Kannan-Type Contractions on New Extended <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="M1" 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 Spaces

Aydi, Hassen; Aslam, Muhammad; Sagheer, Dur-e-Shehwar; Batul, Samina; Ali, Rashid; Ameer, Eskandar

doi:https://doi.org/10.1155/2021/7613684

Journal of Function Spaces

On this page

Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Unique and Non-Unique Fixed Points and their Applications

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 7613684 | https://doi.org/10.1155/2021/7613684

Kannan-Type Contractions on New Extended -Metric Spaces

Hassen Aydi,^1,2,3Muhammad Aslam,⁴Dur-e-Shehwar Sagheer,⁵Samina Batul,⁵Rashid Ali,⁵and Eskandar Ameer⁶

Academic Editor: Liliana Guran

Received16 May 2021

Accepted20 Aug 2021

Published13 Sept 2021

Abstract

This article is focused on the generalization of some fixed point theorems with Kannan-type contractions in the setting of new extended -metric spaces. An idea of asymptotic regularity has been incorporated to achieve the new results. As an application, the existence of a solution of the Fredholm-type integral equation is presented.

1. Introduction and Preliminaries

The existence of fixed points for some operators has a noteworthy contribution in many branches of applied and pure mathematics. The theory of a fixed point provides very valuable and effective tools in mathematics. It has a wide range of implications in nonlinear analysis and has been established in two directions. One is to change the space under consideration (see the works of Bakhtin [1], Jleli and Samet [2], Karapinar [3], Kamran et al. [4], etc.), and the other is to change the contraction conditions (see the works of Ćirić [5], Popescu [6], Rakotch [7], etc.).

In 1922, the Polish mathematician Banach [8] established a remarkable result relevant to a metric fixed point theory, that is known as the Banach contraction principle (BCP). The work of Banach is well regarded and an adaptable consequence in the theory of fixed points. BCP laid a foundation of research in this field, which is further investigated by many researchers from 1922 till now. One of the prominent generalizations of the BCP was presented by Kannan [9]. Additional works on the existence of (common) fixed points can be seen in [10–14].

In 1989, Bakhtin [1] introduced the notion of -metric spaces. Later on, the concept of -metric spaces was further used by Czerwik [15] to establish different fixed point results in -metric spaces. The study of -metric spaces endowed an imperative place in the fixed point theory with multiple aspects. Many mathematicians (Abdeljawad et al. [16, 17], Ali et al. [18], Akkouchi [19], Chifu and Karapinar [20], Kadelburg and Radenović [21], Parvaneh et al. [11], Gupta et al. [12], Mlaiki et al. [22], etc.) led the foundation to improve the fixed point theory in -metric spaces. Another innovative task has been achieved by Kamran et al. [4] in 2017 by introducing the notion of an extended -metric space, which generalizes the notion of a -metric space. Some fixed point results are proved in this new setting; see for instance the work presented in [23–25].

The work of Kannan [9] refined the concept of the Banach contraction mapping by introducing a new contraction, now known as Kannan contraction. The Kannan fixed point result has been further extended and generalized in the setup of -metric spaces [15] and for generalized metric spaces [26].

In 2019, the notion of a new extended -metric space has been initiated by Aydi et al. [27], where the control function depends on three variables. This fact is new since all precedent control functions depend on two variables. The objective of this work is at investigating Kannan-type contractions in the context of new extended - metric spaces by extending the main results of Gornicki [28]. For this purpose, some basic concepts are needed in the sequel.

Definition 1 (see [15]). Let be a nonempty set and be a real number. A function is called a -metric, if it satisfies the following for all : (b1)(b2), if and only if (b3)(b4)

The pair is called a -metric space. If , then, a -metric space becomes a metric space. In 2017, Kamran et al. [4] generalized the -metric space setting to an extended -metric space (in the same direction, see also [29, 30]).

Definition 2 (see [4]). Let be a nonempty set and be a function. The map is called an extended -metric if for all , it satisfies the following axioms: (i)(ii), if and only if (iii)(iv)

In 2019, Aydi et al. [27] introduced the notion of new extended -metric spaces. Here, the control function depends on 3 variables.

