Fixed points of rational type contractions in G-metric spaces

Abstract: We establish three major fixed-point theorems for functions satisfying generalized rational type almost contraction conditions. Firstly we consider the case of a single mapping, secondly we look at the case of a triplet of mappings and we conclude by the case of a family of mappings. The theorems we present generalize similar results already obtained by Abbas, Rhoades, Gaba, and others. The operators we consider are all of the weakly Picard type.


Introduction and preliminaries
Recently, applications of G-metric spaces, in the fields like optimization theory, differential and integral equations, have been discovered and this has generated a lot of interest for these type of spaces (see Mustafa, Obiedat, & Awawdeh, 2008;Mustafa & Sims, 2006;Shoaib, Arshad, & Kazmi, 2017). Their relevance is no more to be demonstrated as it has been extensively discussed in the literature. In this paper, we prove three main fixed point results in that setting. We propose generalizations which ensure existence results for fixed points, and to this goal we investigate the character of the sequence of iterates {T n x} ∞ n=0 (resp. {T i (x i−1) } ∞ i=0 ) where T:X → X (resp. T i :X → X ) is (resp. are) the map (resp. maps) under consideration, x ∈ X and X a complete G-metric space. More precisely, we consider mappings that satisfy a rational type almost contraction and the results we present are comparable to previous ones already obtained in Gaba (2017). The paper is divided in two major sections, a first section which gives an introduction and some preliminaries and a second section which deals with the statements of results. The second section contains three subsections of which the first two present proofs making use of classical arguments (already used in Gaba, 2017), and of which the third one presents a result based on -series, see Sihag et al. (2014). The elementary facts about G-metric spaces can be found in Gaba (2017), Mustafa and Sims (2006) and the references therein. We give here a summary of these prerequisites.

PUBLIC INTEREST STATEMENT
In this paper, we give fixed point results for a certain type of functions (called rational contractions). Roughly speaking, the biggest motivation comes from the fact that, using fixed point theory in metric spaces it is possible to obtain sufficient conditions for studying and solving differential and variational problems arising in the applied sciences. These problems from the applied sciences try to describe our daily activities as mathematical problems. Definition 1.1 (see [Mustafa & Sims, 2006, Definition 3]) Let X be a nonempty set, and let the function G:X × X × X → [0, ∞) satisfy the following properties: (G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = …, (symmetry in all three variables); (G5) for any points x, y, z, a ∈ X Then (X, G) is called a G-metric space.
Definition 1.2 (see [Mustafa & Sims, 2006]) Let (X, G) be a G-metric space, and let (x n ) n≥1 be a sequence of points of X, therefore, we say that the sequence ( that is, for any > 0, there exists N ∈ ℕ such that G(x, x n , x m ) < , for all, n, m ≥ N. We call x the limit of the sequence and write x n → x or lim n→∞ x n = x.
Definition 1.4 (See Mustafa & Sims, 2006) Let (X, G) be a G-metric space. A sequence (x n ) n≥1 is called a G-Cauchy sequence if for any > 0, there is N ∈ ℕ such that G(x n , x m , x l ) < for all n, m, l ≥ N, that is G(x n , x m , x l ) → 0 as n, m, l → +∞. In a G-metric space (X, G), the following are equivalent Theorem 1.8 (see Mustafa & Sims, 2006) A G-metric G on a G-metric space (X, G) is continuous on its three variables.
We conclude this introductory part with: (a, y, z)].

The results
This section on our main results begins with the case of a single map.

