Abstract
We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in space. The fundamental tool in the presentation of our proofs is the measure of noncompactness approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.
1. Introduction
The integral equations have a variety of practical applications in defining specific real-world problems and situations, such as the law of physics, the theory of radioactive transmission, statistical mechanics, and cytotoxic activity (see [1–3]). Fixed point theory and have several applications in solving various types of differential and integral equations [4, 5]. The existence of solutions for the system of integral equations is studied by Aghajani et al. [6]. Mursaleen and Mohiuddin [7] proved the existence theorem for the infinite system of differential equations in the space . Banas [8] studied the solution of nonlinear differential and integral equations. Arab et al. [9] proved the existence of functional integral equations using . Also, Çakan [10] proved the existence of nonlinear integral equations in Banach spaces by using the technique of . The notions of - and - condensing operators were recently developed in [11], and they were used to show some new fixed point results using the technique of measure of noncompactness.
Fractional calculus theory and applications grew rapidly in the nineteenth and early twentieth century, and many contributors provided interpretations for fractional derivatives and integrals. Rezapour et al. [12] studied a fractional-order model for anthrax disease among animals based on the Caputo-Fabrizio derivative. The Erdélyl-Kober fractional integral is utilized in various areas of mathematics, including porous media and viscoelasticity [13–15]. The study of fractional order integral equations has become necessary due to their importance. The Erdélyl-Kober fractional operator was studied by various researchers for differential and integral equations. Darwish [16] studied the existence of a solution for Erdélyl-Kober fractional Urysohn-Volterra quadratic integral equations. Mollapouras and Ostadi [17] investigated the existence and stability of the solution of the functional integral equations of fractional order arising in physics, mechanics, and chemical reactions. Mohammadi et al. [18] and Jleli et al. [19] have been generalized Darbo’s fixed point theorem with the help of a new type of contraction operator. Motivated by these, we have generalized Darbo’s fixed point theorem by using the -type contraction [20] whose extension is also generalized and established by Jleli et al. [19] and Mohammadi et al. [18] and applied it to the existence of solutions for the infinite system of fractional order integral equations in space.
2. Preliminaries
We introduce notations, definitions, and introductory facts in this section, which will be used throughout the paper.
For bounded subset of a metric space , Kuratowski [21] measure of noncompactness is defined as where denotes the diameter of a set .
The Hausdorff (or ball) for is defined as follows:
Throughout this paper, we assumed that (,) is a real Banach space and . Let , we mark by and Conv the closure and the closed convex hull of , respectively. In addition, let denotes the family of nonempty bounded subset of and collection of all nonempty relatively compact subsets of .
Definition 1 (see [22]). A mapping is called the measure of noncompactness in if
(A1). The family ker is nonempty and ker
(A2).
(A3).
(A4).
(A5). for
(A6). If , , and then is nonempty
In the Banach space , the Hausdorff is given by
Now, we will go over the following important theorems in fixed point theory that play a key role.
Theorem 2 (see [23], Schauder). Suppose is a nonempty, bounded, convex, and closed subset of a Banach space . Then, each continuous, compact map has a fixed point.
The following theorem given by Darbo is a generalization of Theorem 2.
Theorem 3 (see [22], Darbo). Suppose be a nonempty, bounded, convex, and closed subset of a Banach space , and let is a continuous operator such that there exists a constant with the property . Then, has a fixed point in the set .
The Darbo fixed point theorem is more effective than the Schauder fixed point theorem. Because in the case of Darbo fixed point theorem, the compactness of the domain of the operator, which is essential in Schauder’s theorem, has been relaxed.
Jleli et al. obtained the following extension of the Darbo fixed point theorem.
Theorem 4 (see [19]). Suppose is a nonempty, bounded, closed, and convex subset of a Banach space , and be continuous operator such that for every nonempty subset of ,
where and , is the class of function satisfying the following condition:
For each , .
Then, has at least one fixed point in .
With a newly defined contraction operator, Mohammadi et al. have established the following generalization of Darbo’s fixed point theorem, which is also generalized the result of Wardowski’s F-contraction [24].
Theorem 5 (see [18]). Suppose is a nonempty, bounded, convex, and closed subset of a Banach space , and be continuous mapping such that for every nonempty subset of ,
where is the arbitrary mnc, is the arbitrary positive constant, , be the set of function such that
(W1). is a strictly increasing and continuous function
(W2). , for each
Then, has at least one fixed point in .
We also recall the concept of a coupled fixed point for a bivariate mapping and an important theorem on the construction of a measure of noncompactness on a finite product space.
Definition 6 (see [25]). An element is called a coupled fixed point of mapping if and .
Theorem 7 (see [26]). Let be in Banach spaces respectively, and the function is a convex function and if and only if . Then,
is a in where is the natural projection of into
Now, we have the following example, which was described in [6], as a result of Theorem 7.