Definition 3 (see [27]). Let be a nonempty set and be a function. The map is called a new extended -metric if for all , it satisfies the following axioms: (i)(ii), if and only if (iii)(iv)

The pair is named to be a new extended -metric space. If (for ), we get Definition 1.

Example 4 (see [27]). Let . Define by where Here, is a new extended -metric space.

On the other hand, by taking , and , we have

It is not possible to find so that holds. Thus, is not a -metric on .

Example 5. Consider . Take as where as Then, is a new extended -metric space.

Proof. The first three conditions are trivially verified. To check the triangular inequality, we proceed as follows: so . Similarly, we can check the other two pairs.
Therefore, for all in ,

Definition 6. Let be a new extended -metric space. (i)A sequence is called convergent to if for , there is such that for all (ii)A sequence is called Cauchy if for , there is such that for all (iii)The space is called complete if every Cauchy sequence in is convergent in

Definition 7. Let be a new extended -metric space. Denote by and the open and closed balls in , respectively. (i)A subset of is called open if for any , there exists an such that (ii)A subset of is called closed if for any such that , then In this paper, we are going to prove some Kannan-type fixed point theorems in the setting of new extended -metric spaces. Some examples are also provided to make effective the obtained results.

2. Main Results

We define a Kannan-type fixed point contraction on new extended -metric spaces.

Definition 8. Let be a new extended -metric space. A mapping is a Kannan-type contraction if there are and such that

Our first main result is as follows:

Theorem 9. Consider a complete new extended -metric space such that is a continuous functional. Let be a mapping such that there are and so that Assume that for each , such that ,
Then, has only one fixed point . Further, for any , the iterative sequence converges to , and

Proof. For an arbitrary point , construct the iterative sequence If for some , , so is a fixed point of . Otherwise, assume that for all . Since one writes Then, That is, Continuing in this way, we have Let be such that . Applying triangular inequality, we get Since one writes Since in view of (8), the series is convergent for each by ratio test.
Let and So, for , the above inequality implies that That is, Hence, the sequence is a Cauchy sequence. By the completeness of , there is such that
We claim that is a fixed point of . We have That is, As , we have in view of the assumption that is continuous, which holds unless , and so,
The uniqueness is as follows:
Let be another fixed point of . We have That is, It is only possible if Thus, is the unique fixed point of . Further, we have Then, Also, Using (15), That is,

The following examples illustrate Theorem 9. We deal with noncompact sets.

Example 10. Let be the space of all bounded sequences of real numbers, that is, where may depend on the sequence but does not depend on Take that are in .
Then, is a complete new extended -metric space with being defined by Consider given as . For each , we have Thus, (7) holds with and . Also, and for all . Hence, (8) holds, and by Theorem 9, has a fixed point.

Example 11. Let be the set of all real-valued continuous functions defined on Define and as Then, is a complete new extended -metric space, consider a mapping given as For all , we have Thus, (7) holds with and . Also, and for all . That is, (8) holds. Since all the conditions of Theorem 9 are satisfied, has a fixed point.

Example 12. Choose Define and by Let be given as Then, for all with neither nor , we have If and , then Thus, (7) is satisfied for and for each Also, . If , then, for the iterative sequence for each , we have . If (say for some ); then, for the iterative sequence for each , we have . Hence, Theorem 9 ensures the existence of a fixed point of .

Remark 13. In the following, we ensure the completeness of the spaces given in precedent examples. (a)Completeness of Let and let be a Cauchy sequence in Define a metric on as where and Since is a Cauchy sequence, for there is such that for all , That is, where is arbitrary. Hence, for every fixed is a Cauchy sequence of complex numbers and it converges, so as Construct a sequence by using these infinitely many limits to show that and From (48) with , we have Since , there is a real number so that . Hence, That is, So, (51) holds for each . It implies that is a bounded sequence of complex numbers. This leads to Also, from (48), we have This implies that ( is endowed with the new extended metric ). (b)Completeness of Let be the function space where is any closed interval in Define Let be a Cauchy sequence in . Then, for each , there exists an such that Hence, for each fixed , we have That is, This shows that is a Cauchy sequence of real numbers; hence, it converges. That is, as In this way, we can associate to each a unique real number This defines a function (pointwise) in Further, we need to show that From (53), we have as Hence, for each and , we have which implies that . Thus, converges uniformly to on the interval Since is continuous and the convergence is uniform; hence, is continuous. This leads to the completeness of with the new extended metric

Theorem 14. Let be a complete new extended -metric space where is a continuous functional and is a nonempty closed subset of . Let satisfy Assume that there exist with and such that for an arbitrary , there is verifying Also, for an arbitrary , assume that the sequence verifies Then, has only one fixed point.

