Some Fixed Point Results in Ordered S-Metric Spaces for Rational Type [620035]

Some Fixed Point Results in Ordered S-Metric Spaces for Rational Type
Expressions

Abstracts: The aim of this paper is to present some fixed point theorems for non-
decreasing map involving rational expression in the framework of S-metric spaces
endowed with a partial order using a class of pairs of functions satisfying certain
assumptions. Our results generalize and extend some known results on metric spaces to
S-metric spaces, which appeared in [ 4-5, 14-15 ]. We give examples to demonstrate the
validity of the results.

Key words: Fixed point, S-metric space, contractions; partially ordered set, altering
distance function.

1. Introduction
Banach’s contraction principle is one of the pivotal results of analysis. Its
significance lies in its vast applicability to a great number of branches of mathematics
and other sciences, for example, theory of existence of solutions for nonlinear
differential, integral, and functional equations, variational inequalities, and optimization
and approximation theory.
Das and Gupta [ 5] were the Pioneers in proving fixed point theorems using
contractive conditions involving rational expressions. They proved the following fixed
point theorem.
Theorem 1 (see [ 5]) Let (/gX85∗, /gX85S) be a complete metric space and /gX858: /gX85∗ → /gX85∗ a mapping
such that there exist /gD∗∗d, /gD∗X∗ ≥ 0 with /gD∗∗d + /gD∗X∗ < 1 satisfying
/gX85S(/gX858/gX8ℝS, /gX858/gX8ℝℝ) ≤ /gD∗∗d/g2∗2X(/g2∗5D,/g2∗22/g2∗5D)[/gD8Sd /gD8ℝ8 /g2∗2X(/g2∗5X,/g2∗22/g2∗5X)]
/gD8Sd /gD8ℝ8 /g2∗2X(/g2∗5X,/g2∗5D)+ /gD∗X∗/gX85S(/gX8ℝS, /gX8ℝℝ) (A)

