Fixed points of rational type contractions in G-metric spaces

Gaba, Cogent Mathematics & Statistics (2018), 5: 1444904 https://doi.org/10.1080/23311835.2018.1444904

PURE MATHEMATICS | RESEARCH ARTICLE

Fixed points of rational type contractions in G-metric spaces Yaé Ulrich Gaba1,2* Received: 03 December 2017 Accepted: 20 February 2018 First Published: 07 March 2018 *Correspondin author: Yaé Ulrich Gaba, African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon; Department of Mathematical Sciences, North West University, Private Bag X2046, Mmabatho 2735, South Africa E-mail: [email protected] Reviewing editor: Lishan Liu, Qufu Normal University, China Additional information is available at the end of the article

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. Subjects: Analysis - Mathematics; Pure Mathematics; Foundations & Theorems Keywords: G-metric; fixed point; rational type contraction AMS subject classifications: Primary 47H05; Secondary 47H09, 47H10 1. 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 n ∞ ∞ sequence of iterates {T x}n=0 (resp. {Ti (xi−1) }i=0 ) where T:X → X (resp. Ti :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 𝛼-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.

ABOUT THE AUTHOR

PUBLIC INTEREST STATEMENT

Yaé Ulrich Gaba is a postdoctoral fellow at NorthWest University (South Africa) and also affiliated to the African Centre for Advanced Studies (ACAS). One of his research orientations deals with Fixed Point Theory in metric like spaces and their applications to physical sciences.

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.

© 2018 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license.

Page 1 of 14


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: G(x, y, z) = 0 if x = y = z whenever x, y, z ∈ X ; (G1) G(x, x, y) > 0 whenever x, y ∈ X with x ≠ y ; (G2) G(x, x, y) ≤ G(x, y, z) whenever x, y, z ∈ X with z ≠ y ; (G3) G(x, y, z) = G(x, z, y) = G(y, z, x) = …, (symmetry in all three variables); (G4)

(G5) for any points x, y, z, a ∈ X G(x, y, z) ≤ [G(x, a, a) + G(a, y, z)].

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 (xn )n≥1 be a sequence of points of X, therefore, we say that the sequence (xn )n≥1 is G-convergent to x ∈ X if lim G(x, xn , xm ) = 0, n,m→∞

that is, for any 𝜀 > 0, there exists N ∈ ℕ such that G(x, xn , xm ) < 𝜀, for all, n, m ≥ N. We call x the limit of the sequence and write xn → x or lim xn = x. n→∞

Proposition 1.3 (Compare [Mustafa & Sims, 2006, Proposition 6]) Let (X, G) be a G-metric space. Define on X the metric dG by dG (x, y) = G(x, y, y) + G(x, x, y) whenever x, y ∈ X . Then for a sequence (xn )n≥1 ⊆ X , the following are equivalent (xn ) is G-convergent to x ∈ X. (i)

(ii) limn,m→∞ G(x, xn , xm ) = 0. (iii) limn→∞ dG (xn , x) = 0. limn→∞ G(x, xn , xn ) = 0. (iv)

(v) limn→∞ G(xn , x, x) = 0. Definition 1.4 (See Mustafa & Sims, 2006) Let (X, G) be a G-metric space. A sequence (xn )n≥1 is called a G-Cauchy sequence if for any 𝜀 > 0, there is N ∈ ℕ such that G(xn , xm , xl ) < 𝜀 for all n, m, l ≥ N, that is G(xn , xm , xl ) → 0 as n, m, l → +∞. Proposition 1.5 (Compare [Mustafa & Sims, 2006, Proposition 9]) In a G-metric space (X, G), the following are equivalent (i) The sequence (xn )n≥1 ⊆ X is G-Cauchy. (ii) For each 𝜀 > 0 there exists N ∈ ℕ such that G(xn , xm , xm ) < 𝜀 for all m, n ≥ N. Definition 1.6 (Compare [Mustafa & Sims, 2006, Definition 4]) A G-metric space (X, G) is said to be symmetric if G(x, y, y) = G(x, x, y), for all x, y ∈ X.

