Abstract

The concept of measure of noncompactness in a Banach space is used in this paper to extend some tripled fixed point theorems. We prove the existence of fractional integral equation solutions using a generalized Darbo fixed point theorem. To demonstrate the validity of the main result, an example is provided.

1. Introduction

Noncompactness measure has ushered in a new branch of nonlinear analysis. It covers a wide range of applications in operator theory. Noncompactness measures have a wide range of applications in FP theory and are particularly useful in differential and integral equations, as well as fractional calculus. Kuratowski [1] investigated the first definition of a noncompactness measure. In 1955, Darbo [2] ensured the existence of fixed points for some mappings using the notion of noncompactness measures, which were obtained by generalizing the Schauder FP theorem [3] and the Banach contraction principle. Many authors use the term “noncompactness measure” to make Darbo FP theorem more general.

The goal of this paper is to extend Darbo’s FP theorem and to apply our findings to determine the existence of solutions of fractional integral equations.

We begin with preliminaries, notations, concepts, and definitions that will be used throughout the paper.

Let us have a real Banach space , and . Let . Also, let(a).(b).(c) the closure of .(d) the convex closure of .(e) the set of all nonempty and bounded subsets of .(f) the set of all relatively compact sets.

We provide the below definition of MNC, which is referenced in [4].

Definition 1. A mapping is said to be a MNC in if it fulfills the following axioms:(i)The family ker and ker .(ii).(iii).(iv).(v) for any .(vi)If for and , then .

Since for all , , and so .

In the theory of fixed points, the Schauder FP principle and Darbo theorem are crucial.

Theorem 1 (see [3]) (Schauder). For a nonempty, bounded, closed, and convex subset (NBCCS) of a Banach space , if is a continuous and compact mapping, it must have at least one FP.

Theorem 2 (see [2]) (Darbo). For a NBCCS of a Banach space , if is a continuous self-mapping withwhere and is an arbitrary MNC on , then has a FP.

In fractional calculus, fixed point theorems have numerous applications. Let us have a look at some of the work that has been done in this area.

In [5], Sahoo et al. developed numerous new inequalities for twice differentiable convex functions that are coupled with the Hermite–Hadamard integral inequality by using an integral equality related to the k-Riemann–Liouville fractional operator. In addition, for various types of convex functions, certain fresh examples of the established conclusions are derived. This fractional integral adds the symmetric properties of Riemann–Liouville and Hermite–Hadamard inequalities. The authors in [6] explored the existence and uniqueness of solutions to two-dimensional Volterra integral equations, Riemann–Liouville integrals, and Atangana–Baleanu integral operators.

Deng et al. [7] examined the existence of mild solutions for a class of impulsive neutral stochastic functional differential equations in Hilbert spaces with noncompact semigroup. The Hausdorff measure of noncompactness and the Mönch fixed point theorem are used to find sufficient conditions for the existence of mild solutions. The presence of an almost periodic solution to a fractional differential equation with impulse and fractional Brownian motion under nonlocal conditions was the subject of the essay [8].

2. Main Result

We now recall some important definitions that are helpful to our work.

Definition 2. Let be the set of all maps satisfyingfor all .

Definition 3 (see [9]). Let be the set of all functions that fulfills the axioms:(1) for .(2) is continuous.(3).

Example 1. is an example of the class .

Using the above two classes of control functions, we prove the following results.

Theorem 3. Let be a NBCCS of a Banach space . Also, let be a continuous mapping withfor all , where is an arbitrary MNC, , and . Also, let be a nondecreasing continuous mapping. So, has at least one FP in .

Proof. We define the sequence as follows:We can easily see through induction thatIf so that , then , that is, is a relatively compact set. So, by Theorem 1, admits a FP in .
Now, we may assume that for each .
On the contrary, we haveSince , then , as .
This implies thatSince , by Definition 1, we obtain that is a nonempty, closed, and convex subset of and is invariant.
So, Theorem 1 concludes that has a FP in .
Hence, we have the completed proof.

The following is a crucial consequence of Theorem 3.

Corollary 1. Let be a NBCCS of a Banach space . Also, let be a continuous mapping withfor all , where is an arbitrary MNC and . Also, let be a nondecreasing continuous mapping. So, has at least one FP in .

Proof. Putting in Theorem 3, we get the above corollary.

Corollary 2. Let be a NBCCS of a Banach space . Also, let be a continuous mapping withfor all , where is an arbitrary MNC and . So, has at least one FP in .

Proof. Setting in Corollary 1, we obtain the above corollary.

Corollary 3. Let be a NBCCS of a Banach space . Also, let be a continuous mapping withfor all , where is an arbitrary MNC and . So, has at least one FP in .

Proof. Setting in Corollary 2, we obtain the above corollary.

Definition 4. (see [10]). A mapping is called to have a tripled fixed point if , and .

Theorem 4 (see [4]). Let be an MNC in , respectively. Additionally, suppose that the mapping is convex with for . Then, will be an MNC in .

Example 2 (see [11]). Let , for . Now, . As is convex which fulfills all conditions of Theorem 4, is an MNC on , where is the natural projection of into for .

Example 3 (see [12]). Let , for . Now, . As is convex which fulfills all conditions of Theorem 4, is an MNC on , where is the natural projection of into for .

Theorem 5. Let be a NBCCS of a Banach space . Also, let be a continuous mapping withfor all , where is an arbitrary MNC and and are as in Theorem 1. Also, let and ; . So, has at least a tripled fixed point in .

Proof. We consider a function byfor all . It is trivial that is continuous.
Since is continuous, assume that is nonempty. We havewhere represent ’s natural projections.
Now, we getWe can conclude from Theorem 1 that has a minimum of one FP in .
Now, from Theorem 1, admits a tripled fixed point.

3. Measure of Noncompactness on

Let be the space of real continuous functions on , where , which is equipped with

Let be bounded. For and , denote by the modulus of the continuity of , i.e.,

Moreover, we set

It is generally known that the mapping is a MNC in , and will be the Hausdorff MNC (see [4]).

4. Solvability of Fractional Integral Equations

In this part, we show how our conclusions concerning the existence of a solution to a fractional integral equation in a Banach space can be applied.

Consider the following fractional integral equation [13]:where .

Let

Assume that(A) is a continuous function and there exists a constant satisfyingAlso,(B) is a continuous function and there exists a nondecreasing function satisfying(C)There exists a positive solution for the following inequality:

Theorem 6. If constraints (A)–(C) hold, equation (18) has at least one solution in .

Proof. Consider the following operator such that

Step 1. We show that maps into . Let , and we now haveAlso,Hence, givesDue to assumption (C), maps into .

Step 2. We show that is continuous on . Let and such that . For all , we havewhereHence, givesAs , we get .
This clearly proves that is continuous on .

Step 3. An estimation of with respect to : now, assume that . Let be arbitrary. Also, choose with such that and .
Now,whereAs , , so we getHence,i.e.,By the uniform continuity of on , we now obtain , as .
Taking and , we getHence, by Corollary 3, has a FP in .
That is, equation (18) has a solution in .

Example 4. Consider the following fractional integral equation:for , which is a particular case of equation (18).
Here,Also, it is trivial that is continuous and satisfiesTherefore, .
If , thenSo,Putting these values in the inequality of assumption (C), we getHowever, assumption (C) is also fulfilled for .
We can see that all of Theorem 5’s assumptions are achieved, from (A) to (C). Equation (37), according to Theorem 5, has a solution in .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

Authors’ Contributions

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