#### 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 Brouwerâ€™s fixed point theorem and its infinite dimensional form, Schauderâ€™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 Â£: 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 Â£ is an MNC defined on Îž. Then, has a fixed point in Î›.*

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 [5â€“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 Â£ 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. GÃ²rnicki [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 Â£ 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 Â£. 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 Î›.*

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 Â£ 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, {Â£ (*Î› _{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 Â£. 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