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# Solving Fractional Differential Equations via Fixed Points of Chatterjea Maps

1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

2 Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif, 21944, Saudi Arabia

* Corresponding Authors: Nawab Hussain. Email: ; Hind Alamri. Email:

(This article belongs to the Special Issue: Computational Aspects of Nonlinear Operator and Fixed Point Theory with Applications)

*Computer Modeling in Engineering & Sciences* **2023**, *135*(3), 2617-2648. https://doi.org/10.32604/cmes.2023.023143

**Received** 10 May 2022; **Accepted** 14 July 2022; **Issue published** 23 November 2022

## Abstract

In this paper, we present the existence and uniqueness of fixed points and common fixed points for Reich and Chatterjea pairs of self-maps in complete metric spaces. Furthermore, we study fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space using the Krasnoselskii-Ishikawa iteration method associated with and consider some applications of our results to prove the existence of solutions for nonlinear integral and nonlinear fractional differential equations. We also establish certain interesting examples to illustrate the usability of our results.## Keywords

Fixed point theory plays an important role in various branches of mathematics as well as in nonlinear functional analysis, and is very useful for solving many existence problems in nonlinear differential and integral equations with applications in engineering and behavioural sciences. Recently, many authors have provided the extended fixed point theorems for the different classes of contraction type mappings, such as Kannan, Reich, Chatterjea and Ćirić-Reich-Rus mappings (see [1–10]).

Let

for each

Kannan [11] established a fixed point theorem for mapping satisfying:

for each

In 1980, Gregus [13] combined nonexpansive and Kannan nonexpansive mappings as follows:

where

In [15], the authors considered the Rhoades mapping satisfying the following condition:

for each

In 1971, Ćirić [20] introduced the notion of orbital continuity. Sastry et al. [21] defined the notion of orbital continuity for a pair of mappings. We now recall some relevant definitions.

Definition 1.1. [20] If

is said to be an orbit of

for some

In addition,

Definition 1.2. [21] Let

is called the

Ćirić in [22] proved that continuity of

Definition 1.3. [23] A mapping

Note that, 1-continuity is equivalent to continuity and for any

On the other hand, the concept of asymptotic regularity has been introduced by Browder et al. [24] in connection with the study of fixed points of nonexpansive mappings. Asymptotic regularity is a fundamentally important concept in metric fixed point theory. A self-mapping

In [25], Gòrnicki proved the following fixed point theorem:

Theorem 1.1. Let

for all

Recently, Bisht [26] showed that the continuity assumption considered in Theorem 1.1 can be weakened by the notion of orbital continuity or

Theorem 1.2. Let

This paper is organised as follows: First, we establish some fixed point theorems for Reich and Chatterjea nonexpansive mappings to include asymptotically regular or continuous mappings in complete metric spaces. After that, we prove some fixed point theorems and common fixed points for Reich and Chatterjea type nonexpansive mappings in Banach space using the Krasnoselskii-Ishikawa method associated with

2 Asymptotic Behaviour of Mappings in Complete Metric Spaces

In this section, we study fixed point and common fixed point theorems for Reich and Chatterjea type nonexpansive mappings in complete metric space.

To start with the following lemma, which is useful to prove the results of this section:

Lemma 2.1. [27] Let

where

(i)

(ii)

(iii)

Then,

Theorem 2.1. Let

for all

Proof. The proof of the theorem is organized in three steps:

Step 1: We shall prove that

for all

As

Using triangle inequality and asymptotic regularity of

Thereafter, suppose that

Let

Step 2: Let

where

Choosing

Thus, we have

As

Further, by asymptotic regularity of

the implication is that

Now, using (6) we get

Taking limit as

from Step 1 and

Step 3: We will prove that

For the uniqueness of the limit, we get

Similarly, let

we obtain

In addition, since

Example 2.2. Consider

Clearly, the two mappings

Case 1: Let

Case 2: Let

Case 3: Let

Case 4: If

Therefore, in all the cases,

In the special case of our result, we can generate the Theorem 1.4 of Gòrnicki [28].

Corollary 2.1. Let

for all

Proof. Take

for all

which implies that

which implies that

Example 2.3. Let

and

Choosing

for some

Obviously, as all the assumptions of Corollary 2.1 hold,

Theorem 2.4. Let

for all

(i) The mapping

(ii) For

Then,

Proof. First, we shall prove condition (i). Let

Using the triangle inequality and asymptotic regularity in (12) we get for any n and

Thus,

as

which is a contradiction. Hence,

Hence,

This shows that

Next, we will consider condition (ii). Choose

Theorem 2.5. Let

Proof. Suppose that there exist

which implies

As

Theorem 2.6. Let

for each

Proof. Since

Thus,

Since

Suppose that

further

The compatibility of

which is a contradiction, thence,

In the next theorem, we establish a common fixed point result on Chatterjea nonexpansive mapping.

