Fixed Point Theorems for Modified Generalized F-Contraction in G-Metric Spaces

In this paper, we introduce new notions of generalized F-contractions of type (S) and type (M) in G-metric spaces. Some fixed point theorems are established using these new notions. A suitable example is also provided to support our results.


Introduction
A metric space is a nonempty set X equipped with a map d of two variables which helps us to measure the distance between two points. In higher mathematics, we need to find the distance not only between numbers and vectors, but also between sequences and functions. In order to find an appropriate concept of a metric space, numerous approaches exist in this sphere. A number of generalizations of a metric space have been discussed by many eminent mathematicians. In 2006, Sims and Mustafa introduced the perception of G-metric space and gave an important generalization of a metric space as follows: Definition 1.1. [3] Let X be a non empty set and G : X 3 → [0, ∞) be a map which satisfies the following properties: (i) G(x, y, z) = 0 if x = y = z; (ii) 0 < G(x, x, y) whenever x = y; (iii) G(x, x, y) ≤ G(x, y, z), y = z; (iv) G(x, y, z) = G(x, z, y) = G(y, x, z) = G(z, x, y) = G(y, z, x) = G(z, y, x); (v) G(x, y, z) ≤ G(x, a, a) + G(a, y, z), ∀x, y, z, a ∈ X. Then, the function G is said to be G-metric on X and the pair (X, G) is known as G-metric space.
In 1922, Banach established a useful result in fixed point theory regarding a contraction mapping, known as the Banach contraction principle. Definition 1.2. [1] Let (X, d) be a complete metric space and let f : X → X be a self-mapping. Let d(f x, f y) < d(x, y) holds for all x, y ∈ X with x = y. Then, f is called a contraction known as Banach contraction.
In 2012, Wardowski [6] gave a new contraction known as F-contraction and proved fixed point theorem concerning F-contractions. In this manner, Wardowski conclude the Banach contraction principle in a different way from the eminent results from the literature. Piri and Kumam [5] also established Wardowski type fixed point theorems in complete metric spaces. Motivated by the perception of Dung and Hang [2], recently Piri and Kumam [5] generalized the concept of generalized F-contraction and established 2 M. Kumar and S. Arora some fixed point theorems for such kind of functions in complete metric spaces, by addition of four terms Wardowski [6] defined the F-contraction as follows: Definition 1.3. [6] Let (X, d) be a metric space and let f : (F2) For every sequence {a n } of natural numbers, lim n→∞ a n = 0 if and only if lim n→∞ F (a n ) = −∞.
Wardowski [6] gave some examples of ℑ as follows: Remark 1.4. Let f : R + → R be defined as F = ln(β), then F ∈ ℑ. Now, F-contraction changes to a Banach contraction. Consequently, the Banach contractions are special case of F-contractions. There are F-contractions which are not Banach contractions(see [6]).
F-weak contraction was established by Wardowski and Dung in 2014, which is defined as follows: Theorem 1.6.
[7] Let (X, d) be a complete metric space and let T : X → X be an F-weak contraction. If F or T is continuous, then T has a unique fixed point x * ∈ X and the sequence {T n x} converges to x * for every x ∈ X, where n varies from 1 to ∞. [2] investigated the concept of generalized F-contraction and proved useful fixed point results for such kind of functions. Definition 1.7. [2] Let (X, d) be a metric space and f : X → X be a self-mapping. Then, f is called a generalized F-contraction on (X, d), if there exist F ∈ ℑ and δ > 0 such that for all x, y ∈ X,
They gave the notation F to denote the class of all maps F : R + → R which fulfil the conditions (F 1), (F 2) and (F 3 ′ ). Piri and Kumam also proved some useful fixed point results for metric spaces. Now, the conditions (F 3) and (F 3 ′ ) are not associated with each other. For example, for q ≥ 1, F (β) = −1 β q , then F meet the conditions (F1) and (F2) but it does not fulfil (F3), while it fulfils the condition (F 3 ′ ). In view of this, it is significant to observe the sequel of Wardowski [6] with the functions F ∈ ℑ rather than F ∈ F.
The goal of our paper is to propose new notions of modified generalized F-contraction of type (S) and type (M) in G-metric spaces and prove fixed point theorems for such functions.

