Some Fixed Point Results in GP-Metric Spaces

Abstract

Following a recent paper of Zand and Nezhad (2011), we establish some fixed point results in GP-metric spaces. The presented theorems generalize and improve several existing results in the literature. Also, some examples are presented.

1. Introduction

Partial metric space is a generalized metric space introduced by Matthews [1] in which each object does not necessarily have to have a zero distance from itself. A motivation is to introduce this space to give a modified version of the Banach contraction principle [2]. Subsequently, several authors studied the problem of existence and uniqueness of a fixed point for mappings satisfying different contractive conditions, see [3–23].

On the other hand, in 2006 Mustafa and Sims [24] introduced a new notion of generalized metric spaces called 𝐺-metric spaces. Based on the notion of a 𝐺-metric space, many fixed point results for different contractive conditions have been presented, for more details see [25–42].

Recently, based on the two above notions, Zand and Nezhad [43] introduced a new generalized metric space as both a generalization of a partial metric space and a 𝐺-metric space. It is given as follows.

Definition 1.1 (see [43]). Let 𝑋 be a nonempty set. A function 𝐺_𝑝∶𝑋×𝑋×𝑋→[0,+∞) is called a 𝐺𝑃-metric if the following conditions are satisfied:  (GP1) 𝑥=𝑦=𝑧 if 𝐺_𝑝(𝑥,𝑦,𝑧)=𝐺_𝑝(𝑧,𝑧,𝑧)=𝐺_𝑝(𝑦,𝑦,𝑦)=𝐺_𝑝(𝑥,𝑥,𝑥);  (GP2) 0≤𝐺_𝑝(𝑥,𝑥,𝑥)≤𝐺_𝑝(𝑥,𝑥,𝑦)≤𝐺_𝑝(𝑥,𝑦,𝑧) for all 𝑥,𝑦,𝑧∈𝑋;  (GP3) 𝐺_𝑝(𝑥,𝑦,𝑧)=𝐺_𝑝(𝑥,𝑧,𝑦)=𝐺_𝑝(𝑦,𝑧,𝑥)=⋯, symmetry in all three variables;  (GP4) 𝐺_𝑝(𝑥,𝑦,𝑧)≤𝐺_𝑝(𝑥,𝑎,𝑎)+𝐺_𝑝(𝑎,𝑦,𝑧)−𝐺_𝑝(𝑎,𝑎,𝑎) for any 𝑥,𝑦,𝑧,𝑎∈X.
Then the pair (𝑋,𝐺) is called a 𝐺𝑃-metric space.

Example 1.2 (see [43]). Let 𝑋=[0,∞) and define 𝐺_𝑝(𝑥,𝑦,𝑧)=max{𝑥,𝑦,𝑧}, for all 𝑥,𝑦,𝑧∈𝑋. Then (𝑋,𝐺_𝑝) is a 𝐺𝑃-metric space.

Proposition 1.3 (see [43]). Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-metric space, then for any 𝑥,𝑦,𝑧 and 𝑎∈𝑋 it follows that (i)𝐺_𝑝(𝑥,𝑦,𝑧)≤𝐺_𝑝(𝑥,𝑥,𝑦)+𝐺_𝑝(𝑥,𝑥,𝑧)−𝐺_𝑝(𝑥,𝑥,𝑥); (ii)𝐺_𝑝(𝑥,𝑦,𝑦)≤2𝐺_𝑝(𝑥,𝑥,𝑦)−𝐺_𝑝(𝑥,𝑥,𝑥); (iii)𝐺_𝑝(𝑥,𝑦,𝑧)≤𝐺_𝑝(𝑥,𝑎,𝑎)+𝐺_𝑝(𝑦,𝑎,𝑎)+𝐺_𝑝(𝑧,𝑎,𝑎)−2𝐺_𝑝(𝑎,𝑎,𝑎); (iv)𝐺_𝑝(𝑥,𝑦,𝑧)≤𝐺_𝑝(𝑥,𝑎,𝑧)+𝐺_𝑝(𝑎,𝑦,𝑧)−𝐺_𝑝(𝑎,𝑎,𝑎).

Proposition 1.4 (see [43]). Every 𝐺𝑃-metric space (𝑋,𝐺_𝑝) defines a metric space (𝑋,𝐷_𝐺𝑝), where 𝐷_𝐺𝑝𝑥,𝑦=𝐺_𝑝𝑥,𝑦,𝑦+𝐺_𝑝𝑦,𝑥,𝑥−𝐺_𝑝(𝑥,𝑥,𝑥)−𝐺_𝑝𝑦,𝑦,𝑦𝑓𝑜𝑟𝑎𝑙𝑙𝑥,𝑦∈𝑋. (1.1)

Definition 1.5 (see [43]). Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-metric space and let {𝑥_𝑛} be a sequence of points of 𝑋. A point 𝑥∈𝑋 is said to be the limit of the sequence {𝑥_𝑛} or 𝑥_𝑛→𝑥 if lim_{𝑚,𝑚→∞}𝐺_𝑝𝑥,𝑥_𝑚,𝑥_𝑛=𝐺_𝑝(𝑥,𝑥,𝑥). (1.2)

Proposition 1.6 (see [43]). Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-metric space. Then, for any sequence {𝑥_𝑛} in 𝑋, and a point 𝑥∈𝑋 the following are equivalent: (A){𝑥_𝑛} is 𝐺𝑃-convergent to 𝑥; (B)𝐺_𝑝(𝑥_𝑛,𝑥_𝑛,𝑥)→𝐺_𝑝(𝑥,𝑥,𝑥) as 𝑛→∞; (C)𝐺_𝑝(𝑥_𝑛,𝑥,𝑥)→𝐺_𝑝(𝑥,𝑥,𝑥) as 𝑛→∞.

