Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations via Common Fixed-Point Techniques in Complex Valued Fuzzy Metric Spaces

Humaira, undefined; Sarwar, Muhammad; Abdeljawad, Thabet

doi:https://doi.org/10.1155/2020/7042715

Mathematical Problems in Engineering

On this page

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

Research Article | Open Access

Volume 2020 | Article ID 7042715 | https://doi.org/10.1155/2020/7042715

Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations via Common Fixed-Point Techniques in Complex Valued Fuzzy Metric Spaces

Humaira,¹Muhammad Sarwar,¹and Thabet Abdeljawad^2,3,4

Academic Editor: Ivan Giorgio

Received22 May 2020

Accepted14 Aug 2020

Published21 Oct 2020

Abstract

The main purpose of this paper is to study the existence theorem for a common solution to a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations. In this respect, we study the fuzzy version of some essential common fixed-point results from metric spaces in the newly introduced notion of complex valued fuzzy metric spaces. Also, we provide an illustrative example to demonstrate the validity of our derived results.

1. Introduction and Preliminaries

Zadeh initiated the concept of fuzzy sets in 1965 [1], which introduced a deep research activity leading to the improvement of attractive theory of fuzzy system. Afterwards, several researchers have contributed towards some basic significant results in fuzzy sets.

The notion of fuzzy metric was established by Kramosil and Michalek [2]. They generalized the concepts of probabilistic metric spaces to the fuzzy situation. George and Veeramani [3] amended the notion of fuzzy metric to derive a Hausdorff topology initiated by fuzzy metric. This obtained a milestone in the existence theory of fixed point in fuzzy metric spaces. Afterwards, a number of different generalizations appeared for the existence theory of fixed point in fuzzy metric. Garbiec [4] established the fuzzy version of Banach contraction principle in fuzzy metric spaces. For some necessary definitions, examples, and basic results, we refer to [5–7] and the references herein.

Fixed-point theory has a broad set of applications in modern mathematics. Banach contraction principle is the most basic and widely used technique in mathematical analysis. Due to the constructive nature of Banach contraction principle, it is the most useful tool to solve several existence problems in mathematics. Several generalizations of the Banach contraction technique are made in many directions satisfying different types of contractive conditions and having many applications in mathematical disciplines [8, 9].

In the meanwhile, researchers realized that due to vector division, rational type contraction is not meaningful in cone metric spaces. Thus, many results cannot be extended to cone metric spaces. To overcome this problem, very recently, in 2011, Azam et al. [10] initiated a new setting of metric fixed-point theory which is known as complex valued metric spaces. Here, they considered the set of complex number instead of set of positive real numbers as a ground set endowing with a partial structure. The authors obtained fixed-point results satisfying rational contraction and discussed its applications in the said setting. Moreover, Rouzkard and Imdad [11] generalized the applications of complex valued metric spaces and wrote some beautiful remarks. Furthermore, Sintunavarat and Kumam obtained existence results of fixed point for single valued mappings involving control functions instead of constants in contractive condition [12].

Very recently, Shukla et al. [13] introduced an innovative concept of complex valued fuzzy metric spaces where they defined several associated topological features for complex valued fuzzy metric spaces. Moreover, they established the fuzzy version of the well-celebrated Banach contraction principle in different directions and discussed its applications.

The theory of impulsive functional differential equations is emerging as an important area of investigation since such equations appear to represent a natural framework for mathematical modeling of many real processes and phenomena studied in optimal control, electronics, economics, and so on. To further study on impulsive functional differential equations, we refer the readers to [14, 15]. The researchers used a set of different techniques to discuss the existence of solution of such models such as the homotopy perturbation method, Laplace transform method, Adomian decomposition method, and different types of approximation methods. Among such techniques, fixed-point theory is one of the main tools to investigate the analytical and numerical solution of mathematical models. Many researchers have utilized the existence results of fixed point to investigate the analytical solution of different types of differential and integral equations in different spaces. For instance, we refer to [16–24].

In our work, we extend fixed-point results under the general contractive condition in [25] to the setting of complex valued fuzzy metric spaces. Moreover, we studied a result of existence and uniqueness of the solution of nonlinear impulsive fractional differential equations.where the notation stands for Caputo fractional derivative of order , and is a continuous function. Further, the nonlinear functions , are also continuous for and and , where represent the right and left-hand limit of the function , respectively. Also, , where is the set of positive integers. Finally, we gave an illustrative example to make our results strong.

Throughout the paper, we have denoted the set of complex numbers by . Let . The elements are denoted by and , respectively.

