RACSAM DOI 10.1007/s13398-016-0305-3 ORIGINAL PAPER
Fixed point results for generalized F-contractive and Roger Hardy type F-contractive mappings in G-metric spaces Deepak Singh1 · Vishal Joshi2 · Poom Kumam3,5 · Naval Singh4
Received: 1 May 2015 / Accepted: 25 April 2016 © Springer-Verlag Italia 2016
Abstract Very recently, Piri and Kumam (Fixed Point Theory Appl 210:11, 2014) improved the concept of F-contraction due to Wardowski (Fixed Point Theory Appl, 2012) by invoking some weaker conditions on mapping F and established some fixed point results in metric spaces. The purpose of this paper is twofold. Firstly, acknowledging the aforesaid idea of Piri and Kumam, a new generalized F-contraction in the framework of G-metric spaces is defined and by emphasizing the role of generalized F-contraction, a fixed point theorem in the structure of G-metric spaces is proved. Secondly, in the setting of G-metric spaces, Roger Hardy type F-contractive mappings are also defined and employing this, certain fixed point results are presented. Recently, Samet et al. (Int J Anal, 2013) and Jleli et al. (Fixed Point Theory Appl 210:7, 2012) observed that the most of the fixed point results in the structure of G-metric spaces can be obtained from existing literature on usual metric space. Countering this, our aforementioned results in the setting of G-metric spaces cannot
B
Poom Kumam
[email protected] Deepak Singh
[email protected] Vishal Joshi
[email protected] Naval Singh
[email protected]
1
Department of Applied Sciences, NITTTR, Under Ministry of HRD, Government of India, Bhopal 462002, India
2
Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, India
3
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
4
Department of Mathematics, Government of Science and Commerce College, Benazeer, Bhopal, India
5
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
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be concluded from the existence work in the milieu of associated metric spaces. Our findings are also authenticated with the aid of some appropriate examples. Keywords Generalized F-Contraction · G-metric spaces · Cauchy sequence · Fixed point · Roger Hardy type contraction condition Mathematics Subject Classification 47H10 · 54H25
1 Introduction and preliminaries The commencement of fixed point theory on complete metric space is associated to Banach Contraction Principle due to Banach [1], published in 1922. Banach Contraction Principle pronounces that any contractive self-mappings on a complete metric space has a unique fixed point. This principle is one of a very supremacy test for the existence and uniqueness of the solution of substantial problems arising in mathematics. Because of its significance for mathematical theory, Banach Contraction Principle has been extended and generalized in many directions (see [2–11]). Recently, one of the most interesting generalization of it was given by Wardowski [12]. He introduced a new contraction called F-contraction and established a fixed point result as a generalization of the Banach contraction principle in an unlike way than in the other recognized results from the literature. Afterwards Secelean [13] changed the condition (F2) of [12] by an equivalent and more simple one. Most recently Piri and Kumam [14] portrayed a large class of functions by replacing condition (F3) by the condition (F3 ) in the definition of F-contraction due to Wardowski [12] . Mustafa and Sims [15] introduced the notion of G-metric spaces as a generalization of metric spaces and investigated the topology of such spaces. In the last decade, the concept of G-metric spaces has attracted extensive attention from researchers, more than ever from fixed point theorists [16– 25]. Samet et al. [16] and Jleli et al. [17] scrutinized that some theorems in the context of a G-metric spaces in the literature can be obtained directly by some existing results in the setting of (quasi-) metric spaces. Further Karapinar et al. [25] countered the approach of [16,17] with the remark that techniques used in [16,17] are inapplicable unless the contraction condition in the statement of the theorem can be reduced into two variables. Recently, Nashine and Kadelburg [26] proved common fixed point results are derived under generalized weakly contractive conditions of HardyRogers-type for two and three mappings defined on an ordered orbitally complete metric space using orbital continuity of one of the involved mappings. The object of our paper is to define some generalized contractive mappings acknowledging the concepts of Wardowaski [12] and Piri and Kumam [14] in G-metric spaces. Utilizing this notion and also the technique mentioned in [25] to get some fixed point results in G-metric spaces that cannot be attained from the existence results in the setting of associated metric spaces. In what follows, we collect the background material to make our presentation as self contained as possible. Definition 1.1 [15] Let X be a nonempty set and let G : X × X × X → R + be a function satisfying the following properties: (G-1) G(x, y, z) = 0 if x = y = z, (G-2) 0 < G(x, x, y), for all x, y ∈ X with x = y, (G-3) G(x, x, y) ≤ G(x, y, z), for all x, y, z ∈ X with y = z,
Fixed point results for generalized F-contractive and...