Definition 1.7 (see [43]). Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-metric space.  (S1) A sequence {𝑥_𝑛} is called a 𝐺𝑃-Cauchy if and only if lim_{𝑚,𝑛→∞}𝐺_𝑝(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚) exists (and is finite). (S2) A 𝐺𝑃-partial metric space (𝑋,𝐺_𝑝) is said to be 𝐺𝑃-complete if and only if every 𝐺𝑃-Cauchy sequence in X is 𝐺𝑃-convergent to 𝑥∈𝑋 such that 𝐺_𝑝(𝑥,𝑥,𝑥)=lim_{𝑚,𝑛→∞}𝐺_𝑝(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚).

Now, we introduce the following.

Definition 1.8. Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-metric space. (M1) A sequence {𝑥_𝑛} is called a 0-𝐺𝑃-Cauchy if and only if lim_{𝑚,𝑛→∞}𝐺_𝑝(𝑥_𝑛,𝑥_𝑚,𝑥_𝑚)=0; (M2) A 𝐺𝑃-metric space (𝑋,𝐺_𝑝) is said to be 0-𝐺𝑃-complete if and only if every 0-𝐺𝑃-Cauchy sequence is GP-convergent to a point 𝑥∈𝑋 such that 𝐺_𝑝(𝑥,𝑥,𝑥)=0.

Example 1.9. Let 𝑋=[0,+∞) and define 𝐺_𝑝(𝑥,𝑦,𝑧)=max{𝑥,𝑦,𝑧}, for all 𝑥,𝑦,𝑧∈𝑋. Then (𝑋,𝐺_𝑝) is a 𝐺𝑃-complete 𝐺𝑃-metric space. Moreover, if 𝑋=ℚ∩[0,+∞) (where ℚ denotes the set of rational numbers), then (𝑋,𝐺_𝑝) is a 0-𝐺𝑃-complete 𝐺𝑃-metric space.

Lemma 1.10. Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-metric space. Then (A) if 𝐺_𝑝(𝑥,𝑦,𝑧)=0, then 𝑥=𝑦=𝑧; (B) if 𝑥≠𝑦, then 𝐺_𝑝(𝑥,𝑦,𝑦)>0.

Proof. By (GP2) we have 𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝𝑥,𝑦,𝑦,𝐺_𝑝𝑥,𝑥,𝑦≤𝐺_𝑝𝑥,𝑦,𝑧=0. (1.3) Then, by Proposition 1.4, we have 𝐷_𝐺𝑝(𝑥,𝑦)=0, that is, 𝑥=𝑦. Similarly, we can obtain that 𝑦=𝑧. The assertion (A) is proved.
On the other hand, if 𝑥≠𝑦 and 𝐺_𝑝(𝑥,𝑦,𝑦)=0, then by (A), 𝑥=𝑦 which is a contradiction and so (B) holds.

In this paper, we establish some fixed point results in 𝐺𝑃-metric spaces analogous to results of Ilić et al. [44] which were proved in partial metric spaces. Also, some examples are provided to illustrate our results. To our knowledge, we are the first to give some fixed point results in 𝐺𝑃-metric spaces, and so is the novelty and original contributions of this paper. This opens the door to other possible fixed (common fixed) point results.

2. Main Results

We start by stating a fixed point result of Ilić et al. [44].

Theorem 2.1 (see [44]). Let (𝑋,𝑝) be a complete partial metric space. Let 𝑓 be a self-mapping on 𝑋. Suppose that for all 𝑥,𝑦,𝑧∈𝑋 the following condition holds: 𝑝𝑓𝑥,𝑓𝑦≤max𝛼𝑝𝑥,𝑦,𝑝(𝑥,𝑥),𝑝𝑦,𝑦, (2.1) where 0≤𝛼<1. Then (1) the set 𝑋_𝑃={𝑦∈𝑋∶𝑝(𝑥,𝑥)=inf_𝑦∈𝑋𝑝(𝑦,𝑦)} is nonempty; (2) there is a unique 𝑢∈𝑋_𝑃 such that 𝑓𝑢=𝑢; (3) for all 𝑥∈𝑋_𝑃 the sequence {𝑓^𝑛𝑥} converges to 𝑢 with respect to the metric 𝑑_𝑝 (where 𝑑_𝑝(𝑥,𝑦)=𝑝(𝑥,𝑦)−𝑝(𝑥,𝑥)−𝑝(𝑦,𝑦) for 𝑥,𝑦∈𝑋).

The analog of Theorem 2.1 in 𝐺𝑃-metric spaces is given as follows.

Theorem 2.2. Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-complete 𝐺𝑃-metric space. Let 𝑓 be a self-mapping on 𝑋. Suppose that for all 𝑥,𝑦,𝑧∈𝑋 the following condition holds: 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧≤max𝑟𝐺_𝑝𝑥,𝑦,𝑧,𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧), (2.2) where 0≤𝑟<1. Then (T1) the set 𝑋_𝐺𝑝={𝑦∈𝑋∶𝐺_𝑝(𝑥,𝑥,𝑥)=inf_𝑦∈𝑋𝐺_𝑝(𝑦,𝑦,𝑦)} is nonempty; (T2) there is a unique 𝑥^∗∈𝑋_𝐺𝑝 such that 𝑓𝑥^∗=𝑥^∗; (T3) for all 𝑥∈𝑋_𝐺𝑃 the sequence {𝑓^𝑛𝑥} converges to 𝑥^∗ with respect to the metric 𝐷_𝐺𝑝.

