Abstract
In this paper, we firstly propose the notion of double controlled partial metric type spaces, which is a generalization of controlled metric type spaces, partial metric spaces, and double controlled metric type spaces. Secondly, our aim is to study the existence of fixed points for Kannan type contractions in the context of double controlled partial metric type spaces. The proposed results enrich, theorize, and sharpen a multitude of pioneer results in the context of metric fixed point theory. Additionally, we provide numerical examples to illustrate the essence of our obtained theoretical results.
1. Introduction and Preliminaries
The study of fixed points of given mappings satisfying certain contractive conditions in various abstract spaces has been at the middle of vigorous research activity. Banach contraction mapping principle has attracted the eye of the many authors to generalize, extend, and improve the metric fixed point theory. For this purpose, the authors considered the extension of metric fixed point theory to different abstract spaces such as symmetric spaces, quasimetric spaces, fuzzy metric spaces, partial metric spaces, probabilistic metric spaces, and spaces with graph.
The notion of b-metric spaces was first presented by Bakhtin [1] and Czerwik [2]. Many writers have since obtained a number of fixed point solutions in b-metric spaces for single and multivalued operators. We reference Kamran et al. [3] (see also [4, 5]), who presented extended b-metric spaces by manipulating the triangle inequality rather than utilizing control functions, as one of the generalizations concerning b-metric spaces. Following that, in 2018, Abdeljawad et al. [6, 7] established the concepts of controlled metric type spaces and double controlled metric type spaces, respectively. Souayah and Mrad [8] proposed a more broad idea of controlled partial metric type spaces in 2019. It is useful to establish the extensions of the contraction principle from metric spaces to b-metric spaces, and therefore the controlled metric type of spaces is useful to prove the existence and uniqueness of theorems for many forms of integral and differential equations. Some interesting applications can be found in the recent papers [4, 9–15]. It is always interesting to find novel applications dealing with engineering science and technology using fixed point technique.
On the other hand, the notion of partial metric space was given by Matthews [16, 17] in 1992, which is the generalization of the usual metric space in which d (x, x) is not zero. After that, many researchers worked on the partial metric type spaces to discover the existence of fixed point and their uniqueness. In 2019, Gu and Shatanawi [18] expounded some coupled fixed point theorems in the context of partial metric spaces for hybrid pairs of mappings satisfying a symmetric type contraction. In 2020, Nguyen and Tram [19] demonstrated various fixed point results involving involution mappings. Recently, in 2021, Javaid et al. [20] propounded fixed point results in the setting orthogonal partial metric spaces with application. Researchers can refer to [14, 21–23] for further information on fixed points in partial type metric spaces.
Taking into consideration the facts mentioned above, in this article, we introduce the concept of double controlled partial metric type space, which is an extension of the controlled metric type spaces, double controlled metric type spaces, and controlled partial metric type spaces. We also look into the existence and uniqueness of fixed point results, which are Kannan contractions’ extensions.
Let us begin by reviewing the definition of double controlled metric space as follows.
Definition 1 (see [6]). Let be a nonempty set and consider the functions .
Let satisfy(1),(2),(3), for all , then is called a double controlled metric type space.
2. Double Controlled Partial Metric Type Spaces
The following is the formal definition of the double controlled partial metric type space which generalizes the notation of controlled metric type spaces, double controlled metric type spaces, and partial metric spaces.
Definition 2. Let be a nonempty set consider be a function.
Let satisfy(1),(2),(3),(4), for all x1, x2, x3 ∈ X, then (X, d) is called a double controlled partial metric type space.Note that double controlled partial metric type space is more extensive than the double controlled metric type space.
Example 1. A double controlled partial metric type space is not necessarily a double controlled metric type space.
Let and take . Consider , whereLet the metric be defined by the following (Table 1).
It is easy to verify that and are true.
We prove condition (3) with different cases, that is, , for all and . Case (i): let , , satisfied for all and . Case (ii): let , , satisfied for all and Case (iii): let , , satisfied for all l and . Case (iv): let , , satisfied for all and . Case (v): let , , satisfied for all and .Now, we will prove the property . Case (i): to satisfy , we have Case (ii): now, we have to satisfy : Case (iii): to prove , we have Case (iv): in order to show , we proceed as follows: Case (v): now, we have to satisfy : Case (vi): for the case , we have Case (vii): to satisfy , we have Case (viii): now, we have to satisfy : Case (ix): for the case , consider the following: Case (x): for the case , we have Case (xi): to satisfy we proceed as follows: Case (xii): next, we have to satisfy : Case(xiii): now, for the case , we consider Case (xiv): now, we have to satisfy : Case (xv): lastly, for the case , we haveTherefore, is a double controlled partial metric type space but is not a double controlled metric type space since is not equal to zero all the time.
We define Cauchy and convergent sequence in double controlled partial metric type spaces as follows.