(G-4) G(x, y, z) = G(x, z, y) = G(y, z, x) = · · · , symmetry in all three variables, (G-5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z), for all x, y, z, a ∈ X. The function G is called a generalized or a G-metric on X and the pair (X, G) is called a G-metric space. Definition 1.2 [15] Let (X, G) be a G-metric space and let {xn } be a sequence of points of X . We say [15] that the sequence {xn } is G-convergent to x ∈ X if lim
n,m→+∞
G(x, xn , xm ) = 0,
that is, for any > 0, there exists N ∈ N such that G(x, xn , xm ) < , for all m, n > N . We call x the limit of the sequence and write x n → x or limn,m→+∞ xn = x. Proposition 1.1 [15] Let (X, G) be a G-metric space. Then the following statements are equivalent: (1) (2) (3) (4)
{xn } is G-convergent to x; G(xn , xn , x) → 0 as n → +∞; G(xn , x, x) → 0 as n → +∞; G(xn , xm , x) → 0 as n, m → +∞.
Definition 1.3 [15] Let (X, G) be a G-metric space. A sequence {xn } is called G-Cauchy if for every > 0, there is N ∈ N such that G(xn , xm , xl ) < , for all n, m, l ≥ N , that is G(xn , xm , xl ) → 0 as n, m, l → +∞. Proposition 1.2 [15] Let (X, G) be a G-metric space. Then the following statements are equivalent: (1) {xn } is G-Cauchy; (2) For every > 0, there is N ∈ N such that G(xn , xn , xm ) < for all n, m ≥ N . Definition 1.4 [15] A G-metric space (X, G) is called G-complete if every G-Cauchy sequence is G-convergent in (X, G). Lemma 1.1 [27] By the rectangle inequality (G-5) together with the symmetry (G-4), we have G(x, y, y) = G(y, y, x) ≤ G(y, x, x) + G(x, y, x) = 2G(y, x, x). Wardowski [12] initiated and considered a new contraction called F-contraction to prove a fixed point result as a generalization of the Banach contraction principle. Definition 1.5 [12] Let F : R+ → R be a mapping satisfying the following conditions: (F1) F is strictly increasing; (F2) for all sequence αn ⊆ R + , limn→∞ αn = 0 if and only if limn→∞ F(αn ) = −∞; (F3) there exists 0 < k < 1 such that limα→0+ α k F(α) = 0. Wordowski [12], defined the class of all functions F : R+ → R by and introduced the notion of F-contraction as follows.
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Definition 1.6 [12] Let (X; d) be a metric space. A self-mapping T on X is called an Fcontraction if there exists τ > 0 such that for x, y ∈ X d(T x, T y) > 0 ⇒ τ + F(d(T x, T y)) ≤ F(d(x, y)), where F ∈ . Afterward Secelean [13] established the following lemma and utilized an equivalent but a more simple condition (F2 ) instead of condition (F2). Lemma 1.2 [13] Let F : R+ → R be an increasing map and αn be a sequence of positive real numbers. Then the following assertions hold: (a) if limn→∞ F(αn ) = −∞ then limn→∞ αn = 0; (b) if in f F = −∞ and limn→∞ αn = 0; then limn→∞ F(αn ) = −∞. He forwarded the following conditions. (F2 ) in f F = −∞ or (F2 ) there exists αn , a sequence of positive real numbers such that limn→∞ F(αn ) = −∞. Very recently Piri and Kumam [14] replaced the condition (F3 ) by (F3 ) in Definition 1.5 due to Wardowski as follows. (F3 ) F is continuous on (0, 1). Consistent with Piri and Kumam [14], throughout our subsequent discussion, We denote by the family of all functions F : R+ → R which satisfy the following conditions. (F1) F is strictly increasing; (F2 ) in f F = −∞; (F3 ) F is continuous on (0, 1). Example 1.1 [14] Let F1 (α) = − α1 , F2 (α) = − α1 + α, F3 (α) = Then F1 , F2 , F3 , F4 ∈ .
1 1−eα ,
F4 (α) =
1 . eα −e−α
2 Generalized F-contraction and fixed point result in G-metric spaces We begin this section by defining the generalized F-contraction in G-metric spaces. Definition 2.1 A mapping T : X → X is said to be a generalized F-contraction in G-metric spaces if F ∈ and there exists τ > 0 such that ∀x, y ∈ X, [G(T x, T 2 x, T y) > 0 ⇒ τ + F(G(T x, T 2 x, T y)) ≤ F(G(x, T x, y))]. (2.1) Example 2.1 Let (X, G) be a G metric space. Define F : R+ → R by F(α) = − α1 for α > 0. Then clearly F ∈ . Each mapping T : X → X satisfying (2.1) is a generalized F-contraction such that G(x, T x, y) G(T x, T 2 x, T y) ≤ , 1 + τ G(x, T x, y) for all x, y ∈ X and G(T x, T 2 x, T y) > 0.