Definition 1.7 (Compare [Mustafa & Sims, 2006, Definition 9]) A G-metric space (X, G) is G-complete if every G-Cauchy sequence of elements of (X, G) is G-convergent in (X, G). 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:

Page 2 of 14


Definition 1.9 (Compare [Sihag et al., 2014, Definition 2.1]) For a sequence (an )n≥1 of nonnegative ∑∞ real numbers, the series n=1 an is an 𝛼-series if there exist 0 < 𝜆 < 1 and n(𝜆) ∈ ℕ such that L ∑

ai ≤ 𝜆L for each L ≥ n(𝜆).

i=1

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

2.1. 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: G(Tx, Ty, Tz) ≤

(

) a.G(Tx, y, z) + b.G(x, Ty, z) + c.G(x, y, Tz) G(x, y, z), (b + c).G(x, Tx, Tx) + b.G(y, Ty, Ty) + c.G(z, Tz, Tz) + 1

(2.1)

for all x, y, z ∈ X , where a, b, c are non-negative reals. Then (a) T has at least one fixed point 𝜉 ∈ X; (b) for any x ∈ X , the sequence (T n x)n≥1 G-converges to a fixed point of T; (c) if 𝜉, 𝜅 ∈ X are two distinct fixed points, then

G(𝜉, 𝜅, 𝜅) = G(𝜉, 𝜉, 𝜅) ≥

1 . a+b+c

Proof We imitate the steps of the proof of [Gaba, 2017 Theorem 2.1]. Let x0 ∈ X be arbitrary and construct the sequence (xn )n≥1 such that xn+1 = Txn . Moreover, we may assume, without loss of generality that xn ≠ xm for n ≠ m. For the triplet (xn , xn+1 , xn+1 ), and by setting dn = G(xn , xn+1 , xn+1 ), we have: 0 < dn = G(xn , xn+1 , xn+1 ) = G(Txn−1 , Txn , Txn ) ) ( a.G(xn , xn , xn ) + b.G(xn−1 , xn , xn+1 ) + c.G(xn−1 , xn , xn+1 ) dn−1 ≤ (b + c)dn−1 + bdn + cdn + 1 ) ( (b + c)dn−1 + (b + c)dn ≤ d , (b + c)dn−1 + (b + c)dn + 1 n−1

since G(xn−1 , xn , xn+1 ) ≤ G(xn−1 , xn , xn ) + G(xn , xn , xn+1 ) = G(xn−1 , xn , xn ) + G(xn , xn+1 , xn+1 ).

If we set 𝛼n =

(b + c)dn−1 + (b + c)dn (b + c)dn−1 + (b + c)dn + 1

,

we get, iteratively dn ≤ 𝛼n dn−1 ≤ 𝛼n 𝛼n−1 dn−2 ⋮ ≤ 𝛼n 𝛼n−1 ⋯ 𝛼1 d0 . Page 3 of 14


Claim: The sequence (𝛼n )n≥1 is a non-increasing sequence of non-negative reals. Indeed, since we have (b + c)dn−1 + (b + c)dn ≤ (b + c)dn−1 + (b + c)dn + 1,

it is very clear that for any natural number n ∈ ℕ, 0 ≤ 𝛼n < 1, and so dn < dn−1. We then have the following consecutive equivalences: dn ≤ dn−1 ⟺ dn + dn+1 ≤ dn−1 + dn ⟺1+ ⟺

1 1 ≤1+ (b + c)dn−1 + (b + c)dn (b + c)dn + (b + c)dn+1

(b + c)dn−1 + (b + c)dn + 1 (b + c)dn−1 + (b + c)dn

≤

(b + c)dn + (b + c)dn+1 + 1 (b + c)dn + (b + c)dn+1

1 1 ⟺ ≤ . 𝛼n 𝛼n+1

Hence 𝛼n 𝛼n−1 ⋯ 𝛼1 ≤ 𝛼1n → 0 as n → ∞.

Therefore lim 𝛼n 𝛼n−1 ⋯ 𝛼1 = 0,

n→∞

hence lim dn = 0.

n→∞

For any m, n ∈ ℕ, m > n, since we have m−n

G(xn , xm , xm ) ≤

∑

G(xn+i , xn+i+1 , xn+i+1 ),

i=0

the above translates to m−n

G(xn , xm , xm ) ≤

∑

dn+i ,

i=0

and we obtain m−n

G(xn , xm , xm ) ≤

∑

[(𝛼n+i ⋯ 𝛼1 )d0 ].

i=0

Put bk = 𝛼k ⋯ 𝛼1 and observe that lim

k→∞

bk+1 bk

= lim 𝛼k+1 = 0 since 𝛼k = k→∞

(b + c)dk−1 + (b + c)dk (b + c)dk−1 + (b + c)dk + 1

and lim dk = 0. k→∞

Hence ∞ ∑

bk < ∞,

k=0

therefore m−n

∑

(𝛼n+i ⋯ 𝛼1 ) → 0 as m → ∞.

i=0

In other words, (xn )n≥1 is a G-Cauchy sequence so G-converges to some 𝜉 ∈ X. Page 4 of 14


Claim: 𝜉 is a fixed point of T. For the triplet (xn+1 , T𝜉, T𝜉) in (2.1), we get G(xn+1 , T𝜉, T𝜉) ≤

(

a.G(xn+1 , 𝜉, 𝜉) + (b + c)G(xn , T𝜉, 𝜉) (b + c)dn + (b + c)G(𝜉, T𝜉, T𝜉) + 1

) G(xn , 𝜉, 𝜉).