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for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ . Then /gX858 has a unique fixed point in /gX85∗.
In [4], Cabrera, Harjani and Sadarangani proved the abov e theorem in the context of
partially ordered metric spaces.
Definition 1 (see [ 4]) Let (/gX85∗, ⪯) is a partially ordered set and /gX858 ∶ /gX85∗ → /gX85∗ is said to be
monotone non-decreasing if for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ ,
/gX8ℝS ⪯ /gX8ℝℝ ⇒ /gX858/gX8ℝS ⪯ /gX858/gX8ℝℝ .
Theorem 2 (see [ 4]) Let (/gX85∗, ⪯) is a partially ordered set. Suppose that there exi st a
metric /gX85S on /gX85∗ such that (/gX85∗, /gX85S) be a complete metric space. Let /gX858: /gX85∗ → /gX85∗ be a
continuous and non-decreasing mapping such that (A) is satisfied for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ with
/gX8ℝS ≤ /gX8ℝℝ . If there exist /gX8ℝS/gD8S8∈ /gX85∗ such that /gX8ℝS/gD8S8⪯ /gX858/gX8ℝS /gD8S8, then /gX858 has a fixed point.
Theorem 3 (see [ 4]) Let (/gX85∗, ⪯) is a partially ordered set. Suppose that there exi st a
metric /gX85S on /gX85∗ such that (/gX85∗, /gX85S) be a complete metric space. Assume that if /g4SS8/gX8ℝS/g2∗4X/g4SSd is non-
decreasing sequence in /gX85∗ such that /gX8ℝS/g2∗4X→ /gX8ℝ2, then /gX8ℝS/g2∗4X⪯ /gX8ℝ2, for all /gX8SS ∈ ℕ . Let /gX858: /gX85∗ → /gX85∗ be
a non-decreasing mapping such that (A) is satisfied for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ with /gX8ℝS ⪯ /gX8ℝℝ . If there
exist /gX8ℝS/gD8S8∈ /gX85∗ such that /gX8ℝS/gD8S8⪯ /gX858/gX8ℝS /gD8S8, then /gX858 has a fixed point.
Theorem 4 (see [ 4]) In addition to the hypothesis of Theorem 2 or Th eorem 3,
suppose that for every /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ , there exist /gX8ℝ2 ∈ /gX85∗ such that /gX8ℝ2 ⪯ /gX8ℝS and /gX8ℝ2 ⪯ /gX8ℝℝ . Then /gX858
has a unique fixed point.
In this paper, we establish some fixed point theore ms for mappings involving rational
expression in the framework of S-metric spaces endo wed with a partial order using a
class of pairs of functions satisfying certain assu mptions.Our result has as particular
cases a great number of interesting consequences wh ich extend and generalize some
results appearing in the literature.
Metric spaces are very important in mathematics and applied sciences. So, some
authors have tried to give generalizations of metri c spaces in several ways. For example,
Gahler [ 11 ] and Dhage [ 7] introduced the concepts of 2-metric spaces and D-metric
spaces, respectively.
In 2006, Mustafa and Sims [ 19 ] introduced a new structure of generalized metric
spaces which are called G-metric spaces as a generalization of metric spaces (/gX85∗, /gX85S) to
develop and introduce a new fixed point theory for various mappings in this new
structure.
Definition 2 (see [ 19 ]) Let /gX85∗ be a non-empty set. Suppose that a mapping /gX822: /gX85∗ × /gX85∗ ×
/gX85∗ → [0, +∞) satisfies the following conditions, for each /gX8ℝS, /gX8ℝℝ, /gX8ℝ8, /gX852 ∈ /gX85∗;
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(G1). /gX822(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= 0 if /gX8ℝS = /gX8ℝℝ = /gX8ℝ8,
(G2). 0 < /gX822 (/gX8ℝS, /gX8ℝS, /gX8ℝℝ ), ∀ /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ with /gX8ℝS ≠ /gX8ℝℝ,
(G3). /gX822(/gX8ℝS, /gX8ℝS, /gX8ℝℝ )≤ /gX822 (/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ) with /gX8ℝℝ ≠ /gX8ℝ8,
(G4). /gX822(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= /gX822 (/gX8ℝS, /gX8ℝ8, /gX8ℝℝ )= /gX822 (/gX8ℝℝ, /gX8ℝS, /gX8ℝ8 )= /gX822 (/gX8ℝ8, /gX8ℝS, /gX8ℝℝ )= /gX822 (/gX8ℝℝ, /gX8ℝ8, /gX8ℝS )= /gX822 (/gX8ℝ8, /gX8ℝℝ, /gX8ℝS ),
(G5). /gX822(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )≤ /gX822 (/gX8ℝS, /gX852, /gX852 )+ /gX822 (/gX852, /gX8ℝℝ, /gX8ℝ8 ),
Then the pair (/gX85∗, /gX822) is called a generalized /gX822-metric space or, more specifically, a
/gX822metric space.
Sedghi et al. [28 ] introduced the notion of a /gX82∗∗-metric space as follows.
Definition 3 (see [ 28 ]) Let X be a non-empty set. A D∗-metric on X is a function
/gX82∗∗: /gX85∗ × /gX85∗ × /gX85∗ → [0, +∞) that satisfies the following conditions, for each /gX8ℝS, /gX8ℝℝ, /gX8ℝ8, /gX852 ∈ /gX85∗;
(D*1). /gX82∗∗(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )≥ 0,
(D*2). /gX82∗∗(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= 0 if and only if /gX8ℝS = /gX8ℝℝ = /gX8ℝ8,
(D*3). /gX82∗∗(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= /gX82∗∗(/gX8ℝS, /gX8ℝ8, /gX8ℝℝ )= /gX82∗∗(/gX8ℝℝ, /gX8ℝS, /gX8ℝ8 )= /gX82∗∗(/gX8ℝ8, /gX8ℝS, /gX8ℝℝ )= /gX82∗∗(/gX8ℝℝ, /gX8ℝ8, /gX8ℝS )=
/gX82∗∗(/gX8ℝ8, /gX8ℝℝ, /gX8ℝS ),
(D*4). /gX82∗∗(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )≤ /gX82∗∗(/gX8ℝS, /gX8ℝℝ, /gX852 )+ /gX82∗∗(/gX852, /gX8ℝ8, /gX8ℝ8 ).
Then /gX82∗∗ is called a /gX82∗∗-metric on /gX85∗ and (/gX85∗, /gX82∗∗) is called a /gX82∗∗-metric space.
Every G-metric space is a D∗-metric space. Indeed conditions (G1), (G2), and (G 3)
imply ( D∗1). Axioms (G1) and ( D∗2) are equivalent. (G4) and ( D∗4) are also equivalent,
whereas (G4) and (G5) imply ( D∗4). The converse, however, is false in general; a D∗-
metric space is not necessarily a G-metric space.
Sedghi et al. [ 29 ] identified condition (G3) as a peculiar limitatio n of the G-metric
space but classified the symmetry condition as a co mmon weakness of both G- metric
and D∗-metric spaces. To overcome these difficulties, Sed ghi et al. [ 29 ] introduced a
new generalized metric space called an S-metric spa ce.
Definition 4 (see [ 29 ]) Let /gX85∗ be a non-empty set. An S-metric on X is a function
/gX845: /gX85∗ × /gX85∗ × /gX85∗ → [0, +∞) that satisfies the following conditions, for each /gX8ℝS, /gX8ℝℝ, /gX8ℝ8, /gX852 ∈ /gX85∗,
(S1). /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )≥ 0,
(S2). /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= 0 if and only if /gX8ℝS = /gX8ℝℝ = /gX8ℝ8,
(S3). /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )≤ /gX845(/gX8ℝS, /gX8ℝS, /gX852 )+ /gX845(/gX8ℝℝ, /gX8ℝℝ, /gX852 )+ /gX845(/gX8ℝ8, /gX8ℝ8, /gX852 ).
Then S is called an S-metric on /gX85∗ and (/gX85∗, /gX845) is called an S-metric space.
The following is the intuitive geometric example fo r S-metric spaces.
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Example 1 (see [ 29 ], Example 2.4) Let /gX85∗ = ℝ/gD8ℝ∗ and d be the ordinary metric on X. Put
/gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= /gX85S (/gX8ℝS, /gX8ℝℝ)+ /gX85S(/gX8ℝS, /gX8ℝ8) + /gX85S(/gX8ℝℝ, /gX8ℝ8) for all /gX8ℝS, /gX8ℝℝ ∈ ℝ/gD8ℝ∗, that is, /gX845 is the perimeter of
the triangle given by /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 . Then /gX845 is an /gX845-metric on /gX85∗.
Lemma 1 (see [ 29 ], Lemma 2.5) Let (/gX85∗, /gX845) be an S-metric space. Then /gX845(/gX8ℝS, /gX8ℝS, /gX8ℝℝ) =
/gX845(/gX8ℝℝ, /gX8ℝℝ, /gX8ℝS) for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ .
Lemma 2 (see [ 10 ], Lemma 1.6) Let (/gX85∗, /gX845) be an S-metric space. Then /gX845(/gX8ℝS, /gX8ℝS, /gX8ℝ8 )≤
2/gX845(/gX8ℝS, /gX8ℝS, /gX8ℝℝ )+ /gX845(/gX8ℝℝ, /gX8ℝℝ, /gX8ℝ8) and /gX845(/gX8ℝS, /gX8ℝS, /gX8ℝ8 )≤ 2/gX845 (/gX8ℝS, /gX8ℝS, /gX8ℝℝ )+ /gX845(/gX8ℝ8, /gX8ℝ8, /gX8ℝℝ) for all /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ /gX85∗ .
Definition 5 (see [ 29 ]) Let /gX85∗ be an S-metric space.
(i). A sequence /g4SS8/gX8ℝS/g2∗4X/g4SSd converges to /gX8ℝS if and only if /gX8S4/gX8SX/gX8S5 /g2∗4X→∞/gX845(/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝS)= 0. That is for
each /gD∗25 > 0 there exists /gX8SS/gD8S8∈ ℕ such that for all /gX8SS ≥ /gX8SS /gD8S8, /gX845(/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝS)< /gD∗25 and we
denote this by /gX8S4/gX8SX/gX8S5 /g2∗4X→∞/gX8ℝS/g2∗4X= /gX8ℝS.
(ii). A sequence /g4SS8/gX8ℝS/g2∗4X/g4SSd is called a Cauchy if /gX8S4/gX8SX/gX8S5 /g2∗4X,/g2∗4∗→∞/gX845(/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝS/g2∗4∗)= 0. That is, for
each /gD∗25 > 0 there exists /gX8SS/gD8S8∈ ℕ such that for all /gX8SS, /gX8S5 ≥ /gX8SS /gD8S8, /gX845(/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝS/g2∗4∗)< /gD∗25.
(iii). /gX85∗ is called complete if every Cauchy sequence in /gX85∗ is a convergent.
From (see [ 29 ], Examples in page 260), we have the following.
Example 2
(a). Let ℝ be the real line. Then /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )=|/gX8ℝS − /gX8ℝ8 |+|/gX8ℝℝ − /gX8ℝ8 | for all /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ ℝ , is an
S-metric on ℝ. This S-metric is called the usual S-metric on ℝ. Furthermore, the
usual S-metric space ℝ is complete.
(b). Let /gX85X be a non-empty set of ℝ. Then /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )=|/gX8ℝS − /gX8ℝ8 |+|/gX8ℝℝ − /gX8ℝ8 | for all
/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ /gX85X , is an S-metric on /gX85X. If /gX85X is a closed subset of the usual metric space ℝ,
then the S-metric space /gX85X is complete.
Lemma 3 (see [ 29 ], Lemma 2.11) Let (/gX85∗, /gX845) be an S-metric space. If the sequence
/g4SS8/gX8ℝS/g2∗4X/g4SSd in /gX85∗ converges to /gX8ℝS, then /gX8ℝS is unique.
Lemma 4 (see [ 29 ], Lemma 2.12) Let (/gX85∗, /gX845) be an S-metric space. If /gX8S4/gX8SX/gX8S5 /g2∗4X→/gD8ℝ8∞/gX8ℝS/g2∗4X=
/gX8ℝS and /gX8S4/gX8SX/gX8S5 /g2∗4X→/gD8ℝ8∞/gX8ℝℝ/g2∗4X = /gX8ℝℝ, then /gX8S4/gX8SX/gX8S5 /g2∗4X→/gD8ℝ8∞/gX845(/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝℝ/g2∗4X)= /gX845(/gX8ℝS, /gX8ℝS, /gX8ℝℝ ).
Remark 1 (see [ 10 ]) It is easy to see that every D∗-metric is S-metric, but in general
the converse is not true, see the following example .
Example 3 (see [ 10 ]) Let /gX85∗ = ℝ/g2∗4X and ‖ .‖ a norm on /gX85∗, then /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )=‖/gX8ℝℝ + /gX8ℝ8 −
2/gX8ℝS‖+‖/gX8ℝℝ − /gX8ℝ8 ‖ is S-metric on /gX85∗, but it is not D∗-metric because it is not symmetric.
The following lemma shows that every metric space i s an S-metric space.
Lemma 5 (see [ 10 ], Lemma 1.10) Let (/gX85∗, /gX85S) be a metric space. Then we have
(1). /gX845/g2∗2X(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= /gX85S(/gX8ℝS, /gX8ℝ8)+ /gX85S(/gX8ℝℝ, /gX8ℝ8) for all /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ /gX85∗ is an S-metric on /gX85∗.
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(2). /gX8S4/gX8SX/gX8S5 /g2∗4X→/gD8ℝ8/gDdd8 /gX8ℝS/g2∗4X= /gX8ℝS in (/gX85∗, /gX85S) if and only if /gX8S4/gX8SX/gX8S5 /g2∗4X→/gD8ℝ8/gDdd8 /gX8ℝS/g2∗4X= /gX8ℝS in (/gX85∗, /gX845 /g2∗2X).
(3). /g4SS8/gX8ℝS/g2∗4X/g4SSd/g2∗4X/gD88∗/gD8Sd/gDdd8 is Cauchy in (/gX85∗, /gX85S) if and only if /g4SS8/gX8ℝS/g2∗4X/g4SSd/g2∗4X/gD88∗/gD8Sd/gDdd8 is Cauchy in (/gX85∗, /gX845 /g2∗2X).
(4). (/gX85∗, /gX85S) is complete if and only if (/gX85∗, /gX845 /g2∗2X) is complete.
The following example proves that the inversion of Lemma 5 does not hold.
Example 4 (see [ 10 ], Example 1.10) Let /gX85∗ = ℝ and let /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )=|/gX8ℝℝ + /gX8ℝ8 − 2/gX8ℝS |+
|/gX8ℝℝ − /gX8ℝ8 | for all /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ /gX85∗ . By ([ 29 ], Example 1, page 260), (/gX85∗, /gX845) is an S-metric space.
Dung et al. [ 9] proved that there does not exist any metric d such that /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )=
/gX85S(/gX8ℝS, /gX8ℝ8)+ /gX85S(/gX8ℝℝ, /gX8ℝ8) for all /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ /gX85∗ . Indeed, suppose to the contrary that there exists a
metric d with /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝ8 )= /gX85S(/gX8ℝS, /gX8ℝ8)+ /gX85S(/gX8ℝℝ, /gX8ℝ8) for all /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ /gX85∗ . Then /gX85S(/gX8ℝS, /gX8ℝ8)=
/gD8Sd
/gD8ℝ∗/gX845(/gX8ℝS, /gX8ℝS, /gX8ℝ8 )= 2|/gX8ℝS − /gX8ℝ8 | and /gX85S(/gX8ℝS, /gX8ℝℝ)=/gD8Sd
/gD8ℝ∗/gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝℝ )= 2|/gX8ℝS − /gX8ℝℝ | for all /gX8ℝS, /gX8ℝℝ, /gX8ℝ8 ∈ /gX85∗ . It is a
contradiction.
In 2012, Sedghi et al. [ 29 ] asserted that an S-metric is a generalization of a G-metric,
that is, every G-metric is an S-metric, see [ 29 , Remarks 1.3] and [ 29 , Remarks 2.2]. The
Example 2.1 and Example 2.2 of Dung et al. [ 9] shows that this assertion is not correct.
Moreover, the class of all S-metrics and the class of all G-metrics are distinct.
Example 5 (see [ 9]) There exists a G-metric which is not an S-metric.
Proof Let X be the G-metric space in (see [1 9], Example 1). Then we have
2 = /gX822 (/gX852, /gX854, /gX854 )> 1 = /gX822 (/gX852, /gX852, /gX854 )+ /gX822 (/gX854, /gX854, /gX854 )+ /gX822(/gX854, /gX854, /gX854) .
This proves that G is not an S-metric on /gX85∗.
Example 6 (see [ 9]) There exists an S-metric which is not a G-metric.
Proof Let (/gX85∗, /gX845) be the S-metric space in Example 4. We have
/gX845(1,0,2 )= |0 + 2 − 2 |+|0 − 2 | = 2 ,
/gX845(2,0,1 )= |0 + 1 − 4 |+|0 − 1 | = 4 .
Then /gX845(1,0,2 )≠ /gX845(2,0,1 ). This proves that S is not a G-metric.
Also in 2012, Jeli and Samet [ 15 ] showed that a G-metric is not a real generalizati on of
a metric. Further, they proved that the fixed point theorems proved in G-metric spaces
can be obtained by usual metric arguments. The simi lar approach may be found in [ 1].
The key of that approach is the lemma.
Lemma 6 (see [ 15 ], Theorem 2.2) Let (/gX85∗, /gX822) be a G-metric space. Then we have
1 /gX85S(/gX8ℝS, /gX8ℝℝ)= /gX8S5/gX852/gX8ℝS /g4SS8/gX822(/gX8ℝS, /gX8ℝℝ, /gX8ℝℝ ), /gX822(/gX8ℝℝ, /gX8ℝS, /gX8ℝS) /g4SSd for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ is a metric on /gX85∗.
2 /gX85S(/gX8ℝS, /gX8ℝℝ)= /gX822(/gX8ℝS, /gX8ℝℝ, /gX8ℝℝ) for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ is a quasi-metric on /gX85∗.
The Lemma 6 does not hold if the G-metric is replac ed by an S-metric space. Then, in
general, arguments in [ 1], [ 15 ] are not applicable to S-metric spaces.
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Lemma 7 (see [ 9], Example 2.4) Let (X, S) be an S-metric space. Then we have
1. /gX85S(/gX8ℝS, /gX8ℝℝ)= /gX8S5/gX852/gX8ℝS /g4SS8/gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝℝ ), /gX845(/gX8ℝℝ, /gX8ℝS, /gX8ℝS) /g4SSd for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ is not a metric on /gX85∗.
2. /gX85S(/gX8ℝS, /gX8ℝℝ)= /gX845(/gX8ℝS, /gX8ℝℝ, /gX8ℝℝ) for all /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ is not a quasi-metric on /gX85∗.
Khan et al. [ 17 ] initiated the use of control function that alter distance between two
points in a metric space, which they called an alte ring distance function
Definition 6 (see [ 17 ]) A function /gD∗2∗: [0, ∞) → [0, ∞) is called altering distance
function if the following conditions are satisfied:
(a1). /gD∗2∗ is monotone increasing and continuous,
(a2). /gD∗28(/gX8ℝD) = 0 if and only if /gX8ℝD = 0 .
In [ 3], Bergiz et al. introduced the notion of pair of g eneralized altering distance
functions as follows.
Definition 7 (see [3]) The pair (/gD∗2∗, /gD∗28) , where /gD∗2∗, /gD∗28: [0, ∞ )→[0, ∞ ), is called a pair of
generalized altering distance functions if the foll owing conditions are satisfied:
(b1). /gD∗2∗ is continuous;
(b2). /gD∗2∗ is non-decreasing;
(b3). /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gD∗28(/gX8ℝD /g2∗4X) = 0 ⇒ /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX8ℝD/g2∗4X= 0.
The condition (b3) was introduced by Moradi and Far ajzadeh in [ 18 ]. Notice that the
above conditions do not determine the values of /gD∗2∗(0) and /gD∗28(0) .
In the recent work, Agarwal et al. [ 2] introduced the following family of function.
Definition 8 (see [ 2]) We will denote by ℱ the family of all pairs (/gD∗2∗, /gD∗28) , where
/gD∗2∗, /gD∗28: [0, ∞ )→[0, ∞ ) are functions satisfying the following three condi tions.
(c1). /gD∗2∗ is non-decreasing;
(c2). if there exists /gX8ℝD/gD8S8∈[0, ∞ ) such that /gD∗28 (/gX8ℝD /gD8S8) = 0 , then /gX8ℝD/gD8S8= 0 and /gD∗2∗/gD8ℝd/gD8Sd(0)=/g4SS80/g4SSd.
(c3). if /g4SS8/gX852/g2∗28/g4SSd,/g4SS8/gX854/g2∗28/g4SSd⊂[0, ∞ ) are sequences such that /g4SS8/gX852/g2∗28/g4SSd→ /gX828, /g4SS8/gX854/g2∗28/g4SSd→ /gX828 and verifying
/gX828 < /g4SS8/gX854/g2∗28/g4SSd and /gD∗2∗(/gX854/g2∗28)≤(/gD∗2∗ − /gD∗28 )(/gX852/g2∗28) for all /gX8S2, then /gX828 = 0 .
In this paper, we consider the following class of p airs of functions /gD2ℝX.
Definition 9 (see [ 24 ]) A pair of functions (/gD∗2∗, /gD∗28) is said to belong to the class /gD2ℝX, if
they satisfy the following conditions:
(e1). /gD∗2∗, /gD∗28: [0, ∞) → [0, ∞);
(e2). for /gX8ℝD, /gX8ℝX ∈ [0, ∞) , /gD∗2∗(/gX8ℝD) ≤ /gD∗28(/gX8ℝX) then /gX8ℝD ≤ /gX8ℝX;
(e3). for /g4SS8/gX8ℝD/g2∗4X/g4SSd and /g4SS8/gX8ℝX/g2∗4X/g4SSd sequence in [0, ∞) such that /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX8ℝD/g2∗4X= /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX8ℝX/g2∗4X= /gX852,
if /gD∗2∗(/gX8ℝD /g2∗4X) ≤ /gD∗28(/gX8ℝX /g2∗4X) for any /gX8SS ∈ ℕ , then /gX852 = 0 .
UNDER PEER REVIEW