Fixed point results for generalized F-contractive and...
Example 2.2 Let (X, G) be a G metric space. Consider F(α) = − α1 + α for α > 0. Then obviously F ∈ . In this case each mapping T : X → X satisfying (2.1) is a generalized F-contraction such that −1 −1 τ+ + G(T x, T 2 x, T y) ≤ + G(x, T x, y) G(T x, T 2 x, T y) G(x, T x, y) or τ G(T x, T 2 x, T y) + [G(T x, T 2 x, T y)]2 − 1 [G(x, T x, y)]2 − 1 ≤ , 2 G(T x, T x, T y) G(x, T x, y) for all x, y ∈ X and G(T x, T 2 x, T y) > 0. The very first result concerning generalized F-contraction in G-metric space runs as follows. Theorem 2.1 Let (X, G) be a G-complete metric space and T : X → X be a generalized F-contraction. Then T has a unique fixed point x ∗ ∈ X and for every x0 ∈ X , the sequence {T n x0 }∞ n=1 converges to x. Proof Taking an arbitrary x0 ∈ X and construct sequence {xn }∞ n=1 of points in X in the following manners x1 = T x0 , x2 = T x1 = T 2 x0 , . . . , xn+1 = T xn = T n+1 x0 ,
f or all n ∈ N .
Notice that if xn 0 = xn 0 +1 , for some n 0 ∈ N . Then obviously T has a fixed point. Thus, we assume that xn = xn+1 , for every n ∈ N . Employing inequality (2.1) repeatedly with x = xn−1 , y = xn , one acquires F(G(T xn−1 , T 2 xn−1 , T xn )) ≤ = ≤ = ≤ .. .
F(G(xn−1 , T xn−1 , xn )) − τ F(G(T xn−2 , T 2 xn−2 , T xn−1 )) − τ F(G(xn−2 , T xn−2 , xn−1 )) − τ − τ F(G(T xn−3 , T 2 xn−3 , T xn−2 )) − 2τ F(G(xn−3 , T xn−3 , xn−2 )) − 3τ
frequent application yields F(G(T xn−1 , T 2 xn−1 , T xn )) ≤ F(G(x0 , T x0 , x1 )) − nτ. Which making on n → ∞, reduces to lim F(G(T xn−1 , T 2 xn−1 , T xn )) = −∞
n→∞
which together with (F2 ) and Lemma (1.2), gives lim G(T xn−1 , T 2 xn−1 , T xn ) = 0
n→∞
or lim G(xn , xn+1 , xn+1 ) = 0.
n→∞
(2.2)
Next, we assert that {xn }∞ n=1 is a G-Cauchy sequence. To the contrary, assume that there exists > 0, and sequences {xn(k) } and {xm(k) } of natural numbers, such that G(xm(k) , T xm(k) , xn(k) ) ≥ with n(k) ≥ m(k) > k.
(2.3)
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Now corresponding to m(k), n(k) can be selected in such a manner that it is the smallest integer with n(k) > m(k) satisfying (2.3). Consequently, we have G(xm(k) , T xm(k) , xn(k)−1 ) < . (2.4) On using Lemma (1.1) and property (G-5), one gets ≤ G(xm(k) , T xm(k) , xn(k) ) ≤ G(xn(k) , xn(k)−1 , xn(k)−1 ) + G(xn(k)−1 , T xm(k) , xm(k) ) ≤ G(xn(k)−1 , T xm(k) , xm(k) ) + 2G(xn(k)−1 , xn(k) , xn(k) ) ≤ G(xm(k) , T xm(k) , xn(k)−1 ) + 2Sn(k)−1 . Thus we obtain ≤ + 2Sn(k)−1 .
(2.5)
where Sn(k)−1 = G(xn(k)−1 , xn(k) , xn(k) ). Letting k → ∞ in (2.5) and utilizing (2.2), we arrive at lim G(xm(k) , T xm(k) , xn(k) ) = .
n→∞
(2.6)
Consider the following and on utilizing Lemma (1.1) and (G-5), G(xm(k) , T xm(k) , xn(k) ) ≤ G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xm(k)−1 , T xm(k) , xn(k) ) = G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xn(k) , xm(k)−1 , T xm(k) ) ≤ G(xm(k) , xm(k)−1 , xm(k)−1 ) + G(xn(k) , xn(k)−1 , xn(k)−1 ) + G(xn(k)−1 , xm(k)−1 , T xm(k) ) ≤ 2G(xm(k)−1 , xm(k) , xm(k) ) + 2G(xn(k)−1 , xn(k) , xn(k) ) + G(xn(k)−1 , xm(k)−1 , T xm(k) ) ≤ 2Sm(k)−1 + 2Sn(k)−1 + G(xn(k)−1 , xm(k)−1 , T xm(k) ).
