Nonunique Fixed Point Results via Kannan <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.06580067pt" id="M1" height="11.4652pt" version="1.1" viewBox="-0.0657574 -11.3994 10.5867 11.4652" width="10.5867pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-71" d="M584 650H137L131 622C214 614 217 612 200 521L125 127C109 41 101 35 23 28L17 0H288L294 28C201 35 193 42 209 128L242 309H348C440 309 442 300 443 226H471L510 422H482C452 354 449 348 357 348H251L295 575C302 609 304 615 338 615H426C502 615 517 604 526 581C534 560 536 524 537 492L565 494C574 554 583 631 584 650Z"/></g></svg>-</span>Contraction on Quasi-Partial <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M2" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.20973 12.533" width="8.20973pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g></svg>-</span>Metric Space

Gautam, Pragati; Kumar, Santosh; Verma, Swapnil; Gupta, Gauri

doi:https://doi.org/10.1155/2021/2163108

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Unique and Non-Unique Fixed Points and their Applications

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Volume 2021 | Article ID 2163108 | https://doi.org/10.1155/2021/2163108

Nonunique Fixed Point Results via Kannan -Contraction on Quasi-Partial -Metric Space

Pragati Gautam,¹Santosh Kumar,²Swapnil Verma,¹and Gauri Gupta¹

Academic Editor: Giovanni Di Fratta

Received25 Jun 2021

Revised17 Jul 2021

Accepted01 Oct 2021

Published02 Nov 2021

Abstract

This paper is aimed at acquainting with a new Kannan -expanding type mapping by the approach of Wardowski in the complete metric space. We establish some fixed point results for Kannan -expanding type mapping and -contractive type mappings which satisfy -contraction conditions. Additionally, some new results are given which generalize several results present in the literature. Moreover, some applications and examples are provided to show the practicality of our results.

1. Introduction and Preliminaries

In 1922, Banach [1] commenced one of the most essential and notable results called the Banach contraction principle, i.e., let be a self-mapping on a nonempty set and be a complete metric, if there exists a constant such that for all . Then, it has a unique fixed point in . Due to its significance, in 1968, Kannan [2] introduced a different intuition of the Banach contraction principle which removes the condition of continuity, i.e., for all , there exists a constant such that

On the other hand, the notion of metric space has been generalized in several directions, and the abovementioned contraction principle has been enhanced in the new settings by considering the concept of convergence of functions. In 1989, Bakhtin [3] introduced the notion of -metric space which was revaluated by Czerwik [4] in 1993.

Definition 1. A -metric space on a nonempty set is a function such that for all and for some real number , it satisfies the following:
(M1) If , then
(M2)
(M3)

Then, the pair is called the -metric space. Motivated by this, many researchers [5–8] generalized the concept of metric spaces and established on the existence of fixed points in the setting of -metric space keeping in mind that, unlike standard metric, -metric is not necessarily continuous due to the modified triangle inequality. In general, a -metric does not induce a topology on .

Partial metric space is one of the attempts to generalize the notion of the metric space. In 1994, Matthews [9] introduced the notion of a partial metric space in which are no longer necessarily zero.

Definition 2. A partial metric on a nonempty set is a function such that for all , it satisfies the following:
(PM1) If , then
(PM2)
(PM3)
(PM4)

Then, the pair is called the partial metric space.

Definition 3. Let be a partial metric space. Then, several topological concepts for partial metric space can be easily defined as follows(1)A sequence in the partial metric space converges to the limit if (2)It is said to be a Cauchy sequence if exists and is finite(3)A partial metric space is called complete if every Cauchy sequence in converges with respect to , to a point such that

For more details, see, for example, [10–12], and the related references therein. The following definition gives room for the lack of symmetry in the spaces under study. In 2013, Karapinar et al. [13] introduced quasi-partial metric space that satisfies the same axioms as metric spaces.

