Generalized Darbo-Type <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 and <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="M2" 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)"><use xlink:href="#g113-71"/></g></svg>-</span>Expansion and Its Applications to a Nonlinear Fractional-Order Differential Equation

Zada, Mian Bahadur; Sarwar, Muhammad; Abdeljawad, Thabet; Jarad, Fahd

doi:https://doi.org/10.1155/2020/4581035

Journal of Function Spaces

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Abstract Introduction and Preliminaries Applications Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 4581035 | https://doi.org/10.1155/2020/4581035

Generalized Darbo-Type -Contraction and -Expansion and Its Applications to a Nonlinear Fractional-Order Differential Equation

Mian Bahadur Zada,¹Muhammad Sarwar,¹Thabet Abdeljawad,^2,3,4and Fahd Jarad⁵

Academic Editor: Adrian Petrusel

Received14 Mar 2020

Accepted30 Apr 2020

Published19 Nov 2020

Abstract

In this work, we introduce various Darbo-type -contractions, and utilizing these contractions, we present some fixed point theorems. Moreover, we introduce a Darbo-type -expanding mapping and prove fixed point theorems under the Darbo-type -expanding mapping. Employing our results, we check the existence of a solution to the nonlinear fractional-order differential equation under the integral type boundary conditions. For its validity, an appropriate example is given.

1. Introduction and Preliminaries

For the sake of completeness, we provide a brief introduction and recollect basic notions, definitions, and fundamental results. In the sequel, we symbolize by the set of all real numbers, by the set of all positive integers, by the closure of and by the convex hull closure of . Additionally, denotes a Banach space , the kernel of function and

Many researchers have been interested in the fixed point theory. This theory is branched into two notable areas. One deals with contraction mappings on metric spaces. In this area, the first important result is the Banach contraction principle. The second deals with continuous operators on convex and compact subsets of a Banach space. In this area, two important results are Brouwer’s fixed point theorem and its infinite dimensional form, Schauder’s fixed point theorem.

Theorem 1. Every continuous mapping from the unit ball of into itself has a fixed point.

Theorem 2 (see [1]). Let . Then, a compact continuous operator has a fixed point.

In both theorems, compactness plays a crucial role. To overcome such hurdle, one of the techniques is to use the notion of a measure of noncompactness (in short MNC). The axiomatic definition of an MNC is as below.

Definition 3 (see [2]). A map £: is an MNC in if for all it satisfies the following axioms: (i)ker and relatively compact in (ii)(iii)(iv)(v)(vi)If is a sequence of closed sets in such that , then

The Kuratowski MNC [3] is the function defined by where is the diameter of a set . Using the notion of an MNC, Darbo [4] published a fixed point result, which determines the existence of a fixed point.

Theorem 4 (see [4]). Let and be a continuous function. If there exists such that where and £ is an MNC defined on Ξ. Then, has a fixed point in Λ.

It generalizes the well-known Schauder fixed point result and includes the existence part of the Banach contraction principle. Many extensions and generalizations of the Darbo fixed point result can be noticed in the existing literature.

The contraction on underlying mappings plays a central role for finding the fixed point. Inspired from this natural idea, the Banach contraction has been improved and extended by several researchers [5–9]. Wardowski [10] proposed a new contraction, called the -contraction and established fixed point theorems.

Definition 5 (see [10]). Let be a map such that
()
() , for any sequence
()

Symbolized by , the family of all maps which fulfill the axioms , and by , the family of functions such that .

Using the specific form of , we deduce other known existing contractions. Many articles concerning -contractions and its generalizations have come into view (see, e.g., [11, 12] and the references cited therein). In particular, Jleli et al. [13] generalized -contraction such that -contraction is an established Darbo-type fixed point result.

Definition 6 (see [13]). Let . Then, the mapping is -contraction if there exists and such that where is a subset of , and £ is an MNC defined in .

Theorem 7 (see [13]). Let . If the mapping is continuous and an -contraction, then has a fixed point in .

Gillespie et al. [14] introduced the concept of expanding mapping. Gòrnicki [15] introduced the idea of -expanding mappings and presented some fixed point results. To find the fixed point of expanding mappings, one needs the following lemma.

Lemma 8 (see [15]). For surjective map , there exists a map such that is an identity map.

This manuscript has two aims. Firstly, we prove various fixed point theorems: the -weak contraction, -weak Suzuki contraction, almost -contraction, Hardy-Rogers-type -contraction and Reich-type -contraction. Secondly, we prove fixed point results under the Darbo-type -expanding mapping. We also observe that several existing results can be concluded from our main results. Furthermore, we check the existence of a solution to the nonlinear fractional-order differential equation via integral type boundary conditions, and for its validity we construct an example.