(2.7)
Which on making k → ∞, reduces to ≤ lim G(xn(k)−1 , xm(k)−1 , T xm(k) ). n→∞
(2.8)
With the similar approach as above, one can get G(xn(k)−1 , xm(k)−1 , T xm(k) ) ≤ Sn(k)−1 + Sm(k)−1 + G(xm(k) , T xm(k) , xm(k) ).
(2.9)
Which on letting k → ∞, gives rise to lim G(xn(k)−1 , xm(k)−1 , T xm(k) ) ≤ .
n→∞
(2.10)
Clubbing (2.8) and (2.10), one attains lim G(xn(k)−1 , xm(k)−1 , T xm(k) ) = .
n→∞
(2.11)
Arguing as above, this follows G(xn(k)−1 , T xm(k)−1 , T xm(k) ) ≤ 2Sm(k) + G(xm(k)−1 , T xm(k)−1 , xn(k)−1 )
(2.12)
G(xm(k)−1 , T xm(k)−1 , xn(k)−1 ) ≤ Sm(k)−1 + Sm(k) + G(xm(k) , T xm(k) , xn(k)−1 ).
(2.13)
and
Fixed point results for generalized F-contractive and...
Taking limit n → ∞ in (2.12) and (2.13) and utilizing (2.11), one gets lim G(xm(k)−1 , T xm(k)−1 , xn(k)−1 ) = .
n→∞
(2.14)
Employing (2.1) with x = xm(k)−1 and y = xn(k)−1 , we have τ + F(G(T xm(k)−1 , T 2 xm(k)−1 , T xn(k)−1 )) ≤ F(G(xm(k)−1 , T xm(k)−1 , xn(k)−1 )) ⇒ τ + F(G(xm(k) , T xm(k) , xn(k) )) ≤ F(G(xm(k)−1 , T xm(k)−1 , xn(k)−1 )). Letting k → ∞ and using (F3 ), (2.6) and (2.14), we obtain τ + F() ≤ F(). This is a contradiction and demonstrates that {xn }∞ n=1 is a Cauchy sequence. ∗ Now by the completeness of G-metric space (X, G), {x n }∞ n=1 converges to some point x in X . Application of continuity of T gives G(x ∗ , T x ∗ , x ∗ ) = lim G(xn , T xn , xn+1 ) n→∞
= lim G(xn , xn+1 , xn+1 ) n→∞
= G(x ∗ , x ∗ , x ∗ ) = 0. This amounts to say that T x ∗ = x ∗. Therefore x ∗ is a fixed point of T . Now to show that T has a unique fixed point. Infact if x ∗ , y ∗ ∈ X be two distinct fixed points of T , i.e. T x ∗ = x ∗ = y ∗ = T y ∗ . therefore τ + F(G(T x ∗ , T 2 x ∗ , T y ∗ )) ≤ F(G(x ∗ , T x ∗ , y ∗ )) ⇒ τ + F(G(x ∗ , x ∗ , y ∗ )) ≤ F(G(x ∗ , x ∗ , y ∗ )).
(2.15)
Which is a contradiction. Then we must have x ∗ = y ∗ . Hence T has a unique fixed point.
Following example substantiates the validity of hypothesis of Theorem 2.1. Example 2.3 Consider the sequence S1 = 1 × 2, S2 = 1 × 2 + 3 × 4, S3 = 1 × 2 + 3 × 4 + 5 × 6, .. .
. Sn = 1 × 2 + 3 × 4+, . . . , +(2n − 1)2n = n(n+1)(4n−1) 3 0, i f and only i f x=y=z Let X = {Sn : n ∈ N } and G(x, y, z) = max{x, y, z}, other wise. Then (X, G) is complete G-metric space. Define the mapping T : X → X by T (S1 ) = S1 , T (Sn ) = Sn−1 ,
f or all n > 1.
Now consider the mapping F(α) = −1 α + α. Clearly F ∈ . We claim that T is F-contraction with τ = 30 > 0.