Definition 4. A quasi-partial metric on a nonempty set is a function that satisfies the following:
(QPM1) If , then
(QPM2)
(QPM3)
(QPM4) for all . Then, the pair is called quasi-partial metric space.

Later on, Gupta and Gautam [14, 15] introduced quasi-partial -metric space.

Definition 5. A quasi-partial -metric on a nonempty set is a function such that for some real number , it satisfies the following:
(QPb₁) If then (indistancy implies equality)
(QPb₂) (small self-distances)
(QPb₃) (small self-distances)
(QPb₄) (triangularity)
for all The infimum over all reals satisfying is called the coefficient of and represented by .

Lemma 6. Let be a quasi-partial -metric space. Then, the following hold(i)If , then (ii)If , then and

Definition 7. Let be a quasi-partial -metric. Then, (i)a sequence converges to if and only if(ii)a sequence is called a Cauchy sequence if and only if(iii)the quasi-partial -metric space is said to be complete if every Cauchy sequence converges with respect to to a point such that(iv)a mapping is said to be continuous at , if for every , there exists such that

The extensive application of the Banach contraction principle has motivated many researchers to study the possibility of its generalization. A great number of generalizations of this famous result have appeared in the literature. In 2012, Wardowski [16] established a new notion of -contraction and proved the fixed point theorem which generalized the Banach contraction principle.

Definition 8 (see [16]). Let be a metric space, and there exists a mapping which satisfies the following condition
(F₁) is strictly increasing
(F₂) For any sequence , if and only if
(F₃) for some
Then, a mapping is said to be Wardowski -contraction if implies for all

Theorem 9 (see [16]). Let be a complete metric space and an -contraction. Then, has a unique fixed point , and for every , the sequence converges to .

In 2012, Samet et al. [17] established the class of -admissible mappings as follows

Definition 10 (see [17]). Let be given mapping where is a nonempty set. A self-mapping is called -admissible if for all , we have

Motivated by this, Aydi et al. [18] extended the notion of -contraction and prove the following result.

Theorem 11 (see [18]). Let be a metric space. A self-mapping is said to be a modified -contraction via -admissible mappings. Suppose that (i) is -admissible(ii)there exists such that (iii) is continuous

Then, has a fixed point. In 2015, Kumam et al. [19] generalized the contraction condition by adding four new values and introduced -Suzuki contraction mappings in complete metric space. The Suzuki-type generalization can be said to have many applications, as in computer science, game theory, biosciences, and in other areas of mathematical sciences such as in dynamic programming, integral equations, and data dependence. Recently, Wardowski [20] proposed the replacement of the positive constant in equation (6) by a function and relaxed the conditions on .

Definition 12 (see [20]). Let be a metric space, and satisfy the following(1) is strictly increasing, i.e., implies for all (2)(3) for all A mapping is called an -contraction on if for all for which .

Consider a function by . Note that with , the -contraction reduces to a Banach contraction. Therefore, the Banach contractions are a particular case of -contractions. Meanwhile, there exist -contractions which are not Banach contractions.

The concept of an -contraction has been generalized in many directions (see, e.g., [21–24]), and as an extension, engaging work was done by many authors [25–34], which enhanced this field. In 2015, Cosentino et al. [35] extended the concept of -contraction in metric space to -contraction in -metric space by introducing the following condition with continuation of Definition 7.

For some and any sequence , we have for all

In 2017, Gonicki [36] established -expanding type mappings.

Definition 13. Let be a metric space. A mapping is called -expanding if for all and , we have

The concept of -expanding type mappings was redefined as Kannan -expanding type mappings by Goswami et al. [37].

Definition 14 (see [37]). A mapping is said to be Kannan -expanding type mapping if there exists such that implies and implies for all . Following this direction, we have established a new type of mapping, i.e., Kannan -expanding type mapping, and proved some fixed point results for -contractive type mappings as well as Kannan -expanding type mappings in the setting of quasi-partial -metric space without using the continuity of mapping. Also, we attain the nonunique fixed point in quasi-partial -metric space which lacks symmetry property.