Single maps
Theorem 2.1 Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition: for all x, y, z ∈ X, where a, b, c are non-negative reals. Then (a) T has at least one fixed point ∈ X; (b) for any x ∈ X, the sequence (T n x) n≥1 G-converges to a fixed point of T; (c) if , ∈ X are two distinct fixed points, then Proof We imitate the steps of the proof of [Gaba, 2017 Theorem 2.1].
Let x 0 ∈ X be arbitrary and construct the sequence (x n ) n≥1 such that x n+1 = Tx n . Moreover, we may assume, without loss of generality that x n ≠ x m for n ≠ m.
For the triplet (x n , x n+1 , x n+1 ), and by setting d n = G(x n , x n+1 , x n+1 ), we have: Claim: The sequence ( n ) n≥1 is a non-increasing sequence of non-negative reals.
Indeed, since we have it is very clear that for any natural number n ∈ ℕ, 0 ≤ n < 1, and so d n < d n−1 . We then have the following consecutive equivalences: Hence Therefore hence For any m, n ∈ ℕ, m > n, since we have the above translates to and we obtain Put b k = k ⋯ 1 and observe that Hence therefore In other words, (x n ) n≥1 is a G-Cauchy sequence so G-converges to some ∈ X.
Claim: is a fixed point of T.
For the triplet (x n+1 , T , T ) in (2.1), we get On taking the limit on both sides of (2.2), and using the fact that the function G is continuous, we have If is a fixed point of T with ≠ , then Therefore,

✷
The following two corollaries, particular cases of Theorem 2.1, are of interest for us, due to our previous work in Gaba (2017).

Corollary 2.2 Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition:
for all x, y, z ∈ X. Then (a) T has at least one fixed point ∈ X; (b) for any x ∈ X, the sequence (T n x) n≥1 G-converges to a fixed point; Let X = 0, 1 2 , 1 and let G: (X, G) is a symmetric G-complete G-metric space.
We have Again, Finally, Therefore T satisfies all the conditions of Theorem 2.2. Also, T has two distinct fixed points {0, 1 2 } and Corollary 2.4 (Compare [Gaba, 2017, Theorem 2

.1]) Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition:
for all x, y, z ∈ X. Then (a) T has at least one fixed point ∈ X; (b) for any x ∈ X, the sequence (T n x) n≥1 G-converges to a fixed point; (c) if , ∈ X are two distinct fixed points, then Proof Apply Theorem 2.1 with a = 1, b = c = 1.

✷
The previous results naturally extend if we consider a partially ordered complex valued G-metric space. Moreover, one can replace the non-negative real constants a, b, c by non-negative real valued functions.
We can define a partial order ≲ on the set ℂ of complex numbers by setting, for any z 1 , z 2 ∈ ℂ, Moreover, on partial ordered G-metric space, the convergence of a sequence is interpreted in the canonical way, i.e. for a sequence (x n ) n≥1 ⊆ (X, G, ⪯) where (X, G, ⪯) is a partial ordered complex valued G-metric space, Similarly for G-Cauchy sequences. Furthermore, a self mapping T defined on a partial ordered Gmetric space (X, G, ⪯) is nondecreasing if Tx ⪯ Ty whenever x ⪯ y, for x, y ∈ X.
We then state the result: Theorem 2.5 Let (X, G, ⪯) be a symmetric, G-complete, complex valued G-metric space. Assume that if (x n ) n≥1 is a nondecreasing sequence of elements of X such that x n G-converges tox * , then x n ⪯ x * for all n ∈ ℕ. Let T:X → X be a nondecreasing mapping such that: a(x, y, z), b: = b(x, y, z), c: = c(x, y, z) are non-negative real valued functions.
If there exists x 0 ∈ X with x 0 ⪯ Tx 0 , then (i) T has at least one fixed point ∈ X; (ii) for any x ∈ X, the sequence (T n x) n≥1 G-converges to a fixed point; (iii) if , ∈ X are two distinct fixed points, then Proof Following the steps of the proof of Theorem 2.1, it is very easy to see that the sequence of iterates T n x 0 , n = 1, 2, ⋯ , is nondecreasing and G-converges to some ∈ X. Therefore x n ⪯ for all n ∈ ℕ. Now applying (2.5) to the triplet (x n+1 , T , T ) we have: Now taking the limit as n → ∞, and using the fact that the function G is continuous, we have: i.e. G( , T , T ) = 0, thus T = .
If is a fixed point of T with ≠ , then Therefore, ✷ Another variant of Theorem 2.1 goes as follows: Theorem 2.6 Let (X, G) be a symmetric G-complete G-metric space and T be a mapping from X to itself. Suppose that T satisfies the following condition: (2.5) G(Tx, Ty, Tz) ≲ a.G (Tx, y, z) G(y, Ty, Ty) + c.G(z, Tz, Tz) + 1  G(x, y, z), (2.6) G(Tx, Ty, Tz) ≤ K(x, y, z)G(x, y, z), for all x, y, z ∈ X, where a: = a(x, y, z), b: = b(x, y, z), c: = c(x, y, z) are non-negative real valued functions and Then T has at least one fixed point ∈ X.
Remark 2.7 In general, the self mapping T in Theorem 2.6 (as well as in Theorem 2.1) is a weakly 1 Picard operator. Moreover, the reader can convince him/her-self that if and are fixed points of T in X, a lower bound can be found for G( , , ) = G( , , ) (see point (c) in Theorem 2.1). Furthermore, Theorem 2.6 can be expressed in a setting of a partially ordered complex valued G-metric space.
We conclude this subsection by proving the following result, which presents a reverse rational type contraction. Actually, this mapping can be classified as an expansion type mapping.