7
Notice that, if a pair (/gD∗2∗, /gD∗28) verifies (c1) and (c2), then the pair (/gD∗2∗, /gD∗28 = /gD∗2∗ − /gD∗28)
satisfies (e1) and (e2). Furthermore, if (/gD∗2∗, /gD∗28 = /gD∗2∗ − /gD∗28) satisfies (e3), then (/gD∗2∗, /gD∗28)
satisfies (c3).
Remark 2 (see [ 24 ], Remark 4) Note that, if (/gD∗2∗, /gD∗28) ∈ /gD2ℝX and /gD∗2∗(/gX8ℝD) ≤ /gD∗28(/gX8ℝD) , then /gX8ℝD = 0 ,
since we can take /gX8ℝD/g2∗4X= /gX8ℝX /g2∗4X= /gX8ℝD for any /gX8SS ∈ ℕ and by (e3) we deduce /gX8ℝD = 0 .
Example 7
(a). The conditions (e1)-(e3) of the above definition ar e fulfilled for the functions
/gD∗2∗, /gD∗28: [0, ∞ )→[0, ∞ ) defined by /gD∗2∗(/gX8ℝD)= /gX8S4/gX8SS /g4SℝD/gD8ℝ2/g2∗4ℝ/gD8ℝ8/gD8Sd
/gD8Sd/gD8ℝ∗ /g4Sℝ2 and /gD∗28(/gX8ℝD) = /gX8S4/gX8SS /g4SℝD/gD8ℝX/g2∗4ℝ/gD8ℝ8/gD8Sd
/gD8Sd/gD8ℝ∗ /g4Sℝ2 for
all /gX8ℝD ∈ [0, ∞) .
(b). The conditions (e1)-(e3) of the above definition ar e fulfilled for the functions
/gD∗2∗, /gD∗28: [0, ∞ )→[0, ∞ ) defined by /gD∗2∗(/gX8ℝD)= /gX8S4/gX8SS /g4SℝD/gX8ℝD +/gD8Sd
/gD8ℝ∗/g4Sℝ2 and /gD∗28(/gX8ℝD) = /gX8S4/gX8SS /g4SℝD/g2∗4ℝ
/gD8ℝ∗+/gD8Sd
/gD8ℝ∗/g4Sℝ2 for
all /gX8ℝD ∈ [0, ∞) .
In the sequel, we present some interesting examples of pairs of functions belonging to
the class /gD2ℝX which will be very important in our study.
Example 8 (see [ 24 ], Example 5) Let /gD∗2∗ ∶ [0, ∞) → [0, ∞) be a continuous and
increasing function such that /gD∗2∗(/gX8ℝD) = 0 if and only if /gX8ℝD = 0 (these functions are known
in the literature as altering distance functions).
Let /gD∗28 ∶ [0, ∞) → [0, ∞) be a non-decreasing function such that /gD∗28(/gX8ℝD) = 0 if and only if
/gX8ℝD = 0 and suppose that /gD∗28 ≤ /gD∗2∗ . Then the pair (/gD∗2∗, /gD∗2∗ − /gD∗28 )∈ /gD2ℝX.
An interesting particular case is when φ is the identity mapping, /gD∗2∗ = 1 [/gD8S8,/gDdd8) and
/gD∗28: [0, ∞) → [0, ∞) is a non-decreasing function such that /gD∗28(/gX8ℝD) = 0 if and only if /gX8ℝD = 0
and /gD∗28(/gX8ℝD) ≤ /gX8ℝD for any /gX8ℝD ∈ [0, ∞) .
Example 9 (see [ 24 ], Example 6) Let S be the class of functions defined by
/gX845 = /g4SS8/gD∗∗d ∶ [0, ∞) → [0, 1) ∶ /gD∗∗d(/gX8ℝD /g2∗4X) → 1 ⇒ /gX8ℝD /g2∗4X→ 0/g4SSd/g4SSd .
Let us consider the pairs of function /g24251[/gD8S8,/gDdd8 ) , /gD∗∗d1 [/gD8S8,/gDdd8 )/g242d, where /gD∗∗d ∈ /gX845 and /gD∗∗d1 [/gD8S8,/gDdd8) is
defined by /g2425/gD∗∗d1 [/gD8S8,/gDdd8) /g242d(/gX8ℝD) = /gD∗∗d (/gX8ℝD)/gX8ℝD, for /gX8ℝD ∈ [0, ∞) . Then (1[/gD8S8,/gDdd8) , /gD∗∗d1 [/gD8S8,/gDdd8) ) ∈ /gD2ℝX .
Remark 3 (see [ 24 ], Remark 7) Suppose that /gX85d ∶ [0, ∞) → [0, ∞) is an increasing
function and (/gD∗2∗, /gD∗28) ∈ /gD2ℝX . Then it is easily seen that the pair (/gX85d ∘ /gD∗2∗, /gX85d ∘ /gD∗28) ∈ /gD2ℝX .

