Pata contractions and coupled type fixed points
© Eshaghi et al.; licensee Springer. 2014
Received: 22 November 2013
Accepted: 8 May 2014
Published: 2 June 2014
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.
Keywordspartial ordered metric space mixed monotone property coupled fixed point
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 . 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  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 ). Motivated by , 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 
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 
Definition 1.3 
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.
for each pair , defines a metric on , which will be denoted for convenience by d, too.
Let be an increasing function vanishing with continuity at zero. Also consider the vanishing sequence depending on , .
is satisfied for every and with , , then F has a coupled fixed point .
for some positive constant .
So by continuing this argument we obtain two sequences and such that and .
From the last inequality and (2.6) 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 .
for a non-decreasing sequence , then for all ,
for a non-increasing sequence , then for all .
Then F has a coupled fixed point .
for some positive constant .
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.
which is valid for every . We can replace ε, for each , by a sequence as which forces .
for every . We can replace ε, for each , by the sequence as , which forces . □
for every .
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