Abstract
Two concepts—one of Darbo-type theorem and the other of Banach sequence spaces—play a very important and active role in ongoing research on existence problems. We first demonstrate the generalized Darbo-type fixed point theorems involving the concept of continuous functions. Keeping one of these theorems into our account, we study the existence of solutions of system of nonlinear integral equations in the setting of tempered sequence space. Moreover, a very interesting and illustrative example is designed to visualize our findings.
1. Introduction and Preliminaries
Darbo [1] constructed the fixed point theorem, and later, researchers called this widely studied theorem by his name, that is, “Darbo fixed point theorem” wherein he enforced the technique of measure of noncompactness (shortly, MNC) while Kuratowski [2] was the first who described the idea of MNC. Many researchers are employing Darbo’s theorem to demonstrate the existence or solvability of several functional equations (linear or nonlinear) in conjunction with different kind of Banach sequence spaces or simply called Banach spaces. Recently, the infinite system of several kinds of differential equations was considered by Banas and Lecko [3], Mursaleen et al. [4, 5], and Mohiuddine et al. [6] to obtain the existence of solutions in the framework of Banach spaces, namely, the spaces , , , and of null, convergent, absolutely -summable, and absolutely summable sequences in conjunction with the Dorbo-type theorem. The reader can refer to the recent monographs [7, 8] on the normed/paranormed sequence spaces and related topics.
The integral equations play a significant contribution in diverse branches of science and engineering as well as this theory is applicable in several real life problems such as gas kinetic theory, neutron transportation, and radiation [9]. Most recently, the researchers used different kinds of integral equations (infinite system) (see [10–12]) to demonstrate existence of solutions by means of the notion of MNC, i.e., in [13] and in Banach space [14–16].
Suppose that is a Banach space, and suppose also that is a closed ball. If , then its closure and convex closure, respectively, will write by symbols and Conv. Further, will be used to denote the family of bounded (nonempty) subsets of as well as its subfamily, , which consists of all relatively compact sets. The MNC is defined in [17] (see also [18]) as follows.
Definition 1. A mapping is called MNC in if (i), which implies gives be relatively compact(ii) Also, (iii)(iv)(v)(vi)(vii) for and , and then, Note that Since for any , we infer that
Banas and Krajewska [19] proposed the generalization of classical spaces , , and with the help of tempering sequence while the tempering sequence means that is positive for any and is nonincreasing, and they defined , , and which are called the tempered sequence space. Inspired by these constructions, very recently, Rebbani et al. [20] defined the tempered space as follows: where is the space of real or complex sequences, or simply, we shall write . Clearly, is a Banach space endowed with
In case of for all , the tempered space coincides with , and, in addition, if , coincides with . In the same paper, they gave the Hausdorff MNC for a nonempty bounded set of () by
We will use to denote the collection of all continuous mappings from () to , and is a Banach space with the norm where . For any nonempty bounded set of and for , one defines and hence, its MNC is given by
Recall the theorem given in [1] as follows:
Theorem 2. Suppose that is a nonempty, closed, bounded, and convex subset of , and suppose also that is a continuous mapping, and there exists satisfying Then, has a fixed point.
2. Dorbo-Type Fixed Point Theorems
In order to discuss our Dorbo-type theorems, we first recall the set of functions which has been recently used in [13] as follows: Consider the function such that (1) for (2) is continuous and nondecreasing(3)hold. We will denote the collection of such functions by . The example of aforesaid kind of function is .
Theorem 3. Consider a Banach space , a nonempty, closed, bounded convex set , and an arbitrary MNC . Also, consider a continuous mapping satisfying the inequality for any , where and are functions such that are continuous on and is lower semicontinuous which satisfies the relations Then,
Proof. Consider a sequence such that and for . One can find that We obtain in a similar way that If there exists satisfying , then is a compact set. With a view of Schauder theorem [21], has a fixed point in .
Further, assume that for Clearly, is nonnegative, decreasing, and bounded below sequence. Therefore, is convergent and
Inequality (7) gives
If possible, assume . Letting in the last inequality, one obtains
which yields
It follows from the inequality (13) that
Consequently, we get So, . Therefore, we have
Using the fact and Definition 1, we fairly have
which is nonempty, convex, closed subset of and is invariant. By taking into account Schauder theorem [21], we conclude that (9) holds.