2. Fixed Point Results
Now, we are ready to present our first result.
UNDER PEER REVIEW

8
Theorem 5 Let (/gX85∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric S
on /gX85∗ such that (/gX85∗, /gX845) be a complete S-metric space. Let /gX858: /gX85∗ → /gX85∗ be a non-decreasing
mapping such that there exists a pair of functions (/gD∗28, /gD∗2∗) ∈ /gD2ℝX satisfying
/gD∗2∗/g2425/gX845 (/gX858/gX8ℝS, /gX858/gX8ℝS, /gX858/gX8ℝℝ )/g242d ≤ /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX8ℝS, /gX8ℝS, /gX8ℝℝ )/g242d, /gD∗28 /g4SℝD/g2∗D∗(/g2∗5D,/g2∗5D,/g2∗22/g2∗5D )[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗5X,/g2∗5X,/g2∗22/g2∗5X )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗5X,/g2∗22/g2∗5X,/g2∗22/g2∗5D )/g4Sℝ2/g4Sℝℝ (1)
for all comparable elements /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ . Assume that if /g4SS8/gX8ℝS/g2∗4X/g4SSd is non-decreasing sequence in
/gX85∗ such that /gX8ℝS/g2∗4X→ /gX8ℝ2, then /gX8ℝS/g2∗4X⪯ /gX8ℝ2, for all /gX8SS ∈ ℕ . If there exist /gX8ℝS/gD8S8∈ /gX85∗ such that /gX8ℝS/gD8S8⪯
/gX858/gX8ℝS/gD8S8, then /gX858 has a fixed point.
Proof If /gX8ℝS/gD8S8= /gX858/gX8ℝS /gD8S8, then the proof is finished. Suppose now that /gX8ℝS/gD8S8≺ /gX858/gX8ℝS /gD8S8. Since /gX858 is
non- decreasing function, we have
/gX8ℝS/gD8S8≺ /gX858/gX8ℝS /gD8S8⪯ /gX858/gD8ℝ∗/gX8ℝS/gD8S8⪯ /gX858/gD8ℝX/gX8ℝS/gD8S8⪯ ⋯ ⪯ /gX858/g2∗4X/gD8ℝd/gD8Sd/gX8ℝS/gD8S8⪯ /gX858/g2∗4X/gX8ℝS/gD8S8⪯ ⋯ (2)
Put
/gX8ℝS/g2∗4X/gD8ℝ8/gD8Sd = /gX858/gX8ℝS /g2∗4X, ∀ /gX8SS ≥ 0 . (3)
For simplicity, we denote /gX845/g2∗4X= /gX845(/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X/gD8ℝ8/gD8Sd)= /gX845(/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X/gD8ℝ8/gD8Sd, /gX8ℝS/g2∗4X/gD8ℝ8/gD8Sd) for all n ∈ ℕ (by
Lemma 1).
We will show that /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX845/g2∗4X= 0.
If there exists /gX8SS ≥ 1 such that /gX8ℝS/g2∗4X= /gX8ℝS /g2∗4X/gD8ℝ8/gD8Sd, then from (2), /gX858/gX8ℝS/g2∗4X= /gX8ℝS /g2∗4X/gD8ℝ8/gD8Sd = /gX8ℝS /g2∗4X, that is /gX8ℝS/g2∗4X
is a fixed point of /gX858 and the proof is finished. Now suppose that /gX8ℝS/g2∗4X≠ /gX8ℝS /g2∗4X/gD8ℝ8/gD8Sd, that is,
/gX845/g2∗4X≠ 0 for all /gX8SS ≥ 1 . Since /gX8ℝS/g2∗4X/gD8ℝd/gD8Sd ≺ /gX8ℝS /g2∗4X for all /gX8SS ≥ 1 , applying the contractive condition
(1), we have
/gD∗2∗(/gX845/g2∗4X)= /gD∗2∗/g2425/gX845 (/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X/gD8ℝ8/gD8Sd)/g242d
= /gD∗2∗/g2425/gX845 (/gX858/gX8ℝS/g2∗4X/gD8ℝd/gD8Sd, /gX858/gX8ℝS /g2∗4X/gD8ℝd/gD8Sd, /gX858/gX8ℝS /g2∗4X)/g242d
≤ /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX8ℝS/g2∗4X/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4X/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4X)/g242d, /gD∗28 /g4SℝD/g2∗D∗(/g2∗5X/g2D8d,/g2∗5X/g2D8d,/g2∗22/g2∗5X /g2D8d)[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗5X/g2D8d/g2XDℝ/g2XXℝ ,/g2∗5X/g2D8d/g2XDℝ/g2XXℝ ,/g2∗22/g2∗5X /g2D8d/g2XDℝ/g2XXℝ )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗5X/g2D8d/g2XDℝ/g2XXℝ ,/g2∗22/g2∗5X /g2D8d/g2XDℝ/g2XXℝ ,/g2∗22/g2∗5X /g2D8d)/g4Sℝ2/g4Sℝℝ
= /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X, /gX8ℝS/g2∗4X/gD8ℝd/gD8Sd)/g242d, /gD∗28 /g4SℝD/g2∗D∗(/g2∗5X/g2D8d,/g2∗5X/g2D8d,/g2∗5X/g2D8d/g2XDS/g2XXℝ )[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗5X/g2D8d/g2XDℝ/g2XXℝ ,/g2∗5X/g2D8d/g2XDℝ/g2XXℝ ,/g2∗5X/g2D8d)]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗5X/g2D8d,/g2∗5X/g2D8d,/g2∗5X/g2D8d/g2XDS/g2XXℝ )/g4Sℝ2/g4Sℝℝ
= /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28 (/gX845/g2∗4X/gD8ℝd/gD8Sd), /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g2XDℝ/g2XXℝ ]
/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g4Sℝ2/g4Sℝℝ (4)
Now, we can distinguish two cases.
Case I. Consider
/gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28 (/gX845/g2∗4X/gD8ℝd/gD8Sd), /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g2XDℝ/g2XXℝ ]
/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g4Sℝ2/g4Sℝℝ = /gD∗28 (/gX845/g2∗4X/gD8ℝd/gD8Sd) (5)
In this case from (4), we have
/gD∗2∗(/gX845/g2∗4X)≤ /gD∗28 (/gX845/g2∗4X/gD8ℝd/gD8Sd) (6)
Since (/gD∗28, /gD∗2∗) ∈ /gD2ℝX , we deduce that /gX845/g2∗4X≤ /gX845 /g2∗4X/gD8ℝd/gD8Sd.
UNDER PEER REVIEW

9
Case II . If
/gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28 (/gX845/g2∗4X/gD8ℝd/gD8Sd), /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g2XDℝ/g2XXℝ ]
/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g4Sℝ2/g4Sℝℝ = /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g2XDℝ/g2XXℝ ]
/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g4Sℝ2 (7)
In this case from (4), we have
/gD∗2∗(/gX845/g2∗4X)≤ /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g2XDℝ/g2XXℝ ]
/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g4Sℝ2 (8)
Since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, we get
/gX845/g2∗4X≤/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g2XDℝ/g2XXℝ ]
/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d
Since /gX845/g2∗4X≠ 0, from the last inequality it follows that /gX845/g2∗4X≤ /gX845 /g2∗4X/gD8ℝd/gD8Sd.
In both cases, we obtain that /gX845/g2∗4X≤ /gX845 /g2∗4X/gD8ℝd/gD8Sd and consequently, the sequence /g4SS8/gX845/g2∗4X/g4SSd is a
decreasing sequence of non-negative real numbers an d is bounded below, there exists
/gX8ℝ∗ ≥ 0 such that
/gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX845/g2∗4X= /gX8ℝ∗. (9)
Now, we shall show that /gX8ℝ∗ = 0 .
Denote
A = /g4SS8/gX8SS ∈ ℕ ∶ n satis/gdℝSies (5)/g4SSd
B = /g4SS8/gX8SS ∈ ℕ ∶ n satis/gdℝSies (7)/g4SSd.
We note that the remarks following.
(1). If /gX8Dd/gX852/gX8ℝ∗/gX85S /gX8Dℝ = ∞, then from (4), we can find infinitely natural numb ers n satisfying
inequality (6) and since /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX845/g2∗4X= /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX845/g2∗4X/gD8ℝd/gD8Sd = /gX8ℝ∗ and (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, by
condition (e3), we have /gX8ℝ∗ = 0 .
(2). If /gX8Dd/gX852/gX8ℝ∗/gX85S /gX8D8 = ∞, then from (4), we can find infinitely many /gX8SS ∈ ℕ satisfying
inequality (8). Since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, we obtain
/gX845/g2∗4X≤/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d/g2XDℝ/g2XXℝ ]
/gD8Sd/gD8ℝ8/g2∗D∗ /g2D8d
for infinitely many /gX8SS ∈ ℕ . Letting the limit as /gX8SS → ∞ and taking into account that
/gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX845/g2∗4X= /gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX845/g2∗4X/gD8ℝd/gD8Sd = /gX8ℝ∗, we deduce that /gX8ℝ∗ ≤ /gX8ℝ∗ (1 + /gX8ℝ∗ ) (1 + /gX8ℝ∗ ) ⁄ and
consequently, we obtain /gX8ℝ∗ = 0 .
In both cases, we obtain that
/gX8S4/gX8SX/gX8S5 /g2∗4X→/gDdd8 /gX845/g2∗4X= /gX8ℝ∗ = 0 . (10)
Now, we will show that /g4SS8/gX8ℝS/g2∗4X/g4SSd is a Cauchy sequence.
Suppose the contrary. Then there exist /gD∗25 > 0 for which we can find two subsequences
/g4SS8/gX8ℝS/g2∗4∗/g2D84/g4SSd and /g4SS8/gX8ℝS/g2∗4X/g2D84/g4SSd of /g4SS8/gX8ℝS/g2∗4X/g4SSd such that /gX8S5/g2∗2S is the smallest index for which
/gX8S5/g2∗2S> /gX8SS /g2∗2S> /gX8SX, /gD∗XD/g2∗2S= /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/g242d ≥ /gD∗25 (11)
UNDER PEER REVIEW