Main Results
Throughout the paper, we use the following notations. S G is the class of all functions F : (0, ∞) → R such that (S1) F is strictly increasing, that means x < y implies that F x < F y, where x, y are positive reals. (S2) lim n→∞ a n = 0 if and only if lim n→∞ F (a n ) = −∞, for every sequence {a n } of positive numbers. (S3) F is continuous on(0,∞). M G is the class of all maps F : (0, ∞) → R such that (M1) F is strictly increasing, that means x < y implies that F x < F y, where x, y are positive reals. (M2) lim n→∞ a n = 0 if and only if lim n→∞ F (a n ) = −∞, for every sequence {a n } of positive numbers. (M3) ∃ m ∈ (0, 1) such that lim a→0 + a m F (a) = 0.

Definition 2.2.
Let (X, G) be a G-metric space and f : X → X be a mapping. Then, f is known as Let f : X → X be defined as f x = 3, when x ∈ [0, 2) and f x = 1 2 , if x = 5. Now, (X, G) is complete metric space. By choosing F x = ln x and λ = ln 1 2 , we get that f is modified generalized F-contraction of type (S) and type (M). Theorem 2.4. Let (X, G) be a complete G-metric space and f : X → X be a modified generalized Fcontraction of type (S). Then, f has a unique fixed point u ∈ X and the sequence {f n (x 0 )}, where n ∈ N, converges to u for each u ∈ X.
Proof. Let x 0 ∈ X and {x n } be the Picard sequence, that is, Therefore, If there exists n ∈ N such that we get a contradiction, because λ > 0. Therefore, Therefore, Using the condition of (S1), F is strictly increasing. Therefore, So, {G(x n+1 , x n , x n )} is non negative decreasing sequence of real numbers, where n ∈ N . Thus, we conclude that lim n→∞ G(x n+1 , x n , x n ) = µ ≥ 0. Now, we claim that µ = 0. Let us suppose that µ > 0. Also, {G(x n+1 , x n , x n )} is non negative decreasing sequence of real numbers, where n ∈ N. Therefore, Again, by using the assumption (S1), F is is strictly increasing. Therefore, F µ ≤ H(G(x n+1 , x n , x n )). Using equation (2.3), we obtain Therefore,  Further, we claim that {x n } ∞ n=1 is a Cauchy sequence. Arguing by contradiction, let us suppose that there exist δ > 0 and sequences {a(n)} ∞ n=1 , {b(n)} ∞ n=1 such that for each n ∈ N, So, the above inequality becomes Now, we claim that Again by contradiction, let us suppose that ∃ q ≥ N , such that Combining (2.6),(2.9),(2.11), we get which is contradiction and hence our supposition is wrong. Thus, we get From (2.10) and assumption of the theorem, we obtain Finally, we show that f u = u. Two cases arise, (i) ∀ n ∈ N ∃ k n ∈ N, k n > k n−1 , k 0 = 1 and x kn+1 = f u.
In the second case, using the assumption of the Theorem 2.4, we get for each n ≥ m 3 . From (S3), (2.13) and taking limit when n → ∞, the above inequality becomes λ + H(G(u, f u, f u)) ≤ H(G(u, f u, f u)), which is contradiction. So our supposition is wrong. Therefore, f u = u. Next, we show that f has atmost one fixed point. On the contrary, we suppose that u and v are two fixed points of f , such that f u . It is a contradiction. Therefore, G(u, v, v) = 0, that means u = v. This establishes that the fixed point of f is unique. Theorem 2.5. Let (X, G) be a complete G-metric space and f : X → X be a continuous modified generalized F-contraction of type (M). Then, f has a unique fixed point u ∈ X and the sequence f n (x 0 ), where n ∈ N converges to u, for each u ∈ X.