Proof. Let 𝑥∈𝑋. By (2.2), we have 𝐺_𝑝𝑓𝑥,𝑓𝑥,𝑓𝑥≤max𝑟𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝(𝑥,𝑥,𝑥)=𝐺_𝑝(𝑥,𝑥,𝑥). (2.3) Hence, {𝐺_𝑝(𝑓^𝑛𝑥,𝑓^𝑛𝑥,𝑓^𝑛𝑥)}_𝑛≥0 is a nonincreasing sequence. Put 𝑆_𝑥∶=lim_𝑛𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑛𝑥,𝑓^𝑛𝑥=inf_𝑛𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑛𝑥,𝑓^𝑛𝑥,Γ_𝑥1∶=𝐺1−𝑟^𝑝𝑥,𝑓𝑥,𝑓𝑥+𝐺_𝑝(𝑥,𝑥,𝑥). (2.4)(2.5) We shall show that 𝐺_𝑝𝑓^𝑖𝑥,𝑓^𝑗𝑥,𝑓^𝑗𝑥≤3Γ_𝑥forall𝑖,𝑗≥0. (2.6) Again, by (2.2), we have for all 𝑚>𝑛≥0𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑚𝑥,𝑓^𝑚𝑥≤max𝑟𝐺_𝑝𝑓^𝑛−1𝑥,𝑓^𝑚−1𝑥,𝑓^𝑚−1𝑥,𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑛𝑥,𝑓^𝑛𝑥. (2.7) At first 𝐺_𝑝𝑥,𝑓^𝑗𝑥,𝑓^𝑗𝑥≤𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+𝐺_𝑝𝑓𝑥,𝑓^𝑗𝑥,𝑓^𝑗𝑥≤𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+max𝑟𝐺_𝑝𝑥,𝑓^𝑗−1𝑥,𝑓^𝑗−1𝑥,𝐺_𝑝(𝑥,𝑥,𝑥). (2.8)
Similarly 𝐺_𝑝𝑥,𝑓^𝑗−1𝑥,𝑓^𝑗−1𝑥≤𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+max𝑟𝐺_𝑝𝑥,𝑓^𝑗−2𝑥,𝑓^𝑗−2𝑥,𝐺_𝑝(𝑥,𝑥,𝑥). (2.9)
Then we have 𝐺_𝑝𝑥,𝑓^𝑗𝑥,𝑓^𝑗𝑥≤𝐺_p𝑥,𝑓𝑥,𝑓𝑥+max𝑟𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+𝑟²𝐺_𝑝𝑥,𝑓^𝑗−2𝑥,𝑓^𝑗−2𝑥,𝐺_𝑝(𝑥,𝑥,𝑥). (2.10) By continuing this process, we get 𝐺_𝑝𝑥,𝑓^𝑗𝑥,𝑓^𝑗𝑥≤𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+max𝑟𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+⋯+𝑟^𝑗−1𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥,𝐺_𝑝(𝑥,𝑥,𝑥)≤𝐺_𝑝𝑟𝑥,𝑓𝑥,𝑓𝑥+max𝐺1−𝑟^𝑝𝑥,𝑓𝑥,𝑓𝑥,𝐺_𝑝(𝑥,𝑥,𝑥)≤𝐺_𝑝+𝑟𝑥,𝑓𝑥,𝑓𝑥𝐺1−𝑟^𝑝𝑥,𝑓𝑥,𝑓𝑥+𝐺_𝑝(𝑥,𝑥,𝑥)=Γ_𝑥. (2.11)
Now, by Proposition 1.3 (ii), we have 𝐺_𝑃(𝑓^𝑖𝑥,𝑥,𝑥)≤2𝐺_𝑃(𝑥,𝑓^𝑖𝑥,𝑓^𝑖𝑥)≤2Γ_𝑥. Hence 𝐺_𝑝𝑓^𝑖𝑥,𝑓^𝑖𝑥,𝑓^𝑗𝑥≤𝐺_𝑝𝑓^𝑖𝑥,𝑥,𝑥+𝐺_𝑝𝑥,𝑓^𝑗𝑥,𝑓^𝑗𝑥≤3Γ_𝑥, (2.12) that is, (2.6) holds. On the other hand, by (GP2), we have 𝑆_𝑥≤𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑛𝑥,𝑓^𝑛𝑥≤𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑚𝑥,𝑓^𝑚𝑥. (2.13) Given any 𝜖>0, by (2.4), there exists 𝑛₀∈ℕ such that 𝐺_𝑝(𝑓^𝑛0𝑥,𝑓^𝑛0𝑥,𝑓^𝑛0𝑥)<𝑆_𝑥+𝜖. Since 0≤𝑟<1, so without loss of generality, we have 3Γ_𝑥𝑟^𝑛0<𝑆_𝑥+𝜖. Therefore, for all 𝑚,𝑛≥2𝑛₀𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑚𝑥,𝑓^𝑚𝑥≤max𝑟𝐺_𝑝𝑓^𝑛−1𝑥,𝑓^𝑚−1𝑥,𝑓^𝑚−1𝑥,𝐺_𝑝𝑓^𝑛−1𝑥,𝑓^𝑛−1𝑥,𝑓^𝑛−1𝑥,𝐺_𝑝𝑓^𝑚−1𝑥,𝑓^𝑚−1𝑥,𝑓^𝑚−1𝑥𝑟≤max²𝐺_𝑝𝑓^𝑛−2𝑥,𝑓^𝑚−2𝑥,𝑓^𝑚−2𝑥,𝐺_𝑝𝑓^𝑛−2𝑥,𝑓^𝑛−2𝑥,𝑓^𝑛−2𝑥,𝐺_𝑝𝑓^𝑚−2𝑥,𝑓^𝑚−2𝑥,𝑓^𝑚−2𝑥𝑟≤⋯≤max^𝑛0𝐺_𝑝𝑓^𝑛−𝑛0𝑥,𝑓^𝑚−𝑛0𝑥,𝑓^𝑚−𝑛0𝑥,𝐺_𝑝𝑓^𝑛−𝑛0𝑥,𝑓^𝑛−𝑛0𝑥,𝑓^𝑛−𝑛0𝑥,𝐺_𝑝𝑓^𝑚−𝑛0𝑥,𝑓^𝑚−𝑛0𝑥,𝑓^𝑚−𝑛0𝑥<𝑆_𝑥+𝜖. (2.14) Then, 𝑆_𝑥=lim_{𝑚,𝑛→∞}𝐺_𝑝(𝑓^𝑛𝑥,𝑓^𝑚𝑥,𝑓^𝑚𝑥) and so {𝑓^𝑛𝑥} is a 𝐺𝑃-Cauchy sequence. Since (𝑋,𝐺_𝑝) is 𝐺𝑃-complete, then there exists 𝑥^∗∈𝑋 such that {𝑓^𝑛𝑥}  𝐺𝑃-converges to 𝑥^∗, that is, 𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗=lim_{𝑚,𝑛→∞}𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑚𝑥,𝑓^𝑚𝑥. (2.15) Since {𝑓^𝑛𝑥}𝐺𝑃-converges to 𝑥^∗, then Proposition 1.6 yields that 𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗=lim_𝑛→∞𝐺_𝑝𝑥^∗,𝑓^𝑛𝑥,𝑓^𝑛𝑥=lim_𝑛→∞𝐺_𝑝𝑓^𝑛𝑥,𝑥^∗,𝑥^∗. (2.16) We obtain that 𝑆_𝑥=𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗=lim_𝑛→∞𝐺_𝑝𝑥^∗,𝑓^𝑛𝑥,𝑓^𝑛𝑥=lim_𝑛→∞𝐺_𝑝𝑓^𝑛𝑥,𝑥^∗,𝑥^∗=lim_{𝑚,𝑛→∞}𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑚𝑥,𝑓^𝑚𝑥. (2.17)(2.18) For