Define a partial ordering on by if . The relations and indicate that , and , , respectively. A sequence is monotonic with respect to if either or . Let the unit closed complex interval be denoted by , the open unit complex interval be denoted by , and the set be denoted by . Clearly, for if .

Let . If there exists such that it is the lower bound of , that is and for every lower bound of , then is called the greatest lower bound of . In the same way, we define , (lub) the least upper bound of .

Definition 1. (see [13]). Let be a nonempty set. A complex fuzzy set is characterized by a mapping with domain and the range in the closed unit complex interval , which assigns each element , a grade of membership in , and is thus of the formwhere . The complex fuzzy set may be written as

Definition 2. (see [13]). A binary equation is said to be complex valued t-norm if the following conditions hold:(1)(2) whenever (3)(4)for all .

Example 1. Let be three binary operations defined, respectively, by(1), for all (2), for all (3), for all Then, are complex valued t-norms.
Indeed if is the real unit closed interval and are two t-norms, then defined byis a complex valued t-norm.

Definition 3. (see [13]). Let be a nonempty set, be a continuous complex valued t-norm, and be a complex fuzzy set on satisfying the following conditions:(1)(2) for every if and only if (3)(4)(5) is continuous for all and Then, the triplet is said to be a complex valued fuzzy metric space and is called a complex valued fuzzy metric on . The functions denote the degree of nearness and the degree of non-nearness between and with respect to the complex parameter , respectively.

Example 2. Consider any metric space . Define by , for all . Let the complex fuzzy set defined be given byfor all . Then, is a complex valued fuzzy metric space.
Indeed in above example, if is continuous and nondecreasing function, that is, implies that , then is a complex valued fuzzy metric space, whereSimilarly, it is obvious that for the fuzzy set defined by is a complex valued fuzzy metric space.

Definition 4. (see [13]). Let be a complex valued fuzzy metric space. A sequence converges to , if for each and , there exists with

Definition 5. (see [13]). Let be a complex valued fuzzy metric space. A sequence in is known as a Cauchy sequence ifThe complex valued fuzzy metric space is called complete if every Cauchy sequence is convergent in .

Lemma 1 (see [13]). Let be a complex valued fuzzy metric space. If and , then .

Lemma 2 (see [13]). Let be complex valued fuzzy metric space. A sequence in converges to if holds .

Remark 1. (see [13]). Let .(a)If the sequence is monotonic with respect to and there exist with , then there exists such that .(b)Although the partial ordering is not a linear order on , the pair is a lattice.(c)If and there exists with , then and both exist.

Remark 2. (see [13]). Let , .(a)If and , then .(b)If and , then .(c)If and , then .

2. Common Fixed-Point Results

Theorem 1. Let be a complete complex valued fuzzy metric space and let be self-mappings, such thatfor all , where .

Then, the pair of mappings has a unique common fixed point.

Proof. Let . Define a sequence in byBy using (10), we havewhich yieldsIf , thenBy Lemma 1, this leads to a contradiction; therefore, letwhich implies thatLet ; we shall show that . Therefore, by definition, we haveHence, is monotonic sequence in , and using Remark 1 and (18), there exists such thatInequality (13) suggests thatBy using (19), we obtainSince and utilizing Remark 2, we must obtain . Thus,To show that is a Cauchy sequence, definefor and fixed . Since for all , using Remark 1, we obtain that for all the infimum exists. For , by (10), we have now for each positive integer ,It follows thatTherefore,Therefore, from (26), we have showed that is a Cauchy sequence in . Since is complete, by Lemma 2, there exists an element such thatFor and for any , we obtain from (10) thatwhich implies thatNow, for any ,By taking limit as and using Remark 2 and (27), we haveThus, we obtain that for all , that is, . Similarly, it follows that and so . Hence, the pair has a common fixed point. Assume that is any other common fixed point of and there exists with ; then,This leads to a contradiction. Therefore, for all , that is, .

Theorem 2. Let be a complete complex valued fuzzy metric space with for all , where is a sequence in . Let be self-mappings, such thatfor all , where .

Then, the pair of mappings has a unique common fixed point.