Theorem 2.7. Let

for each

Proof. We follow the lines of Theorem 2.1 to prove this theorem. The proof is divided into three steps as follow:

Step 1: We shall prove that

for each

which implies

As

By the asymptotic regularity of

Next, suppose that

obtain

which implies

Let

Step 2: Let

Assume that

Choosing

Now, we have that

As

Furthermore, by asymptotic regularity of

implying that

Then, using (14), we have

as

from (15) and

Step 3: We prove the uniqueness of the common fixed point of

Let

Since the limit is unique, it implies

Similarly, suppose that

implies that

Example 2.8. Let

where

and

If we choose

In fact, we have the following four cases:

Case 1: Let

Case 2: If

Therefore,

Case 3: If

Thus,

Case 4: If

However,

Therefore, in all cases,

Corollary 2.2. Let

for all

Proof. Take

According to Theorem 2.7, f and g have a unique common fixed point

implies that

Theorem 2.9. Let

for all

(i) The mapping

(ii) For

Then

Proof. First, we will proof the condition (i) holds. Let

Using the triangle inequality and asymptotic regularity in (21) we obtain for any n and

Then,

as

which is a contradiction. Hence,

Hence,

This shows that

We next prove the condition (ii) holds. Suppose that

Finally, we show that

Theorem 2.10. Let

Proof. Let

This implies

As

Theorem 2.11. Let

for each

Proof. Since

Thus,

Since

Assuming that

further,

Then, compatibility of

that is,

Example 2.12. Let

Then

Similarly, if take

If

Otherwise, if

3 Fixed Point and Common Fixed Point Results in Banach Spaces

In this section, we present some fixed point and common fixed point theorems for Reich and Chatterjea nonexpansive mappings in a Banach space.

Consider a fixed point iteration, which is given by

with an arbitrary

Define

for each

However, if

In the following, we prove basic lemmas for the Reich nonexpansive mapping which in turn are useful to proving the results of this section.

Lemma 3.1. Let

for all

where

Proof. Let us choose

and

for all

Since

Using (28), (29) and (31), we obtain

Since

Let

which implies that

such that

Remark 3.1. It follows from Lemma 3.1 that if

Lemma 3.2. Let

holds for each

Proof. Let

we see that

Then, (35) implies that

Let n be any positive integer. From Remark 3.1, there exists

Applying a triangle inequality and Lemma 3.1, we obtain

Therefore,

Since n was arbitrary, the proof is completed.

Now, we state and prove our main results of this section:

Theorem 3.2. Let

converges to a unique fixed point

Proof. Following a similar lines of the proof of Lemma 3.1, we have

where

Let

We shall show that the sequence of iterates

where

Again, by Lemma 3.1, we have

which implies that

Moreover, by (38) we get

Continuing this process, we obtain

and it follows from Lemma 3.2 that

Taking limit as

Hence,

Since

which is a contradiction, hence,

In the next theorem, we present a common fixed point result for Reich mappings.

Theorem 3.3. Let

for all

Proof. Suppose that

Now, using the operator defined by (25), we obtain

From (39) and (41), we get the following:

such that

where

Similarly,

Continuing the process, we get the following:

Now, we show that

Since

where we have

this is a contradiction, hence

Theorem 3.4. Let

for all

Proof. Let

Following similar lines of the proof of Theorem 3.2, we obtain

Let us choose

where

Similarly, we obtain

Continuing the process, we get the following:

Next, we show that

Since

Since

which is a contradiction, hence

Example 3.5. Let

Take

Let

Case 1: Let

Case 2: Let

Case 3: Let

Therefore, in all the cases,

4 Application to Nonlinear Integral Equations

Let

We consider the following integral equations formulated as a common fixed point problem of the following nonlinear mappings:

such that

(i)

(ii) The two mappings G and

Theorem 4.1. Under the assumption (i) and (ii), then the system of (48) has a common fixed point in E.

Proof. We have

We also have,

Suppose that

which implies that

Therefore,

Now, since

By Theorem 2.1 there exists a common fixed point of

5 Application to Nonlinear Fractional Differential Equation

Fractional differential equations have applications in various fields of engineering and science including diffusive transport, electrical networks, fluid flow and electricity. Many researchers have studied this topic because it has many applications. Related to this matter, we suggest the recent literature [29–35] and the references therein.