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First, we observe that G(T Sn , T 2 Sn , T Sm ) > 0 ⇔ ((1 = n < 2 < m) ∨ (1 = m < 2 < n) ∨ (1 < n < m) ∨ (1 < m < n)). Case-I for 1 = n ∧ m > 2. We have G(T S1 , T 2 S1 , T Sm ) = G(S1 , S1 , Sm−1 ) = Sm−1 , since S1 = Sm−1 and G(S1 , T S1 , Sm ) = G(S1 , S1 , Sm ) = Sm . Since m > 2, so we have −1 −1 < . 1 × 2 + 3 × 4 + · · · + (2m − 3)(2m − 2) 1 × 2 + 3 × 4 + · · · + (2m − 1)(2m) Now consider 30 −
1 Sm−1
1 1 × 2 + 3 × 4 + · · · + (2m − 3)(2m − 2) + 1 × 2 + 3 × 4 + · · · + (2m − 3)(2m − 2) 1 < 30 − 1 × 2 + 3 × 4 + · · · + (2m − 1)(2m) + 1 × 2 + 3 × 4 + · · · + (2m − 3)(2m − 2) 1 2, then consider G(T Sn , T 2 Sn , T S1 ) = G(Sn−1 , T Sn−2 , S1 ) = Sn−1 and G(Sn , T Sn , S1 ) = G(Sn , T Sn−1 , S1 ) = Sn . Calculating as above, one can show that τ−
1 1 + Sn−1 ≤ − + Sn , Sn−1 Sn
f or τ = 30.
Thus τ + F(G(T Sn , T 2 Sn , T S1 )) ≤ F(G(Sn , T Sn , S1 )).
Fixed point results for generalized F-contractive and...
Case-III For m > n > 1, we have G(T Sn , T 2 Sn , T Sm ) = G(Sn−1 , T Sn−2 , Sm−1 ) = Sm−1 and G(Sn , T Sn , Sm ) = G(Sn , Sn−1 , Sm ) = Sm , Sn = Sm . similarly as in Case-I, we have τ−
1 Sm−1
+ Sm−1 ≤ −
1 + Sm , Sm
f or τ = 30.
As desired. Case-IV For n > m > 1, we have G(T Sn , T 2 Sn , T Sm ) = G(Sn−1 , T Sn−2 , Sm−1 ) = Sn−1 and G(Sn , T Sn , Sm ) = G(Sn , Sn−1 , Sm ) = Sn . Thus we easily obtain τ−
1 −1 + Sn−1 ≤ + Sn , Sn−1 Sn
f or τ = 30.
As required. Therefore τ + F(G(T Sn , T 2 Sn , T Sm )) ≤ F(G(Sn , T Sn , Sm )), for every n, m ∈ N . Hence T is an F-contractive mapping. Thus all the conditions of Theorem 2.1 are satisfied and S1 is a fixed point of T and is indeed unique.
3 Hardy–Rogers type generalized F-contractive mappings in G-metric spaces In this section, firstly, Hardy–Rogers type generalized F-contractive mappings in the framework of G-metric spaces is defined as follows. Definition 3.1 A mapping T : X → X is said to be a generalized Hardy–Rogers type F-contractive mappings in G-metric spaces if F ∈ and there exists τ > 0, such that G(T x, T y, T 2 y) > 0 ⇒ τ + F(G(T x, T y, T 2 y)) ≤ F(αG(x, y, T y) + βG(x, T x, T y) + γ G(y, T y, T 2 y) + δG(y, T 2 x, T 2 y) + ηG(x, T x, T 2 x)). (3.1) For all x, y ∈ X and α, β, γ , δ, η ≥ 0 with α + β + γ + δ + η < 1. Example 3.1 Let (X, G) be a G metric space. Define F : R+ → R by F(α) = − α1 for α > 0. Then clearly F ∈ . Each mapping T : X → X satisfying (3.1) is a Hardy–Rogers type generalized F-contraction such that G(T x, T 2 x, T y) ≤
F(αG(x,y,T y)+βG(x,T x,T y)+γ G(y,T y,T 2 y)+δG(y,T 2 x,T 2 y)+ηG(x,T x,T 2 x)) , 1+F(αG(x,y,T y)+βG(x,T x,T y)+γ G(y,T y,T 2 y)+δG(y,T 2 x,T 2 y)+ηG(x,T x,T 2 x))
for all x, y ∈ X and G(T x, T 2 x, T y) > 0.