Definition 3. Let be a double controlled partial metric type space; the sequence converges to some in , if ; in this case, we write .
Definition 4. The sequence in a double controlled partial metric type space is said to be Cauchy sequence, if exists and is finite.
Definition 5. A double controlled partial metric type space is said to be complete if every Cauchy sequence in converges to a point , that is, .
Definition 6. Let be a double controlled partial metric type space. Let and .(i)The open ball is(ii)The mapping is said to be continuous at if for all , there exists such thatTherefore, if is continuous at x in the double controlled partial metric type space , then implies that as
3. Some Novel Results
This section is devoted to discuss some fixed point results in double controlled partial metric type space . The main result of this article is given by the following theorem.
Theorem 1. Let be a complete double controlled partial metric type space by the functions . Suppose that satisfiesfor all , where . For , take , assuming that
Furthermore, assume that for every , , , , and exist and are finite. Then, the sequence converges to some ; moreover, if and satisfy the following assumptions,then has a unique fixed point.
Proof. Consider , let be arbitrary, and let and let be chosen.
By using (19), we getThen,By repeating the same procedure in inequality (23), we obtainNow, we have to show that is Cauchy sequence. Since is a double controlled partial metric type space, for all natural numbers with , we acquireAssume thatThen, we obtainUsing ratio test, we haveTaking , (27) becomesThis implies that is a Cauchy sequence in a complete double controlled metric type space , so converges to some . Now, we have to prove that is a fixed point of , so we need to verify thatFrom the , we haveHence, for proving , it is sufficient to prove that and . The triangular inequality yields thatTaking limit as , we obtainUtilizing condition (21), we getOn the other hand,Hence, we getFrom (31)–(36), we obtainUniqueness: assume that there are two fixed points and of T, thenFurthermore, we haveThen, . Since , then . Therefore, , which gives and has a unique fixed point.
Definition 7. Let be complete double controlled partial type metric space; a mapping is sequentially convergent. For every sequence , if is convergent, then also converges. Also, is said to be subsequentially convergent. For every sequence , if is convergent, then has a convergent subsequence.
Theorem 2. Let be a complete double controlled partial metric type space and be mapping such that is continuous, one-to-one, and subsequentially convergent
For all , where . For , take , assuming that
Furthermore, assume that for every , , , , and exist and are finite. Then, has a unique fixed point.
Proof. Let be an arbitrary point in and consider the sequence defined in the hypothesis of the theorem. From (41), we obtainBy induction, we getwhereNow, we have to show that is a Cauchy sequence. Since is double controlled partial metric type space for all natural numbers with , we getAssume thatThen, we obtainUsing ratio test, we haveTaking inequality, (48) reduces toThis amounts to say that is a Cauchy sequence in a complete double controlled partial metric type space , hence there exists such thatSince is convergent, the sequence has a convergent subsequence denoted by such thatUsing the continuity of , we obtainFrom (51) and (53), we conclude that . Making use of triangular inequality, we getProceeding the , we obtainwhich proves that . From the triangular inequality, we haveSuppose that , thenOn the other hand,Note that if , then and we get . Thus, from (57), we obtainFrom (55) and (57), we deduce that . To check the property , i.e.,It is easy to see thatThus, , since f is one-to-one, . Therefore, is a fixed point of .
Uniqueness: let be two fixed points of , then and . From the condition , we haveOn the other hand, using the triangular inequality, we getSince , then and we get . Therefore, from (62) and (63), we obtain thatUtilizing the property of the double controlled partial metric type space, we obtain . Hence, is one-to-one so that . Finally, by replacing with , we conclude that converges to as . Thus, the sequence converges to the unique fixed point .
Corollary 1 (Banach contraction).. Let be a complete double controlled partial metric type space by the functions . Suppose that satisfiesfor all , where . For , take , assuming that
Furthermore, assume that for every , , , , and exist and are finite. Then, the sequence converges to some ; moreover, if and satisfy the following assumptions,then has a unique fixed point.
Remark 1. Results presented in this manuscript generalize, enrich, and theorize the prominent results due to Kannan [24] and Bojor [25] in the framework of double controlled partial metric type spaces.
Example 2. Let ; consider the function given as follows: (Table 2)
Given is defined asIt is easy to verify that given equipped with is double controlled partial metric type space but not double controlled metric type space because for all . Now, we define a mapping by the following:Choose and , then by using (19), we acquireSince , we choose ; taking and , it is clear that condition (20) is satisfied as follows:Since inequality (20) is satisfied for every , additionally, for each , we haveTherefore, all the hypotheses of Theorem 1 are contended and 1 is the unique fixed point of .
4. Conclusions
We launched a new concept of double controlled partial metric type spaces which expands the ideas of certain variants of metric spaces, viz., controlled metric type spaces, double controlled metric type spaces, and partial metric spaces. The introduced results sum up and broaden some previous writing, and some illustrative examples are investigated to show the potency of our work.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.