(2.2)

On taking the limit on both sides of (2.2), and using the fact that the function G is continuous, we have G(𝜉, T𝜉, T𝜉) ≤

(

) (b + c)G(𝜉, T𝜉, 𝜉) G(𝜉, 𝜉, 𝜉), (b + c)G(𝜉, T𝜉, T𝜉) + 1

i.e. G(𝜉, T𝜉, T𝜉) = 0, thus T𝜉 = 𝜉. If 𝜅 is a fixed point of T with 𝜅 ≠ 𝜉, then G(𝜉, 𝜅, 𝜅) = G(T𝜉, T𝜅, T𝜅) ≤ [a.G(𝜉, 𝜅, 𝜅) + (b + c)G(𝜉, 𝜅, 𝜅)]G(𝜉, 𝜅, 𝜅) ≤ (a + b + c)[G(𝜉, 𝜅, 𝜅)]2 .

Therefore, 1 . G(𝜉, 𝜅, 𝜅) = G(𝜉, 𝜉, 𝜅) ≥ a+b+c

✷

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:

G(Tx, Ty, Tz) ≤

(

G(Tx, y, z) + 12 (G(x, Ty, z) + G(x, y, Tz)) G(x, Tx, Tx) + 12 (G(y, Ty, Ty) + G(z, Tz, Tz)) + 1

)

G(x, y, z),

(2.3)

for all x, y, z ∈ X . Then (a) T has at least one fixed point 𝜉 ∈ X; (b) for any x ∈ X , the sequence (T n x)n≥1 G-converges to a fixed point; (c) if 𝜉, 𝜅 ∈ X are two distinct fixed points, then G(𝜉, 𝜅, 𝜅) = G(𝜉, 𝜉, 𝜅) ≥

1 . 2

Proof Apply Theorem 2.1 with a = 1, b = c = 12 .

✷

Example 2.3 (Compare [Gaba, 2017, Example 2.2]) { } Let X = 0, 12 , 1 and let G:X 3 → [0, ∞) be defined by ) ) ( ( 1 1 1 =4=G , 0, 0 G(0, 1, 1) = 6 = G(1, 0, 0), G 0, , 2 2 2 ( ) ( ) ( ) 1 1 1 1 15 G , 1, 1 = 5 = G 1, , , G 0, , 1 = 2 2 2 2 2 G(x, x, x) = 0 ∀x ∈ X. Page 5 of 14


(X, G) is a symmetric G-complete G-metric space. Let T:X → X be defined by T(0) = 0,

T

( ) 1 2

= 12 ,

T(1) = 0.

( ( ) ( ) ( )) 1 1 1 1 ,T = G 0, , = 4; G(T0, T1, T1) = G(0, 0, 0) = 0; G T0, T 2 2 2 2 ( ( ) ) ( ) ( ( ) ) ( ) 1 1 1 1 G T , T1, T1 = G , 0, 0 = 4; G T0, T , T1 = G 0, , 0 = 4. 2 2 2 2

We have ( ) ( ( ) ( )) 1 1 1 1 ,T = G 0, , 4 = G T0, T 2 2 2 2 ( ( ) ) ( ( )) ) ( G T0, 12 , 12 + 12 G 0, T 12 , 12 + 12 G 0, 12 , T 12 ≤ ( ( ) ( )) G(0, T0, T0) + G 12 , T 12 , T 12 +1 ( ) 1 1 × G 0, , 2 2 ) ) ) ( ( ( ) ( G 0, 12 , 12 + 12 G 0, 12 , 12 + 12 G 0, 12 , 12 1 1 × G 0, , = ( ) 2 2 G(0, 0, 0) + G 12 , 12 , 12 + 1 =

4+2+2 4 = 32. 1

Again, 0 = G(T0, T1, T1) = G(0, 0, 0) ≤ =

G(T0, 1, 1) + 12 G(0, T1, 1) + 12 G(0, 1, T1) G(0, T0, T0) + G(1, T1, T1) + 1 G(0, 1, 1) + 12 G(0, 0, 1) + 12 G(0, 1, 0)

