Abstract

In this paper, we consider an auxiliary function to combine and unify several existing fixed point theorems in the setting of the complete partial -metric space. We consider also some examples to support the observed main results.

1. Introduction and Preliminaries

The notion of the distance has been investigated and improved from the beginning of the mathematics sciences. The first formal definition was given by Hausdorff and Frechet under the name of metric spaces. The formal definition was extended, improved, and generalized in several ways. In this paper, we shall consider the combination of notions of partial metric space and -metric space. Partial metric space, defined by Matthews [1, 2] is the most economical way to calculate the distance in computer science. So, it is important in the setting of theoretical computer science. On the other hand, -metric is the most interesting and real generalization of metric spaces; in this case, the triangle inequality is replaced by a modified version of triangle inequality.For more details on the advances of fixed point theory in the setting of b-metric spaces, see e.g. [13]-[27].

In this paper, we shall propose a fixed point theorem by using an auxiliary function to combine, generalize, and unify several fixed point results in the setting of the complete partial -metric spaces.

In [3], the authors proposed a new fixed point theorem in the setting of metric spaces.

We consider the follow sets of functions: (1) be the set of the functions that satisfy the following conditions:

(f₁) is continuous,

(f₂) ,

(f₃) , for all

In [3], some examples of such a function were given. (i)(ii)(iii)(2) be the set of functions that satisfy the following conditions:

(b₁) is nondecreasing,

(b₂) for each . (Here, by , we denote the th iterate oh.)

We mention that the functions are called -comparison functions. Moreover, it is not difficult to check that for every (3)

Theorem 1 (see [3]). Let be a complete metric space, a lower semicontinuous function , and a self-mapping . If there exist and such that for every , then has a unique fixed point.

Let be a nonempty set. (i)A function is a -metric on if for a given real number and for all the following conditions hold:

The triplet is called a -metric space. (ii)A function is a partial metric on if for all the following conditions hold:

The pair is said to be a partial metric space.

Combining these two concepts, Shukla [4] introduced the notion of partial -metric space as follows. (iii)A function is a partial -metric on if for all the following conditions hold:

The triplet is said to be a partial -metric space.

On a partial -metric space a sequence is said to be (i)convergent to if (the limit of a convergent sequence is not necessarily unique)(ii)Cauchy if exists and its finite

Moreover, the partial -metric space is complete if for every Cauchy sequence there exists such that

Let be a partial -metric space. We say that a self-mapping on is continuous if for every sequence in which converges to a point we have

In [5], the authors introduced the following new notions. (i)On a partial -metric space, a sequence is a -Cauchy sequence if (ii)The space is said to be -complete if for each 0-Cauchy sequence in , there exists a point such that

Moreover, they proved that if the partial -metric space is complete, then it is -complete.

For a better understanding of the connections between these spaces (partial metric space, -metric space, and partial -metric space), we mention some papers that can be consulted [6–12].

Let be the set of functions that satisfy the following conditions:

is nondecreasing,

for each . (Here, by , we denote the th iterate oh .)

2. Main Results

The following is the main result of the paper.

Theorem 2. Letbe a 0-complete partial-metric space, a function, , and a self-mapping. If there existssuch thatfor every. Ifis continuous oris continuous, thenhas a unique fixed point.

Proof. Starting with a point , we consider the sequence defined by , . Without losing the generality, we can assume that for any , we have . Indeed, on the contrary, if there exists a positive integer such that , we get that is a fixed point of , because due to the way the sequence was defined, it follows that . Moreover, using this remark, we can easily see that Again supposing that for some from , we have which is a contradiction. Taking and in (8) we get There are two possibilities, namely, which leads us (since for any ) to But, this is a contradiction, and then Therefore, by (11) and taking into account , we have Consequently, for every , we obtain Let such that . By applying the (triangle-type inequality) , we have and (17) leads us to where . Keeping in mind , we deduce that there exists as , and from (18), we get Consequently, is a 0-Cauchy sequence in a 0-complete partial -metric space, and then there exists such that Moreover, by together with (16), we have and using Plus, by and (20), We claim that this point is in fact a fixed point of the mapping . If the mapping is continuous, then by (6), we have Thus, applying the triangle inequality , and together with (20) and (24), letting , we get that is, is a fixed point of .
Let assume now that is continuous, that is, Replacing by and by in (8), we have (for every ) Letting and taking into account , we have Consequently, But, taking into account, we get which means Thus,
As a last step, we claim that is the unique fixed point of . Supposing on the contrary, that there exists another point such that . First of all, applying (8) with , we have which implies that . Let now and in (8). We have This is a contradiction. Therefore, , that is, admits a unique fixed point.