all 𝑛∈ℕ𝐺_𝑝𝑥^∗,𝑓𝑥^∗,𝑓𝑥^∗≤𝐺_𝑝𝑥^∗,𝑓^𝑛𝑥,𝑓^𝑛𝑥+𝐺_𝑝𝑓^𝑛𝑥,𝑓𝑥^∗,𝑓𝑥^∗−𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑛𝑥,𝑓^𝑛𝑥. (2.19) By taking the limit as 𝑛→∞ in the above inequality, we get 𝐺_𝑝𝑥^∗,𝑓𝑥^∗,𝑓𝑥^∗≤lim_𝑛→∞𝐺_𝑝𝑓^𝑛𝑥,𝑓𝑥^∗,𝑓𝑥^∗. (2.20) On the other hand, from (2.2), we have lim_𝑛→∞𝐺_𝑝𝑓^𝑛𝑥,𝑓𝑥^∗,𝑓𝑥^∗≤max𝑟lim_𝑛→∞𝐺_𝑝𝑓^𝑛−1𝑥,𝑥^∗,𝑥^∗,lim_𝑛→∞𝐺_𝑝𝑓^𝑛−1𝑥,𝑓^𝑛−1𝑥,𝑓^𝑛−1𝑥,𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗=𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗. (2.21) Thus, 𝐺_𝑝(𝑥^∗,𝑓𝑥^∗,𝑓𝑥^∗)≤𝐺_𝑝(𝑥^∗,𝑥^∗,𝑥^∗). By (GP2), we deduce that 𝐺_𝑝𝑥^∗,𝑓𝑥^∗,𝑓𝑥^∗=𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗. (2.22) Now we show that 𝑋_𝐺𝑝 is nonempty. Let Ω=inf_𝑦∈𝑋𝐺_𝑝(𝑦,𝑦,𝑦). For all 𝑘∈ℕ, pike 𝑥_𝑘∈𝑋 with 𝐺_𝑝(𝑥_𝑘,𝑥_𝑘,𝑥_𝑘)<Ω+1/𝑘. Define 𝑥^∗_𝑛=𝑓^𝑛𝑥^∗ for all 𝑛≥1. Let us show that lim_{𝑚,𝑛→∞}𝐺_𝑝𝑥^∗_n,𝑥^∗_𝑚,𝑥^∗_𝑚=Ω. (2.23) Given 𝜖>0, put 𝑛₀∶=[3/𝜖(1−𝑟)]+1. If 𝑘≥𝑛₀, then we have Ω≤𝐺_𝑝𝑓𝑥^∗_𝑘,𝑓𝑥^∗_𝑘,𝑓𝑥^∗_𝑘≤𝐺_𝑝𝑥^∗_𝑘,𝑥^∗_𝑘,𝑥^∗_𝑘=𝑆_𝑥𝑘≤𝐺_𝑝𝑥_𝑘,𝑥_𝑘,𝑥_𝑘1<Ω+𝑘1<Ω+𝑛₀3<Ω+𝜖(1−𝑟)≤𝐺_𝑝𝑓𝑥^∗_𝑘,𝑓𝑥^∗_𝑘,𝑓𝑥^∗_𝑘+3𝜖().1−𝑟 (2.24) Therefore, we have 𝑇_𝑘∶=𝐺_𝑝𝑥^∗_𝑘,𝑥^∗_𝑘,𝑥^∗_𝑘−𝐺_𝑝𝑓𝑥^∗_𝑘,𝑓𝑥^∗_𝑘,𝑓𝑥^∗_𝑘<3𝜖().1−𝑟 (2.25) On the other hand, if 𝑘≥𝑛₀, then 𝐺_𝑝(𝑥^∗_𝑘,𝑥^∗_𝑘,𝑥^∗_𝑘)=𝑆_𝑥𝑘<Ω+1/𝑛₀. It follows that 𝐺_𝑝𝑥^∗_𝑘,𝑥^∗_𝑘,𝑥^∗_𝑘3<Ω+𝜖().1−𝑟 (2.26) By (GP4), (2.22), and (2.25), we can obtain 𝐺_𝑝x^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚≤𝐺_𝑝𝑥^∗_𝑛,𝑓𝑥^∗_𝑛,𝑓𝑥^∗_𝑛+𝐺_𝑝𝑓𝑥^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚−𝐺_𝑝𝑓𝑥^∗_𝑛,𝑓𝑥^∗_𝑛,𝑓𝑥^∗_𝑛=𝑇_𝑛+𝐺_𝑝𝑓𝑥^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚,𝐺_𝑝𝑓𝑥^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚≤𝐺_𝑝𝑓𝑥^∗_𝑛,𝑓𝑥^∗_𝑚,𝑓𝑥^∗_𝑚+𝐺_𝑝𝑓𝑥^∗_𝑚,𝑥^∗_𝑚,𝑥^∗_𝑚−𝐺_𝑝𝑓𝑥^∗_𝑚,𝑓𝑥^∗_𝑚,𝑓𝑥^∗_𝑚=𝐺_𝑝𝑓𝑥^∗_𝑛,𝑓𝑥^∗_𝑚,𝑓𝑥^∗_𝑚+𝑇_𝑚. (2.27) Thus 𝐺_𝑝𝑥^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚≤𝑇_𝑚+𝑇_𝑛+𝐺_𝑝𝑓𝑥^∗_𝑛,𝑓𝑥^∗_𝑚,𝑓𝑥^∗_𝑚≤𝑇_𝑚+𝑇_𝑛+max𝑟𝐺_𝑝𝑥^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚,𝐺_𝑝𝑥^∗_𝑛,𝑥^∗_𝑛,𝑥^∗_𝑛,𝐺_𝑝𝑥^∗_𝑚,𝑥^∗_𝑚,𝑥^∗_𝑚. (2.28) Now, by (2.25) and (2.26), we have Ω≤𝐺_𝑝𝑥^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚2≤max32𝜖,3𝜖(1−𝑟)+𝐺_𝑝𝑥^∗_𝑛,𝑥^∗_𝑛,𝑥^∗_𝑛,23𝜖(1−𝑟)+𝐺_𝑝𝑥^∗_𝑚,𝑥^∗_𝑚,𝑥^∗_𝑚2≤max3𝜖,Ω+𝜖(1−𝑟)<Ω+𝜖, (2.29) that is, (2.23) holds. Again, Since (𝑋,𝐺_𝑝) is 𝐺𝑃-complete, then there exists 𝑦∈𝑋 such that 𝐺_𝑝𝑦,𝑦,𝑦=lim_𝑛→∞𝐺_𝑝𝑦,𝑥^∗_𝑛,𝑥^∗_𝑛=lim_𝑛→∞𝐺_𝑝𝑥^∗_𝑛,𝑦,𝑦=lim_{𝑚,𝑛→∞}𝐺_𝑝𝑥^∗_𝑛,𝑥^∗_𝑚,𝑥^∗_𝑚=Ω. (2.30) This leads that 𝑦∈𝑋_𝐺𝑝, so 𝑋_𝐺𝑝 is nonempty.
Let 𝑥∈𝑋. By (2.22), we get Ω≤𝐺_𝑝𝑓𝑥^∗,𝑓𝑥^∗,𝑓𝑥^∗≤𝐺_𝑝𝑥^∗,𝑓𝑥^∗,𝑓𝑥^∗=𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗=𝑆_𝑥=Ω. (2.31) From (GP1), it follows that 𝑥^∗=𝑓𝑥^∗. By (2.17), we have lim_𝑛→∞𝐷_𝐺𝑝𝑓^𝑛𝑥,𝑥^∗=lim_𝑛→∞𝐺_𝑝𝑓^𝑛𝑥,𝑥^∗,𝑥^∗+lim_𝑛→∞𝐺_𝑝𝑥^∗,𝑓^𝑛𝑥,𝑓^𝑛𝑥−lim_𝑛→∞𝐺_𝑝𝑓^𝑛𝑥,𝑓^𝑛𝑥,𝑓^𝑛𝑥−𝐺_𝑝𝑥^∗,𝑥^∗,𝑥^∗=0. (2.32) Therefore, for all 𝑥∈𝑋_𝑃 the sequence {𝑓^𝑛𝑥} converges with respect to the metric 𝐷_𝐺𝑝 to 𝑥^∗. The uniqueness of the fixed point follows easily from (2.2).