G(0, 0, 0) + G(1, 0, 0) + 1 6+3+3 = 6. 7

× G(0, 1, 1)

× G(0, 1, 1)

) ) ( ( ( ) 1 1 , T1, T1 = G , 0, 0 Also,4 = G T 2 2 ( ) ( ) ) ( ( ) G T 12 , 1, 1 + 12 G 12 , T1, 1 + 12 G 12 , 1, T1 ≤ ( ( ) ( )) G 12 , T 12 , T 12 + G(1, T1, T1) + 1 ( ) 1 ×G , 1, 1 2 ) ( ) ( ) ( ) ( G 12 , 1, 1 + 12 G 12 , 0, 1 + 12 G 12 , 1, 0 1 , 1, 1 ×G = ( ) 2 G 12 , 12 , 12 + G(1, 0, 0) + 1 5+ =

7

15 2

5.

Finally,

Page 6 of 14


) ) ( ( ( ) 1 1 , T1 = G 0, , 0 4 = G T0, T 2 2 ) ( ( ) ) ( ) ( G T0, 12 , 1 + 12 G 0, T 12 , 1 + 12 G 0, 12 , T1 ≤ ( ( ) ( )) G(0, T0, T0) + 12 G 12 , T 12 , T 12 + 12 G(1, T1, T1) + 1 ( ) 1 × G 0, , 1 2 ) ( ) ( ) ( ( ) G 0, 12 , 1 + 12 G 0, 12 , 1 + 12 G 0, 12 , 0 1 × G 0, , 1 = ( ) 2 G(0, 0, 0) + 12 G 12 , 12 , 12 + 12 G(1, 0, 0) + 1 =

15 + 4 15 × . 7 2

Therefore T satisfies all the conditions of Theorem 2.2. Also, T has two distinct fixed points {0, 12 } and ) ( ) ( 1 1 1 1 1 , 0, 0 = 4 ≥ =G G 0, , = . 2 2 2 2 a+b+c

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: G(Tx, Ty, Tz) ≤

(

) G(Tx, y, z) + G(x, Ty, z) + G(x, y, Tz) G(x, y, z), 2G(x, Tx, Tx) + G(y, Ty, Ty) + G(z, Tz, Tz) + 1

(2.4)

for all x, y, z ∈ X . Then (a) T has at least one fixed point 𝜉 ∈ X; (b) for any x ∈ X , the sequence (T n x)n≥1G-converges to a fixed point; (c) if 𝜉, 𝜅 ∈ X are two distinct fixed points, then G(𝜉, 𝜅, 𝜅) = G(𝜉, 𝜉, 𝜅) ≥

1 . 3

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 z1 , z2 ∈ ℂ,

z1 ≲ z2 ⟺ Re(z1 ) ≤ Re(z2 ) and Im(z1 ) ≤ Im(z2 ) ⟺ z2 ≳ z1 . Moreover, on partial ordered G-metric space, the convergence of a sequence is interpreted in the canonical way, i.e. for a sequence (xn )n≥1 ⊆ (X, G, ⪯) where (X, G, ⪯) is a partial ordered complex valued G-metric space,

(xn )n≥1 G-converges tox∗ ⟺ ∀c ∈ ℂ, with0 ≲ c, ∃n0 ∈ ℕ:∀n > n0 G(x∗ , xn , xn ) ≲ c. 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:

Page 7 of 14


Theorem 2.5 Let (X, G, ⪯) be a symmetric, G-complete, complex valued G-metric space. Assume that if (xn )n≥1 is a nondecreasing sequence of elements of X such that xn G-converges tox∗, then xn ⪯ x∗ for all n ∈ ℕ. Let T:X → X be a nondecreasing mapping such that: G(Tx, Ty, Tz) ≲

(

) a.G(Tx, y, z) + b.G(x, Ty, z) + c.G(x, y, Tz) G(x, y, z), (b + c).G(x, Tx, Tx) + b.G(y, Ty, Ty) + c.G(z, Tz, Tz) + 1

(2.5)

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. If there exists x0 ∈ X with x0 ⪯ Tx0, then (i) T has at least one fixed point 𝜉 ∈ X; (ii) for any x ∈ X , the sequence (T n x)n≥1G-converges to a fixed point; (iii) if 𝜉, 𝜅 ∈ X are two distinct fixed points, then G(𝜉, 𝜅, 𝜅) = G(𝜉, 𝜉, 𝜅) ≳

