Common Fixed Point Theorems for Contractive Mappings of Integral Type in 
 G
 -Metric Spaces and Applications

Two common fixed point theorems for weakly compatible mappings satisfying contractive conditions of integral type in 
 
 G
 
 -metric spaces are demonstrated. The results obtained in this paper generalize and differ from a few results in the literature and are used to prove the existence and uniqueness of common bounded and continuous solutions for certain functional equations and nonlinear Volterra integral equations. A nontrivial example is included.


Introduction
The Banach fixed point theorem which was first presented by Banach in 1922 is a significant result in fixed point theory. Because of its importance in proving the existence of solutions for functional equations, nonlinear Volterra integral equations and nonlinear integro-differential equations, this result has been extended in many different directions (see, e.g.,  and the references cited therein). In particular, Rhoades [12] and Branciari [4] generalized the Banach fixed point theorem and gave the following fixed point theorems, respectively.
Theorem 1 (see [12]). Let f be a mapping from a complete metric space ðX, dÞ into itself satisfying where φ ∈ Φ 4 . Then, f has a unique fixed point in X.
Theorem 2 (see [4]). Let ðX, dÞ be a complete metric space and f : X → X be a mapping satisfying where φ ∈ Φ 1 and c ∈ ½0, 1Þ is a constant. Then, f has a unique fixed point a ∈ X such that lim n→∞ f n x = a for each x ∈ X.
In 2013, Gupta and Mani [21] obtained the existence and uniqueness of a fixed point for contractive mappings of an integral type in complete metric spaces by using iterative approximations. In 2007, Kumar et al. [6] proved a common fixed point theorem for a pair of compatible mappings satisfying a contractive inequality of integral type, which improves Theorem 2.
Theorem 3 (see [6]). Let ðX, dÞ be a complete metric space and f , g : X → X be compatible mappings such that f X ð Þ ⊆ g X ð Þ, g is continuous, where φ ∈ Φ 1 and c ∈ ½0, 1Þ is a constant. Then, f and g have a unique common fixed point in X.
In 2006, Mustafa and Sims [9] introduced a new concept of generalized metric space called G-metric space. From then on, lots of research works have been carried out on generalizing contractive conditions for different contractive mappings satisfying various known properties in G-metric spaces [1-3, 5, 10, 11, 13, 15, 19, 20]. In 2018, Gupta et al. [19] proved some fixed point theorems for the functions satisfying ϕ -contraction and mixed g-monotone property in G-metric spaces. In 2015, Gupta and Deep [20] gave a few common fixed point theorems using the property E.A. in the setting of G-metric and fuzzy metric spaces by taking a set of three conditions for self-mappings. In 2011, Aydi [1] proved a fixed point theorem for mappings satisfying a ðψ, ϕÞ-weakly contractive condition in G-metric spaces.
Theorem 4 (see [1]). Let ðX, GÞ be a complete G-metric space and f be a mapping from X into itself satisfying where ψ, ϕ ∈ Φ 2 . Then, f has a unique fixed point u ∈ X and f is G-continuous at u.
In 2012, Aydi [2] obtained the following common fixed point theorem for a pair of mappings involving a contractive condition of integral type in G-metric spaces.
Theorem 5 (see [2]). Let ðX, GÞ be a G-metric space and f , g be two mappings from X into itself such that where φ ∈ Φ 1 and α ∈ ½0, 1Þ is a constant. If f ðXÞ ⊆ gðXÞ and gðXÞ is a complete subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point.
The objective of this paper is both to introduce two new classes of contractive mappings of integral type in the setting of G-metric spaces and to prove the existence and uniqueness of points of coincidence and common fixed points for these mappings. Our results extend Theorem 5, are different from Theorem 4, and are used to show solvability of the functional equations arising in dynamic programming and nonlinear Volterra integral equations. A nontrivial example is given.
(1) A point x ∈ X is said to be a fixed point of T if Tx = x.
(2) A point x ∈ X is said to be a coincidence point of S and T if Tx = Sx and w = Sx = Tx is said to be a point of coincidence of S and T.
(3) A point x ∈ X is said to be a common fixed point of S and T if x = Tx = Sx.
Definition 15. A pair of self-mappings f and g in a G-metric space ðX, GÞ are said to be weakly compatible if for any x ∈ X, the equality f x = gx gives that f gx = gf x.
Lemma 16 (see [14]). Let X be a nonempty set and f , g : X → X be weakly compatible mappings. If f and g have a unique point of coincidence w ∈ X, then w is the unique common fixed point of f and g.