10
This means that
/gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d < /gD∗25 (12)
By Lemma 1, Lemma 2, and using (11) and (12), we h ave
/gD∗25 ≤ /gD∗XD /g2∗2S
= /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/g242d
= /gX845/g2425/gX8ℝS /g2∗4∗/g2D84, /gX8ℝS/g2∗4∗/g2D84, /gX8ℝS/g2∗4X/g2D84/g242d
≤ 2/gX845/g2425/gX8ℝS /g2∗4∗/g2D84, /gX8ℝS/g2∗4∗/g2D84, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d + /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d
≤ 2/gX845/g2425/gX8ℝS /g2∗4∗/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4∗/g2D84/g242d + /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d
≤ /gD∗25 + 2/gX845 /g2∗4∗/g2D84/gD8ℝd/gD8Sd (13)
On letting the limit as /gX8SX → ∞ in (13) and using (10), we obtain
/gX8S4/gX8SX/gX8S5 /g2∗2S→/gDdd8 /gD∗XD/g2∗2S= /gD∗25 (14)
Let /gD∗XX/g2∗2S= /gX845/g2425/gX8ℝS /g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d. Notice that
|/gD∗XX/g2∗2S− /gD∗XD /g2∗2S|= /g2SDℝ/gX845/g2425/gX8ℝS /g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d − /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/g242d/g2SDℝ
≤ 2/gX845 /g2∗4X/g2D84/gD8ℝd/gD8Sd+ /gX845/g2425/gX8ℝS /g2∗4∗/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4X/g2D84/g242d − /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/g242d
≤ 2/gX845 /g2∗4X/g2D84/gD8ℝd/gD8Sd+ 2/gX845 /g2∗4∗/g2D84/gD8ℝd/gD8Sd+ /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/g242d − /gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/g242d
= 2/gX845 /g2∗4X/g2D84/gD8ℝd/gD8Sd+ 2/gX845 /g2∗4∗/g2D84/gD8ℝd/gD8Sd (15)
On making the limit as /gX8SX → ∞ , we immediately obtain that /gX8S4/gX8SX/gX8S5 /g2∗2S→/gDdd8|/gD∗XX/g2∗2S− /gD∗25|= 0 and
consequently
/gX8S4/gX8SX/gX8S5 /g2∗2S→/gDdd8 /gD∗XX/g2∗2S= /gD∗25 (16)
Now using contractive condition (1), we get
/gD∗2∗(/gD∗XD/g2∗2S)= /gD∗2∗ /g4SℝD/gX845/g2425/gX8ℝS /g2∗4X/g2D84, /gX8ℝS/g2∗4X/g2D84, /gX8ℝS/g2∗4∗/g2D84/g242d/g4Sℝ2
= /gD∗2∗ /g4SℝD/gX845/g2425/gX858/gX8ℝS /g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX858/gX8ℝS /g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX858/gX8ℝS /g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d/g4Sℝ2
≤ /gX8S5/gX852/gX8ℝS /g4S8D/gD∗28 /g4SℝD/gX845 /g2425/gX8ℝS/g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d/g4Sℝ2 , /gD∗28 /g4Sℝ8/g2∗D∗/g4SℝD/g2∗5X /g2D8d/g2D84/g2XDℝ/g2XXℝ,/g2∗5X/g2D8d/g2D84/g2XDℝ/g2XXℝ,/g2∗22/g2∗5X /g2D8d/g2D84/g2XDℝ/g2XXℝ/g4Sℝ2/g4Sℝ4/gD8Sd/gD8ℝ8/g2∗D∗/g4SℝD/g2∗5X /g2D88/g2D84/g2XDℝ/g2XXℝ,/g2∗5X/g2D88/g2D84/g2XDℝ/g2XXℝ,/g2∗22/g2∗5X /g2D88/g2D84/g2XDℝ/g2XXℝ/g4Sℝ2/g4Sℝ5
/gD8Sd/gD8ℝ8/g2∗D∗/g4SℝD/g2∗22/g2∗5X /g2D8d/g2D84/g2XDℝ/g2XXℝ,/g2∗22/g2∗5X /g2D8d/g2D84/g2XDℝ/g2XXℝ,/g2∗22/g2∗5X /g2D88/g2D84/g2XDℝ/g2XXℝ/g4Sℝ2/g4Sℝd/g4S82
= /gX8S5/gX852/gX8ℝS /g4S8D/gD∗28 /g4SℝD/gX845/g2425/gX8ℝS /g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4X/g2D84/gD8ℝd/gD8Sd, /gX8ℝS/g2∗4∗/g2D84/gD8ℝd/gD8Sd/g242d/g4Sℝ2 , /gD∗28 /g4Sℝ8/g2∗D∗/g4SℝD/g2∗5X /g2D8d/g2D84/g2XDℝ/g2XXℝ,/g2∗5X/g2D8d/g2D84/g2XDℝ/g2XXℝ,/g2∗5X/g2D8d/g2D84/g4Sℝ2/g4Sℝ4/gD8Sd/gD8ℝ8/g2∗D∗/g4SℝD/g2∗5X /g2D88/g2D84/g2XDℝ/g2XXℝ,/g2∗5X/g2D88/g2D84/g2XDℝ/g2XXℝ,/g2∗5X/g2D88/g2D84/g4Sℝ2/g4Sℝ5
/gD8Sd/gD8ℝ8/g2∗D∗/g4SℝD/g2∗5X /g2D8d/g2D84,/g2∗5X/g2D8d/g2D84,/g2∗5X/g2D88/g2D84/g4Sℝ2/g4Sℝd/g4S82
= /gX8S5/gX852/gX8ℝS /g4S8D/gD∗28 (/gD∗XX/g2∗2S), /gD∗28 /g4Sℝ8/g2∗D∗/g2D8d/g2D84/g2XDℝ/g2XXℝ/g4Sℝ4/gD8Sd/gD8ℝ8/g2∗D∗ /g2D88/g2D84/g2XDℝ/g2XXℝ/g4Sℝ5
/gD8Sd/gD8ℝ8/g2∗82 /g2D84/g4Sℝd/g4S82 (17)
Let us put
/gX8Dd = /g4SS8/gX8SX ∈ ℕ ∶ /gD∗2∗ (/gD∗XD/g2∗2S)≤ /gD∗28 (/gD∗XX/g2∗2S)/g4SSd
UNDER PEER REVIEW

11
/gX82∗ = /g4S8D/gX8SX ∈ ℕ ∶ /gD∗2∗ (/gD∗XD/g2∗2S)≤ /gD∗28 /g4Sℝ8/g2∗D∗/g2D8d/g2D84/g2XDℝ/g2XXℝ/g4Sℝ4/gD8Sd/gD8ℝ8/g2∗D∗ /g2D88/g2D84/g2XDℝ/g2XXℝ/g4Sℝ5
/gD8Sd/gD8ℝ8/g2∗82 /g2D84/g4Sℝd/g4S82.
By (17), we have /gX8Dd/gX852/gX8ℝ∗/gX85S /gX8Dd = ∞ or /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82∗ = ∞ . Let us suppose that /gX8Dd/gX852/gX8ℝ∗/gX85S /gX8Dd = ∞ . Then
there exists infinitely many /gX8SX ∈ ℕ satisfying inequality /gD∗2∗(/gD∗XD/g2∗2S)≤ /gD∗28 (/gD∗XX/g2∗2S). Since (/gD∗28, /gD∗2∗ )∈
/gD2ℝX, we infer from (14), (16) and condition (e3) that /gD∗25 = 0 . This is a contradiction.
On the other hand, if /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82∗ = ∞ , then we can find infinitely many /gX8SX ∈ ℕ satisfying
inequality
/gD∗2∗(/gD∗XD/g2∗2S)≤ /gD∗28 /g4Sℝ8/g2∗D∗/g2D8d/g2D84/g2XDℝ/g2XXℝ/g4Sℝ4/gD8Sd/gD8ℝ8/g2∗D∗ /g2D88/g2D84/g2XDℝ/g2XXℝ/g4Sℝ5
/gD8Sd/gD8ℝ8/g2∗82 /g2D84/g4Sℝd
and since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, we obtain
/gD∗XD/g2∗2S≤/g2∗D∗/g2D8d/g2D84/g2XDℝ/g2XXℝ/g4Sℝ4/gD8Sd/gD8ℝ8/g2∗D∗ /g2D88/g2D84/g2XDℝ/g2XXℝ/g4Sℝ5
/gD8Sd/gD8ℝ8/g2∗82 /g2D84
Taking the limit as /gX8SX → ∞ in above inequality and using (10) and (14), we ob tain
/gD∗25 ≤ 0 , which is a contradiction.
Therefore, since in both possibilities /gX8Dd/gX852/gX8ℝ∗/gX85S /gX8Dd = ∞, and /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82∗ = ∞, we obtain a
contradiction, we deduce that /g4SS8/gX8ℝS/g2∗4X/g4SSd is a Cauchy sequence in X. Since /gX85∗ is complete,
there exists /gX8ℝ2 ∈ /gX85∗ such that lim /g2∗4X→/gDdd8 /gX8ℝS/g2∗4X= /gX8ℝ2.
Next, we will show that /gX8ℝ2 is a fixed point of /gX858.
Since /g4SS8/gX8ℝS/g2∗4X/g4SSd is non-decreasing sequence in /gX85∗ such that /gX8ℝS/g2∗4X→ /gX8ℝ2, then /gX8ℝS/g2∗4X⪯ /gX8ℝ2. Applying
contractive condition (1), for any /gX8SS ∈ ℕ, we obtain
/gD∗2∗/g2425/gX845 (/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX858/gX8ℝS /g2∗4X)/g242d ≤ /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝS /g2∗4X)/g242d, /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗22/g2∗48 )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗48,/g2∗22/g2∗48,/g2∗22/g2∗5X /g2D8d)/g4Sℝ2/g4Sℝℝ (18)
Let us denote
/gX82X = /g24Xd/gX8SS ∈ ℕ ∶ /gD∗2∗/g2425/gX845 (/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX858/gX8ℝS /g2∗4X)/g242d ≤ /gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝS /g2∗4X)/g242d/g24D2,
/gX82D = /g4SℝS/gX8SS ∈ ℕ ∶ /gD∗2∗/g2425/gX845 (/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX858/gX8ℝS /g2∗4X)/g242d ≤ /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗22/g2∗48 )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗48,/g2∗22/g2∗48,/g2∗22/g2∗5X /g2D8d)/g4Sℝ2/g4Sℝℝ
By (18), we have /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82X = ∞ or /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82D = ∞ . Let us suppose that /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82X = ∞ .
Then there exists infinitely natural numbers /gX8SS satisfying inequality
/gD∗2∗/g2425/gX845 (/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX858/gX8ℝS /g2∗4X)/g242d ≤ /gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝS /g2∗4X)/g242d and since lim /g2∗4X→/gDdd8 /gX8ℝS/g2∗4X= /gX8ℝ2 and (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, letting
the limit as /gX8SS → ∞ in above inequality, we obtain lim /g2∗4X→/gDdd8 /gX845(/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX858/gX8ℝS /g2∗4X)= 0. Thus
lim /g2∗4X→/gDdd8 /gX858/gX8ℝS/g2∗4X= /gX858/gX8ℝ2 ,
where, to simplify our assumptions, we will denote the subsequence by the same
symbol /gX858/gX8ℝS /g2∗4X. By (3), we have
lim /g2∗4X→/gDdd8 /gX8ℝS/g2∗4X/gD8ℝ8/gD8Sd = lim /g2∗4X→/gDdd8 /gX858/gX8ℝS/g2∗4X= /gX858/gX8ℝ2 . (19)
UNDER PEER REVIEW