1 . a+b+c

Proof Following the steps of the proof of Theorem 2.1, it is very easy to see that the sequence of iterates T n x0 , n = 1, 2, ⋯ , is nondecreasing and G-converges to some 𝜉 ∈ X . Therefore xn ⪯ 𝜉 for all n ∈ ℕ. Now applying (2.5) to the triplet (xn+1 , T𝜉, T𝜉) we have: G(xn+1 , T𝜉, T𝜉) = G(Txn , T𝜉, T𝜉) ( ) aG(xn+1 , 𝜉, 𝜉) + (b + c)G(𝜉, T𝜉, 𝜉) ≲ G(xn , 𝜉, 𝜉). (b + c)G(xn , xn+1 , xn+1 ) + (b + c)G(𝜉, T𝜉, T𝜉) + 1

Now taking the limit as n → ∞, and using the fact that the function G is continuous, we have: G(𝜉, T𝜉, T𝜉) ≲

(

) (b + c)G(𝜉, T𝜉, 𝜉) G(𝜉, 𝜉, 𝜉), (b + c)G(𝜉, T𝜉, T𝜉) + 1

i.e. G(𝜉, T𝜉, T𝜉) = 0, thus T𝜉 = 𝜉. If 𝜅 is a fixed point of T with 𝜅 ≠ 𝜉, then G(𝜉, 𝜅, 𝜅) = G(T𝜉, T𝜅, T𝜅) ≲ [a.G(𝜉, 𝜅, 𝜅) + (b + c)G(𝜉, 𝜅, 𝜅)]G(𝜉, 𝜅, 𝜅) ≲ (a + b + c)[G(𝜉, 𝜅, 𝜅)]2 .

Therefore, 1 . G(𝜉, 𝜅, 𝜅) = G(𝜉, 𝜉, 𝜅) ≳ a+b+c

✷

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: G(Tx, Ty, Tz) ≤ K(x, y, z)G(x, y, z),

(2.6)

Page 8 of 14


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 ⎛ ⎞ a.G(x, Ty, Tz) + b.G(Tx, y, Tz) + c.G(Tx, Ty, z) ⎜ ⎟ K(x, y, z) = ⎜ � � � � ⎟. ⎜ a.G(x, Tx, Tx) + a + b .G(y, Ty, Ty) + a + c .G(z, Tz, Tz)] + 1 ⎟ 2 2 ⎝ ⎠

Then T has at least one fixed point 𝜉 ∈ X. Remark 2.7 In general, the self mapping T in Theorem 2.6 (as well as in Theorem 2.1) is a weakly1 Picard operator. Moreover, the reader can convince him/her-self that if 𝜉 and 𝜅 are fixed points of T in X, a lower bound can be found for G(𝜉, 𝜉, 𝜅) = G(𝜉, 𝜅, 𝜅) (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: G(Tx, Ty, Tz) ≥ A(x, y, z)G(x, y, z),

(2.7)

for all x, y, z ∈ X, x ≠ y , where a, b, c are non-negative reals and � � � � ⎛ ⎞ a a + b .G(y, Ty, Ty) + + c .G(z, Tz, Tz) + 1 ⎟ a.G(x, Tx, Tx) + ⎜ 2 2 A(x, y, z) = ⎜ ⎟. a.G(x, Ty, Tz) + b.G(Tx, y, Tz) + c.G(Tx, Ty, z) ⎜ ⎟ ⎝ ⎠

Then T has at least one fixed point 𝜉 ∈ X. Proof Let Tx = Ty , then 0 = G(Tx, Ty, Ty) ≥ A(x, y, z)G(x, y, y).

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 G(x, y, z) = G(T(Hx), T(Hy), T(Hz)) ≥ A(Hx, Hy, Hz)G(Hx, Hy, Hz).

Hence for all x, y, z ∈ X, x ≠ y G(Hx, Hy, Hz) ≤

1 G(x, y, z). A(Hx, Hy, Hz)

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.

2.2. 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:

Page 9 of 14


G(Tx, Py, Qz) ≤

(

) a.G(Tx, y, z) + b.G(x, Py, z) + c.G(x, y, Qz) G(x, y, z), (b + c).G(x, Tx, Tx) + (a + b + 2c).G(y, Py, Py) + c.G(z, Qz, Qz) + 1

(2.8)

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 functions. Then T, P and Q have a common fixed point, i.e. ∃ u ∈ X such that Tu = Pu = Qu = u. Proof For any initial point x0 ∈ X, we construct the sequence (xn )n≥1 by setting x3n+1 = Tx3n , x3n+2 = Px3n+1 , x3n+3 = Qx3n+2 , n ≥ 0.

