Definition 1 (see [22].)A mapping $$ is called the measure of noncompactness (mnc) in $$ if

(A₁). The family ker $$ is nonempty and ker $$

(A₂). $$

(A₃). $$

(A₄). $$

(A₅). $$ for 0 ≤ s ≤ 1

(A₆). If $$, $$, $$ and $$ then $$ is nonempty

In the Banach space $$, the Hausdorff mncχ 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 ζ be a nonempty, bounded, convex, and closed subset of a Banach space $$, and let $$ is a continuous operator such that there exists a constant m ∈ [0, 1) with the property $$. Then, $$ has a fixed point in the set ζ.

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 ζ is a nonempty, bounded, closed, and convex subset of a Banach space $$, and $$ be continuous operator such that for every nonempty subset Ω of ζ,

where k ∈ (0, 1) and θ ∈ Δ, Δ is the class of function θ : (0, ∞)⟶(1, ∞) satisfying the following condition:

For each {t_n} ⊂ (0, ∞), $$.

Then, $$ has at least one fixed point in ζ.

Theorem 5 (see [18].)Suppose ζ is a nonempty, bounded, convex, and closed subset of a Banach space $$, and $$ be continuous mapping such that for every nonempty subset Ω of ζ,

where ν is the arbitrary mnc, τ is the arbitrary positive constant, $$, Γ be the set of function $$ such that

(W₁). $$ is a strictly increasing and continuous function

(W₂). $$⇔$$, for each {t_k} ⊂ ℝ⁺

Then, $$ has at least one fixed point in ζ.

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 (w₁, w₂) ∈ Z × Z is called a coupled fixed point of mapping $$ if $$ and $$.

Theorem 7 (see [26].)Let ν₁, ν₂, ⋯ν_k be mnc in Banach spaces $$ respectively, and the function $$ is a convex function and h(r₁, r₂, ⋯, r_k) = 0 if and only if r_j = 0 ∀  j = 1, 2, ⋯, k. Then,

is a mnc in $$ where ω_i is the natural projection of ω into $$

Now, we have the following example, which was described in [6], as a result of Theorem 7.

Example 1. Considering ν be a mnc on a Banach space $$, let G(w₁, w₂) = max{w₁, w₂} for any $$, as we can see that G is convex and G(w₁, w₂) = 0 if and only if w₁ = w₂ = 0; hence, all the conditions of Theorem 7 are satisfied. Therefore, $$ defines a mnc in the space $$ where E_i, i = 1, 2 denote the natural projection of E. Similarly, by letting G(w₁, w₂) = w₁ + w₂ for any $$, we have $$ defines a mnc in the space $$ where E_i, i = 1, 2 denote the natural projection of E.

In 2016, Liu et al. [20] introduced Φ, the set of function ϕ : ℝ₊⟶ℝ₊ such that

(ф₁). ϕ is nondecreasing

(ф₂). $$ if and only if $$, for each sequence {t_k} ⊂ (0, ∞)

(ф₃). ϕ is continuous

Berinde [27] introduced the class of function known as comparison function. Let Ψ be the family of all function ψ : ℝ₊⟶ℝ₊ satisfying the following conditions:

(ѱ₁). ψ is monotone increasing

(ѱ₂). $$

Theorem 8. Suppose $$ is a nonempty, bounded, convex, and closed subset of a Banach space $$, and $$ be continuous mapping such that

for all subset Q of $$, where ϕ ∈ Φ and ψ ∈ Ψ and ν is an arbitrary mnc. Then, $$ has at least one fixed point.

Proof. By induction, we define a sequence $$ by letting $$ and $$k ≥ 1. We have $$, $$; therefore, by continuing this process, we obtain

Let there exists an integer K ≥ 0 such that $$. So, $$ is relatively compact and $$; hence, Theorem 2 yields that $$ has a fixed point. So we suppose that $$ for k ∈ ℕ. By Equation (7), we have

Thus, we have

Letting k⟶∞ in (10) and applying (Ψ₂), we have

By using (Φ₂), we obtained

Since $$ and $$ for all k ≥ 1, then from (A₆), $$ is nonempty convex closed set, invariant under $$ and belong to ker ν. Now, from Theorem 2, $$ has a fixed point.

Remark 9. We can get the Darbo’s fixed point Theorem 3, if we take ψ(s) = λs, λ ∈ (0, 1), s > 0 and ϕ(s) = s : (0, ∞)⟶(0, ∞) in above theorem.

Remark 10. We get Theorem 4 if we take ψ(s) = λs, λ ∈ (0, 1), s > 0ϕ(s) = lnθ(s): (0, ∞)⟶(0, ∞) in Theorem 8.

Remark 11. We get Theorem 5 if we take ψ(s) = e^−τs, s, τ > 0 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 ω₁, ω₂ of $$, where ν arbitrary mnc and ϕ and ψ are as in Theorem 8. Then, $$ has at least a coupled fixed point.

Proof. From Example 1, we have $$ defines a mnc in the space $$ where ω_i, i = 1, 2 denote the natural projection of ω. 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.