Proof. Let be an arbitrary element of Consider the sequence in . We have Since we have Let such that . Applying triangular inequality, we get Using (63), we get Since , the series is convergent for each by ratio test.
Let So, for , the above inequality implies that Letting , the sequence is a Cauchy sequence. By the completeness of , there is such that
We will prove that is a fixed point of . By using (58), we get As , we have Hence, .
The uniqueness is as follows:
Assume on contrary that there is so that ; then, That is, , which is a contradiction. Thus, is the unique fixed point of .

Remark 15. To prove Theorem 9 in new extended -metric spaces, by using the following conditions: we proceed as follows:
For any , take . Then, where by assumption and . Now, for arbitrary , we can inductively define a sequence . By Theorem 14, this sequence is convergent. So, Thus, .

Also, for each ,

Theorem 16. Let be a complete new extended -metric space such that is a continuous functional. Let satisfy where are nonnegative real numbers such that and . Assume that for an arbitrary , we have where and .
Then, has only one fixed point.

Proof. For an arbitrary , take the sequence . Substituting and in (74), we obtain That is, Thus, Moreover, Thus, we reach By assumption on the parameters , , and , one has .
Following the same steps as given in Theorem 14, one can show that is a Cauchy sequence. By the completeness of , there is such that To prove replace and in (74). We have Then, That is, We have , which holds unless
The uniqueness is as follows:
Let and be two fixed points, such that Then, using inequality (74), we get which is a contradiction. Hence, has only one fixed point.

Now, we use the concept of an asymptotically regular mapping [31, 32] in new extended -metric spaces.

Definition 17. Let be a new extended -metric space. A mapping satisfying the condition is called asymptotically regular.

Example 18. Let . Define by and Consider and We claim that satisfies condition (7). Indeed,

Case 1. If , then, (7) gives , which is true for all and

Case 2. If , then, (7) gives , which is true for all and

Case 3. If , then, (7) implies that , which is true for all and Notice that is fixed point free. The iterative sequence is not convergent, so is not asymptotically regular.

Theorem 19. Let be a complete new extended -metric space such that is a continuous functional. Let be an asymptotically regular self mapping such that there is so that Then, has only one fixed point .

Proof. Let and take be defined inductively. Let such that ; then, according to asymptotic regularity, Thus, the sequence is a Cauchy sequence. By the completeness of , there is such that To prove that is a fixed point of , we proceed as follows: That is, Taking limit and using the asymptotic regularity of , one obtains which implies that
To prove uniqueness, let be another fixed point of . We have This is true unless , and so, . Hence, is the only fixed point of . Further, for each , the iterative sequence converges to .

Remark 20. It is noteworthy that if the mapping is asymptotically regular, then, the condition on can be relaxed.

Theorem 21. Let be a complete new extended -metric space such that is a continuous functional. Let be an asymptotically regular mapping such that there is so that Then, has only one fixed point provided that exists for and is arbitrary in .

Proof. Let and take defined inductively. Let such that ; then, using (93), we have Thus, the sequence is a Cauchy sequence. By the completeness of , there is such that Now, by using triangular inequality and (93), we get so At the limit, Thus, , which is possible only if .
The uniqueness is as follows:
Suppose that there is so that then which a contradiction. Hence, has only one fixed point. Thus, for each , converges to .

3. Application

Let be the set of all real-valued continuous functions on , and let be defined as

One can easily verify that is a complete new extended -metric space. Consider the Fredholm integral equation where and are continuous.

Theorem 22. Let and let the operator be defined by Assume that the following condition holds for each for all , where . Then, the integral equation (102) has a solution, provided that for every iterative sequence , for each , we have

Proof. It is required to prove that the operator satisfies the conditions of Theorem 9. For this, we will use the following inequality for : For , consider That is, This implies that (7) holds for . Hence, by Theorem 9, the operator has a fixed point, provided that for every iterative sequence , for each , we have , that is, the Fredholm integral equation (102) has a solution.