Without loss of generality, assume that xn ≠ xm for n ≠ m. Plugging in (x3n+1 , x3n+2 , x3n+3 ) = (Tx3n , Px3n+1 , Qx3n+2 ) in (2.8), we have: G(x3n+1 , x3n+2 , x3n+3 ) = G(Tx3n , Px3n+1 , Qx3n+2 ) ≤ Hn .G(x3n , x3n+1 , x3n+2 ),

where Hn =

a.G(x3n+1 , x3n+1 , x3n+2 ) + b.G(x3n , x3n+2 , x3n+2 ) + c.G(x3n , x3n+1 , x3n+3 ) (b + c).G(x3n , x3n+1 , x3n+1 ) + (a + b + 2c).G(x3n+1 , x3n+2 , x3n+2 ) + c.G(x3n+2 , x3n+3 , x3n+3 ) + 1

.

Each of the term in the numerator of Hn can be bounded as follows: G(x3n+1 , x3n+1 , x3n+2 ) = G(x3n+1 , x3n+2 , x3n+2 ) G(x3n , x3n+2 , x3n+2 ) ≤ G(x3n , x3n+1 , x3n+1 ) + G(x3n+1 , x3n+2 , x3n+2 ) G(x3n , x3n+1 , x3n+3 ) ≤ G(x3n , x3n+1 , x3n+1 ) + G(x3n+1 , x3n+1 , x3n+3 ) ≤ G(x3n , x3n+1 , x3n+1 ) + G(x3n+1 , x3n+2 , x3n+2 ) + G(x3n+1 , x3n+2 , x3n+3 ) ≤ G(x3n , x3n+1 , x3n+1 ) + G(x3n+1 , x3n+2 , x3n+2 ) + G(x3n+1 , x3n+2 , x3n+2 ) + G(x3n+2 , x3n+2 , x3n+3 ) = G(x3n , x3n+1 , x3n+1 ) + G(x3n+1 , x3n+2 , x3n+2 ) + G(x3n+1 , x3n+2 , x3n+2 ) + G(x3n+2 , x3n+3 , x3n+3 ).

By setting dn = G(xn , xn+1 , xn+1 ), Hn is bounded as Hn ≤

a.d3n+1 + b.(d3n + d3n+1 ) + c.(d3n + 2d3n+1 + d3n+2 ) d3n + d3n+1 + d3n+2 + 1

i.e. Hn ≤

(b + c).d3n + (a + b + 2c).d3n+1 + c.d3n+2 (b + c).d3n + (a + b + 2c).d3n+1 + c.d3n+2 + 1

.

So if we denote 𝛼n : =

(b + c).d3n + (a + b + 2c).d3n+1 + c.d3n+2 (b + c).d3n + (a + b + 2c).d3n+1 + c.d3n+2 + 1

,

plugging in (x3n+1 , x3n+2 , x3n+3 ) = (Tx3n , Px3n+1 , Qx3n+2 ) in (2.8) implies G(x3n+1 , x3n+2 , x3n+3 ) ≤ 𝛼n G(x3n , x3n+1 , x3n+2 ).

If we inspire ourselves from the proof of Theorem 2.1, one can easily establish that the sequence (𝛼n )n≥1 is a non-increasing sequence of non-negative real numbers and that for any natural number n ∈ ℕ, 0 ≤ 𝛼n < 1. Moreover, it is readily seen that (xn )n≥1 is a G-Cauchy sequence so G-converges to some 𝜉 ∈ X.

Page 10 of 14


By substituting (T𝜉, P𝜉, Q𝜉) for (x, y, z) in (2.8), we get G(T𝜉, P𝜉, Q𝜉) ≤ 𝛾.G(𝜉, 𝜉, 𝜉) = 0,

where 𝛾 ≥ 0 can easily be recovered from (2.8). Hence G(T𝜉, P𝜉, Q𝜉) = 0 i.e. T𝜉 = P𝜉 = Q𝜉.