The classical Caputo fractional derivative is defined by

where

Now, we consider the following fractional differential equation:

where

Assume that

for all

Theorem 5.1. Let

and

for all

Proof. It is easy to see that

Let

where

Example 5.2. Consider the following fractional differential equation:

The exact solution of the above problem (51) is given by

The operator

and

By taking

On the other hand, we obtain that

which implies that

By Theorem 2.7 there exists a common fixed point of

This paper contains the study of Reich and Chatterjea nonexpansive mappings on complete metric and Banach spaces. The existence of fixed points of these mappings which are asymptotically regular or continuous mappings in complete metric space is discussed. Furthermore, we provide some fixed point and common fixed point theorems for Reich and Chatterjea nonexpansive mappings by employing the Krasnoselskii-Ishikawa iteration method associated with

Author Contribution: N. H. and H. A. introduced contractive inequalities, obtained the solution and wrote the paper. S. A. read and analyzed the paper; all authors read and approved the final manuscript.

Acknowledgement: The authors wish to express their appreciation to the reviewers for their helpful suggestions which greatly improved the presentation of this paper.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

## References

1. Banach, S. (1922). Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fundamenta Mathematicae, 3(1), 133–181. DOI 10.4064/FM-3-1-133-181. [Google Scholar] [CrossRef]

2. Abbas, M., Anjum, R., Berinde, V. (2021). Equivalence of certain iteration processes obtained by two new classes of operators. Mathematics, 9(18), 2292. DOI 10.3390/math9182292. [Google Scholar] [CrossRef]

3. Berinde, V., Păcurar, M. (2021). Fixed point theorems for enriched Ćirić-Reich-Rus contractions in Banach spaces and convex metric spaces. Carpathian Journal of Mathematics, 37(2), 173–184. DOI 10.37193/CJM.2021.02.03. [Google Scholar] [CrossRef]

4. Berinde, V., Pacurar, M. (2021). Fixed points theorems for unsaturated and saturated classes of contractive mappings in Banach spaces. Symmetry, 13(4), 713. DOI 10.3390/sym13040713. [Google Scholar] [CrossRef]

5. Debnath, P., Konwar, N., Radenovic, S. (2021). Metric fixed point theory: Applications in science, engineering and behavioural sciences. 1st edition. Singapore: Springer Verlag. DOI 10.1007/978-981-16-4896-0. [Google Scholar] [CrossRef]

6. Jungck, G. (1986). Compatible mappings and common fixed points. International Journal of Mathematics and Mathematical Sciences, 9 (4), 771–779. DOI 10.1155/S0161171286000935. [Google Scholar] [CrossRef]

7. Khan, A. R., Oyetunbi, D. M. (2020). On some mappings with a unique common fixed point. Journal of Fixed Point Theory and Applications, 22(2), 1–7. DOI 10.1007/s11784-020-00781-w. [Google Scholar] [CrossRef]

8. Hussain, N., Alsulami, S. M., Alamri, H. (2022). A Krasnoselskii-Ishikawa iterative algorithm for monotone Reich contractions in partially ordered Banach spaces with an application. Mathematics, 10(1), 76. DOI 10.3390/math10010076. [Google Scholar] [CrossRef]

9. Pathak, H. K. (2018). An introduction to nonlinear analysis and fixed point theory. Singapore: Springer. DOI 10.1007/978-981-10-8866-7. [Google Scholar] [CrossRef]

10. Radenović, S., Kadelburg, Z., Jandrlić, D., Jandrlć, A. (2012). Some results on weakly contractive maps. Bulletin of the Iranian Mathematical Society, 38(3), 625–645. [Google Scholar]

11. Kannan, R. (1979). Some results on fixed points. II. The American Mathematical Monthly, 76(4), 405–408. [Google Scholar]

12. Kannan, R. (1973). Fixed point theorems in reflexive Banach spaces. Proceedings of the American Mathematical Society, 38(1), 111–118. DOI 10.1090/S0002-9939-1973-0313896-2. [Google Scholar] [CrossRef]

13. Gregus, M. (1980). A fixed point theorem in Banach spaces. Bollettino dell’Unione Matematica Italiana, 5(17), 193–198. [Google Scholar]

14. Reich, S. (1971). Kannan’s fixed point theorem. Bollettino dell’Unione Matematica Italiana, 4(4), 1–11. [Google Scholar]

15. Rhoades, B. E. (1977). A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society, 226, 257–290. DOI 10.2307/1997954. [Google Scholar] [CrossRef]

16. Reich, S. (1971). Some remarks concerning contraction mappings. Canadian Mathematical Bulletin, 14(1), 121–124. DOI 10.4153/CMB-1971-024-9. [Google Scholar] [CrossRef]

17. Mustafa, Z., Parvaneh, V., Jaradat, M. M., Kadelburg, Z. (2019). Extended rectangular b-metric spaces and some fixed point theorems for contractive mappings. Symmetry, 11(4), 594. DOI 10.3390/sym11040594. [Google Scholar] [CrossRef]

18. Roshan, J. R., Shobkolaei, N., Sedghi, S., Parvaneh, V., Radenović, S. (2014). Common fixed point theorems for three maps in discontinuous Gb-metric spaces. Acta Mathematica Scientia, 34(5), 1643–1654. DOI 10.1016/S0252-9602(14)60110-7. [Google Scholar] [CrossRef]

