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Pata contractions and coupled type fixed points
Fixed Point Theory and Applications volume 2014, Article number: 130 (2014)
Abstract
A new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered complete metric spaces is established. It is shown that the coupled fixed point can be unique under some extra suitable conditions involving mid point lower or upper bound properties. Also the corresponding convergence rate is estimated when the iterates of our function converge to its coupled fixed point.
MSC:47H10, 34B15, 54H25.
1 Introduction and preliminaries
The well-known Banach contraction principle, which guarantees the existence of a unique fixed point for a mapping defined on a complete metric space satisfying the contraction condition, was introduced in 1922 by Banach [1]. After this a great deal of effort has gone into the theory and application of the Banach contraction theorem. Some authors generalized this theorem from the single-valued case to the multivalued [2–4]. Some extensions are to fixed point theorems for contraction mappings in generalized form of metric spaces, especially a metric space endowed with a partial order. Single, coupled, tripled and other types of fixed point theorems are investigated in many works, for instance, [5–9] and references cited therein. Such fixed point theorems are applied to establishing the existence of a unique solution to periodic boundary value problems, matrix equations, ordinary differential equations, and integral equations [10–13]. Recently Pata in [14] introduced a fixed point theorem with weaker hypotheses than those of the Banach contraction principle with an explicit estimate of the convergence rate (see also [15]). Motivated by [16], we establish a new coupled fixed point theorem related to the Pata contraction for mappings having the mixed monotone property in partially ordered metric spaces. We prove that the coupled fixed point can be unique under some suitable conditions. Also the corresponding convergence rate is estimated when the iterates of our function converge to its coupled fixed point.
Definition 1.1 [16]
Let be a partially ordered set. The product space can be endowed with a partial order such that for we can define .
Definition 1.2 [16]
Let be a partially ordered set and . We say that F has the mixed monotone property on X if is monotone non-decreasing in x and is monotone non-increasing in y, that is, for any ,
Definition 1.3 [16]
An element is called a coupled fixed point of the mapping if
2 Main results
In this section we prove a coupled type fixed point theorem with the convergence rate estimation. Also an example is given as an application of the main theorem.
Definition 2.1 Let be a metric space. The mapping given by
for each pair , defines a metric on , which will be denoted for convenience by d, too.
For a metric space , selecting an arbitrary , we denote
Let be an increasing function vanishing with continuity at zero. Also consider the vanishing sequence depending on , .
Theorem 2.2 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on X and let , , and be fixed constants. Suppose that there exist such that
If the inequality
is satisfied for every and with , , then F has a coupled fixed point .
Furthermore, we denote (n times),
for some positive constant .
Proof Set and . So
Now from letting , and choosing the notations
with the mixed monotone property of F we have
So by continuing this argument we obtain two sequences and such that and  .
Furthermore for we set
Since (2.1) is true for every , setting we have the following relations:
Using (2.5), (2.3), (2.4), and (2.1) we have
But from the definition of we have
so for there are some such that
Accordingly,
which holds by hypothesis for any taken for each . If there is a subsequence , then the choice leads to the following contradiction:
Then the sequence is bounded. Also the sequences and are Cauchy sequences:
But since
we have
For fixed m, set
and . So
Setting we have
Therefore
Taking limits as , we get . This implies that and are Cauchy sequences in X. Since X is a complete metric space, there are such that
Using (2.7) and Definition 2.1 we have
and from the continuity of F we have and . Also, the convergence rate estimate stated in (2.2) is obtained from the following relations:
which implies that
From the last inequality and (2.6) we have . □
Example 2.3 Let and be a metric on X defined as for all which is a complete metric space. Now consider with the partial order defined in Definition 1.1. The metric on the product metric space is defined by
Consider the mapping F defined as
for all fixed constants , , and , and by using (2.1) we have
Since and are positive, we have , which forces . Also there exists such that and . Thus F satisfies all conditions of Theorem 2.2. The pair is the coupled fixed point for the mapping F.
3 Supplementary results
In this section we prove some supplementary results. The first theorem is about replacing the continuity condition of the function F in Theorem 2.2 with a new one. The second is about the uniqueness of the coupled fixed point. At last we show that the hypothesis of Theorem 2.2 is weaker than those of Theorem 2.1 in [16].
Theorem 3.1 Let be a partially ordered set and suppose that there is a metric d on X such that is a complete metric space. Let be a mapping having the mixed monotone property on X such that there are two elements with
Let , , and be fixed constants. Suppose that the inequality
is satisfied for every and with , . Furthermore, suppose that X has the following properties:
-
(i)
for a non-decreasing sequence , then for all ,
-
(ii)
for a non-increasing sequence , then for all .
Then F has a coupled fixed point .
Furthermore, calling (n times), we have
for some positive constant .
Proof We only need to prove that and . Consider the following relations:
Now since
and since the contractive condition (3.1) holds for any real constant , we can replace ε, for each , by a sequence as . Then by letting as , we have . □
Definition 3.2 Suppose that X is a partially ordered metric space with a metric d. A pair of has either a mid point lower bound or a mid point upper bound if there are comparable to and such that . The space has the mid point lower bound or the mid point upper bound property if any pair in has a mid point lower bound or a mid point upper bound.
Theorem 3.3 Adding the condition of the above definition to the space in the hypothesis of Theorem 3.1, we obtain the uniqueness of the coupled fixed point of F.
Proof If is another coupled fixed point of F where , , then we have two cases.
Case (1): If is comparable to then
for some . Putting , we have
which is valid for every . We can replace ε, for each , by a sequence as which forces .
Case (2): If is not comparable to then there exists a mid upper bound or mid lower bound for and . So is comparable to and . It follows that
for some . Set . Hence,
Now by Definition 3.2 we have
for every . We can replace ε, for each , by the sequence as , which forces . □
Remark 3.4 Note that Theorem 2.2 is stronger than Theorem 2.1 in [16]. Indeed with the hypothesis of Theorem 2.1 in [16], for each , we have
Thus for , , and , for arbitrary , we get
for every .
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Eshaghi, M., Mohseni, S., Delavar, M.R. et al. Pata contractions and coupled type fixed points. Fixed Point Theory Appl 2014, 130 (2014). https://doi.org/10.1186/1687-1812-2014-130
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DOI: https://doi.org/10.1186/1687-1812-2014-130