Proof. Let be any arbitrary point in . Define a sequence in by

Case 1. When , for by using (33) with and , we haveNow suppose that if , then using (36), we haveBy Lemma 1, this leads to a contradiction; therefore, letand this implies thatTo show that is a Cauchy sequence, definefor and fixed . Since for all , using Remark 1, we obtain that for all the infimum, (say), exists. For and with , we obtain the following from (39) and Lemma 1:which yieldsTherefore, by definition, we haveHence, is monotonic sequence in , and using Remark 1 and (43), there exists such thatAgain from (39), we have the following for and :Similarly, we get the following for :Hence, for all and ,Since , by (44) and by the hypothesis, we haveFrom (41) and (48), we obtainHence, is a Cauchy sequence in . Since is complete and using Lemma 2, there exists such thatFor any , it follows from (33) thatBy taking limit as and using Remark 2 and (50), we obtain that for all , that is, . Similarly, it follows that , and so . Hence, the pair has a common fixed point. Assume that is any other common fixed point of and there exists with ; then,for all . Applying and , it follows that from the above inequality, we have . This leads to a contradiction. Therefore, for all , that is, .

Case 2. If . It implies that the sequence is constant and so convergent. The rest of the proof can be completed on the steps of Case 1. This completes the proof.

Example 3. Let with the metric defined byDefine a t-norm “” by where . Let be the complex valued fuzzy set given byfor . Clearly, is a complex valued fuzzy metric space.
Certainly, for any sequence with and for each fixed , we have, , for all , thenTherefore,so thatDefine . Note that with and by routine calculation, one can easily verify that satisfy condition (33). Hence, all the assumptions of Theorem 2 are satisfied. Moreover, remains fixed under ; therefore, they have a common fixed point.

Remark 3. In Theorem 1, the contraction condition (10) for the pair of self-mappings can be replaced by the following one, with analogous proof:for all , where is a real valued function .

Corollary 1. Let be a complete complex valued fuzzy metric space and let be self-mapping, such thatfor all , where .Then, the mapping has a unique fixed point.

Proof. The proof is immediate from Theorem 1 by putting .

Corollary 2. Let be a complete complex valued fuzzy metric space and let be self-mapping, such thatfor all , where .

Then, the mapping has a unique fixed point.

Proof. From Corollary 1, we obtain that that there exists such thatSo,This yieldsSince satisfies all the conditions of Corollary 1, has a unique common fixed . But implies that is another fixed point of . As common fixed point is unique, we have . Since the fixed point of is also a fixed point of , the fixed point of is unique.

3. Existence Theorem for Nonlinear Impulsive Fractional Differential Equations

In this section, we present a situation where our obtained results can be applied. Precisely, we study the existence of solution for a class of nonlinear three-point implicit boundary value problems of impulsive fractional differential equations (1). This problem is equivalent to the integral equationwhere

For simplicity, we use the notation .

If necessary, the reader can refer to [16] for a more detailed explanation of the background of the problem. Here, we shall prove our result by establishing the existence of a common fixed point for a pair of integral operators , defined on as

Theorem 3. Assume the following hypotheses are satisfied for .(i) The nonlinear function is continuous.(ii) Let us have constants , which satisfy(iii)() , with .(iv) , with .(v)() and

Then, the pair of nonlinear integral equationshas a common solution in .

Proof. Consider with the metricDefine t-norm asfor and defined byfor . It is obvious that is a complex valued fuzzy metric space. Let and ; then, one hasWith the use of , one hasTherefore, using (77) in (76), we obtainNow,Thus, Theorem 1 applies to , which have a common fixed point , that is, is a common solution of (72).
As an immediate consequence of Theorem 3, in the case , we find that integral equation (67) has a solution in , and hence the nonlinear implicit boundary value problems of impulsive fractional differential equations (1) have a solution.