Again, from (2.8), we can write that: (2.9)

G(T𝜉, x3n+2 , x3n+3 ) = G(T𝜉, Px3n+1 , Qx3n+2 ) ≤ 𝛾1 .G(𝜉, x3n+1 , x3n+2 ), G(x3n+1 , P𝜉, x3n+3 ) = G(Tx3n , P𝜉, Qx3n+2 ) ≤ 𝛾2 .G(x3n , 𝜉, x3n+2 ),

(2.10)

G(x3n+1 , x3n+2 , Q𝜉) = G(Tx3n , Px3n+1 , Q𝜉) ≤ 𝛾3 .G(x3n , x3n+1 , 𝜉),

(2.11)

where 𝛾1 , 𝛾2 and 𝛾3 can easily be recovered from (2.8). Since lim G(𝜉, x3n+1 , x3n+2 ) = lim G(x3n , 𝜉, x3n+2 ) = lim G(x3n , x3n+1 , 𝜉) = G(𝜉, 𝜉, 𝜉)2 = 0,

n→∞

n→∞

n→∞

the relations (2.9), (2.10) and (2.11) respectively give that G(T𝜉, 𝜉, 𝜉) = 0, G(𝜉, P𝜉, 𝜉) = 0 and G(𝜉, 𝜉, Q𝜉) = 0, i.e. T𝜉 = P𝜉 = Q𝜉 = 𝜉.

This completes the proof.

✷

Remark 2.10 The reader can convince him(her)-self that if we replace the condition (2.8) by G(Tx, Py, Qz) ≤

(

) a.G(x, Py, Qz) + b.G(Tx, y, Qz) + c.G(Tx, Py, z) G(x, y, z), d.G(x, Tx, Tx) + e.G(y, Py, Py) + f .G(z, Qz, Qz) + 1

(2.12)

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.

2.3. 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. We make use of the following special class Φ of homogeneous functions. Let Φ be the class of continuous, non-decreasing, sub-additive and homogeneous functions F:[0, ∞) → [0, ∞) such that F −1 (0) = {0} and F(1) ≤ 13. Theorem 2.11 Let (X, G) be a symmetric G-complete G-metric space and {Tn } be a family of self mappings on X such that

F(G(Ti x, Tj y, Tk z)) ≤ F

(

k Δi,j

ai .G(Ti x, y, z) + aj .G(x, Tj y, z) + ak .G(x, y, Tk z) y

(aj + ak ).Γxi + (ai + aj + 2ak ).Γj + ak .Γzk + 1

)

.F(G(x, y, z)),

(2.13)

where Γxi : = G(x, Ti x, Ti x), and ai : = ai (x, y, z) are non-negative functions, the constants k Δi,j are such that 0 ≤k Δi,j < 1;i, j, k = 1, 2, ⋯ , and some F ∈ Φ homogeneous with degree s. Page 11 of 14


If ∞ ∑ (i+2 Δi,i+1 )s . i=1

is an 𝛼-series, then {Tn} has a common fixed point in X. Proof For any x0 ∈ X , we construct the sequence (xn )n≥1 by setting xn = Tn (xn−1 ), n = 1, 2, ⋯ . We may assume without loss of generality that xm ≠ xn for all n ≠ m ∈ ℕ. We observe that, by setting di = G(xi , xi+1 , xi+1 ), i ≥ 1, and plugging in the triplet (xi , xi+1 , xi+2 ) we have F(G(xi , xi+1 , xi+2 )) = F(G(Ti xi−1 , Ti+1 xi , Ti+2 xi+1 )) ≤ (i+2 Δi,i+1 )s F(𝛼i )F(G(xi−1 , xi , xi+1 )),

where 𝛼i =

(ai+1 + ai+2 )di−1 + (ai + ai+1 + 2ai+2 )di + ai+2 di+1 (ai+1 + ai+2 )di−1 + (ai + ai+1 + 2ai+2 )di + ai+2 di+1 + 1

.

When we write the above for the triplet (x1 , x2 , x3 ), we obtain F(G(x1 , x2 , x3 )) ≤ (3 Δ1,2 )s F(𝛼1 )F(G(x0 , x1 , x2 ).

Also we get F(G(x2 , x3 , x4 )) ≤ (4 Δ2,3 )s F(𝛼2 )F(G(x1 , x2 , x3 ) ≤ (4 Δ2,3 )s (3 Δ1,2 )s F(𝛼2 )F(𝛼1 )F(G(x0 , x1 , x2 ).