We illustrate Theorem 2.2 by the following examples.

Example 2.3. Let 𝑋=[0,∞) and define 𝐺_𝑝(𝑥,𝑦,𝑧)=max{𝑥,𝑦,𝑧}, for all 𝑥,𝑦,𝑧∈𝑋. Then (𝑋,𝐺_𝑝) is a complete 𝐺𝑃-metric space. Clearly, (𝑋,𝐺) is not a 𝐺-metric space. Consider 𝑓∶𝑋→𝑋 defined by 𝑓𝑥=𝑥²/(1+𝑥). Without loss of generality, take 𝑥≤𝑦≤𝑧. We have 𝐺_𝑝=𝑧𝑓𝑥,𝑓𝑦,𝑓𝑧²𝐺1+𝑧≤𝑧=max_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧)=max𝑟𝐺_𝑝𝑥,𝑦,𝑧,𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧), (2.33) for all 𝑟∈[0,1). So, (2.2) holds. Here, 𝑢=0 is the unique fixed point of 𝑓.

Example 2.4. Let 𝑋=[0,∞). Define 𝐺_𝑝∶𝑋³→[0,∞) by 𝐺_𝑝(𝑥,𝑦,𝑧)=max{𝑥,𝑦,𝑧}. Clearly, (𝑋,𝐺_𝑝) is a 𝐺𝑃-metric space. Define 𝑓∶𝑋→𝑋 by 𝑥𝑓𝑥=²1,0≤𝑥≤3(1−𝑥)2,131<𝑥≤21()12𝑥+1𝑥>2. (2.34)
Then, the inequality (2.2) of Theorem 2.2 holds. Here, 𝑢=0 is the unique fixed point of 𝑓.