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Example 3.2 Let (X, G) be a G metric space. Consider F(α) = ln(α) + α. Then obviously F ∈ . In this case each mapping T : X → X satisfying (3.1) is a generalized Fcontraction such that 2
2
2
2
2
G(T x, T y, T 2 y)e G(T x,T y,T y)−{αG(x,y,T y)+βG(x,T x,T y)+γ G(y,T y,T y)+δG(y,T x,T y)+ηG(x,T x,T x)} ≤ e−τ αG(x, y, T y) + βG(x, T x, T y) + γ G(y, T y, T 2 y) + δG(y, T 2 x, T 2 y) + ηG(x, T x, T 2 x)
for all x, y ∈ X and G(T x, T 2 x, T y) > 0. Next theorem is demonstrated for the Hardy–Rogers type generalized F-contractive mappings in G-metric spaces. Theorem 3.1 Let (X, G) be a G-complete metric space and T : X → X be a Hardy–Rogers type generalized F-contractive mapping, that is, if F ∈ and there exists τ > 0, such that τ + F(G(T x, T y, T 2 y)) ≤ F(αG(x, y, T y) + βG(x, T x, T y) + γ G(y, T y, T 2 y) + δG(y, T 2 x, T 2 y) + ηG(x, T x, T 2 x)). (3.2) For all x, y ∈ X , G(T x, T y, T 2 y) > 0, where α, β, γ , δ, η ≥ 0 with α + β + γ + δ + η < 1. Then T has a fixed point in X . Furthermore, if α + 2β + δ ≤ 1, then fixed point of T is unique. Proof Let x0 ∈ X be an arbitrary point in X . Now constructing the sequence {x n } in X such that xn+1 = T xn . Clearly xn = xn−1 , if so then we immediately find a fixed point. Consequently, one can get (G(T xn−1 , T xn , T 2 xn )) > 0. Then from (3.2) with x = xn−1 , y = xn , we have τ + F(G(xn , xn+1 , xn+2 )) = τ + F(G(T xn−1 , T xn , T 2 xn )) ≤ F(αG(xn−1 , xn , T xn ) + βG(xn−1 , T xn−1 , T xn ) + γ G(xn , T xn , T 2 xn ) + δG(xn , T 2 xn−1 , T 2 xn ) + ηG(xn−1 , T xn−1 , T 2 xn−1 )) ≤ F(αG(xn−1 , xn , xn+1 ) + βG(xn−1 , xn , xn+1 ) + γ G(xn , xn+1 , xn+2 ) + δG(xn , xn+1 , xn+2 ) + ηG(xn−1 , xn , xn+1 )).
(3.3)
Taking G n = G(xn , xn+1 , xn+2 ) then (3.3) becomes τ + F(G n ) ≤ F((α + β + η)G n−1 + (γ + δ)G n ). Since F is strictly increasing, one gets G n < (α + β + η)G n−1 + (γ + δ)G n ⇒ G n
0. Then on utilizing the Property (F1), Lemma 1.1 and Condition (3.2) with x = u and y = w, one can get τ + F(G(u, w, w)) = τ + F(G(T u, T w, T 2 w)) ≤ F(αG(u, w, T w) + βG(u, T u, T w) + γ G(w, T w, T 2 w) + δG(w, T 2 u, T 2 w) + ηG(u, T u, T 2 u)) = F(αG(u, w, w) + βG(u, u, w) + δG(w, u, w)) ≤ F(αG(u, w, w) + 2βG(u, w, w) + δG(u, w, w)) = F((α + 2β + δ)G(u, w, w)). This is a contradiction, in view of α + 2β + δ ≤ 1, and therefore u = w. This concludes the proof.
Choosing α = β = γ = 0 in Theorem 3.1, resulting subsequent Corollary. Corollary 3.1 Let (X, G) be a G-complete metric space and T be a self mapping on X . Assume that there exist F ∈ and τ > 0 such that τ + F(G(T x, T y, T 2 y)) ≤ F(δG(y, T 2 x, T 2 y) + ηG(x, T x, T 2 x)), For all x, y, ∈ X , G(T x, T y, T 2 y) > 0, where δ, η ≥ 0 with δ + η < 1. Then T has a unique fixed point in X . Taking β = δ = 0 in Theorem 3.1, then another corollary is obtained. Corollary 3.2 Let (X, G) be a G-complete metric space and T be a self mapping on X . Assume that there existsF ∈ and τ > 0 such that τ + F(G(T x, T y, T 2 y)) ≤ F(αG(x, y, T y) + γ G(y, T y, T 2 y) + ηG(x, T x, T 2 x)). For all x, y, ∈ X , G(T x, T y, T 2 y) > 0, where α, γ , η ≥ 0 with α + γ + η < 1. Then T has a unique fixed point in X . Remark 3.1 On choosing different combinations of suitable values of constants α, β, γ , η and δ in Theorem 3.1, host of the corollaries can be attained which include new versions of Chatterjea Type result [28], Kannan Type theorem [2] and Reich type result [29] in the context of G-metric spaces. e.g. Corollary 3.2 is the Reich type result [29] in the framework of G-metric spaces. Next, we furnish an illustrative example which demonstrates the validity of the hypotheses and degree of generality of Theorem 3.1.
Fixed point results for generalized F-contractive and...
Example 3.3 Consider the subsequent sequence {Sn }n∈N as S1 = 2.1, S2 = 2.1 + 22 .2, S3 = 2.1 + 22 .2 + 23 .3, .. .