Hence, we derive, iteratively, that [ n ] ][ n ∏ ∏ s F(G(xn , xn+1 , xn+2 )) ≤ [(i+2 Δi,i+1 ) F(𝛼i ) F(G(x0 , x1 , x2 )). i=1

i=1

Therefore, for all l > m > n > 2, since G(xn , xm , xl ) ≤ G(xn , xn+1 , xn+1 ) + G(xn+1 , xn+2 , xn+2 ) + ⋯ + G(xl−1 , xl−1 , xl ) ≤ G(xn , xn+1 , xn+2 ) + G(xn+1 , xn+2 , xn+3 ) + ⋯ + G(xl−2 , xl−1 , xl ),

using the fact that F is sub-additive, we write ([ n ] ][ n ∏ ∏ s F(G(xn , xm , xl )) ≤ [(i+2 Δi,i+1 ) F(𝛼i ) i=1

[ n+1 ∏

+

i=1

] ][ n+1 ∏ s [(i+2 Δi,i+1 ) F(𝛼i )

i=1

i=1

+ ⋯+ ]) [ l−2 ][ l−2 ∏ ∏ [(i+2 Δi,i+1 )s F(𝛼i ) F(G(x0 , x1 , x2 )) + i=1

i=1

]) (l−n−2 [ n+k ][ n+k ∑ ∏ ∏ s (i+2 Δi,i+1 ) F(𝛼i ) F(G(x0 , x1 , x2 )) = k=0

i=1

i=1

( l−2 [ k ]) ][ k ∑ ∏ ∏ s = (i+2 Δi,i+1 ) F(𝛼i ) F(G(x0 , x1 , x2 )). k=n

i=1

i=1

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We already know that the sequence (𝛼i )n≥1 is a sequence of non-negative reals and that for any natural number n ∈ ℕ, 0 ≤ 𝛼n < 1. Therefore for any natural number n ∈ ℕ, F(𝛼n ) ≤ 1. Hence [

k ∏

]

F(𝛼i ) ≤ 1.

i=1

Now, let 𝜆 and n(𝜆) as in Definition 1.9, then for n ≥ n(𝜆) and using the fact that the geometric mean of non-negative real numbers is at most their arithmetic mean, it follows that [ ( k )]k l−2 ∑ 1 ∑ s F(G(xn , xm , xl )) ≤ ( Δ ) F(G(x0 , x1 , x2 )) k i=1 i+2 i,i+1 k=n ( l−2 ) ∑ k = 𝜆 F(G(x0 , x1 , x2 )) k=n

𝜆n ≤ F(G(x0 , x1 , x2 )). 1−𝜆

As n → ∞, we deduce that G(xn , xm , xl ) → 0. Thus (xn )n≥1 is a G-Cauchy sequence and since X is complete there exists 𝜉 ∈ X such that (xn )n≥1 G-converges to 𝜉. Furthermore, for any i ≥ 1 F(G(Ti 𝜉, xi+1 , xi+2 )) ≤ 𝛾. F(G(𝜉, xi , xi+1 )),

for some 𝛾 ≥ 0. Now taking the limit using the fact that the function G is continuous, we obtain F(G(Ti 𝜉, 𝜉, 𝜉)) ≤ F(G(𝜉, 𝜉, 𝜉)) = 0, i.e.Ti 𝜉 = 𝜉 for any i ≥ 1.

This terminates the proof. Funding The author received no direct funding for this research. Author details Yaé Ulrich Gaba1,2 E-mail: [email protected] ORCID ID: http://orcid.org/0000-0001-8128-9704 1 African Center for Advanced Studies, P.O. Box 4477, Yaounde, Cameroon. 2 Department of Mathematical Sciences, North West University, Private Bag X2046, Mmabatho 2735, South Africa. Citation information Cite this article as: Fixed points of rational type contractions in G-metric spaces, Yaé Ulrich Gaba, Cogent Mathematics & Statistics (2018), 5: 1444904. Notes 1. This means that the sequence of iterates T n x0 , n = 1, 2, ⋯, for any initial point x0, converges to a fixed and this fixed point might surely not be unique 2. See Theorem 1.8

3. The function F(x) =

√ x is an example of such function.

References Gaba, Y. U. (2017). Fixed point theorems in G-metric spaces. Journal of Mathematical Analysis and Applications, 455, 528–537. Mustafa, Z., Obiedat, H., & Awawdeh, F. (2008). Some fixed point theorem for mappings on a complete G- metric space. Fixed Point Theory and Applications, 2008, 12, Article ID 189870. Mustafa, Z., & Sims, B. (2006). A new approach to generalized metric spaces. Journal of Nonlinear Convex Analysis, 7, 289–297. Shoaib, A., Arshad, M., & Kazmi, S. H. (2017). Fixed point results for Hardy Roger type contraction in ordered complete dislocated Gd metric space. Turkish Journal of Analysis and Number Theory, 5(1), 5–12. Sihag, V., Vats, R. K., & Vetro, C. (2014). A fixed point theorem in G-metric spaces via α-series. Quaestiones Mathematicae, 37(3), 429–434.

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