Proof. Clearly, 𝑀(𝑥,𝑦,𝑧)∶=max{𝑟𝐺_𝑝(𝑥,𝑦,𝑧),𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝(𝑦,𝑦,𝑦),𝐺_𝑝(𝑧,𝑧,𝑧)}=max{𝑥,𝑦,𝑧}, for all 𝑟∈(0,1). We have the following cases.
Case  1 (0≤𝑥,𝑦,𝑧≤1/3).
Consider the following: 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧=max𝑥²,𝑦²,𝑧²≤max𝑥,𝑦,𝑧=𝑀𝑥,𝑦,𝑧. (2.35)Case  2 (1/3<𝑥,𝑦,𝑧≤1/2).
Consider the following: 𝐺_𝑝()𝑓𝑥,𝑓𝑦,𝑓𝑧=max1−𝑥2,1−𝑦2,(1−𝑧)2≤max𝑥,𝑦,𝑧=𝑀𝑥,𝑦,𝑧. (2.36)Case  3 (𝑥,𝑦,𝑧>1/2).
Consider the following: 𝐺_𝑝1𝑓𝑥,𝑓𝑦,𝑓𝑧=max(),12𝑥+1,12𝑦+1(2𝑧+1)≤𝑀𝑥,𝑦,𝑧. (2.37)Case  4 (0≤𝑥≤1/3, 1/3<𝑦≤1/2 and 𝑧>1/2).
Consider the following: 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧=max𝑥²,1−𝑦2,1(2𝑧+1)≤𝑀𝑥,𝑦,𝑧. (2.38)Case  5 (0≤𝑥,𝑦≤1/3, 1/3<𝑧≤1/2).
Consider the following: 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧=max𝑥²,𝑦²,(1−𝑥)2≤𝑀𝑥,𝑦,𝑧. (2.39)Case  6 (0≤𝑥,𝑦≤1/3, 𝑧>1/2).
Consider the following: 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧=max𝑥²,𝑦²,1().2𝑥+1≤𝑀𝑥,𝑦,𝑧 (2.40)Case  7 (1/3<𝑥,𝑦≤1/2, 0≤𝑧≤1/3).
Consider the following: 𝐺_𝑝()𝑓𝑥,𝑓𝑦,𝑓𝑧=max1−𝑥2,1−𝑦2,𝑧²≤𝑀𝑥,𝑦,𝑧. (2.41)Case  8 (1/3≤𝑥,𝑦≤1/2, 𝑧>1/2).
Consider the following: 𝐺_𝑝()𝑓𝑥,𝑓𝑦,𝑓𝑧=max1−𝑥2,1−𝑦2,1(2𝑧+1)≤𝑀𝑥,𝑦,𝑧. (2.42)Case  9 (𝑥,𝑦>1/2, 0≤𝑧≤1/3).
Consider the following: 𝐺_𝑝1𝑓𝑥,𝑓𝑦,𝑓𝑧=max(),12𝑥+12𝑦+1,𝑧²≤𝑀𝑥,𝑦,𝑧. (2.43)Case  10 (𝑥,𝑦>1/2, 1/3<𝑧≤1/2).
Consider the following: 𝐺_𝑝1𝑓𝑥,𝑓𝑦,𝑓𝑧=max(),12𝑥+1,()2𝑦+11−𝑧2≤𝑀𝑥,𝑦,𝑧. (2.44)
Thus, the inequality (2.2) holds. Applying Theorem 2.2, we get 𝑢=0 is the unique point fixed point of 𝑓.

Also, Ilić et al. [44] proved the following result.

Theorem 2.5 (see [44]). Let (𝑋,𝑃) be a complete partial metric space. Let 𝑓 be a self-mapping on 𝑋. Suppose that for all 𝑥,𝑦,𝑧∈𝑋 the following condition holds: 𝑝,𝑝()𝑓𝑥,𝑓𝑦≤max𝛼𝑝𝑥,𝑦𝑥,𝑥+𝑝𝑦,𝑦2, (2.45) where 0≤𝛼<1. Then (i) the set 𝑋_𝑃={𝑦∈𝑋∶𝑝(𝑥,𝑥)=inf_𝑦∈𝑋𝑝(𝑦,𝑦)} is nonempty; (ii) there is a unique 𝑥^∗∈𝑋_𝑃 such that 𝑓𝑥^∗=𝑥^∗; (iii) for all 𝑥∈𝑋_𝑃, the sequence {𝑓^𝑛𝑥} converges to 𝑥^∗ with respect to the metric 𝑑_𝑝.

The analog of Theorem 2.5 in 𝐺𝑃-metric spaces is stated as follows.

Theorem 2.6. Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-complete 𝐺𝑃-metric space. Let 𝑓 be a self-mapping on 𝑋. Suppose that for all 𝑥,𝑦,𝑧∈𝑋 the following condition holds: 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧≤max𝑟𝐺_𝑝,𝐺𝑥,𝑦,𝑧^𝑝(𝑥,𝑥,𝑥)+𝐺_𝑝𝑦,𝑦,𝑦+𝐺_𝑝(𝑧,𝑧,𝑧)3, (2.46) where 0≤𝑟<1. Then (R1) the set 𝑋_𝑃={𝑦∈𝑋∶𝐺_𝑝(𝑥,𝑥,𝑥)=inf_𝑦∈𝑋𝐺_𝑝(𝑦,𝑦,𝑦)} is nonempty; (R2) there is a unique 𝑥^∗∈𝑋_𝑃 such that 𝑓𝑥^∗=𝑥^∗; (R3) for all 𝑥∈𝑋_𝐺𝑃, the sequence {𝑓^𝑛𝑥} converges to 𝑥^∗ with respect to the metric 𝐷_𝐺𝑝.

Proof. Since 𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧)≤max𝑟𝐺_𝑝𝑥,𝑦,𝑧,𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧), (2.47) then 𝐺_𝑝(𝑥,𝑥,𝑥)+𝐺_𝑝𝑦,𝑦,𝑦+𝐺_𝑝(𝑧,𝑧,𝑧)3≤max𝑟𝐺_𝑝𝑥,𝑦,𝑧,𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧). (2.48) Thus 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧≤max𝑟𝐺_𝑝𝑥,𝑦,𝑧,𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧). (2.49) Then, the conditions of Theorem 2.1 hold. Hence, it follows that (R1), (R2), and (R3) hold.

Example 2.7. Let 𝑋=[0,1] and define 𝐺_𝑝(𝑥,𝑦,𝑧)=max{𝑥,𝑦,𝑧}, for all 𝑥,𝑦,𝑧∈𝑋. We have (𝑋,𝐺_𝑝) is a complete 𝐺𝑃-metric space. Take 𝑓𝑥=𝑥²/2 and 𝑟=1/2. For all 𝑥≤𝑦≤𝑧, we have 𝐺_𝑝=𝑧𝑓𝑥,𝑓𝑦,𝑓𝑧²2≤𝑟𝐺_𝑝𝑥,𝑦,𝑧≤max𝑟𝐺_𝑝𝑥,𝑦,𝑧,𝐺_𝑝(𝑥,𝑥,𝑥),𝐺_𝑝𝑦,𝑦,𝑦,𝐺_𝑝(𝑧,𝑧,𝑧), (2.50) that is, (2.2) holds. Here, 𝑢=0 is the unique fixed point of 𝑓.