4. Conclusion

(i)The idea of new extended -metric spaces was elaborated with examples(ii)Some results involving Kannan-type contractions on new extended -metric spaces are provided(iii)Results presented by Gornicki [28] are generalized and modified

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

All authors contributed equally to the writing of this paper.

Acknowledgments

M. Aslam extends his appreciation to the deanship of scientific research at King Khalid University, Abha 61413, Saudi Arabia, for funding this work through the research group program under grant number R.G. P-1/23/42.

References

I. A. Bakhtin, “The contraction mapping principle in quasi metric spaces,” Funct. Anal. Unianowsk Gos. Ped. Inst., vol. 30, pp. 26–37, 1989.
View at: Google Scholar
M. Jleli and B. Samet, “A generalized metric space and related fixed point theorems,” Fixed Point Theory and Applications, vol. 2015, no. 1, Article ID 61, 2015.
View at: Publisher Site | Google Scholar
E. Karapinar, “A short survey on the recent fixed point results on -metric spaces,” Constructive Mathematical Analysis, vol. 1, no. 1, pp. 15–44, 2018.
View at: Publisher Site | Google Scholar
T. Kamran, M. Samreen, and Q. UL Ain, “A generalization of b-metric space and some fixed point theorems,” Mathematics, vol. 5, no. 2, p. 19, 2017.
View at: Publisher Site | Google Scholar
L. B. Ćirić, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, no. 2, pp. 267–273, 1974.
View at: Publisher Site | Google Scholar
O. Popescu, “Some new fixed point theorems for -Geraghty contraction type maps in metric spaces,” Fixed Point Theory and Applications, vol. 2014, no. 1, Article ID 190, 2014.
View at: Publisher Site | Google Scholar
E. Rakotch, “A note on contractive mappings,” Proceedings of the American Mathematical Society, vol. 13, no. 3, pp. 459–465, 1962.
View at: Publisher Site | Google Scholar
S. Banach, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
View at: Publisher Site | Google Scholar
R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp. 71–76, 1968.
View at: Google Scholar
H. Aydi, E. Karapinar, and W. Shatanawi, “Coupled fixed point results for -weakly contractive condition in ordered partial metric spaces,” Computers and Mathematics with Applications, vol. 62, no. 12, pp. 4449–4460, 2011.
View at: Publisher Site | Google Scholar
V. Parvaneh, M. R. Haddadi, and H. Aydi, “On best proximity point results for some type of mappings,” Journal of Function Spaces, vol. 2020, Article ID 6298138, 6 pages, 2020.
View at: Publisher Site | Google Scholar
V. Gupta, E. Ozgur, S. Rajani, and M. De la Sen, “Various fixed point results in complete -metric spaces,” Dynamic System and Applicatios, vol. 30, no. 2, pp. 277–293, 2021.
View at: Google Scholar
Y. C. Kwun, A. A. Shahid, W. Nazeer, S. I. Butt, M. Abbas, and S. M. Kang, “Tricorns and multicorns in noor orbit with s-convexity,” IEEE Access, vol. 7, pp. 95297–95304, 2019.
View at: Publisher Site | Google Scholar
L. Luo, R. Ullah, G. Rahmat, S. I. Butt, and M. Numan, “Approximating common fixed points of an evolution family on a metric space via Mann iteration,” Journal of Mathematics, vol. 2021, Article ID 6764280, 7 pages, 2021.
View at: Publisher Site | Google Scholar
S. Czerwik, “Contraction mappings in b-metric spaces,” Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 1, pp. 5–11, 1993.
View at: Google Scholar
T. Abdeljawad, R. P. Agarwal, E. Karapınar, and P. S. Kumari, “Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space,” Symmetry, vol. 11, no. 5, p. 686, 2019.
View at: Publisher Site | Google Scholar
T. Abdeljawad, E. Karapınar, S. K. Panda, and N. Mlaiki, “Solutions of boundary value problems on extended-Branciari -distance,” Journal of Inequalities and Applications, vol. 2020, no. 1, Article ID 103, 2020.
View at: Publisher Site | Google Scholar
