Abstract

In this research article, by introducing a mapping defined on , with some axioms, we define two generalized contractions called -contractions and -contractions. We investigate their mutual relation and establish an existence theorem addressing -contractions with some applications.

1. Introduction

Frechet gave an abstraction to the notion of distance in Euclidean spaces by introducing metric spaces. Partial metrics (denoted by ) were introduced in [1] as a generalization of the notion of metric to allow nonzero self-distance for the purpose of modeling partial objects in reasoning about data flow networks. The self-distance is to be understood as a quantification of the extent to which is unknown. Matthews [1] proved an analogue of Banach’s fixed point theorem in partial metric spaces. This remarkable fixed point theorem led many researchers to investigate fixed points of self-mappings in partial metric spaces (see [2–7]).

The investigation of fixed points of multivalued or set-valued mappings was started by Nadler [8]. For this purpose, Nadler introduced a metric function to measure distance between two nonempty closed and bounded sets. This metric function is also known as the Hausdorff metric in literature. Aydi et al. [5] generalized the Hausdorff metric to the partial Hausdorff metric and hence generalized the Nadler fixed point theorem. Nazam et al. [7] established various fixed point results using the partial Hausdorff metric. Recently, Pathak et al. [9] introduced another metric function to measure the distance between two nonempty closed and bounded sets and hence proved some fixed point results. Nashine et al. [10] also proved some fixed points theorems on -multivalued contractions and their application to homotopy theory. For recent research in this direction, see [11–13].

In 1922, Banach introduced the Banach Contraction Principle in his PhD thesis. Since then, there has been a trend to generalize and apply it to show the existence of the solutions to various mathematical models (both linear and nonlinear). A large number of research articles contain many useful generalizations of Banach Contraction Principle. In one such attempt, Wardowski [14] introduced -contractions, where represents the class of nonlinear real-valued functions satisfying three axioms (). The concept of -contractions proved to be a useful addition in fixed point theory (see for instant [15–18] and references therein). The advancement in the study of -contraction is in progress, and in this direction recently, Abbas et al. [19] introduced the Presic-type -contraction and established a fixed point theorem for such kind of mappings. Tomar et al. [20] provided an existence theorem for six self-mappings under the notion of -contraction. Durmaz et al. [17] studied -contraction under the effect of a partial order. Sgroi et al. [21] extended the notion of -contraction to multivalued -contraction by combining the ideas of Wardowski and Nadler. Durmaz et al. [22] generalized the results given in [16, 17, 21] by introducing -contraction. Similarly, Piri et al. [23] proved some theorems on the -Suzuki type inequalities under some weaker conditions, and Shukla et al. [24] established a common fixed point theorem for weak -contraction under 0-complete partial metric spaces. Recently, Karapinar et al. [25] presented a survey paper which encompasses almost all the results addressing -contractions.

The motivation to write this article is the contents of the article [26]. In [26], authors introduced a function (called auxiliary function) defined on satisfying some axioms and used it to establish a fixed point theorem. It was then shown that the Banach fixed point theorem, Kannan fixed point theorem, Chatterjea fixed point theorem, Reich fixed point theorem, Hardy and Rogers fixed point theorem, and Círíc type fixed point theorems are particular cases of this fixed point theorem. Since all the above mentioned fixed point theorems have been generalized using the notion of -contraction both in metric spaces and partial metric spaces (see [25]), we develop a general fixed point theorem, representing them all, in the Hausdorff partial metric spaces.

This article is organized as follows. In Section 2, some basic notions are given. In Section 3, we give highlights of Hausdorff p-ms. In Section 4, we introduce the -contractions and -contractions and investigate the relations between them. We also study the existence theorem and its consequences. And in Section 5, we derive two results regarding applications by applying the existence theorem given in Section 4. The presented existence theorem generalizes, improves, and extends the results established by Pathak et al. [9].

2. Basic Notions

Let partial metric spaces be denoted by p-m-s.

Matthews [1], while working on networking topologies, noticed the nonzero self-distance (loop is the best example to understand his point). The self-distance played a key role in introduction of p-m-s. Matthews [1] defined the p-m-s as follows: let be a nonempty set, and the function is said to be a partial metric (p-m) on if for all , the axioms (p₁)-(p₄) are satisfied.