Similarly, we have the following.

Theorem 2.8. Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-complete 𝐺𝑃-metric space. Let 𝑓 be a self-mapping on 𝑋. Suppose that for all 𝑥,𝑦,𝑧∈𝑋 the following condition holds: 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑧³≤𝑟𝐺_𝑝(𝑥,𝑥,𝑥)𝐺_𝑝𝑦,𝑦,𝑦𝐺_𝑝(z,𝑧,𝑧), (2.51) where 0≤𝑟<1. Then (N1) the set 𝑋_𝐺𝑝={𝑦∈𝑋∶𝐺_𝑝(𝑥,𝑥,𝑥)=inf_𝑦∈𝑋𝐺_𝑝(𝑦,𝑦,𝑦)} is nonempty; (N2) there is a unique 𝑥^∗∈𝑋_𝐺𝑝 such that 𝑓𝑥^∗=𝑥^∗; (N3) for all 𝑥∈𝑋_𝐺𝑝, the sequence {𝑓^𝑛𝑥} converges with respect to the metric 𝐷_𝐺𝑝 to 𝑥^∗.

The following lemma is useful.

Lemma 2.9. Let (𝑋,𝐺_𝑝) be a 𝐺𝑃-metric space and {𝑥_𝑛} be a sequence in 𝑋. Assume that {𝑥_𝑛}𝐺𝑃 converges to a point 𝑥∈𝑋 with 𝐺_𝑝(𝑥,𝑥,𝑥)=0. Then lim_𝑛→+∞𝐺_𝑝(𝑥_𝑛,𝑦,𝑦)=𝐺_𝑝(𝑥,𝑦,𝑦) for all 𝑦∈𝑋. Moreover, lim_{𝑚,𝑛→+∞}𝐺_𝑝(𝑥_𝑛,𝑥_𝑚,𝑥)=0.

Proof. By (GP4), we have 𝐺_𝑝𝑥,𝑦,𝑦−𝐺_𝑝𝑥,𝑥_𝑛,𝑥_𝑛≤𝐺_𝑝𝑥_𝑛,𝑦,𝑦≤𝐺_𝑝𝑥_𝑛,𝑥,𝑥+𝐺_𝑝𝑥,𝑦,𝑦, (2.52) and so lim_𝑛→+∞𝐺_𝑝(𝑥_𝑛,𝑦,𝑦)=𝑃(𝑥,𝑦,𝑦). Again by (GP4), we get 𝐺_𝑝𝑥_𝑛,𝑥_𝑚,𝑥≤𝐺_𝑝𝑥_𝑛,𝑥,𝑥+𝐺_𝑝𝑥,𝑥,𝑥_𝑚, (2.53) and hence lim_{𝑚,𝑛→+∞}𝐺_𝑝(𝑥_𝑛,𝑥_𝑚,𝑥)=0.

Theorem 2.10. Let (𝑋,𝐺_𝑝) be a 0-𝐺𝑃-complete 𝐺𝑃-metric space and 𝑓 be a self-mapping on 𝑋. Assume that (1/3)𝐺_𝑝(𝑥,𝑓𝑥,𝑓𝑥)<𝐺_𝑝(𝑥,𝑦,𝑦) implies 𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑦≤𝛼𝐺_𝑝𝑥,𝑦,𝑦+𝛽𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+𝛾𝐺_𝑝𝑦,𝑓𝑦,𝑓𝑦 (2.54) for all 𝑥,𝑦∈𝑋, where 𝛼,𝛽,𝛾≥0 with 𝛼+𝛽+𝛾<1. Then 𝑓 has a unique fixed point.