M. U. Ali, H. Aydi, and M. Alansari, “New generalizations of set valued interpolative Hardy-Rogers type contractions in -metric spaces,” Journal of Function Spaces, vol. 2021, Article ID 6641342, 8 pages, 2021.
View at: Publisher Site | Google Scholar
M. Akkouchi, “A common fixed point theorem for expansive mappings under strict implicit conditions on -metric spaces,” Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, vol. 50, pp. 5–15, 2011.
View at: Google Scholar
C. Chifu and E. Karapinar, “On contractions via simulation functions on extended b-metric spaces,” Miskolc Mathematical Notes, vol. 21, no. 1, pp. 127–141, 2020.
View at: Publisher Site | Google Scholar
Z. Kadelburg and S. Radenović, “Notes on some recent papers concerning -contractions in -metric spaces,” Constructive Mathematical Analysis, vol. 1, no. 2, pp. 108–112, 2018.
View at: Publisher Site | Google Scholar
N. Mlaiki, N. Souayah, T. Abdeljawad, and H. Aydi, “A new extension to the controlled metric type spaces endowed with a graph,” Advances in Difference Equations, vol. 2021, no. 1, Article ID 94, 2021.
View at: Publisher Site | Google Scholar
M. Anwar, D. Shehwar, and R. Ali, “Fixed point theorems on -contractive mapping in extended -metric spaces,” Journal of Mathematical Analysis, vol. 11, pp. 43–51, 2020.
View at: Google Scholar
H. A. Hammad, H. Aydi, and N. Mlaiki, “Contributions of the fixed point technique to solve the 2D Volterra integral equations, Riemann–Liouville fractional integrals, and Atangana–Baleanu integral operators,” Advances in Difference Equations, vol. 2021, no. 1, 2021.
View at: Publisher Site | Google Scholar
M. Samreen, T. Kamran, and M. Postolache, “Extended -metric space, extended -comparison function and nonlinear contractions,” UPB Scientific Bulletin, Series A, vol. 80, no. 4, 2018.
View at: Google Scholar
A. Azam and M. Arshad, “Kannan fixed point theorem on generalized metric spaces,” Journal of Nonlinear Sciences and Applications, vol. 1, no. 1, pp. 45–48, 2008.
View at: Publisher Site | Google Scholar
H. Aydi, A. Felhi, T. Kamran, E. Karapınar, and M. U. Ali, “On nonlinear contractions in new extended -metric spaces,” Applications & Applied Mathematics, vol. 14, no. 1, 2019.
View at: Google Scholar
J. Gornicki, “Fixed point theorems for Kannan type mappings,” Journal of Fixed Point Theory and Applications, vol. 19, no. 3, pp. 2145–2152, 2017.
View at: Publisher Site | Google Scholar
T. Abdeljawad, N. Mlaiki, H. Aydi, and N. Souayah, “Double controlled metric type spaces and some fixed point results,” Mathematics, vol. 6, no. 12, p. 320, 2018.
View at: Publisher Site | Google Scholar
N. Mlaiki, H. Aydi, N. Souayah, and T. Abdeljawad, “Controlled metric type spaces and the related contraction principle,” Mathematics, vol. 6, no. 10, p. 194, 2018.
View at: Publisher Site | Google Scholar
F. E. Browder and W. V. Petryshyn, “The solution by iteration of nonlinear functional equations in Banach spaces,” Bulletin of the American Mathematical Society, vol. 72, no. 3, pp. 571–576, 1966.
View at: Publisher Site | Google Scholar
R. E. Bruck and S. Reich, “Nonexpansive projections and resolvents of accretive operators in Banach spaces,” Houston Journal of Mathematics, vol. 3, pp. 459–470, 1977.
View at: Google Scholar
A. Branciari, “A fixed point theorem of Banach-Caccippoli type on a class of generalized metric spaces,” Universitatis Debreceniensis, vol. 57, pp. 31–37, 2000.
View at: Google Scholar
S. Kakutani, “A generalization of Tychonoff’s fixed point theorem,” Duke Math, vol. 8, pp. 457–459, 1968.
View at: Google Scholar
R. Kannan, “Some results on fixed points-II,” The American Mathematical Monthly, vol. 76, no. 4, pp. 405–408, 1969.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Hassen Aydi 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1321

Downloads

814

Citations