The main motive behind this study is that today, this field of research has vast literature. The significance of the Kannan type mapping is that it characterizes completeness which the Banach contraction does not; also, it does not require continuous mapping. In this paper, some examples and applications for the solution of a certain integral equation and the existence of a bounded solution of the functional equation are also given to represent the practicality of the results obtained. The application shows the role of fixed point theorems in dynamic programming, which is used in computer programming and optimization.

The future aspect of this study is to prove the existence of a unique fixed point in Kannan -expanding type mapping. Another field of research can be the existence of a common fixed point for the same. The notion of interpolative -contraction as well as interpolation for Kannan -expanding type mapping can also be future studies concerning the present manuscript.

2. Fixed Point for -Contractive Type Mappings

In this section, the existence of a fixed point for -contractive type mappings in a quasi-partial -metric space is obtained.

Definition 15. For a quasi-partial -metric space , a mapping is said to be an -contractive type mapping if there exists such that, if , then and if , then for all and .

Definition 16. Let be a quasi-partial -metric space. A self-mapping on is called an -contraction if there exist such that for all with .

Example 17. Let be given by . Here, satisfies (F1)-(F3) for any . Each mapping satisfying Definition 16 is an -contraction such that for all .

It is clear that for such that , the previous inequality also holds, and hence, is a contraction as shown in Figure 1.

Example 18. Consider a function where satisfies (F1)-(F3) for any . In this case, a mapping satisfies for all . Hence, is a contraction as shown in Figure 2.

Theorem 19. Let be a quasi-partial -metric space and be an -contractive type mapping. Then, has a unique fixed point , and for every , a sequence is convergent to .

Proof. Let be an arbitrary and fixed point in , and we assume a sequence such that . To prove has a fixed point, we need to show that if , then for all . Suppose that for every , then , and using equation (6), we have which implies Using , we get Also, using , there exists such that Let us denote by . From inequality (18), the following holds which implies Also, if there exists such that , we have To prove is a Cauchy sequence, let us consider such that . From the definition of quasi-partial -metric space and equation (24), we have Using the convergence of series, we get that is a Cauchy sequence. Since is complete, there exists such that , and the continuity of implies Hence, has a unique fixed point.☐

Theorem 20. For a quasi-partial -metric space , we say is complete if for every closed subset of , is an -contractive type mapping having a fixed point.

Proof. Suppose that there does not exist any Cauchy sequence in which has a convergent subsequence and we have a sequence for all where for . Also, we consider a subsequence such that for any with and for all . Then, is a closed subset of . Define by for all , which implies has no fixed point. Now, By definition, which implies for some . Hence, it proves that is an -contractive type mapping on a closed subset of which has no fixed point. Thus, this is a contradiction and is complete.☐

Theorem 21. Let be a quasi-partial -metric space and be a closed -contraction. Then, has a fixed point.

Proof. Let be an arbitrary point of , and we have . If , then is a fixed point of , and hence, the proof is completed. Now, assume that . Since is a -contraction, there exists such that where and . Also, there exists such that and . With the recurrence of the same process, we get for all . It implies Assume that for all . By equation (37), we have for all . Letting , property implies Let such that By equation (38), the following holds for all . Letting , we get This implies and is convergent. Hence, is Cauchy sequence. Since is complete, there exists such that as . Since is closed, , we get , and hence, is the fixed point of .☐

Corollary 22. Let be a quasi-partial -metric space and be an upper semicontinuous -contraction. Then, has a fixed point.

Example 23. Consider the quasi-partial -metric space where and is given by which is also shown in Figure 3, and is defined by Now, we show that satisfies Definition 16, where and for each . Let for all with , we have . Here, iff and . If it is true, then This implies Hence, for all and . Then, by Theorem 21, has a fixed point.

3. Fixed Point for Kannan -Expanding Type Mapping

In this section, we prove the fixed point results for Kannan -expanding type mappings in a quasi-partial -metric space.