Theorem 4. Consider a Banach space , a nonempty, closed, bounded convex set , and an arbitrary MNC . Also, consider a continuous mapping satisfying the inequality for , where are functions such that are continuous on and is lower semicontinuous satisfies relation (8). Then, (9) holds.
Proof. This result can be obtained by considering the function in Theorem 3.
Theorem 5. Consider a Banach space , a nonempty, closed, bounded convex set , and an arbitrary MNC . Also, consider a continuous mapping satisfies the inequality where are two functions such that is continuous and is nondecreasing satisfying Then, (9) holds.
Proof. Consider such that Then, we see that Continuing in this way, we obtain If there exists satisfying the condition , then the set is compact. By taking into account Schauder theorem [21], we conclude that (9) holds.
We now assume (). Consequently, a sequence is decreasing and bounded below. Thus, is convergent and so
With a view of (18), one writes
Suppose that (if possible). We obtain by letting together with (19) and (23) in the inequality (24) that which yields
We therefore have , so . With the help of (22), we obtain nonempty, convex, closed set which is invariant. Hence, by Schauder theorem [21], we reach to the desired result.
Theorem 6. Consider a Banach space , a nonempty, closed, bounded convex set , and an arbitrary MNC . Also, consider a continuous mapping satisfies the inequality where a function is continuous. Then, (9) holds.
Proof. This can be easily obtained by considering in Theorem 5, above.
Theorem 7. Consider a Banach space , a nonempty, closed, bounded convex set , and an arbitrary MNC . Also, consider a continuous mapping having the property where is a continuous function. Then, (9) holds.
Proof. By using the function , the proof is obtained as an immediate consequence of Theorem 6.
3. Existence of Solutions for Integral Equation
We are studying the existence of solutions for an infinite system of the nonlinear integral equation which is considered as follows: where ,
To discuss the result of this section, our assumptions are as below: (1)For , the functions are continuous with where Moreover, these exist continuous functions such that the inequality holds, where (2)For , the functions are continuous. Also, there exists satisfying Further, (3)Define an operator on to as follows (4)Let such that and
Theorem 8. Under assumptions (1)–(4), the system has at least one solution in , where
Proof. For arbitrary fixed ,
which yields
It follows from (42) that
and hence,
Let us define nonempty set
which is closed, bounded, and convex subset of . By assumption (3) and for arbitrary fixed , we write
Also,
Hence,
Since
we have that maps into . We are now claiming that is continuous on . For this, suppose and satisfying
For arbitrary fixed ,
Considering the fact and is continuous, we get
and so,
It follows from (50) and (52) that
which gives
Therefore,
Hence, is continuous on
Now, for arbitrary fixed and , we write
or
Operating on both sides of (57), we obtain
We thus have
As , so applying Theorem 7 for gives that has at least one fixed point on , i.e., the considered system admits a solution in
Example 1. In order to demonstrate Theorem 8, we consider an infinite system of integral equation as follows: for and . For this demonstration, write Further, take and let , . If for some fixed , then as the series Therefore, for arbitrary fixed , one has and hence, Let . Then, Here Also, Again, We can find that Moreover, we have The functions and are continuous for all as well as the conditions (1)–(4) are fulfilled so with a view of Theorem 8, we reach to our conclusion that the considered system (60) admits a solution in .
4. Concluding Remarks
In this work, we linked three different disciplines such as the concept of measure of noncompactness (MNC), the theory of existence of solutions for functional equations, and the Banach space theory, particularly, in tempered sequence spaces. We first discussed some generalized Dorbo-type fixed point theorems by considering the arbitrary MNC and then discussed the existence of solutions for nonlinear integral equation (infinite system) by taking aforesaid newly investigated Dorbo-type theorem in tempered sequence spaces. Finally, we constructed an illustrative example by taking an integral equation to validate our result.
It is worth noting to the reader that one can obtain the results of Section 2 by taking into account another suitable function instead of and consider two dimensional integral (or fraction integral) equation to extend the results of Section 3.
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.
Acknowledgments
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (G: 735-156-1441). The authors, therefore, acknowledge with thanks DSR for technical and financial support.