Modified proof of Caristi’s fixed point theorem on partial metric spaces
Journal of Inequalities and Applications volume 2013, Article number: 210 (2013)
In this paper, lower semi-continuous functions are used to modify the proof of Caristi’s fixed point theorems on partial metric spaces. We prove a new type of fixed point theorems in complete partial metric spaces, and then generalize them to metric spaces. Some more general results are also obtained on partial metric spaces.
The number of extensions of the Banach contraction principle have appeared in literature. One of its most important extensions is known as Caristi’s fixed point theorem because Caristi’s theorem is a variety of Ekeland’s ϵ-variational principle . In 1976, Caristi  proved the following fixed point theorem.
Theorem 1.1 Let be a complete metric space and , and let ϕ be a lower semicontinuous function from X into . Assume that for all . Then f has a fixed point in X.
Caristi’s fixed point theorem was generalized by several authors. For example, Bae  generalized Caristi’s theorem to prove the fixed point theorem for weakly contractive set-valued mappings. Downing and Kirk  generalized Caristi’s theorem to prove the surjectivity theorem for a nonlinear closed mapping. See also  and others [6, 7].
In 1977, Siegel  found that based on the work of Brondsted  (see also ), Caristi  had given a significant generalization of the contraction theorem. However, Caristi’s proof, as well as other more recent ones , lacked the constructive aspect of the original proof. Therefore, he presented a version of Caristi’s theorem which offered a construction of the fixed point as a countable iteration of application of suitable operators in a complete metric space.
In recent years many works on domain theory have been made in order to equip semantics domain with a notion of distance, see . In particular, Matthews  introduced the notion of a partial metric space as a part of the study of denotational semantics of dataflow network, showing that the Banach contraction mapping theorem can be generalized to the partial metric context for applications in program verification.
In this paper we present a version of Caristi’s theorem which offers a construction of the fixed point as a countable iteration of application of suitable operators in a complete partial metric space. The theorem we present is proved by a very simple argument which is in the spirit of the original contraction theorem.
First, we start with some preliminaries on partial metric spaces. For more details, we refer the reader to .
Definition 2.1 Let X be a nonempty set. The mapping satisfies:
for all .
if and only if .
for all .
for all .
Then p is called a partial metric on X and is called a partial metric space.
Note that the self-distance of any point need not be zero, hence the idea of generalizing metrics so that a metric on a non-empty set X is precisely a partial metric p on X such that for any .
Definition 2.2 Let be a partial metric space. For any and , we define respectively the open and closed ball for the partial metric p by setting
Contrary to the metric space case, some open balls may be empty. As an example, in a partial metric space , the open balls are empty for any . For example, consider the function defined by for any . Then the pair is a partial metric space. In a partial metric space , the set of open balls is the basis of a topology on X, called the partial metric topology and denoted by .
Lemma 2.3 Let be a partial metric space, and let be defined by
Then is a metric space.
Definition 2.4 A sequence in a partial metric space converges to , and we write if, for any such that , there exists so that for any , .
Definition 2.5 If is a sequence in a partial metric space , then is a proper limit of written (properly) if in . If a sequence has a proper limit, then one says that the sequence is properly convergent.
Definition 2.6 A sequence in a partial metric space is called a Cauchy sequence if exists and is finite. A partial metric space is said to be complete if every Cauchy sequence in X converges with respect to to a point such that .
Lemma 2.7 Let be a partial metric space. Then
a sequence in a partial metric space converges to a point if and only if ,
a sequence in a partial metric space converges properly to a point if and only if ,
a sequence in a partial metric space is a Cauchy sequence if and only if it is a Cauchy sequence in the metric ,
a partial metric space is complete if and only if the metric space is complete. Moreover,
Definition 2.8 A function is called lower semi-continuous if for any sequence in X and such that converges to x, we have .
3 Main results
Let be a complete partial metric space. Let and be not necessarily continuous functions such that
Given a sequence of functions , ,
If it exists, call it the countable composition of the .
Now, we let , , and then we prove a simple lemma as follows.
Lemma 3.1 Both Φ and are closed under compositions. Moreover, if ϕ is lower semi-continuous, then Φ and are closed under countable compositions.
Proof We first show that Φ is closed under compositions.
Let and be in Φ, then we have
implies that .
Furthermore, we have is closed under compositions.
Indeed, we let , which gives that , and we now consider
So, we obtain from the above inequalities that
and then . This implies that .
Therefore Φ and are closed under compositions. □
To show the classes are closed under countable composition, we use the following lemma.
Lemma 3.2 Let be a sequence in a partial metric space such that
then and , for each n, where ϕ is a lower semi-continuous function.
Proof We will prove that and
We note that
and we see that
It follows that
and then the sequence of value is decreasing and bounded below, which implies that is a convergent sequence in ℝ, that is, a Cauchy sequence. Thus, for , there exists such that for all , we have
By the triangle inequality, we have
It follows that . Thus is a Cauchy sequence in X, and the completeness of the space implies that converges and so there exists such that . Moreover, we have
This last inequality obtained by ϕ is lower semi-continuous.
Therefore . □
The remainder of the proof of Lemma 3.1 amounts to the observation that for each , the sequence satisfies the conditions of Lemma 3.2. Before starting the main theorem of this paper, we also need the following.
Definition 3.3 Let be a partial metric space.
For , define the diameter of A, written , by
Let ; note implies .
Let . For each , define .
Lemma 3.4 .
Proof Let , be in , then we have
Thus . □
Theorem 3.5 Let be closed under compositions. Let .
Let be closed under countable compositions. Then there exists an such that and for all .
Let the elements of be continuous functions. Then there exists a sequence of functions and such that for all .
Proof Let be a positive sequence converging to 0. Choose such that
Since is closed under compositions, we have and
Again, choose such that
Since is closed under compositions, we have and
Continuing this procedure, we obtain a sequence of such that
Next, to show that under hypothesis 1. Let and . Since is closed under compositions, then . Since , it implies that for each i. On the other hand, since , we have . Since , we obtain that . Thus . Finally, to show that under hypothesis 2. Let . First, since , for each i we have that , the closure of . Since , we have that .
To verify that , observe that for each i. Hence, for any , there exists such that , (here we need g to be continuous). Therefore, for ,
And so implies that
Therefore . The proof is completed. □
Remark In Theorem 3.5(2) one may choose , the set consisting of g and its finite iterates. For this choice of , one has as in the contraction theorem.
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The author is grateful to the referees for precise remarks allowing us to improve the presentation of the paper and would like to thank the faculty of Science, Naresuan University, Phitsanulok for the financial support.
The author declares that they have no competing interests.
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Klin-eam, C. Modified proof of Caristi’s fixed point theorem on partial metric spaces. J Inequal Appl 2013, 210 (2013). https://doi.org/10.1186/1029-242X-2013-210