Existence of Solutions for Nonlinear Integral Equations in Tempered Sequence Spaces via Generalized Darbo-Type Theorem

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 ﬁ rst demonstrate the generalized Darbo-type ﬁ xed 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 ﬁ ndings.


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 c 0 , c, ℓ p , and ℓ 1 of null, convergent, absolutely p-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][11][12]) to demonstrate existence of solutions by means of the notion of MNC, i.e., in ℓ p [13] and in Banach space [14][15][16].
Suppose that E is a Banach space, and suppose also that Bðθ,rÞ = fx ∈ E : kx − θk ≤rg is a closed ball. If Xð≠ ∅Þ ⊆ E, then its closure and convex closure, respectively, will write by symbols X and ConvX. Further, M E will be used to denote the family of bounded (nonempty) subsets of E as well as its subfamily, N E , which consists of all relatively compact sets. The MNC is defined in [17] (see also [18]) as follows.
GðJ j Þ = 0, and then, Since GðJ ∞ Þ ≤ GðJ j Þ for any j, we infer that GðJ ∞ Þ = 0: Banas and Krajewska [19] proposed the generalization of classical spaces c 0 , c, and ℓ ∞ with the help of tempering sequence α = ðα i Þ ∞ i=1 while the tempering sequence means that α i is positive for any i ∈ ℕ and ðα i Þ is nonincreasing, and they defined c α 0 , c α , and ℓ α ∞ which are called the tempered sequence space. Inspired by these constructions, very recently, Rebbani et al. [20] defined the tempered space ℓ α p as follows: where w is the space of real or complex sequences, or simply, we shall write L∶≡ℓ α p . Clearly, ℓ α p is a Banach space endowed with In case of α n = 1 for all n ∈ ℕ, the tempered space ℓ α p coincides with ℓ p , and, in addition, if p = 1, ℓ α p coincides with ℓ 1 . In the same paper, they gave the Hausdorff MNC χ ℓ α p for a nonempty bounded set B α of ℓ α p (1 ≤ p < ∞) by We will use CðI, ℓ α p Þ to denote the collection of all continuous mappings from I = ½0, a (a > 0) to ℓ α p , and CðI, ℓ α p Þ is a Banach space with the norm where ρðsÞ = ðρ n ðsÞÞ ∞ n=1 ∈ CðI, ℓ α p Þ. For any nonempty bounded set E α of CðI, ℓ α p Þ and for s ∈ I, one defines E α ðsÞ = fρðsÞ: ρðsÞ ∈ E α g and hence, its MNC is given by Recall the theorem given in [1] as follows: Suppose that J is a nonempty, closed, bounded, and convex subset of E, and suppose also that S : J ⟶ J is a continuous mapping, and there exists κ ∈ ½0, 1Þ satisfying Then, S has a fixed point.

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 M : (2) M is continuous and nondecreasing hold. We will denote the collection of such functions by M.
Theorem 3. Consider a Banach space E, a nonempty, closed, bounded convex set D ⊆ E, and an arbitrary MNC G. Also, consider a continuous mapping T : D ⟶ D satisfying the inequality for any Xð≠ ∅Þ ⊆ D, where M ∈ M and α, β, γ : ℝ + ⟶ ℝ + are functions such that α, γ are continuous on ℝ + and β is lower semicontinuous which satisfies the relations Then, T has at least one fixed point in D: Proof. Consider a sequence fD n g ∞ n=1 such that D 1 = D and D n+1 = ConvðT D n Þ for n ∈ ℕ. One can find that If there exists n 0 ∈ ℕ satisfying GðD n 0 Þ = 0, then D n 0 is a compact set. With a view of Schauder theorem [21], T has a fixed point in D ⊆ E.