(p₁)

(p₂)

(p₃)

(p₄)

Some examples of are as follows. The function defined by (1) for all is a (2) is a (3), is a .

It is noted that implies . The p-m function is continuous. If is a p-m then the function defined by defines a metric on . A topology can be defined on with -open balls being its elements. The -open ball centered at having radius is defined by . A set is said to be bounded in if there exist and such that for all . Also it is easy to write (closure of ) and is closed in if and only if . If ; then we say that converges to and conversely. If is finite, then the sequence is said to be Cauchy, and in particular, if this Cauchy sequence converges in , then we say that the p-m-s is complete. Lemma 1 provides fundamental rules to work in the p-m-s.

Lemma 1 [1]. (1)If the sequence is Cauchy sequence in , then it is Cauchy sequence in the metric space and conversely(2)The completeness of implies the completeness of and conversely(3), provided is complete.

Remark 1. There are sequences which converge in p-m-s but not in metric spaces. Indeed, for the sequence in and p-m defined by , it is easy to check that the sequence converges to with respect to but does not converge to with respect to metric defined by if and otherwise.

3. Hausdorff Partial Metric

Let the set of nonempty closed and bounded subsets of be denoted by . Let , . Let be defined by . Let be defined by

Let be defined by

Since , for all . A comprehensive study of the distance with reference to metric was presented by Pathak et al. in [9]. We claim that (a) and are topological equivalent(b)the mapping defines a p-m on (c)if the p-m-s is complete then is also complete and vice versa(d)the mapping is continuous.

Proposition 1 [7]. Let be p-m-s. For any , we have the following: (1)(2)(3) (4)

Proposition 2. Let be p-m-s. For any , we have the following: (1) implies (2)(3)(4)

Proof. Following the arguments given in ([5], Proposition 2.2 and Proposition 2.3), we get the result. We omit its details.
-contraction: Let be a p-m-s, the mapping is called an -contraction, if there exists such that for all (see [7]).
-contraction: Let be a p-m-s; the mapping is called an -contraction, if (1) there exists such that for all ; (2) for all , , there exists such that (see [27]).
Since for all , -contraction implies -contraction but not conversely (see Example 1).

Example 1. Let. Define the functionby

Then is a p-m-s. Let be defined by

We have three cases (Case 1: , Case 2: , and Case 3: ).

Case 1. If , then , , and . This clearly shows that whereas

Case 2. If , then , , and . This clearly shows that whereas

Case 3. If , then , , and . This clearly shows that whereas

Note: the inequality also holds for each case, and for all , , .

4. Fixed Points of -Contraction

Let be a self-mapping defined on nonempty set . The problem “to find such that is called fixed point problem. If , then the fixed point problem turns into the form “to find such that .” For the solution of fixed point problem, generally, a Picard iterative sequence is proved to be a Cauchy sequence subject to contractive condition and completeness of the underlying abstract metric space leads to such . In this section, at first, we introduce and compare -contraction and -contraction, and secondly, we obtain a theorem assuring unique fixed point of -contraction. We proceed with definitions of functions and associated with some axioms.

Wardowski [14] considered a nonlinear function with the following axioms: (): is strictly increasing. (): For each sequence of positive numbers, if and only if (): For each sequence of positive numbers , there exists such that . Let .

The collection is nonempty: , , , and are members of this collection.

Let us consider the function satisfying the following axioms:

(C₁) is continuous and non-decreasing in each coordinate

(C₂) if there exist such that then

(C₃) if there exists such that then .

Let . The following examples show that the set is nonempty: (1)(2)(3)(4)(5)(6)(7)(8)(9)

Definition 1. Let and be two functions. A mapping is said to be strictly -admissible if for each and with , there exists such that .

Definition 2. Let be a p-m-s and let be a function. The space is said to be strictly -regular if for any sequence such that for all and as, we have for all

Definition 3. Let be a p-m-s. A mapping is said to be a -contraction if there exist and such that for all .

Let

Definition 4. Let be a p-m-s. A mapping is said to be an -contraction if (a)there exist , and such thatfor all . (b)For every , , and , there exists such that