Abstract

In this paper, we introduce the concept of new type of -contractive type for quasipartial b-metric spaces and some definitions and lemmas. Also, we will prove a new fixed-point theorem in quasipartial -metric spaces for -contractive type mappings. In addition, we give an application which illustrates a situation when Banach’s fixed-point theorem for complete quasipartial -metric spaces cannot be applied, while the conditions of our theorem are satisfying.

1. Introduction

In 1922, Banach [1] introduced the most significant outcome in spaces identified with metric spaces. This principle has been used in proving fixed-point theorems in different spaces and has been used successfully in proving the existence of the solutions of some nonlinear differential equations, integral equations, nonlinear integral differential equations, etc. A great deal of extensions of Banach’s principle have been done, for the most part by generalizing the contraction operator and also by expanding the necessity of completeness or even both (see, for example, [2–4]). The concept of partial metric was introduced by Matthews [5] in 1994.

Furthermore, Matthews introduced the concept of metric spaces via putting self-distances which are not necessarily equal to zero. In the year 1994, Czerwik [6] introduced the notion of a b-metric space as another generalization of metric space. Karapinar et al. [7] defined the concept of quasipartial metric space. This notion is a unification of both quasi metric spaces and partial metric spaces under the same substructure. In the following, we present some results that are related to our paper and show the motivation of our concern.

Wardowski [8] in 2012 Introduced the definition of contraction called F-contraction. After that, in 2017, Górnicki [9] called F-contraction to be Wardowski contraction and also extended fixed-point theorems established by Ran and Reurings [10] and Nieto and Rodríguez-López [11] to CJM contractions on preordered metric spaces, where a preordered binary relation is weaker than a partial order. Recently, in 2019, Goswami et al. [12] defined F-contractive type mappings in b-metric space; they proved fixed-point theorems with suitable examples. After that, they had given F-expanding type mappings and a fixed-point result was obtained. In this paper, we introduce the concept of new type of -contractive type for quasipartial b-metric space and some definitions and lemmas. Also, we state and prove some fixed-point results in new concept and generalize a notion of -contractive type mappings in quasipartial b-metric spaces with application.

2. Preliminaries and Definitions

We begin the section with some basic definitions and concepts.

Definition 1. (see [7]). Let be a set. A function is said to be a quasipartial metric on a nonempty set , such that for all , the following conditions hold:A quasipartial metric space is a pair such that is a nonempty set and is a quasipartial metric on . Then, is a metric on .

Lemma 1. For a quasipartial metric on ,and then is a partial metric on .

In 2015, Gupta and Gautam [13] introduced the concept of quasipartial b-metric space, lemma, example, and some other definitions.

Definition 2. Let be a set. A function is said to be a quasipartial b-metric on a nonempty set , such that for all , the following conditions hold:A quasipartial b-metric space is a pair such that is a nonempty set and is a quasipartial b-metric space on . The number is called the coefficient of . Further, is a quasipartial b-metric space on the set . Then,is a metric on .

Lemma 2. Every is a . But the opposite does not have to be true.

Example 1. Let ; then, . Hence, is quasipartial b-metric space with .

Definition 3. Let be a quasipartial b-metric space which satisfies the following:(i)A sequence converges to iff(ii)A sequence is said to be a Cauchy sequence iff exist and are finite.(iii) is called complete if every Cauchy sequence converges with respect to to a point so that(iv)A mapping is called continuous at if, for every , there is , so that .

Remark 1. Let be a quasipartial b-metric space. Then, the following holds:(i)If .(ii)If and .

Definition 4. (see [8]). Let be a map satisfying the following conditions:  F is strictly increasing.  , for each sequence of positive numbers.  If there exists , then . Let be a metric space. A mapping is called a Wardowski -contraction if there exists such that for all .  After that, in 2015, Cosentino et al. [14] extended the notion of quasipartial b-metric space with the following condition added to these concepts.  Let be a real number. For each sequence of positive numbers,for all and some . Alsulami et al. in 2015, [15] defined a generalized -Suzuki type in a b-metric space as a mapping ; then, there exists with , such thatfor all , since and under condition with and satisfying conditions and .

Lemma 3 (see [16]). Let be a b-metric space and be a convergent sequence in with l. Therefore, , and we have

The present paper aims to establish a similar type of result for F-contractive type mappings in a quasipartial b-metric space. In our results, we consider that a quasipartial b-metric is not a continuous functional.

3. Main Convergent Results

Lemma 4. Let be a quasipartial b-metric space and be a convergent sequence in with l, for all ; we have

Proof. We will apply twice the relaxed triangle inequality ; then, for every , we obtainand then we haveTherefore, we take on the left-hand side inequality and on the right-hand side inequality, and we obtain equation (12); in the same way, we get equation (13).

Definition 5. Let be a quasipartial b-metric space, and a mapping is said to be an -contractive type mapping if there exists such that and implyAlso, if and , we have the same equations ((16) and (17)) for all .

Lemma 5. Let be a quasipartial b-metric space and let be the corresponding b-metric space. Then, the following statements are equivalent:(i) is complete.(ii) is complete.

Moreover,since , for all and .

Definition 6. Let be a quasipartial b-metric space and be a mapping; then, there exists with , for all , such thatHence, we haveSince and under conditionsince with , satisfies each conditions between .
Now, we establish a new type of result for -contractive type mappings in a quasipartial b-metric space by appealing to Lemma 4.

Theorem 1. Let be a complete quasipartial b-metric space and be an F-contractive type mapping. Then, satisfies a Picard operator.

Proof. We start our proof by supposing that is arbitrary (but fixed) and considering the sequence ; since , . Suppose that for all and denote , by ; then, we haveFrom (21), we can writeBy condition , we haveThen, by induction, we getBy taking a limit as , we haveTherefore, from condition , there exists such thatMultiplication of (25) with yieldsBy taking a limit as , we haveNow, we can show that is a Cauchy sequence. Since is a complete quasipartial b-metric space, there exists such thatBy applying Lemma 4, we haveOn the other hand, let and be natural numbers with ; then, using , we getOn generalization, we haveBy taking limit as in (33), since holding fixed, we haveBut, we knowThus,Similarly, we can show thatDue to (18) in Lemma 5 and (36), we getAlso, using (16), we have for all ,since. Then, by taking a limit as in last equation, we getThis impliesSince the sequence converges to both and , we have . Finally to show the uniqueness of the fixed point, let be another fixed point of with ; then, from (16),orThis is a contradiction. Hence, we proved the theorem.