2. Generalization of Darbo-Type Results via -Contractions

In this section, we introduce various types of -contractions of Darbo type, and then, we prove fixed point results for mappings satisfying such contractive condition in the Banach space endowed with an MNC. We first give the definition of the -weak contraction.

Definition 9. Let . Then, the mapping is a weak contraction if there exists and such that where and are subsets of and £ is an MNC defined in and

In the light of -weak contraction, we present the first result.

Theorem 10. Let . If the mapping is continuous and an -weak contraction, then has a fixed point in .

Proof. Define a sequence such that

We need to prove that . For the first inclusion, we use induction. If , then by (6), we get and . Next, for , we assume that

Then

Using (6), we get the first inclusion

With the help of inclusion (9), we obtained the second inclusion as

Now, we discuss two cases subject to £. If we can find an integer such that , then is compact. But since , by Theorem 2, has a fixed point in Instead, if we take , then we have to testify that First, we need to show that . From inclusion (9), we write , that is is a decreasing sequence and hence converges to with . Now, since and , so by assumption on , we can find and such that . Using contraction condition (4) with and , we write where

Thus, from (11), we obtain

From here, we write

Consequently

Thus

Clearly, and using property , we can write. Thus, by Definition 3 and as . Also, since _n for all , so by Definition 3(ii), . Thus, , that is , and hence, is bounded. But is closed such that is compact. Therefore, by Theorem 2, has a fixed point in .

For the support of Theorem 10, we construct the following example.

Example 1. Let be a subset of a Banach space . Then, clearly, . Define by and , respectively. One can easily check that is continuous, and . Also, define an MNC, by
Now, let and be two subsets of . Then, , , and hence Thus, from (4), we write That is is an -weak contraction. Hence, by Theorem 10, has a fixed point .

From Theorem 10, we can deduce several pivotal results. We demonstrate some preferable corollaries that cover and extend several well-known results in the existing literature. The special case, if we take , we deduce the following corollary.

Corollary 11. Let . If the mapping is continuous such that where and are subsets of , then has a fixed point in .

If we take , in Theorem 10, we deduce the following corollary.

Corollary 12. Let . If the mapping is continuous such that where and are subsets of , then has a fixed point in Λ.

If we take , in Theorem 10, we deduce the following corollary.

Corollary 14. Let . If the mapping is continuous such that where and are subsets of , then has a fixed point in .

Definition 13. Let . Then, the mapping is an -weak Suzuki contraction if there exist and such that where and are subsets of , and £ is an MNC defined on and

In the light of the -weak Suzuki contraction, we provide the following result. Since the proof is very easy, we omit it.

Theorem 14. Let . If the map is continuous and an -weak Suzuki contraction, then has a fixed point in .

Definition 15. Let . Then, the mapping is almost an -contraction if there exist and such that

In the light of an almost -contraction, we present the following result.

Theorem 16. Let . If the functionis continuous and an almost -contraction, then has a fixed point in .

Proof. Construct a sequence such that Then, and .
If we take an integer such that , then is a compact, and by Theorem 2, we can find a fixed point of in . Let us take . Then, {£ (Λ_n)} is a decreasing sequence and hence converges to with . Now, since and , so by assumption on , we can find and such that . Now, assume that Then, setting in (24), we have From here, we write Consequently Thus Clearly , and using the property , we can write .
Following the same steps as in Theorem 10, we can easily show that has a fixed point in .

Definition 17. Let . Then, the mapping is the Hardy-Rogers -contraction, if we can find , and such that for all , where with

Remark 18. (1)If , then (31) is a Reich-type -contraction(2)If and, then the contraction (31) becomes a Darbo-type -contraction.

Theorem 19. Let . If the function is continuous and a Hardy-Rogers-type -contraction, then has a fixed point in .

Proof. Construct a sequence such that Then, and .
Now, we discuss two cases subject to £. If we consider as a nonnegative integer with , then is a compact. But , so by Theorem 2, has a fixed point in Instead, let us take . From (9), we write , that is is a decreasing sequence and hence converges with . Now since and , so by assumption on , we can find and such that . Using (31) with and , we have but and is nondecreasing; thus, we have The rest of the proof is analogous to that of Theorem 10.

We provide the following result, the proof is easy, so we omit it.

Theorem 20. Let . If the function is continuous and a Reich -contraction then has a fixed point in .

3. Darbo-Type Result via -Expansion

In this section, we introduce the Darbo-type -expanding mapping and establish some fixed point results.

Definition 21. Let . Then the mapping is -expanding if there exist and such that where .

In the light of -expanding mapping, we present the following result.

Theorem 22. Let . If the mapping is continuous, surjective, and -expanding, then has a fixed point in .