Sn = 2.1 + 22 .2 + 23 .3 + 24 .4 + · · · + 2n .n = (n − 1)2n+1 + 2. 0, i f and only i f x = y = z Let X = {Sn : n ∈ X } and G(x, y, z) = max{x, y, z}, other wise. Then (X, G) is a complete G-metric space. Define the mapping T : X → X by T (S1 ) = S1 and T (Sn ) = Sn−1 for every n > 1. Now we claim that T is a F-contraction with F(α) = ln α + α., clearly F ∈ . First of all, we observe that G(T Sn , T Sm , T 2 Sm ) > 0 ⇔ (n = 1 ∧ m > 2) ∨ (m = 1 ∧ n > 2) ∨ (1 < n < m) ∨ (1 < m < n). In all the possible cases we will show that 2
2
2
2
2
G(T x, T y, T 2 y)e G(T x,T y,T y)−{αG(x,y,T y)+βG(x,T x,T y)+γ G(y,T y,T y)+δG(y,T x,T y)+ηG(x,T x,T x)} αG(x, y, T y) + βG(x, T x, T y) + γ G(y, T y, T 2 y) + δG(y, T 2 x, T 2 y) + ηG(x, T x, T 2 x) ≤ e−τ
(3.5)
for x = Sn , y = Sm , n, m ∈ N and for τ = 6 > 0 with α = 21 , β, γ , δ, η ≥ 0 such that α + β + γ + δ + η < 1 and α + 2β + δ ≤ 1. It is also to notice that according to the structure of {Sn }, it is clear that Sn = (n − 1)2n + 1 2 = (n − 2)2n + 2 + (2n − 1) = Sn−1 + 2n − 1. Consequently we conclude that Sn−1
2. Then G(T x, T y, T 2 y) = G(T Sn , T Sm , T 2 Sm ) = Sm−1 , G(x, y, T y) = G(S1 , Sm , T Sm ) = Sm , G(x, T x, T y) = G(S1 , T S1 , T Sm ) = Sm−1 , G(y, T y, T 2 y) = G(Sm , T Sm , T 2 Sm ) = Sm , G(y, T 2 x, T 2 y) = G(Sm , T 2 S1 , T 2 Sm ) = Sm , G(x, T x, T 2 x) = G(S1 , T S1 , T 2 S1 ) = 0.
D. Singh et al.
Then by utilizing (3.5) and also keep in notice that Sn−1 < αSn , ∀n ∈ N , we have Sm−1 .e Sm−1 −(αSm +β Sm−1 +γ Sm +δSm ) ≤ e Sm−1 −(αSm +β Sm−1 +γ Sm +δSm ) αSm + β Sm−1 + γ Sm + δSm < e Sm−1 −αSm , =e
f or m > 2
−(αSm −Sm−1 )
< e−6 ,
since (0.5)Sm − Sm−1 > 6 ( f or m > 2).
Thus in this case T is a Hardy–Roger type generalized F contractive mapping with τ = 6. Case-II When m = 1, n > 2. G(T x, T y, T 2 y) = G(T Sn , T S1 , T 2 S1 ) = Sn−1 , G(x, y, T y) = G(Sn , S1 , T S1 ) = Sn , G(x, T x, T y) = G(Sn , T Sn , T S1 ) = Sn , G(y, T y, T 2 y) = G(S1 , T S1 , T 2 S1 ) = 0, G(y, T 2 x, T 2 y) = G(S1 , T 2 Sn , T 2 S1 ) = 0 or Sn−2 , G(x, T x, T 2 x) = G(Sn , T Sn , T 2 Sn ) = Sn . Thus from (3.5), we have Sn−1 .e Sn−1 −(αSn +β Sn +δSn−2 +ηSn ) < e−(αSn −Sn−1 ) < e−6 . αSn + β Sn + δSn−2 + ηSn Case-III When 1 < n < m. Then consider G(T x, T y, T 2 y) = G(T Sn , T Sm , T 2 Sm ) = Sm−1 , G(x, y, T y) = G(Sn , Sm , T Sm ) = Sm , G(x, T x, T y) = G(Sn , T Sn , T Sm ) = Sm−1 , G(y, T y, T 2 y) = G(Sm , T Sm , T 2 Sm ) = Sm , G(y, T 2 x, T 2 y) = G(Sm , T 2 Sn , T 2 Sm ) = Sm , G(x, T x, T 2 x) = G(Sn , T Sn , T 2 Sn ) = Sn . As in Case-I, we get the desired. Case-IV When 1 < m < n. Then G(T x, T y, T 2 y) = G(T Sn , T Sm , T 2 Sm ) = Sn−1 , G(x, y, T y) = G(Sn , Sm , T Sm ) = Sn , G(x, T x, T y) = G(Sn , T Sn , T Sm ) = Sn , G(y, T y, T 2 y) = G(Sm , T Sm , T 2 Sm ) = Sm , G(y, T 2 x, T 2 y) = G(Sm , T 2 Sn , T 2 Sm ) = Sm or Sn−2 , G(x, T x, T 2 x) = G(Sn , T Sn , T 2 Sn ) = Sn . Calculating as in Case-II, we get the required. Thus T is a Hardy–Roger type F-contraction mapping and S1 is a fixed point which is indeed unique.