Theorem 2.8 Let (X, G) be a symmetric G-complete G-metric space and T be an onto self mapping on X. Suppose that T satisfies the following condition:
for all x, y, z ∈ X, x ≠ y, where a, b, c are non-negative reals and Then T has at least one fixed point ∈ X.
Proof Let Tx = Ty, then Hence G(x, y, y) = 0, which implies that x = y. So T is injective and invertible.
If H is the inverse mapping of T, then for x, y, z ∈ X, x ≠ y, we have Hence for all x, y, z ∈ X, x ≠ y From Theorem 2.6, the inverse mapping H has a fixed point u ∈ X, i.e. Hu = u. But u = T(H(u)) = T(u). Thus u is also a fixed point of T.

✷
In the next subsection, we consider the case of a triplet of functions and we state an analogue of Theorem 2.1.

Triplets of maps
Theorem 2.9 Let (X, G) be a symmetric G-complete G-metric space and T, P, Q be three self mappings on X. Suppose that T, P, Q satisfy the following condition: (x, y, z)G(x, y, y).
This completes the proof.

✷
Remark 2.10 The reader can convince him(her)-self that if we replace the condition (2.8) by where the non-negative functions a, b, c, d, e and f are well chosen, then P, Q and T have a common fixed point.
We conclude this article with the case of a family of mappings.

Families of maps
Here, in this last subsection of the manuscript, we consider the case of a family of functions and we state an analogue of Theorem 2.9.
Theorem 2.11 Let (X, G) be a symmetric G-complete G-metric space and {T n } be a family of self mappings on X such that where Γ x i : = G(x, T i x, T i x), and a i : = a i (x, y, z) are non-negative functions, the constants k Δ i,j are such that 0 ≤ k Δ i,j < 1;i, j, k = 1, 2, ⋯ , and some F ∈ Φ homogeneous with degree s.
Proof For any x 0 ∈ X, we construct the sequence (x n ) n≥1 by setting x n = T n (x n−1 ), n = 1, 2, ⋯ . We may assume without loss of generality that x m ≠ x n for all n ≠ m ∈ ℕ. We observe that, by setting d i = G(x i , x i+1 , x i+1 ), i ≥ 1, and plugging in the triplet (x i , x i+1 , x i+2 ) we have where When we write the above for the triplet (x 1 , x 2 , x 3 ), we obtain Also we get Hence, we derive, iteratively, that Therefore, for all l > m > n > 2, since using the fact that F is sub-additive, we write F( i ) F(G(x 0 , x 1 , x 2 )).