Measure of noncompactness and a generalized Darbo’s fixed point theorem and its applications to a system of integral equations

Işık et al. (Mathematics 7:862, 2019) presented an interesting generalization of the Banach contraction principle. In this paper, motivated by Işık et al., we give a new extension of the well-known Darbo inequality in a Banach space. Our results provide several generalizations of the Darbo inequality. As an application, we study the existence of solutions for a system of functional integral equations in C[0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C[0,T]$\end{document}. Finally, we expose a genuine example to support the effectiveness of our results.


Introduction and preliminaries
The notion of a measure of noncompactness (MNC) was introduced by Kuratowski [13] in 1930. This concept is a very useful tool in functional analysis, for example, in metric fixed point theory and operator equation theory in Banach spaces. This notion is also applied in the study of existence of solutions for ordinary and partial differential equations, integral, and integro-differential equations. For more details, we refer the reader to [2-6, 8, 9].
We collect some notations and definitions applied throughout this paper. Let R denote the set of real numbers, and let R + = [0, +∞). Let (Λ, · ) be a real Banach space. Moreover, by B(ι, r) we denote the closed ball centered at ι with radius r, and by B r the ball B(0, r). For a nonempty subset of Λ, we denote by and Conv( ) the closure and the closed convex hull of , respectively. Furthermore, we denote by M(Λ) the family of nonempty bounded subsets of Λ and by N (Λ) its subfamily consisting of all relatively compact subsets of Λ.  Işık et al. [12] introduced the following generalization of the Banach contraction principle, where substituting different functions f , we obtain a variety number of contractive inequalities. Theorem 1.4 Let ( , d) be a complete metric space, and let T : → be a continuous self-mapping. Suppose that there exists a function f : for all x, y ∈ . Then T has a unique fixed point.
Let χ : M(Λ) − → R + be a mapping defined by χ(X) = diam X, where diam X = sup{ xy : x, y ∈ X} is the diameter of X. We easily see that χ is a measure of noncompactness in a Banach space E (see [7]). According to this measure of noncompactness, we can easily observe that the Darbo fixed point theorem is a generalization of the Banach fixed point theorem. Using this idea, we want to generalize the result of Işık et al. to a Banach space E.

Main results
Now we state one of the main results in this paper, which extends and generalizes the Darbo fixed point theorem. In fact, motivated by the current work of Işık et al. [12], we give a new extension of the well-known Darbo fixed point theorem in a Banach space. Our results provide several inequalities, which all are generalizations of the Darbo fixed point theorem via substituting appropriate mappings instead of the control function f .

Theorem 2.1
Let Ω be a nonempty, bounded, closed, and convex (NBCC) subset of a Banach space Λ, and let Υ : Ω → Ω be a continuous operator such that is such that lim t→0 + (t) = 0, (0) = 0, and χ is an arbitrary MNC. Then Υ has at least one fixed point in Ω.
If there exists an integer N ∈ N such that χ(Ω N ) = 0, then Ω N is relatively compact, and Theorem 1.2 implies that Υ has a fixed point. So, we assume that χ(Ω N ) > 0 for each n ∈ N .
It is clear that {Ω n } n∈N is a sequence of NBCC sets such that Thus the sequence { (χ(Ω n ))} is nonincreasing. Since is bounded below, there exists L ∈ R + such that lim n→∞ (χ(Ω n )) = L. We know that {χ(Ω n )} n∈N is a positive decreasing and bounded below sequence of real numbers. Thus {χ(Ω n )} n∈N is a convergent sequence. Let lim n→∞ χ(Ω n ) = r.
In view of condition (3), we have Taking the limsup in this inequality, we have Therefore lim n→∞ χ(Ω n ) = 0. According to axiom (6 • ) of Definition 1.1, we have that the set Ω ∞ = ∞ n=1 Ω n is an NBCC set and is invariant under the operator Υ and belongs to ker χ . Then in view of the Schauder theorem, Υ has a fixed point.

Remark 2.3 Note that Theorem 2.1 is a generalization of the Darbo fixed point theorem.
Since Υ : → is a Darbo mapping, there exists k ∈ [0, 1) such that and so Therefore .
Thus the Darbo inequality is a particular case of the contractive inequality of Theorem 2.1.
In the following corollaries, we provide examples of the function for equation (3) in Theorem 2.1 (the contractive inequality from Theorem 2.1) that have no Darbo constant k.
Taking (t) = te t , for all t ≥ 0, we deduce the following corollary.

Corollary 2.4
Let Ω be an NBCC subset of a Banach space Λ, and let Υ : Ω → Ω be a continuous operator such that for all ⊆ Ω, where χ is an arbitrary MNC. Then Υ has at least one fixed point in Ω.
Taking (t) = sinh t for t ≥ 0, we deduce the following corollary.

Corollary 2.5
Let Ω be an NBCC subset of a Banach space Λ, and let Υ : Ω → Ω be a continuous operator such that for all ⊆ Ω, where χ is an arbitrary MNC. Then Υ has at least one fixed point in Ω.

Corollary 2.6
Let Ω be an NBCC subset of a Banach space Λ, and let Υ : Ω → Ω be a continuous operator such that for all ⊆ Ω, where χ is an arbitrary MNC. Then Υ has at least one fixed point in Ω.

Corollary 2.7
Let Ω be an NBCC subset of a Banach space Λ, and let Υ : Ω → Ω be a continuous operator such that for all ⊆ Ω, where χ is an arbitrary MNC. Then Υ has at least one fixed point in Ω.

Coupled fixed point
Bhaskar and Lakshmikantham [11] have introduced the notion of a coupled fixed point and proved some coupled fixed point theorems for some mappings and discussed the existence and uniqueness of solutions for periodic boundary value problems.

Theorem 3.3
Let Ω be an NBCC subset of a Banach space Λ, and let Υ : Ω × Ω → Ω be a continuous function such that for any subset 1 , 2 of Ω, where χ is an arbitrary MNC, and is as in Theorem 2.1. In addition, we assume that is a subadditive mapping. Then Υ has at least one coupled fixed point.

Proof 3.4
We define the mapping Υ : It is clear that Υ is continuous. We show that Υ satisfies all the conditions of Theorem 2.1. Let ⊂ Ω 2 be a nonempty subset. We know that χ( ) = χ( 1 ) + χ( 2 ) is an MNC [7], where 1 and 2 denote the natural projections of into Λ. From (9) we have Now, from Theorem 2.1 we deduce that Υ has at least one fixed point, which implies that Υ has at least one coupled fixed point.
Consequently, all the conditions of Theorem 4.1 are satisfied. Hence the system of integral equations (18) has at least one solution that belongs to the space C[0, 1] × C[0, 1].