Definition 24. Let us consider a mapping ; it is said to be Kannan -expanding type mapping if there exists such that implies and implies for all .

Lemma 25. Let be a quasi-partial -metric space and be surjective. Then, there exists a mapping such that is the identity map on .

Proof. For any point , let be any point such that . Let for all . Then, for all .☐

Theorem 26. Let be a quasi-partial -metric space and be surjective and a Kannan -expanding type mapping. Then, has a unique fixed point .

Proof. Assume that there exists a mapping such that is the identity map on . Let be arbitrary points of such that and which also implies that . Applying equation (48) on , we have for and for . Since and , we get for and for , which implies is Kannan -contractive type mapping. Also, we know that has a unique fixed point , and for every , the sequence converges to . In particular, is also a fixed point of since implies that Finally, if is another fixed point, then from equation (49), which is not possible, and hence, has a unique fixed point.☐

4. Applications of -Contraction

In this section, we discuss the applications of the results obtained to prove the existence of the solution of an integral equation and a functional equation.

4.1. Existence of Solution of Integral Equation

Now, we study the existence of solution of the following Volterra type integral equation

where . Let denote space of all continuous functions on , and for an arbitrary , we define where is taken arbitrary. Clearly, is endowed with quasi-partial -metric defined by for all is a Banach space and for all .

Theorem 27. Let us consider that for the integral equation (56), the following conditions are satisfied: (i) and are continuous where and (ii) is increasing(iii) for some (iv)There exists such thatfor all and . Then, integral equation (56) has a solution.

Proof. Here, is a quasi-partial -metric space, where is the quasi-partial -metric given by equation (58). Let us define by From (iv) we have, It implies or Taking logarithm in both sides, we get which on solving reduces to Now, we observe that the function defined by for each is -contraction. Clearly, from (iii), we have and hence, Theorem 19 applies to , which has a fixed point . Hence, is a solution of integral equation (56).☐

4.2. Existence of Bounded Solutions of Functional Equations

Fixed point theory is widely used in the field of dynamic programming which is the most commonly used tool for mathematical optimization. With this approach, the problem of the dynamic programming process reduces to solving the functional equations.

Let us consider that and are Banach spaces, is a state space, i.e., the set of the initial state of process, and is a decision space, i.e., the set of possible actions that are allowed for the process.

Here, we will prove the existence of the bounded solution of the following functional equation: where . Let denote the set of all bounded real valued functions and for an arbitrary , define . Clearly, endowed with quasi-partial -metric defined by for all is a Banach space. Thus, if we consider a Cauchy sequence in , then converges uniformly to a function, let that is bounded and so . Also, we have defined by

for all and . Hence, is well defined if and are bounded.

Theorem 28. Let be an upper semicontinuous operator defined by (70), and we assume that the following conditions are satisfied: (i) and are bounded and continuous where and (ii)There exists such thatfor all . Then, the functional equation (68) has a bounded solution.

Proof. Clearly, is a quasi-partial -metric given by equation (69). Let be an arbitrary positive number, , , then there exist such that From equations (72) and (75), It implies, Similarly, from equations (73) and (74), From equations (77) and (78), we get i.e., Hence, we conclude that Now, we observe that the function defined by for each is -contractive function, and hence, operator is -contractive.
Since any upper semicontinuous -contractive function has a fixed point , it implies that there exists a bounded solution of functional equation (68).☐

5. Conclusion

In this manuscript, we established a new type of mappings that is Kannan -expanding mappings and obtained fixed point theorems for contractive mappings in the framework of quasi-partial -metric spaces. Moreover, we provided examples that demonstrate the usability of our results. As an application of our result, we also studied a system of integral and functional inclusions. It would be more engaging to work on the obtained results to prove the uniqueness of the fixed point in the future.

Data Availability

This clause is not applicable to this paper.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Copyright © 2021 Pragati Gautam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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