Main Results
Now, we study the existence and uniqueness of points of coincidence and common fixed points for contractive mappings (12) and (51) below in G-metric spaces, respectively.
Theorem 19. Let ðX, GÞ be a G-metric space, f and g : X → X be two mappings satisfying where ðφ, ψ, ϕÞ If f ðXÞ ⊆ gðXÞ and gðXÞ is a complete subset of X, then f and g have a unique point of coincidence in X. Furthermore, if f and g are weakly compatible mappings, then f and g have a unique common fixed point in X.
In light of (G1), (G3), (G5), and (13)-(17), we get that Now we assert that G n ≤ G n−1 , ∀n ∈ ℕ. Suppose that there exists some n 0 ∈ ℕ satisfying G n 0 > G n 0 −1 . It follows from (12), which is a contradiction. Therefore, G n ≤ G n−1 for all n ∈ ℕ and It is apparent that the sequence fG n g n∈ℕ 0 is nonincreasing and bounded, which implies that there exists r with Now, we demonstrate that r = 0. Suppose that r > 0. On account of (12), (20), and (21), ðφ, ψ, ϕÞ ∈ Φ 1 × Φ 2 × Φ 3 and Lemma 17, we deduce that 4 Journal of Function Spaces which is impossible. Thus, r = 0. That is, It follows from (G3), (G4), and (23) that which yield that Next, we verify that f f x n g n∈ℕ 0 is a G-Cauchy sequence. Suppose that f f x n g n∈ℕ 0 is not a G-Cauchy sequence. It follows from Lemma 10 that there exist a constant ε > 0 and two subsequences f f x mðkÞ g k∈ℕ and f f x nðkÞ g k∈ℕ of f f x n g n∈ℕ 0 such that nðkÞ is minimal in the sense that which means that By means of (G3)-(G5) and Lemma 13, we deduce that Letting k → ∞ in (27)-(32) and using (23) and (25), we obtain that
Since gðXÞ is complete, it follows that there exists w ∈ gðXÞ such that In light of Lemma 8 and w ∈ gðXÞ, there exists a ∈ X satisfying ga = w and Next, we prove ga = f a. Suppose that ga ≠ f a. In view of (12), (13), (25), (46), ðφ, ψ, ϕÞ ∈ Φ 1 × Φ 2 × Φ 3 , and Lemmas 12 and 17, we obtain that 7 Journal of Function Spaces which is absurd. Consequently, w = ga = f a, that is, w is a point of coincidence of f and g.
Lastly, we certify that f and g have a unique point of coincidence in X. Assume that there exists b ∈ X with f b = gb ≠ f a. In terms of (13), (G2), and Lemma 13, we receive that According to (12), (49), ðφ, ψ, ϕÞ ∈ Φ 1 × Φ 2 × Φ 3 , and Lemma 18, we gain that which is contradictive. Therefore, f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible mappings, by Lemma 16, we know that f and g have a unique common fixed point in X. This completes the proof. Similar to the argument of Theorem 19, we derive the following result and omit its proof. 8 Journal of Function Spaces Theorem 20. Let ðX, GÞ be a G-metric space, f and g : X → X be two mappings satisfying where ðφ, ψ, ϕÞ ∈ Φ 1 × Φ 2 × Φ 3 and If f ðXÞ ⊆ gðXÞ and gðXÞ is a complete subset of X, then f and g have a unique point of coincidence in X. Furthermore, if f and g are weakly compatible mappings, then f and g have a unique common fixed point in X.
Remark 21. In case ψðtÞ = t, ϕðtÞ = ð1 − λÞt, ∀t ∈ R + and λ ∈ ð0, 1Þ is a constant, then Theorems 19 and 20 reduce to results, which include Theorem 5 as a special case. The following example shows that Theorems 19 and 20 generalize substantially Theorem 5 and differ from Theorem 4.

Journal of Function Spaces
It is obvious that