Proof. Since is surjective, so by Lemma 8, we can find a function such that is the identity function on . Let and be any subsets of such that . Assume that ; then on using (35), we can write Since , then (36) becomes Now, if is fixed point of , then . Thus, to show that has a fixed point, it is sufficient to show that has a fixed point. To do this, construct a sequence such that Then, we can easily show that and .
Next, if we take an integer with , then is compact. So, by Theorem 2, has a fixed point in . Now, let us take , for all . We have to prove that is a nonempty, bounded, closed, and convex subset of . For this, since the sequence is decreasing, it converges to . Now, assume that . Then that is, and , so by assumption on , inf , we can find and such that for all . Using (37) with , we write From here, we write Consequently, By routine calculation, one can easily obtain Clearly, and using the property , we can write . Thus, by Definition 3(vi), is nonempty and as . Also, since , for all , so by Definition 3(ii), . Thus, , and hence, , that is is bounded. But is closed such that is compact. Therefore, by Theorem 2, has a fixed point in . Consequently, has a fixed point in .

From Theorem 22, we can deduce several pivotal results. We demonstrate some preferable corollaries that cover and extend several known theorems in the literature. The special case, if we take , in Theorem 22, we deduce the following corollary.

Corollary 23. Let and be a surjective and continuous mapping such that Then has a fixed point in .

If we take , in Theorem 22, we get the following corollary.

Corollary 24. Let and be a surjective and continuous mapping such that Then, has a fixed point in .

If we take , in Theorem 22, we deduce the following corollary.

Corollary 25. Let and be a surjective and continuous mapping such that Then has a fixed point in the set .

If we take with , in Theorem 22, we deduce the following corollary.

Corollary 26. Let and be a surjective and continuous mapping such that Then, has a fixed point in .

4. Applications

This section deals with some practicing of our fixed point results. Let be a Banach space having the zero element 0. Let be the closed ball with center and radius and be the ball . Our aim is to illustrate sufficient conditions for the existence of a solution of a nonlinear fractional-order differential equation: under the integral type boundary conditions: where , is the Caputo fractional derivative, are continuous functions, and and () are positive real numbers.

Lemma 27 (see [16]). For , we have

To proceed further, we convert the nonlinear fractional-order differential equation (48) to an integral equation. For this, we prove the following lemma.

Lemma 28. A solution of the fractional-order boundary value problem (48) is

Proof. First of all, apply the Riemma-Liouville fractional integrable operator of order to equation (48), and using Lemma 27, we can easily deduce that and by differentiating (53), we get But , , and .
Substituting the values of and in (49), we get Similarly, substituting the values of and in (50), we get Putting the value of in (55), we deduce Thus, by switching the values of and in (53) and by routine calculations, we get equation (52).

Notice that Lemma (49) indicates that the solution of differential equation (48) is equivalent to the solution of integral equation (52). Now, we are in a position to present the existence result.

Theorem 29. Let and are continuous mapping such that where and . Then, there exists a solution of the fractional-order integral equation (52) in . Accordingly, there exists a solution of the nonlinear fractional-order differential equation (48) in .

Proof. Let Then, is a nonempty, closed, bounded, and convex subset of . Define with >1 and and the operator by First, we will show that is well defined. Let , for some . Then, for all , we have That is for all , which implies that , and hence, is well defined.
Now, we have to show that is continuous. For this, consider Since , and are continuous, so for , there exist such that If and , then we conclude that , whenever .
Hence, is continuous. Next, we have to show that is an -weak contraction. Let and be any two subsets of and . Then, by conditions (58), (59), and (60), we have From here, we write Consequently, which implies that That is is an -weak contraction. Thus, by Theorem 10, has a fixed point in . Consequently, equation (52) has a solution in .

To illustrate the existence result (Theorem 29), we present an example.

Example 2. Consider the nonlinear fractional-order differential equation where is the Caputo fractional-order derivative and are continuous functions defined as

Now, we have to verify the conditions of Theorem 29. Let and ; then, for, and , we first show condition (58) of Theorem 29. To do this, we have

Next, to show condition (59) of Theorem 29, we have

Also, to show condition (60) of Theorem 29, we have

Finally, we have to verify condition (75) of Theorem 29. For this, since , we get

Thus, all the conditions of Theorem 29 are satisfied. Hence the nonlinear fractional-order differential equation (70) has a solution in .

5. Conclusion

Through measure of noncampactness, various new -contraction and -expanding mappings have been presented. In the Banach spaces, fixed point results were established, from which several existing results can be extracted. For the accuracy of our results, we have checked the existence of a solution to the nonlinear fractional-order differential equation under the integral-type boundary conditions with an example.

Data Availability

Not applicable.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The third author thanks Prince Sultan University for funding this research through the group: Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RGDES-2017-01-17.

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Copyright

Copyright © 2020 Mian Bahadur Zada 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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