Fixed point results for generalized F-contractive and... Acknowledgments The authors are grateful to the learned referees for their accurate reading and their helpful suggestions. Dr. Poom Kumam was supported by the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU58 Grant No. 59000399).
References 1. Banach, B.: Sur les oprations dons les ensembles abstraits et leur application aux quations intgrales. Fundam. Math. 3, 133–181 (1922) 2. Kannan, R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1965) 3. Khan, M.S.: On fixed point theorem. Math. Jpn. 23, 201–204 (1978) 4. Kalinda, A.K.: On a fixed point theorem for Kannan type mappings. Math. Jpn. 33(5), 721–723 (1988) 5. Azam, A., Arshad, M., Beg, I.: Common fixed points of two maps in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 57, 433–441 (2008) 6. Bellman, R.: Methods of Nonliner Analysis, vol. II, vol. 61 of Mathematics in Science and Engineering. Academic Press, New York (1973) 7. Bellman, R., Lee, E.S.: Functional equations in dynamic programming. Aequ. Math. 17, 118 (1978) 8. Rhoades, B.E.: Some fixed point theorems for pair of mappings. Jnanabha 15, 151–156 (1985) 9. Damjanovic, B., Samet, B., Vetro, C.: Common fixed point theorems for multi-valued maps. Acta Math. Sci. Ser. B 32, 818824 (2012) 10. Rehman, F.U., Ahmad, B.: Some fixed point theorems in complete metric space. Math. Jpn. 36(2), 239–243 (1991) 11. Jr, Nadler: S B: Multi-valued contraction mappings. Pac. J. Math. 30, 475488 (1969) 12. Wardowski, D: Fixed point theory of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 6 (2012) 13. Secelean, NA: Iterated function systems consisting of F-contractions. Fixed Point Theory Appl. 13 (2013) 14. Piri, H., Kumam, P.: Some fixed point theorems concerning F-contraction in complete metric spaces. Fixed Point Theory Appl. 210, 11 (2014) 15. Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7, 289–297 (2006) 16. Samet, B., Vetro, C., Vetro, F.: Remarks on G-metric spaces. Int. J. Anal. Article ID 917158 (2013) 17. Jleli, M., Samet, B.: Remarks on G-metric spaces and fixed point theorems. Fixed Point Theory Appl. 210, 7 (2012) 18. Mustafa, Z., Sims, B.: Fixed point theorems for contractive mappings in complete G-metric spaces. Fixed Point Theory Appl., Article ID 917175 (2009) 19. Mustafa, Z., Obiedat, H., Awawdeh, F.: Some fixed point theorem for mapping on complete G-metric spaces. Fixed Point Theory Appl., Article ID 189870 (2008) 20. Shatanawi, W., Mustafa, Z.: On coupled random fixed point results in partially ordered metric spaces. Mat. Vesn. 64, 139–146 (2012) 21. Aydi, H., Postolache, M., Shatanawi, W.: Coupled fixed point results for (ψ, ϕ)-weakly contractive mappings in ordered G-metric spaces. Comput. Math. Appl. 63(1), 298–309 (2012) 22. Aydi, H., Damjanovic, B., Samet, B., Shatanawi, W.: Coupled fixed point theorems for nonlinear contractions in partially ordered G-metric spaces. Math. Comput. Model. 54, 2443–2450 (2011) 23. Agarwal, R., Karapınar, E.: Remarks on some coupled fixed point theorems in G-metric spaces. Fixed Point Theory Appl. 2, 33 (2013). doi:10.1186/1687-1812-2013-2 24. Asadi, M., Karapinar, E., Salimi, P.: A new approach to G-metric and related fixed point theorems. J. Inequal. Appl. 254, 14 (2013) 25. Karapınar, E., Agarwal, R.: Further fixed point results on G-metric spaces. Fixed Point Theory Appl. 154, 14 (2013) 26. Nashine, H.K., Kadelburg, Z.: Common fixed point theorems under weakly Hardy–Rogerstype contraction conditions in ordered orbitally complete metric spaces, Revista de la Real Academia de Ciencias Exactas. Fisicas y Naturales. Serie A. Matematicas 108(2), 377–395 (2014) 27. Mustafa, Z.: A new structure for generalized metric spaces with applications to fixed point theory. PhD thesis, The University of Newcastle, Australia (2005) 28. Chatterjea, S.K.: Fixed-point theorems. Comptes Rendus de lAcademie Bulgare des Sciences 25, 727–730 (1972) 29. Reich, S.: Some remarks concerning contraction mappings. Can. Math. Bull. 14, 121124 (1971)