Journal of Function Spaces
Further, assume that GðD n Þ > 0 for n ∈ ℕ: Clearly, fGðD n Þg ∞ n=1 is nonnegative, decreasing, and bounded below sequence. Therefore, fGðD n Þg ∞ n=1 is convergent and Inequality (7) gives If possible, assume r > 0. Letting limsup n⟶∞ in the last inequality, one obtains which yields It follows from the inequality (13) that Consequently, we get β½Mðr, γðrÞÞ = 0: So, γðrÞ = r = 0. Therefore, we have Using the fact D n ⊇ D n+1 and Definition 1, we fairly have which is nonempty, convex, closed subset of D and D ∞ is T invariant. By taking into account Schauder theorem [21], we conclude that (9) holds.
Theorem 5. Consider a Banach space E, a nonempty, closed, bounded convex set D ⊆ E, and an arbitrary MNC G. Also, consider a continuous mapping T : D ⟶ D satisfies the inequality where γ, η : ℝ + ⟶ ℝ + are two functions such that γ is continuous and η is nondecreasing satisfying Then, (9) holds.
Proof. Consider fD n g ∞ n=1 such that Then, we see that Continuing in this way, we obtain If there exists n 0 ∈ ℕ satisfying the condition GðD n 0 Þ = 0, then the set D n 0 is compact. By taking into account Schauder theorem [21], we conclude that (9) holds.
We now assume GðD n Þ > 0 (n ∈ ℕ). Consequently, a sequence fGðD n Þg ∞ n=1 is decreasing and bounded below. Thus, fGðD n Þg ∞ n=1 is convergent and so With a view of (18), one writes

Journal of Function Spaces
Suppose that r > 0 (if possible). We obtain by letting n ⟶ ∞ together with (19) and (23) in the inequality (24) that which yields We therefore have γðrÞ = r = 0, so lim n⟶∞ GðD n Þ = 0. With the help of (22), we obtain nonempty, convex, closed set D ∞ ⊆ D which is T invariant. Hence, by Schauder theorem [21], we reach to the desired result.
Theorem 6. Consider a Banach space E, a nonempty, closed, bounded convex set D ⊆ E, and an arbitrary MNC G. Also, consider a continuous mapping T : D ⟶ D satisfies the inequality where a function γ : ℝ + ⟶ ℝ + is continuous. Then, (9) holds.
Proof. This can be easily obtained by considering in Theorem 5, above.

Theorem 7.
Consider a Banach space E, a nonempty, closed, bounded convex set D ⊆ E, and an arbitrary MNC G. Also, consider a continuous mapping T : D ⟶ D having the property where γ is a continuous function. Then, (9) holds.
Proof. By using the function Mðx, yÞ = x + y, the proof is obtained as an immediate consequence of Theorem 6.

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 ΩðξÞ = ðΩ n ðξÞÞ ∞ n=1 , ξ ∈ I = ½0, a, a > 0: To discuss the result of this section, our assumptions are as below: (1) For n ∈ ℕ, the functions F n : where Moreover, these exist continuous functions A n , B n : I ⟶ ℝ + such that the inequality (2) For n ∈ ℕ, the functions G n : I × I × CðI, ℓ α p Þ ⟶ ℝ are continuous. Also, there exists L k satisfying Further, sup n∈ℕ L n = L and lim n⟶∞ L k = 0 ð35Þ (3) Define an operator H on I × CðI, ℓ α p Þ to CðI, ℓ α p Þ as follows such that 0 < 2Â 1/p < 1 and Journal of Function Spaces Theorem 8. Under assumptions (1)-(4), the system has at least one solution in CðI, ℓ α p Þ, where Proof. For arbitrary fixed ξ ∈ I, which yields It follows from (42) that and hence, Let us define nonempty set which is closed, bounded, and convex subset of CðI, ℓ α p Þ. By assumption (3) and for arbitrary fixed ξ ∈ I, we write Also, Hence, ðHΩÞðξÞ ∈ ℓ α p : Since we have that H maps B into B. We are now claiming that H is continuous on B. For this, suppose ϵ > 0 and ΩðξÞ = ðΩ n ðξÞÞ ∞ n=1 , For arbitrary fixed ξ ∈ I, Considering the fact kΩ − Ωk CðI,ℓ α p Þ < δ and G n is continuous, we get and so, We can find that Moreover, we have The functions F n and G n are continuous for all n ∈ ℕ 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 CðI, ℓ α p Þ.

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 M : ℝ + × ℝ + ⟶ ℝ + 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.