Example 1. Considering be a on a Banach space , let for any , as we can see that is convex and if and only if ; hence, all the conditions of Theorem 7 are satisfied. Therefore, defines a in the space where denote the natural projection of . Similarly, by letting for any , we have defines a in the space where denote the natural projection of
In 2016, Liu et al. [20] introduced , the set of function such that
(ф1). is nondecreasing
(ф2). if and only if , for each sequence
(ф3). is continuous
Berinde [27] introduced the class of function known as comparison function. Let be the family of all function satisfying the following conditions:
(ѱ1). is monotone increasing
(ѱ2).
3. Main Result
Theorem 8. Suppose is a nonempty, bounded, convex, and closed subset of a Banach space , and be continuous mapping such that for all subset of , where and and is an arbitrary . Then, has at least one fixed point.
Proof. By induction, we define a sequence by letting and We have , ; therefore, by continuing this process, we obtain Let there exists an integer such that . So, is relatively compact and ; hence, Theorem 2 yields that has a fixed point. So we suppose that for . By Equation (7), we have Thus, we have Letting in (10) and applying , we have By using , we obtained Since and for all , then from , is nonempty convex closed set, invariant under and belong to ker . Now, from Theorem 2, has a fixed point.
Remark 9. We can get the Darbo’s fixed point Theorem 3, if we take and in above theorem.
Remark 10. We get Theorem 4 if we take in Theorem 8.
Remark 11. We get Theorem 5 if we take and in Theorem 8.
Theorem 12. Let be a nonempty, closed, and convex subset of a Banach space , and be continuous operator such that for all subset of , where arbitrary and and are as in Theorem 8. Then, has at least a coupled fixed point.
Proof. From Example 1, we have defines a in the space where denote the natural projection of . Now, consider defined by which is clearly continuous. We show that satisfies all the conditions of Theorem 8. Let be nonempty subset. From Equation (13) we have Therefore, all the conditions of Theorem 8 are satisfied. Hence, has a fixed point which implies that has a coupled fixed point.
4. Application
The fractional integral of a function (the space of Lebesgue measurable functions) of order with respect to monotone, continuous derivative function is defined by [28]
Nieto and Samet [29] discussed the existence of solutions of the generalized implicit fractional integral equation
where and .
Also, the existence of a solution for an infinite system of implicit fractional integral equations in the Banach space and was studied by Das et al. [30].
Motivated by these, we discuss the existence of a solution for the infinite system of implicit fractional integral equations in the Banach space .
Consider the following infinite system of fractional integral equations
where and
4.1. Existence of Solution on
Assume that
M.1. The functions are continuous and satisfy
for and are continuous functions. Also,
converge to zero for all , where such that for all
M.2. The functions are continuous and there exists
where Also,
M.3. The function is in and nondecreasing
M.4. Define an operator from to as follows:
where
M.5. Let
Also, , is convergent and
Assume .
Theorem 13. Under the assumption (M.1)-(M.5), Equation (18) has at least one solution in .
Proof. For any ,
Therefore,
implies
Hence,
Let us take an operator defined by
where .
Since for each , we have by assumption (M.4) that
Hence, .
Let and with .
For , and by using Hölder inequality with , we have
Since are continuous for all , hence for , we have
Therefore,
Therefore, when ; hence, is continuous on .
Now,
Hence,
Let , so .
Therefore,
Thus, from assumption (M.5) and Remark 9, operator has at least one fixed point in . Hence, Equation (18) has a solution in . This completes the proof.
4.2. Particular Cases
Consider the following infinite system of fractional integral equations involving the Riemann-Liouville fractional integral
where and
Taking in Theorem 13, we obtain the following existence result for the integral Equation (37).
Corollary 14. Under the assumption (M.1)-(M.5), Equation (37) has at least one solution in
Consider the following infinite system of fractional integral equation involving the Hadamard fractional integral
where and
Taking in Theorem 13, we obtain the following existence result for the integral Equation (38).
Corollary 15. Under the assumption (M.1)-(M.5), Equation (38) has at least one solution in
4.3. Numerical Example
The following example illustrates the result of Theorem 13
Example 2. Let us take the following infinite system of fractional integral equations:
where
Here, and also take .
It is easy to see that is continuous for all and
Here,
The function is in and nondecreasing and are continuous and So, and .
If , then
Hence, (M.4) is satisfied. Thus, all conditions of Theorem 13 hold; therefore the system (39) has a solution in .
5. Conclusion
In this article, we established a generalization of Darbo’s fixed point theorem and applied it to the infinite system of fractional order integral equation by another function which is the generalization of many other fractional integrals.
The main contribution of this paper is that it extends the results of Jlei et al. and Mohammadi et al. to established a generalization of Darbo’s fixed point and applied it to investigate the existence of the solution of an infinite system of fractional integral equations in , which is a more general space than .
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 authors contributed equally to writing this paper. All authors read and approved the manuscript.