we have from (3.38) that
Therefore
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.
Let be a cone sequentially complete metric space, a map having the mixed monotone property on , and . Assume that there exists a function such that for any with ,
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.
Let be a cone sequentially complete metric space, a map having the mixed monotone property on , and . Assume that there exists a nonnegative number such that for any with ,
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 .
Remark 3.8.
(a) Theorems 3.5 and 3.6 all generalize and improve [4, Theorem ] and some results in [2, 9, 11].
(b) Theorems 3.5–3.7 all generalize BhaskarLakshmikantham's coupled fixed points theorem (i.e., Theorem BL).
Finally, we focus our research on cone metric spaces.
Theorem 3.9.
Let be a cone complete metric space, a map, and . Assume that there exists a function of contractive factor such that for any
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.
For any , by (3.45) and Theorem 2.1, we obtain
From (3.46), we know that is continuous on . Following the same argument as in the proof of Theorem 3.5, we can prove that conclusions (a) and (b) hold and there exists , such that cone converges to and is a coupled fixed point of . To complete the proof, it suffices to show the uniqueness of the coupled fixed point of . On the contrary, suppose that there exists , such that and . By (3.46), we have
So, it follows from (3.47) that
which leads to a contradiction. The proof is completed.
The following results are immediate from Theorem 3.9.
Theorem 3.10.
Let be a cone complete metric space, a map, and . Assume that there exists a function such that for any ,
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 be a cone complete metric space and a map. Assume that there exists a nonnegative number such that for any ,
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 .
Remark 3.12.
(a) Theorems 3.9 and 3.10 all generalize and improve [4, Theorem ].
(b) Theorems 3.9–3.11 all generalize some results in [2, 9, 11].
Acknowledgment
This research was supported by the National Science Council of the Republic of China.
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