4. Conclusion

By successful applications of our derived results, we have studied the existence and uniqueness of common solution to the proposed class of implicit impulsive differential equations. Further, some useful results were also obtained that ensure the generalization of some essential results from metric spaces to complex valued fuzzy metric spaces. We finally hope to study classes of fuzzy fractional differential problems in future works.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All the authors contributed equally to the writing of this manuscript. All the authors read and approved the final version.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
I. Kramosil and J. Michalek, “Fuzzy metric and statistical metric spaces,” Kybernetica, vol. 11, pp. 336–344, 1975.
View at: Google Scholar
A. George and P. Veeremani, “On some results of analysis for fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 90, no. 3, pp. 365–368, 1994.
View at: Publisher Site | Google Scholar
M. Grabiec, “Fixed points in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 27, no. 3, pp. 385–389, 1988.
View at: Publisher Site | Google Scholar
T. Dosenović, D. Rakić, B. Carić, and S. Radenović, “Multivalued generalizations of fixed point results in fuzzy metric spaces,” Nonlinear Analysis: Modelling and Control, vol. 2, pp. 211–222, 2016.
View at: Google Scholar
H. V. Long, J. J. Nieto, and N. T. K. Son, “New approach for studying nonlocal problems related to differential systems and partial differential equations in generalized fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 331, pp. 1–31, 2018.
View at: Publisher Site | Google Scholar
W. Onsod and P. Kumam, “Common fixed point results for (φ − ψ) weak contracction mappings via a f − θ dmissible mappings in intutionistic fuzzy metric spaces,” Communications in Mathematics and Applications, vol. 7, no. 3, pp. 167–178, 2016.
View at: Publisher Site | Google Scholar
S. Chandok, A. Mukheimer, A. Hussain, and L. Paunovic, “Picard-Jungck operator for a pair of mappings and simulation type functions,” Mathematics, vol. 7, no. 5, 2019.
View at: Publisher Site | Google Scholar
A. Hussain and M. Adeel, “Remarks on “new fixed point theorems for α-HΘ-contractions in ordered metric spaces”,” Journal of Fixed Point Theory and Applications, vol. 21, p. 63, 2019.
View at: Publisher Site | Google Scholar
A. Azam, B. Fisher, and M. Khan, “Common fixed point theorems in complex valued metric spaces,” Numerical Functional Analysis and Optimization, vol. 32, no. 3, pp. 243–253, 2011.
View at: Publisher Site | Google Scholar
F. Rouzkard and M. Imdad, “Some common fixed point theorems on complex valued metric spaces,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1866–1874, 2012.
View at: Publisher Site | Google Scholar
W. Sintunavarat and P. Kumam, “Generalized common fixed point theorems in complex valued metric spaces and applications,” Journal of Inequalities and Applications, vol. 2012, p. 84, 2012.
View at: Publisher Site | Google Scholar
S. Shukla, R. Rodríguez-López, and M. Abbas, “Fixed point results for contractive mappings in complex valued fuzzy metric spaces,” Fixed Point Theory, vol. 19, no. 2, pp. 1–22, 2018.
View at: Publisher Site | Google Scholar
M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, NY, USA, 2006.
V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
A. Ali, K. Shah, and D. Baleanu, “Ulam stability results to a class of nonlinear implicit boundary value problem of impulsive fractional differential equations,” Advances in Difference Equations, vol. 2019, p. 5, 2019.
View at: Publisher Site | Google Scholar
A. E. Al-Mazrooei, A. Hussain, M. Ishfaq, and J. Ahmad, “Simulative Suzuki-Gerghaty type contraction with C-class functions and application to nonlinear integral equation,” IEEE Access, vol. 7, pp. 75284–75291, 2019.
View at: Publisher Site | Google Scholar
P. Balasubramaniam and S. Muralisankar, “Existence and uniqueness of fuzzy solution for the nonlinear fuzzy integrodifferential equations,” Applied Mathematics Letters, vol. 14, no. 4, pp. 455–462, 2001.
View at: Publisher Site | Google Scholar
J. J. Buckely, “Fuzzy complex numbers,” in Proceedings of the ISFK, pp. 579–700, Guangzhou, China, 1987.
View at: Google Scholar
J. J. Buckely, “Fuzzy complex numbers,” Fuzzy Sets and Systems, vol. 33, no. 3, pp. 333–345, 1989.
View at: Publisher Site | Google Scholar
J. J. Buckely and Y. Qu, “Fuzzy complex analysis I, Differentiation,” Fuzzy Sets and Systems, vol. 41, no. 3, pp. 269–284, 1991.
View at: Publisher Site | Google Scholar
A. Hussain and T. Kanwal, “Existence and uniqueness for a neutral differential problem with unbounded delay via fixed point results,” Transactions of A. Razmadze Mathematical Institute, vol. 172, no. 3, pp. 480–490, 2018.
View at: Publisher Site | Google Scholar
O. Kalva, “Fuzzy differential equations,” Fuzzy Sets and Systems, vol. 24, no. 3, pp. 301–317, 1987.
View at: Publisher Site | Google Scholar
H. V. Long, N. T. K. Son, and N. V. Hoa, “Fuzzy fractional partial differential equations in partially ordered metric spaces,” Iranian Journal of Fuzzy Systems, vol. 14, no. 2, pp. 107–126, 2017.
View at: Google Scholar
A. Hasanen Hammad and M. De la Sen, “Analytical solution of urysohn integral equations by fixed point technique in complex valued metric spaces,” Mathematics, vol. 7, no. 9, p. 852, 2019.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Humaira 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