12
The uniqueness of the limit, we have /gX858/gX8ℝ2 = /gX8ℝ2 .
On the other hand, if /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82D = ∞, then from (18), we can find infinitely many /gX8SS ∈ ℕ
satisfying inequality
/gD∗2∗/g2425/gX845 (/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX858/gX8ℝS /g2∗4X)/g242d ≤ /gD∗28 /g4SℝD/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗22/g2∗48 )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗48,/g2∗22/g2∗48,/g2∗22/g2∗5X /g2D8d)/g4Sℝ2.
Again to simplify our considerations, we will denot e the subsequence by the same
symbol /gX858/gX8ℝS/g2∗4X. Since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, we deduce that
/gX845(/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX8ℝS /g2∗4X/gD8ℝ8/gD8Sd)≤/g2∗D∗/g2D8d[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗22/g2∗48 )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗48,/g2∗22/g2∗48,/g2∗5X /g2D8d/g2XDS/g2XXℝ )
for infinitely many /gX8SS ∈ ℕ . Letting the limit as /gX8SS → ∞ and taking into account that
lim /g2∗4X→/gDdd8 /gX845/g2∗4X= 0, we deduce that lim /g2∗4X→/gDdd8 /gX845(/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX8ℝS /g2∗4X/gD8ℝ8/gD8Sd)= 0 and consequently, we
obtain lim /g2∗4X→/gDdd8 /gX8ℝS/g2∗4X/gD8ℝ8/gD8Sd = /gX858/gX8ℝ2 . The uniqueness of the limit, we have /gX858/gX8ℝ2 = /gX8ℝ2 .
This result finishes the proof.
In what follows, we prove a sufficient condition fo r the uniqueness of the fixed point
in Theorem 5.
Theorem 6 Suppose that:
(a) Hypothesis of Theorem 5 hold,
(b) For each /gX8ℝS, /gX8ℝℝ ∈ /gX85∗, there exists /gX8ℝ8 ∈ /gX85∗ that is comparable to /gX8ℝS and /gX8ℝℝ. Then /gX858 has a
unique fixed point.
Proof: As in the proof of Theorem 5, we see that /gX858 has a fixed point. Now we prove
that the uniqueness of the fixe point of /gX858. Let /gX8ℝ2 and /gX8ℝ4 be two fixed points of /gX858.
We consider the following two cases:
Case.1 /gX8ℝ2 is comparable to /gX8ℝ4. Then /gX858/g2∗4X/gX8ℝ2 is comparable to /gX858/g2∗4X/gX8ℝ4 for all /gX8SS ∈ ℕ . For all
/gX852 ∈ /gX85∗, applying contractive condition (1), we have
/gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d
= /gD∗2∗/g2425/gX845 (/gX858/g2∗4X/gX8ℝ2, /gX858/g2∗4X/gX8ℝ2, /gX858/g2∗4X/gX8ℝ4)/g242d
≤ /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX858/g2∗4X/gD8ℝd/gD8Sd/gX8ℝ2, /gX858/g2∗4X/gD8ℝd/gD8Sd/gX8ℝ2, /gX858/g2∗4X/gD8ℝd/gD8Sd/gX8ℝ4)/g242d, /gD∗28 /g4SℝD/g2∗D∗/g2425/g2∗22/g2D8d/g2XDℝ/g2XXℝ/g2∗4d,/g2∗22/g2D8d/g2XDℝ/g2XXℝ/g2∗4d,/g2∗22/g2D8d/g2∗4d/g242d/g24Dℝ/gD8Sd/gD8ℝ8/g2∗D∗ /g2425/g2∗22/g2D8d/g2XDℝ/g2XXℝ/g2∗48,/g2∗22/g2D8d/g2XDℝ/g2XXℝ/g2∗48,/g2∗22/g2D8d/g2∗48/g242d/g242X
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2D8d/g2∗48,/g2∗22/g2D8d/g2∗48,/g2∗22/g2D8d/g2∗4d)/g4Sℝ2/g4Sℝℝ
= /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d, /gD∗28 /g4SℝD/g2∗D∗(/g2∗4d,/g2∗4d,/g2∗4d )[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗48 )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗4d )/g4Sℝ2/g4Sℝℝ
= /gX8S5/gX852/gX8ℝS/g24Xd/gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d, /gD∗28(0)/g24D2 (20)

Consider
/gX8S5/gX852/gX8ℝS/g24Xd/gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d, /gD∗28(0)/g24D2 = /gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d
Then from (20), we have
UNDER PEER REVIEW

13
/gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d ≤ /gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d.
Since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, it follows from Remark 2 that /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )= 0 and so by (S2), /gX8ℝ2 = /gX8ℝ4 .
Consider
/gX8S5/gX852/gX8ℝS/g24Xd/gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d, /gD∗28(0)/g24D2 = /gD∗28 (0)
Then from (20), we have
/gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )/g242d ≤ /gD∗28 (0).
Then since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, we have by condition (e2), /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )≤ 0 and consequently
/gX8ℝ2 = /gX8ℝ4 .
Therefore, in both cases we proved that /gX8ℝ2 = /gX8ℝ4 .
Case.2 /gX8ℝ2 is not comparable to /gX8ℝ4. Then there exists /gX8ℝ8 ∈ /gX85∗ that is comparable to /gX8ℝ2 and /gX8ℝ4.
Now, we can define the sequence /g4SS8/gX8ℝ8/g2∗4X/g4SSd in /gX85∗ as follows: /gX8ℝ8/gD8S8= /gX8ℝ8, /gX858/gX8ℝ8 /g2∗4X= /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd, ∀ /gX8SS ∈ ℕ .
Since /gX858 is non-decreasing we have
/gX8ℝ8/gD8S8≤ /gX8ℝ8 /g2∗4X≤ /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd, lim /g2∗4X→/gDdd8 /gX845(/gX8ℝ8/g2∗4X, /gX8ℝ8/g2∗4X, /gX8ℝ8/g2∗4X/gD8ℝ8/gD8Sd)= 0. (20)
As /gX8ℝ2 ≤ /gX8ℝ8 /g2∗4X, putting /gX8ℝS = /gX8ℝ2 and /gX8ℝℝ = /gX8ℝ8 /g2∗4X in the contractive condition (1), we get
/gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)/g242d = /gD∗2∗/g2425/gX845 (/gX858/gX8ℝ2, /gX858/gX8ℝ2, /gX858/gX8ℝ8 /g2∗4X)/g242d
≤ /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X)/g242d, /gD∗28 /g4SℝD/g2∗D∗(/g2∗52/g2D8d,/g2∗52/g2D8d,/g2∗52/g2D8d/g2XDS/g2XXℝ )[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗22/g2∗48 )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗48,/g2∗22/g2∗48,/g2∗22/g2∗52 /g2D8d)/g4Sℝ2/g4Sℝℝ
= /gX8S5/gX852/gX8ℝS /g4SℝS/gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X)/g242d, /gD∗28 /g4SℝD/g2∗D∗(/g2∗52/g2D8d,/g2∗52/g2D8d,/g2∗52/g2D8d/g2XDS/g2XXℝ )
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗52 /g2D8d/g2XDS/g2XXℝ )/g4Sℝ2/g4Sℝℝ (21)
Let us denote
/gX825 = /g24Xd/gX8SS ∈ ℕ ∶ /gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)/g242d ≤ /gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X)/g242d/g24D2,
/gX82S = /g4SℝS/gX8SS ∈ ℕ ∶ /gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)/g242d ≤ /gD∗28 /g4SℝD/g2∗D∗(/g2∗52/g2D8d,/g2∗52/g2D8d,/g2∗52/g2D8d/g2XDS/g2XXℝ )
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗52 /g2D8d/g2XDS/g2XXℝ )/g4Sℝ2/g4Sℝℝ.
Now we remark following again.
(1). If /gX8Dd/gX852/gX8ℝ∗/gX85S /gX825 = ∞, then from (21), we can find infinitely natural num bers /gX8SS satisfying
inequality
/gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)/g242d ≤ /gD∗28/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X)/g242d.
Since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, it follows that the sequence /g4SS8/gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)/g4SSd is non-increasing and
it has a limit /gX8S4 ≥ 0 . Since
lim /g2∗4X→/gDdd8 /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)= lim /g2∗4X→/gDdd8 /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X)= /gX8S4
and (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, using condition (e3), we obtain /gX8S4 = 0 .
(2). If /gX8Dd/gX852/gX8ℝ∗/gX85S /gX82S = ∞, then from (21), we can find infinitely natural num bers /gX8SS satisfying
inequality
UNDER PEER REVIEW

14
/gD∗2∗/g2425/gX845 (/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)/g242d ≤ /gD∗28 /g4SℝD/g2∗D∗(/g2∗52/g2D8d,/g2∗52/g2D8d,/g2∗52/g2D8d/g2XDS/g2XXℝ )
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗52 /g2D8d/g2XDS/g2XXℝ )/g4Sℝ2.
Then since (/gD∗28, /gD∗2∗ )∈ /gD2ℝX, we have
/gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)≤/g2∗D∗(/g2∗52/g2D8d,/g2∗52/g2D8d,/g2∗52/g2D8d/g2XDS/g2XXℝ )
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗48,/g2∗48,/g2∗52 /g2D8d/g2XDS/g2XXℝ )
Since lim /g2∗4X→/gDdd8 /gX845(/gX8ℝ8/g2∗4X, /gX8ℝ8/g2∗4X, /gX8ℝ8/g2∗4X/gD8ℝ8/gD8Sd)= 0, on making the limit as /gX8SS → ∞, we have
lim /g2∗4X→/gDdd8 /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)= 0.
Therefore, in both cases we proved that lim /g2∗4X→/gDdd8 /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)= 0. In the same way it
can be deduced that lim /g2∗4X→/gDdd8 /gX845(/gX8ℝ4, /gX8ℝ4, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)= 0. Therefore passing to the limit in
/gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ4 )≤ /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)+ /gX845(/gX8ℝ2, /gX8ℝ2, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)+ /gX845(/gX8ℝ4, /gX8ℝ4, /gX8ℝ8 /g2∗4X/gD8ℝ8/gD8Sd)
as /gX8SS → ∞, we obtain /gX8ℝ2 = /gX8ℝ4 . That is, the fixed point is unique.
By Theorem 5, we obtain the following corollaries.
Corollary 1 Let (/gX85∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric /gX845
on /gX85∗ such that (/gX85∗, /gX845) be a complete S-metric space. Let /gX858: /gX85∗ → /gX85∗ be a non-decreasing
mapping satisfying
/gX845(/gX858/gX8ℝS, /gX858/gX8ℝS, /gX858/gX8ℝℝ )≤ /gD∗∗d/g2∗D∗(/g2∗5D,/g2∗5D,/g2∗22/g2∗5D )[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗5X,/g2∗5X,/g2∗22/g2∗5X )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗5X,/g2∗22/g2∗5X,/g2∗22/g2∗5D )+ /gD∗X∗/gX845 (/gX8ℝS, /gX8ℝS, /gX8ℝℝ ) (22)
for all comparable elements /gX8ℝS, /gX8ℝℝ ∈ /gX85∗ , where /gD∗∗d, /gD∗X∗ > 0 and /gD∗∗d + /gD∗X∗ < 1 . Assume that if
/g4SS8/gX8ℝS/g2∗4X/g4SSd is non-decreasing sequence in /gX85∗ such that /gX8ℝS/g2∗4X→ /gX8ℝ2, then /gX8ℝS/g2∗4X⪯ /gX8ℝ2, for all /gX8SS ∈ ℕ . If
there exist /gX8ℝS/gD8S8∈ /gX85∗ such that /gX8ℝS/gD8S8⪯ /gX858/gX8ℝS /gD8S8, then /gX858 has a fixed point.
Proof: Since
/gX845(/gX858/gX8ℝS, /gX858/gX8ℝS, /gX858/gX8ℝℝ )≤ /gD∗∗d/g2∗D∗(/g2∗5D,/g2∗5D,/g2∗22/g2∗5D )[/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗5X,/g2∗5X,/g2∗22/g2∗5X )]
/gD8Sd/gD8ℝ8/g2∗D∗ (/g2∗22/g2∗5X,/g2∗22/g2∗5X,/g2∗22/g2∗5D )+ /gD∗X∗/gX845 (/gX8ℝS, /gX8ℝS, /gX8ℝℝ )
≤ /gD∗Xd max /g46ℝ6/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ ),/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝℝ
= max /g46ℝ6/g2∗19/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ ), /g2∗19/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝℝ
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ , where /g2∗19 = /g2∗∗9 + /g2∗1∗ < 1 . This condition is a
particular case of the contractive condition appear ing in Theorem 5 with the pair of
functions (/g2∗2∗, /g2∗28 )∈ /g22ℝ1, given by /g2∗2∗ = 1 /g46ℝ∗/g2868,/g2998 ) and /g2∗28 = /g2∗191 /g46ℝ∗/g2868,/g2998) , (see Example 9).
Furthermore, we relaxed the requirement of the cont inuity of mapping to prove the
results.
By Theorem 5, we obtain the following corollaries.
Corollary 2 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists a pair of functions (/g2∗28, /g2∗2∗) ∈ /g22ℝ1 satisfying
UNDER PEER REVIEW