19. Som, S., Petrusel, A., Garai, H., Dey, L. K. (2019). Some characterizations of Reich and Chatterjea type nonexpansive mappings. Journal of Fixed Point Theory and Applications, 21(4), 1–21. DOI 10.1007/s11784-019-0731-x. [Google Scholar] [CrossRef]

20. Ćirić, L. B. (1974). A generalization of Banach’s contraction principle. Proceedings of the American Mathematical Society, 45(2), 267–273. DOI 10.2307/2040075. [Google Scholar] [CrossRef]

21. Sastry, K., Naidu, S., Rao, I., Rao, K. (1984). Common fixed points for asymptotically regular mappings. Indian Journal of Pure and Applied Mathematics, 15(8), 849–854. [Google Scholar]

22. Ćirić, L. B. (1971). On contraction type mappings. Mathematical Balkanica, 1, 52–57. [Google Scholar]

23. Pant, A., Pant, R. P. (2017). Fixed points and continuity of contractive maps. Filomat, 31(11), 3501–3506. DOI 10.2298/FIL1711501P. [Google Scholar] [CrossRef]

24. Browder, F. E., Petryshyn, W. (1966). The solution by iteration of nonlinear functional equations in Banach spaces. Bulletin of the American Mathematical Society, 72(3), 571–575. DOI 10.1090/S0002-9904-1966-11544-6. [Google Scholar] [CrossRef]

25. Gòrnicki, J. (2019). Remarks on asymptotic regularity and fixed points. Journal of Fixed Point Theory and Applications, 21(1), 29. DOI 10.1007/s11784-019-0668-0. [Google Scholar] [CrossRef]

26. Bisht, R. K. (2019). A note on the fixed point theorem of Gòrnicki. Journal of Fixed Point Theory and Applications, 21(2), 54. DOI 10.1007/s11784-019-0695-x. [Google Scholar] [CrossRef]

27. Xu, H. K. (2003). An iterative approach to quadratic optimization. Journal Optimization Theory and Applications, 116(3), 659–678. DOI 10.1023/A:1023073621589. [Google Scholar] [CrossRef]

28. Gòrnicki, J. (2020). On some mappings with a unique fixed point. Journal of Fixed Point Theory Applications, 22, 8. DOI 10.1007/s11784-019-0741-8. [Google Scholar] [CrossRef]

29. Abbas, M., Asghar, M. W., de la Sen, M. (2022). Approximation of the solution of delay fractional differential equation using AA-iterative scheme. Mathematics, 10(2), 273. DOI 10.3390/math10020273. [Google Scholar] [CrossRef]

30. Ashraf, M. S., Ali, R., Hussain, N. (2020). New fuzzy fixed point results in generalized fuzzy metric spaces with application to integral equations. IEEE Access, 8, 91653–91660. DOI 10.1109/ACCESS.2020.2994130. [Google Scholar] [CrossRef]

31. Asif, A., Alansari, M., Hussain, N., Arshad, M., Ali, A. (2021). Iterating fixed point via generalized Mann’s iteration in convex b-metric spaces with application. Complexity, 2021, 8534239. DOI 10.1155/2021/8534239. [Google Scholar] [CrossRef]

32. Derbazi, C., Baitiche, Z., Abdo, M. S., Shah, K., Abdalla, B. et al. (2022). Extremal solutions of generalized Caputo-type fractional order boundary value problems using monotone iterative method. Fractal and Fractional, 6(3), 146. DOI 10.3390/fractalfract6030146. [Google Scholar] [CrossRef]

33. Khatoon, S., Uddin, I., Baleanu, D. (2021). Approximation of fixed point and its application to fractional differential equation. Journal of Applied Mathematics and Computing, 66(1), 507–525. DOI 10.1007/s12190-020-01445-1. [Google Scholar] [CrossRef]

34. Tariboon, J., Samadi, A., Ntouyas, S. K. (2022). Nonlocal boundary value problems for Hilfer generalized proportional fractional differential equations. Fractal and Fractional, 6(3), 154. DOI 10.3390/fractalfract6030154. [Google Scholar] [CrossRef]

35. Uddin, I., Garodia, C., Abdeljawad, T., Mlaiki, N. (2022). Convergence analysis of a novel iteration process with application to a fractional differential equation. Advances in Continuous and Discrete Models, 2022(1), 1–20. DOI 10.1186/s13662-022-03690-z. [Google Scholar] [CrossRef]

## Cite This Article

**APA Style**

*Computer Modeling in Engineering & Sciences*,

*135*

*(3)*, 2617-2648. https://doi.org/10.32604/cmes.2023.023143

**Vancouver Style**

**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 135, no. 3, pp. 2617-2648. 2023. https://doi.org/10.32604/cmes.2023.023143

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