Abstract

This paper is introduced as a survey of result on some generalization of Banach’s fixed point and their approximations to the fixed point and error bounds, and it contains some new fixed point theorems and applications on dualistic partial metric spaces.

1. Introduction

The partial metric spaces were introduced in [1] as a part of the study of denotational semantics of dataflow networks. He established the precise relationship between partial metric spaces and the weightable quasi-metric spaces, and proved a partial metric generalization of Banach contraction mapping theorem.

A partial metric [1] on a set is a function such that for all (1); (2); (3); (4).

A partial metric space is a pair , where is a partial metric on . If is a partial metric on , then the function given by is a (usual) metric on .

Each partial metric on induces a topology on which has as a basis the family of open -balls , where for all and . Similarly, closed -ball is defined as .

A sequence in a partial metric space is called a Cauchy sequence if there exists (and is finite) [1].

Note that is a Cauchy sequence in if and only if it is a Cauchy sequence in the metric space [1].

A partial metric space is said to be complete if every Cauchy sequence in converges, with respect to to a point such that [1].

A mapping is said to be continuous at , if for , there exists such that [2].

Definition 1 (see [1]). An open ball for a partial metric is a set of the form for each and .

In [3], O’Neill proposed one significant change to Matthews definition of the partial metrics, and that was to extend their range from to . In the following, partial metrics in the O’Neill sense will be called dualistic partial metrics and a pair such that is a nonempty set and is a dualistic partial metric on will be called a dualistic partial metric space.

A dualistic partial metric on a set is a partial metric . A dualistic partial metric space is a pair , where is a dualistic partial metric on .

A quasi-metric on a set is a nonnegative real-valued function on such that for all , , (i),(ii).

Lemma 2 (see [1]). If is a dualistic partial metric space, then the function defined by , is a quasi-metric on such that .

Lemma 3 (see [1]). A dualistic partial metric space is complete if and only if the metric space is complete. Furthermore if and only if .

Before stating our main results, we establish some (essentially known) correspondences between dualistic partial metrics and quasi-metric spaces. Also refer to definition of -Fixed point and the existence of -Fixed point for . Our basic references for quasi-metric spaces are [4, 5] and for -Fixed point is [6].

If is a quasi-metric on , then the function defined on by , is a metric on .

Definition 4 (see [6]). Let be a dualistic partial metric space and be a map. Then is -fixed point for if We say has the -fixed point property if for every , where

Theorem 5 (see [6]). Let be a dualistic partial metric space and be a map, and . If , as for some , then has an -fixed point.

Definition 6 (see [7]). Let , be continues map such that , and . We define diameter by

Theorem 7 (see [1]). The partial metric contraction mapping theorem. Let be a complete partial metric space and be a map such that for all then has a unique fixed point , and as for each .

2. Some Result Fixed Point on Partial Metric

In this section, we give some result on fixed point and -fixed point in dualistic partial metric space and its diameter.

Definition 8. An open ball for a dualistic partial metric is a set of the form for each and .

Definition 9. Let be a mapping of a complete dualistic partial metric into itself , then is called a partial metric contraction mapping if there exists a constant , , such that for all .

Theorem 10. Every contraction mapping defined on a complete dualistic partial metric into itself has a unique fixed point . Moreover, if is any point in and the sequence is defined by then and

Proof. Existence of a fixed point. Let be an arbitrary point in , and we defined by , , . Then,
If , say , .
Then
Continuing this process times, we have for , and all .
However,
Therefore, we see that
Hence,
As , , from (12), we see that ; that is, is a Cauchy sequence in the metric space . Hence, must be convergent, say .
Since is continuous, we have or . Thus, is a fixed point of .
Uniqueness of the fixed point .
Let be another fixed point of . Then . We also have . But which implies that where . This is possible only when , that is, . This proves that the fixed point of is unique.

Corollary 11. Let be a complete dualistic partial metric space and . Let be a partial metric contraction mapping If , then has a fixed point.

Proof. Choose so that . We show that maps the closed ball into itself; for if , then Since is complete and satisfy in (5) thus by Theorem 10, has a fixed point.

Theorem 12. If is a complete dualistic partial metric space, and is such that is contraction for some integer , then has an unique fixed point.

Proof. Since , where is a positive integer, is a contraction mapping by Theorem 10, there exists an unique fixed point of , that is, . We want to show that ia a fixed point of , that is, . Let ; therefore, . This implies that and so say as . Thus, and we have .

Theorem 13. Let is a complete dualistic partial metric space, and let and be two maps contraction. If for every , , chosen suitably. Then for every ,

Proof. The relation is true for . We use the principle of induction in order to prove this relation. Let it be true for all . Then Thus, the relation is true for .

Corollary 14. Let is a complete dualistic partial metric space and let be a partial contractive map on . Moreover, the iteration sequence , , with arbitrary converges to the unique fixed point of . Error estimate is the following estimate (prior estimate): and the posterior estimate

Proof. The First statement is obvious by Thus
Now, inequality (20) follows from (23) by letting . We have
Now, for inequality (21) taking and writing for and for , we have from (23) Setting , we have and obtain (21).

Corollary 15. Let be a complete dualistic partial metric space and let be a contraction on a closed ball . Moreover, assume that . Then, prior error estimate is the following estimate: and the posterior estimate

Proof. By Corollary 11 the iteration sequence as (5) is converges to the unique fixed point of ; hence, by Corollary 14, we have since and , we have Therefore . Also, by Corollary 14 we have since and , we have Therefore .

3. Applications of Banach Contraction Principle on Complete Dualistic Partial Metric Space

In this section, we apply Theorem 10 to prove existence of the solutions a system of linear algebraic equations with unknowns, and we show that applied of Corollaries 14 and 15 in numerical analysis.

3.1. Application 3.1

Suppose we want to find the solution of a system of linear algebraic equations with unknowns, then

This system can be written as By assuming , where Equation (33) can be written in the following equivalent form:

If then (35) can be written in the form , where is defined by

Finding solutions of the system described by (32) or (35) is thus equivalent to finding the fixed point of the operator equation, (36). In order to find a unique solution of , that is, a unique solution of (32), we apply Theorem 10. In fact, we prove the following result. Equation (32) has a unique solution, if For and , we have where If , then . Therefore, Since , we have , , that is, is a contraction mapping of the complete dualistic partial metric space into itself. Hence, by Theorem 10, there exists a unique fixed point of in , that is, is a unique solution of (32).