15
/g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429 ≤ /g2∗28/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429, (23)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.
Corollary 3 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists a pair of functions (/g2∗28, /g2∗2∗) ∈ /g22ℝ1 satisfying
/g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429 ≤ /g2∗28 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2 (24)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.
Taking into account Example 8, we have the followin g corollary.
Corollary 4 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists a pair of functions (/g2∗28, /g2∗2∗) ∈ /g22ℝ1 satisfying
/g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429 ≤ /g1865/g1852/g18ℝ6/g2419/g2∗2∗/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429 − /g2∗28/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429,/g1
/g1/g2∗2∗ /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2 − /g2∗28 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2/g46ℝℝ (25)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.
Corollary 4 has the following consequences.
Corollary 5 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists a pair of functions (/g2∗28, /g2∗2∗) ∈ /g22ℝ1 satisfying
/g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429 ≤ /g2∗2∗/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429 − /g2∗28/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429 (26)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.
Corollary 6 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists a pair of functions (/g2∗28, /g2∗2∗) ∈ /g22ℝ1 satisfying
UNDER PEER REVIEW

16
/g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429 ≤ /g2∗2∗ /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2 − /g2∗28 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2 (27)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.
Taking into account Example 9, we have the followin g corollary.
Corollary 7 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists /g2∗∗9 ∈ /g1845 (see Example 9) satisfying
/g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )≤ max/g2419/g2∗∗9/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429/g1845(/g18ℝ6, /g18ℝ6, /g18ℝℝ ),/g1
/g1/g2∗∗9 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2/g46ℝℝ (28)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.
A consequence of Corollary 7 is the following corol lary.
Corollary 8 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists /g2∗∗9 ∈ /g1845 (see Example 9) satisfying
/g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )≤ /g2∗∗9/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429/g1856/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429 (29)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.
Corollary 9 Let (/g185∗, ⪯) is a partially ordered set. Suppose that there exi st a S-metric
/g1845 on /g185∗ such that (/g185∗, /g1845) be a complete S-metric space. Let /g1858: /g185∗ → /g185∗ be a non-decreasing
mapping such that there exists /g2∗∗9 ∈ /g1845 (see Example 9) satisfying
/g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ ) ≤ /g2∗∗9 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2 (30)
for all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ . Assume that if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing sequence in
/g185∗ such that /g18ℝ6/g2∗41→ /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2, for all /g1866 ∈ ℕ . If there exist /g18ℝ6/g2868∈ /g185∗ such that /g18ℝ6/g2868⪯
/g1858/g18ℝ6/g2868, then /g1858 has a fixed point.

3. Example
UNDER PEER REVIEW

17
We give examples to demonstrate the validity of the above results. The following
example shows that Corollary 1 is a proper generalization o f Theorem 3.1 of [ 29 ].
Example 10 Let /g185∗ = /g4668−3, −1,0,2,4 /g4669 with the /g1845-metric defined by
/g1845(/g18ℝ6, /g18ℝℝ, /g18ℝ8 )=|/g18ℝ6 − /g18ℝ8 |+|/g18ℝℝ − /g18ℝ8 |, ∀ /g18ℝ6, /g18ℝℝ, /g18ℝ8 ∈ /g185∗ .
Consider the function /g1858 ∶ /g185∗ → /g185∗ given as
/g1858 = /g46ℝ2−3 −1 0 2 4
0 0 0 −1 −3/g46ℝ2.
We have
/g1845(/g18582, /g18582, /g18584 )= /g1845(−1, −1, −3 )= 4 = /g1845 (2,2,4 ).
Then Theorem 3.1 of [ 29 ] is not applicable to /g1858.
On the other hand, define the partial order on /g185∗ as follows: /g18ℝ6 ⪯ /g18ℝℝ if and only if
/g18ℝ6, /g18ℝℝ ∈ /g4668−3, −1,0 /g4669 and /g18ℝ6 ≥ /g18ℝℝ .
Then /g1858 is non-decreasing, /g18ℝ6/g2868= 0 ⪯ /g1858/g18ℝ6 /g2868= /g18580 and if /g4668/g18ℝ6/g2∗41/g4669 is non-decreasing and
lim /g2∗41→/g2998 /g18ℝ6/g2∗41= /g18ℝ2, then /g18ℝ6/g2∗41⪯ /g18ℝ2. We also have /g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )= 0 for all /g18ℝ6, /g18ℝℝ ∈
/g4668−3, −1, 0 /g4669. Thus contractive condition (22) is satisfied for all /g18ℝ6, /g18ℝℝ ∈ /g4668−3, −1,0, 2, 4 /g4669
with /g2∗∗9 =/g28ℝ1
/g28ℝ2 and /g2∗1∗ =/g2869
/g28ℝ2. Then, Corollary 1 is applicable to /g1858.
Example 11 Let /g185∗ = /g46681, 2, 3 /g4669 and let /g1845 be defined as follows.
/g1845(1, 1, 1 )= /g1845 (2, 2, 2 )= /g1845 (3,3, 3 )= 0,
/g1845(1, 2, 3 )= /g1845(1, 3, 2 )= /g1845(2, 1, 3 )
= /g1845(3, 1, 2 )= 4,
/g1845(2, 3, 1 )= /g1845 (3, 2, 1 )= /g1845 (1, 1, 2 )
= /g1845 (1, 1, 3 )= /g1845 (2, 2, 1 )= /g1845 (3, 3, 1 )= 2,
/g1845(2, 2, 3 )= /g1845 (3, 3, 2 )= 6,
/g1845(2, 3, 2 )= /g1845 (3, 2, 2 )= /g1845 (3, 2, 3 )
= /g1845 (2, 3, 3 )= 3,
/g1845(1, 2, 1 )= /g1845 (2, 1, 1 )= /g1845 (1, 3, 1 )
= /g1845 (3, 1, 1 )= /g1845 (2, 1, 2 )= /g1845 (1, 2, 2 )
= /g1845 (3, 1, 3 )= /g1845 (1, 3, 3 )= 1.
We have /g1845(/g18ℝ6, /g18ℝℝ, /g18ℝ8) ≥ 0 for all /g18ℝ6, /g18ℝℝ, /g18ℝ8 ∈ /g185∗ and /g1845(/g18ℝ6, /g18ℝℝ, /g18ℝ8) = 0 if and only if /g18ℝ6 = /g18ℝℝ = /g18ℝ8 .
By simple calculations, we see that the inequality
/g1845(/g18ℝ6, /g18ℝℝ, /g18ℝ8 )≤ /g1845(/g18ℝ6, /g18ℝ6, /g1852 )+ /g1845(/g18ℝℝ, /g18ℝℝ, /g1852 )+ /g1845(/g18ℝ8, /g18ℝ8, /g1852 )
holds for all /g18ℝ6, /g18ℝℝ, /g18ℝ8, /g1852 ∈ /g185∗ . Then /g1845 is an /g1845-metric on /g185∗ with the usual.
Consider the function /g1858 ∶ /g185∗ → /g185∗ given as
UNDER PEER REVIEW

18
/g1858 = /g46ℝ21 2 3
1 1 1/g46ℝ2.
Define the functions /g2∗2∗, /g2∗28: /g46ℝ∗0, ∞ )→/g46ℝ∗0, ∞ ) as follows: for all /g18ℝ2 ∈ /g46ℝ∗0, ∞ ),
/g2∗2∗(/g18ℝ2)= ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ2/g2∗4ℝ
/g2869/g28ℝ∗ /g46ℝ2 and /g2∗28(/g18ℝ2)= ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ1/g2∗4ℝ
/g2869/g28ℝ∗ /g46ℝ2.
Then all assumptions of Theorem 5 are satisfied. Th en Theorem 5 is applicable to /g1858 on
(/g185∗, /g1845).
If we define /g2∗∗9(/g18ℝ2)=/g2∗22/g212ℝ/g2295
/g2869/g28ℝ8/g2∗4ℝ, ∀ /g18ℝ2 ∈ /g46ℝ∗0, ∞ ), then /g2∗∗9 ∈ /g1845 and
min /g2∗51,/g2∗52∈/g2∗25 /g2419/g2∗∗9/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429/g2422
= min /g46ℝ61,/g2869
/g28ℝ∗/g2∗22,/g2869
/g28ℝ1/g2∗22/g2118,/g2869
/g28ℝ2/g2∗22/g2119,/g2869
/g28ℝ2/g2∗22/g212∗ ,/g2869
/g28ℝ5/g2∗22/g2122/g46ℝℝ
=/g2869
/g28ℝ5/g2∗22/g2122,
min /g2∗51,/g2∗52∈/g2∗25 /g4668/g1845(/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g4669
= min /g46680,1,2,3,4,6 /g4669= 0,
max /g2∗51,/g2∗52∈/g2∗25 /g4668/g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g4669= 0.
Hence
/g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )≤ /g2∗∗9/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429/g1845(/g18ℝ6, /g18ℝ6, /g18ℝℝ ), ∀/g18ℝ6, /g18ℝℝ ∈ /g185∗ .
Therefore Corollary 8 is also applicable to /g1858 on (/g185∗, /g1845).
Example 12 Let /g185∗ = /g46ℝ∗0,1/g46ℝ1 be equipped with the usual order and complete /g1845-metric
defined by
/g1845(/g18ℝ6, /g18ℝℝ, /g18ℝ8 )=|/g18ℝ6 − /g18ℝ8 |+|/g18ℝℝ − /g18ℝ8 |, ∀ /g18ℝ6, /g18ℝℝ, /g18ℝ8 ∈ /g185∗ .
Consider the mapping /g1858: /g185∗ → /g185∗ defined by /g1858/g18ℝ6 =/g2869
/g2869/g28ℝ4 /g18ℝ6/g28ℝ∗/g185ℝ/g2∗51/g212ℝ/g2118 for each /g18ℝ6 ∈ /g185∗ and the
function /g2∗∗9 is given by /g2∗∗9(/g18ℝ2)=/g2869
/g28ℝ2 for all /g18ℝ2 ∈ /g46ℝ∗0, ∞ ). It is easy to see that /g1858 is an
increasing function and 0 ⪯ /g18580 = 0. For all comparable elements /g18ℝ6, /g18ℝℝ ∈ /g185∗ , by the mean
value theorem, we have
/g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )
= /g1845 /g46ℝ2/g2869
/g2869/g28ℝ4 /g18ℝ6/g28ℝ∗/g185ℝ/g2∗51/g212ℝ/g2118,/g2869
/g2869/g28ℝ4 /g18ℝ6/g28ℝ∗/g185ℝ/g2∗51/g212ℝ/g2118,/g2869
/g2869/g28ℝ4 /g18ℝℝ/g28ℝ∗/g185ℝ/g2∗52/g212ℝ/g2118/g46ℝ2
=/g2869
/g28ℝ6/g262ℝ/g18ℝ6/g28ℝ∗/g185ℝ/g2∗51/g212ℝ/g2118− /g18ℝℝ/g28ℝ∗/g185ℝ/g2∗52/g212ℝ/g2118/g262ℝ
≤/g2869
/g28ℝ6|/g18ℝ6 − /g18ℝℝ |
≤/g2869
/g28ℝ2/g1845(/g18ℝ6, /g18ℝ6, /g18ℝℝ )
= /g2∗∗9/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429/g1845(/g18ℝ6, /g18ℝ6, /g18ℝℝ )
Then, Corollary 8 is applicable to /g1858.
UNDER PEER REVIEW