Proof. If 𝑥=𝑓𝑥, then 𝑥 is a fixed point for 𝑓. Assume that 𝑥≠𝑓𝑥. So by Lemma 1.10, it follows that 𝐺_𝑝(𝑥,𝑓𝑥,𝑓𝑥)>0. Therefore, (1/3)𝐺_𝑝(𝑥,𝑓𝑥,𝑓𝑥)<𝐺_𝑝(𝑥,𝑓𝑥,𝑓𝑥) and so from (2.54), we have 𝐺_𝑝𝑓𝑥,𝑓²𝑥,𝑓²𝑥≤𝑟𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥forall𝑥∈𝑋,𝑟=𝛼+𝛽1−𝛾<1. (2.55) Let 𝑥₀∈𝑋 and define a sequence {𝑥_𝑛} by 𝑥_𝑛=𝑓^𝑛𝑥₀ for all 𝑛∈ℕ. Now by (2.55), we can obtain that 𝐺_𝑝𝑓^𝑛𝑥₀,𝑓^𝑛+1𝑥₀,𝑓^𝑛+1𝑥₀≤𝑟𝐺_𝑝𝑓^𝑛−1𝑥₀,𝑓^𝑛𝑥₀,𝑓^𝑛𝑥₀≤⋯≤𝑟^𝑛𝐺_𝑝𝑥₀,𝑓𝑥₀,𝑓𝑥₀. (2.56) Then, for any 𝑚>𝑛, by (2.56), we get 𝐺_𝑝𝑓^𝑛𝑥₀,𝑓^𝑚𝑥₀,𝑓^𝑚𝑥₀≤𝐺_𝑝𝑓^𝑛𝑥₀,𝑓^𝑛+1𝑥₀,𝑓^𝑛+1𝑥₀+𝐺_𝑝𝑓^𝑛+1𝑥₀,𝑓^𝑚𝑥₀,𝑓^𝑚𝑥₀≤𝐺_𝑝𝑓^𝑛𝑥₀,𝑓^𝑛+1𝑥₀,𝑓^𝑛+1𝑥₀+𝐺_𝑝𝑓^𝑛+1𝑥₀,𝑓^𝑛+2𝑥₀,𝑓^𝑛+2𝑥₀+𝐺_𝑝𝑓^𝑛+2𝑥₀,𝑓^𝑚𝑥₀,𝑓^𝑚𝑥₀≤𝐺_𝑝𝑓^𝑛𝑥₀,𝑓^𝑛+1𝑥₀,𝑓^𝑛+1𝑥₀+𝐺_𝑝𝑓^𝑛+1𝑥₀,𝑓^𝑛+2𝑥₀,𝑓^𝑛+2𝑥₀+𝐺_𝑝𝑓^𝑛+2𝑥₀,𝑓^𝑛+3𝑥₀,𝑓^𝑛+3𝑥₀+⋯+𝐺_𝑝𝑓^𝑚−1𝑥₀,𝑓^𝑚𝑥₀,𝑓^𝑚𝑥₀≤𝑟^𝑛𝐺1−𝑟^𝑝𝑥₀,𝑓𝑥₀,𝑓𝑥₀. (2.57) It implies that lim_{𝑚,𝑛→∞}𝐺_𝑝(𝑓^𝑛𝑥₀,𝑓^𝑚𝑥₀,𝑓^𝑚𝑥₀)=0; that is, {𝑥_𝑛} is a 0-𝐺𝑃-Cauchy sequence. Since 𝑋 is 0-𝐺𝑃-complete, so {𝑥_𝑛}  𝐺𝑃 converges to some point 𝑧∈𝑋 with 𝐺_𝑝(𝑧,𝑧,𝑧)=0, that is, lim_𝑛→+∞𝐺_𝑝𝑥_𝑛,𝑧,𝑧=lim_𝑛→∞𝐺_𝑝𝑥_𝑛,𝑥_𝑛,𝑧=𝐺_𝑃(𝑧,𝑧,𝑧)=0. (2.58)
Now, we suppose that the following inequality holds: 13𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥≥𝐺_𝑝,1𝑥,𝑦,𝑦3𝐺_𝑝𝑓𝑥,𝑓²𝑥,𝑓²𝑥≥𝐺_𝑝𝑓𝑥,𝑦,𝑦, (2.59) for some 𝑥,𝑦∈𝑋. Then, by Proposition 1.3 (iii) and (2.55), we have 𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥≤𝐺_𝑝𝑥,𝑦,𝑦+2𝐺_𝑝≤1𝑓𝑥,𝑦,𝑦3𝐺_𝑝+2𝑥,𝑓𝑥,𝑓𝑥3𝐺_𝑝𝑓𝑥,𝑓²𝑥,𝑓²𝑥≤13𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥+2𝑟3𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥<𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥 (2.60) which is a contradiction. Thus, for all 𝑥,𝑦∈𝑋, either 13𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥<𝐺_𝑝1𝑥,𝑦,𝑦or3𝐺_𝑝𝑓𝑥,𝑓²𝑥,𝑓²𝑥<𝐺_𝑝𝑓𝑥,𝑦,𝑦 (2.61) holds. Therefore, either 13𝐺_𝑝𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1≤𝐺_𝑝𝑥_2𝑛1,𝑧,𝑧or3𝐺_𝑝𝑥_2𝑛+1,𝑥_2𝑛+2,𝑥_2𝑛+2≤𝐺_𝑝𝑥_2𝑛+1,𝑧,𝑧 (2.62) holds for every 𝑛∈ℕ.
On the other hand, by (2.54), it follows that 𝐺_𝑝𝑥_2𝑛+1,𝑓𝑧,𝑓𝑧≤𝛼𝐺_𝑝𝑥_2𝑛,𝑧,𝑧+𝛽𝐺_𝑝𝑥_2𝑛,𝑥_2𝑛+1,𝑥_2𝑛+1+𝛾𝐺_𝑝,𝐺𝑧,𝑓𝑧,𝑓𝑧_𝑝𝑥_2𝑛+2,𝑓𝑧,𝑓𝑧≤𝛼𝐺_𝑝𝑥_2𝑛+1,𝑧,𝑧+𝛽𝐺_𝑝𝑥_2𝑛+1,𝑥_2𝑛+2,𝑥_2𝑛+2+𝛾𝐺_𝑝𝑧,𝑓𝑧,𝑓𝑧. (2.63) If we take the limit as 𝑛→∞ in each of these inequalities, having in mind (2.58), (2.62), and applying Lemma 2.9, then we get (1−𝛾)𝐺_𝑝(𝑧,𝑓𝑧,𝑓𝑧)≤0, that is, 𝑧=𝑓𝑧. The uniqueness of the fixed point follows easily from (2.54).
As a consequence of Theorem 2.10, we may state the following corollaries.

Corollary 2.11. Let (𝑋,𝐺_𝑝) be a 0-𝐺𝑃-complete 𝐺𝑃-metric space and 𝑓 be a self-mapping on 𝑋. Assume that 13𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥<𝐺_𝑝𝑥,𝑦,𝑦𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑦≤𝑟𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥 (2.64) for all 𝑥,𝑦∈𝑋, where 0≤𝑟<1. Then 𝑓 has a unique fixed point.

Corollary 2.12. Let (𝑋,𝐺_𝑝) be a 0-𝐺𝑃-complete 𝐺𝑃-metric space and 𝑓 be a self-mapping on 𝑋. Assume that 13𝐺_𝑝𝑥,𝑓𝑥,𝑓𝑥<𝐺_𝑝𝑥,𝑦,𝑦𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝐺_𝑝𝑓𝑥,𝑓𝑦,𝑓𝑦≤𝑟𝐺_𝑝𝑦,𝑓𝑦,𝑓𝑦 (2.65) for all 𝑥,𝑦∈𝑋, where 0≤𝑟<1. Then 𝑓 has a unique fixed point.

3. Conclusion

In [43], Zand and Nezhad initiated the notion of a 𝐺𝑃-metric space. Also, they studied fully its topology. Based on this new space, in this paper we present some fixed point results for self mappings involving different contractive conditions. They are illustrated by some examples. The presented theorems are the first results in fixed point theory on 𝐺𝑃-metric spaces.

Acknowledgments

The authors thank the editor and referees for their valuable comments and suggestions.