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Example 12 Let /g185∗ = /g46ℝ40,/g2∗95
/g28ℝ2/g46ℝ5 with the /g1845-metric defined by
/g1845(/g18ℝ6, /g18ℝℝ, /g18ℝ8 )=/g2869
/g28ℝ∗(|/g18ℝ6 − /g18ℝ8 |+|/g18ℝℝ − /g18ℝ8 |), ∀ /g18ℝ6, /g18ℝℝ, /g18ℝ8 ∈ /g185∗ .
Define the partial order on /g185∗ by
/g18ℝ6 ⪯ /g18ℝℝ if and only if /g18ℝ6 ≥ /g18ℝℝ ,
where ≤ is the usual order on ℝ. Then (/g185∗, ⪯, /g1845) is a complete partially ordered /g1845-metric
space.
Suppose /g1858: /g185∗ → /g185∗ such that /g1858/g18ℝ6 =/g2869
/g28ℝ2sin /g18ℝ6 for each /g18ℝ6 ∈ /g185∗ and define the functions
/g2∗2∗, /g2∗28: /g46ℝ∗0, ∞ )→/g46ℝ∗0, ∞ ) as follows: for all /g18ℝ2 ∈ /g46ℝ∗0, ∞ ),
/g2∗2∗(/g18ℝ2)= ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ2/g2∗4ℝ
/g2869/g28ℝ∗ /g46ℝ2 and /g2∗28(/g18ℝ2)= ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ1/g2∗4ℝ
/g2869/g28ℝ∗ /g46ℝ2.
For all /g18ℝ6 ≠ /g18ℝℝ, we have
/g1845(/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )= /g1845 /g46ℝ2/g2869
/g28ℝ2sin /g18ℝ6 ,/g2869
/g28ℝ2sin /g18ℝ6 ,/g2869
/g28ℝ2sin /g18ℝℝ/g46ℝ2
=/g2869
/g28ℝ2|sin /g18ℝ6 − sin /g18ℝℝ |
We have that for all /g18ℝ6 ⪯ /g18ℝℝ,
/g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429 = /g2∗2∗ /g46ℝ2/g2869
/g28ℝ2|sin /g18ℝ6 − sin /g18ℝℝ |/g46ℝ2
= ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g2869
/g2869/g28ℝ∗ |sin /g18ℝ6 − sin /g18ℝℝ |/g46ℝ2
/g2∗28/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429 = /g2∗28 (|/g18ℝ6 − /g18ℝℝ |)
= ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ1
/g2869/g28ℝ∗ |/g18ℝ6 − /g18ℝℝ |/g46ℝ2
and /g2∗28 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2
= /g2∗28 /g46ℝ2|/g28ℝ2/g2∗52/g28ℝ9/g2929/g2919/g2924 /g2∗52 |(/g28ℝ2/g28ℝ8|/g28ℝ2/g2∗51/g28ℝ9/g2929/g2919/g2924 /g2∗51 |)
/g28ℝ2/g28ℝ8|/g2929/g2919/g2924 /g2∗51/g28ℝ9/g2929/g2919/g2924 /g2∗52 |/g46ℝ2
= ln /g46ℝ8/g2869
/g2869/g28ℝ∗ +/g28ℝ1
/g2869/g28ℝ∗ /g46ℝ2|/g28ℝ2/g2∗52/g28ℝ9/g2929/g2919/g2924 /g2∗52 |(/g28ℝ2/g28ℝ8|/g28ℝ2/g2∗51/g28ℝ9/g2929/g2919/g2924 /g2∗51 |)
/g28ℝ2/g28ℝ8|/g2929/g2919/g2924 /g2∗51/g28ℝ9/g2929/g2919/g2924 /g2∗52 |/g46ℝ2/g46ℝ9
Since |sin /g18ℝ6 − sin /g18ℝℝ |≤|/g18ℝ6 − /g18ℝℝ |, therefore
/g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429 = ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g2869
/g2869/g28ℝ∗ |sin /g18ℝ6 − sin /g18ℝℝ |/g46ℝ2
≤ max /g4682ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ1
/g2869/g28ℝ∗ |/g18ℝ6 − /g18ℝℝ |/g46ℝ2 , ln /g46ℝ8/g2869
/g2869/g28ℝ∗ +/g28ℝ1
/g2869/g28ℝ∗ /g46ℝ2|/g28ℝ2/g2∗52/g28ℝ9/g2929/g2919/g2924 /g2∗52 |(/g28ℝ2/g28ℝ8|/g28ℝ2/g2∗51/g28ℝ9/g2929/g2919/g2924 /g2∗51 |)
/g28ℝ2/g28ℝ8|/g2929/g2919/g2924 /g2∗51/g28ℝ9/g2929/g2919/g2924 /g2∗52 |/g46ℝ2/g46ℝ9/g4682
= max /g46ℝ6/g2∗28/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429, /g2∗28 /g46ℝ2/g2∗2∗(/g2∗52,/g2∗52,/g2∗22/g2∗52 )/g46ℝ∗/g2869/g28ℝ8/g2∗2∗ (/g2∗51,/g2∗51,/g2∗22/g2∗51 )/g46ℝ1
/g2869/g28ℝ8/g2∗2∗ (/g2∗22/g2∗51,/g2∗22/g2∗51,/g2∗22/g2∗52 )/g46ℝ2/g46ℝℝ
Therefore, Theorem 5 is applicable to /g1858.
Also
UNDER PEER REVIEW

20
/g2∗2∗/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429 − /g2∗28/g2425/g1845 (/g18ℝ6, /g18ℝ6, /g18ℝℝ )/g2429
= /g2∗2∗ (|/g18ℝ6 − /g18ℝℝ |)− /g2∗28 (|/g18ℝ6 − /g18ℝℝ |)
= ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ2
/g2869/g28ℝ∗ |/g18ℝ6 − /g18ℝℝ |/g46ℝ2 − ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g28ℝ1
/g2869/g28ℝ∗ |/g18ℝ6 − /g18ℝℝ |/g46ℝ2
= ln /g46ℝ2/g2869/g28ℝ8/g28ℝ2 |/g2∗51/g28ℝ9/g2∗52 |
/g2869/g28ℝ8/g28ℝ1 |/g2∗51/g28ℝ9/g2∗52 |/g46ℝ2
= ln /g46ℝ21 +/g28ℝ∗|/g2∗51/g28ℝ9/g2∗52 |
/g2869/g28ℝ8/g28ℝ1 |/g2∗51/g28ℝ9/g2∗52 |/g46ℝ2
≥ ln /g46ℝ2/g2869
/g2869/g28ℝ∗ +/g2869
/g2869/g28ℝ∗ |sin /g18ℝ6 − sin /g18ℝℝ |/g46ℝ2
= /g2∗2∗/g2425/g1845 (/g1858/g18ℝ6, /g1858/g18ℝ6, /g1858/g18ℝℝ )/g2429
Hence, Corollary 2 and Corollary 5 are also appli cable to /g1858.

4. Discussion
This article can be considered as a continuation of the remarkable works of [ 29 ] and
[15 ]. In 2012, Sedghi et al. [ 29 ] asserted that an S-metric is a generalization of a G-
metric, that is, every G-metric is an S-metric, see [ 29 , Remarks 1.3] and [ 29 , Remarks
2.2]. The Example 2.1 and Example 2.2 of Dung et al . [ 9] shows that this assertion is
not correct. Moreover, the class of all S-metrics a nd the class of all G-metrics are
distinct. Also in 2012, Jeli and Samet [ 15 ] showed that a G-metric is not a real
generalization of a metric. Further, they proved th at the fixed point theorems proved in
G-metric spaces can be obtained by usual metric arg uments (see [ 15 ], Theorem 2.2). On
the other hand, Dung et al. [ 9] proved that Lemma 6 ([ 15 ], Theorem 2.2) does not hold
if the G-metric is replaced by an S-metric space. T hen, in general, arguments in [ 1], [ 15 ]
are not applicable to S-metric spaces. Also Corolla ry 1 is a proper generalization of
Theorem 3.1 of [ 15 ]. Our result has as particular cases a great numbe r of interesting
consequences which extend and generalize some resul ts appearing in the literature.

5. Conclusion
In this article, motivated by Das and Gupta [ 5], we introduced a new contractive type
condition involving rational expression in S-metric spaces. Inspired by the classical
result of Cabrera et al. [ 4] and Rocha et al. [ 24 ], we established some fixed point
theorems for non-decreasing map involving rational expression in the framework of S-
metric spaces endowed with a partial order using a class of pairs of functions satisfying
certain assumptions. The presented theorems extend, generalize and improve many
UNDER PEER REVIEW

21
existing results on metric spaces to S-metric spaces, which appeared in [ 4-5, 14-15 ].
Our results may be the motivation to other authors for extending and improving these
results to be suitable tools for their applications.
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